Electromagnetic Frontier Theory Exploration 9783110527407, 9783110525083

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Changhong Liang, Xi Chen Electromagnetic Frontier Theory Exploration

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Changhong Liang, Xi Chen

Electromagnetic Frontier Theory Exploration

Physics and Astronomy Classification Scheme 2010 Primary: Electromagnetism, Optics, Acoustics, Heat Transfer, Classical Mechanics and Fluid Dynamics; Secondary: Electromagnetism Authors Prof. Changhong Liang Xidian University School of Electronic Engineering Xi’an, Shaanxi Province China Doc. Xi Chen Xidian University School of Electronic Engineering Xi’an, Shaanxi Province China

ISBN 978-3-11-052508-3 e-ISBN (PDF) 978-3-11-052740-7 e-ISBN (EPUB) 978-3-11-052541-0 Library of Congress Control Number: 2018951334 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck Cover image: dem10 / iStock / Getty Images Plus www.degruyter.com

Content Summary This book is a summary of what the author has experienced in teaching practices for dozens of years, 18 notes on electromagnetic theory together with one note on reflections and one note on teaching, 20 notes in all. In particular, the book covers extensive contents ranging from the self-acting energy in electrostatic field, general electromagnetic inertia to electromagnetic symmetry. Questions are first raised, then discussed and finally refined to make a captivating summary of electromagnetic theory. In addition, the ideas, concepts, methods and engineering applications of electromagnetic theory are highlighted in the book. These notes allow the author to communicate with vast numbers of electromagnetic workers equally and sincerely, to profess his own difficulties and doubts in learning and research and to advance his basic views on the development of electromagnetic theory. The notes provide the author with an opportunity to share learning experience and lessons with many young teachers. In the book, the question-and-answer interview is more vocal and interactive. The book is also a reference book for large numbers of graduate students, undergraduates and junior college students around the 1990s. The book is suitable not only for teachers, graduate students and undergraduate students specializing in electronic information and communication at universities as a primary teaching book and reference book, but also for engineering and technical staff as a reference book.

https://doi.org/10.1515/9783110527407-201

An Epigram for Electromagnetic Field Theory Teaching Series by Professor Liang Changhong Electromagnetic theory develops explosively, and covers extensive contents immense to apply to the universe, subtle to attend to minute details. Erudite scholar, Professor Liang has his new work for publication, a frontier science exploration. An endless learning process, a great pleasure to read it first. Huang Hongjia April 2012

https://doi.org/10.1515/9783110527407-202

Author’s Biography Liang Changhong, born in Shanghai in December 1943, joined the Communist Party of China in 1962. He graduated and received the Bachelor of Science degree in microwave in July 1967 in the department of physics at Xi’an Military Telecommunications Engineering Institute. And then he completed the graduate courses in antenna in the department of physics at the institute in July 1967 and later taught there. In 1978, he became a lecturer. He was a visiting scholar at Syracuse University, New York State, USA, from August 1980 to August 1982. Professor Liang is currently a senior member of IEEE. In September 1984, he was selected to be a senior member of the Chinese Institute of Electronics (now changed to a fellow). In May 1986, he was exceptionally promoted to professorship and in July authorized as a doctoral advisor. In 1989, he served as the vice chairman of the Microwave Society of the Chinese Institute of Electronics. In January 1989, he became the vice president of Xidian University and from 1992 to 2002 was the university’s president. Professor Liang’s main professional interests and series of studies are within the areas of microwave and electromagnetism. He has engaged in them for more than 40 years and made a string of innovative achievements. He makes unique studies on ferrite phase shifter, pore coupling, analysis of network system and numerical calculation of electromagnetic field, and is particularly outstanding for his researches into computational microwave, nonlinear electromagnetics and microwave network theory. He has obtained tens of awards for scientific and technological achievements and has had some 300 papers published in academic journals both at home and abroad. He has published 11 books including Microwave Network and Its Application, Discussions about Measurement Data, Computational Microwave and its collector’s edition, Soliton Theory and Its Applications, Concise Microwave, Discussion of Limit, Discussion of Symmetry, Notes on Vector Calculus, Notes on Complex Function, Random Thoughts on Science and Informal Talk on Science, and one translation Method of No Coordinates. Of these works, Concise Microwave containing millions of words, which won the first prize of superior college teaching materials in Shaanxi in 2011, takes the author 12 years before its publication. This book is well received among teachers and students, so it has presently been selected by many colleges, such as Shanghai Jiao Tong University, South China University of Technology and Hangzhou Dianzi University, as a textbook or reference book. His 45-hour in-class teaching videos have been widely watched on the web and are favorably commented by both teachers and students. Professor Liang is the first person in China to present the conception of computational microwave, which preliminary forms computational microwave theory, develops the CAD method for microwave broadband coupler orientation and solves the optimization problem of Landstofer’s bent antenna array. He is also the first in the https://doi.org/10.1515/9783110527407-203

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Author’s Biography

country to promote the study of nonlinear electromagnetics and lossy electromagnetics and to create China’s first electromagnetic soliton experimental system. Professor Liang has undertaken and accomplished various major engineering projects, many of which have obtained the certification of qualified type and been applied in practice. A series of studies of pore coupling was awarded the first prize for scientific and technological progress by the Ministry of Electronics, ship-borne 381phased array radar was awarded the first prize for scientific and technological progress by the Ministry of Electronics in 1984, a series of studies of method of moment was awarded the second prize for scientific and technological progress by the Ministry of Electronics in 1987 and the study of computational microwave was awarded the second prize for scientific and technological progress by the State Educational Commission in 1989. Professor Liang has a high reputation for rigorous scholarship and as a paragon of virtue. He continues to teach undergraduates when he serves as the president with a demanding administrative schedule. He often reminds himself of a quotation from Lu Xun, “Time is like the water in a sponge: if you squeeze, you can always get some.” He often says, “If you do not want to teach, let alone be a principal, you don’t have to teach when you are a dean. If you want to teach, you can do it whatever you are. The same is true with anything else.” More than ten of the doctoral students whom he has tutored become doctoral advisors and one is named the Distinguished Professor of Changjiang Scholars Program. He was awarded the National People’s Teacher Medal in 1985, the National Electronics Industry Model in 1987, the National Expert with Outstanding Contribution in 1988, the National Outstanding Returned Overseas Chinese Talents in 1991, the Excellent Party Member Expert of Shanxi Province in 1992 and the first National Teaching Celebrities Title in 2003. The course Fundamentals of Microwave Technology taught by him was voted among the first national essential courses. Professor Liang expresses his thanks to many top experts in China and across the world for their marvelous works such as Time–Harmonic Electromagnetic Field and Moment Methods by R. F. Harrington, Field Theory of Guided Waves by R. E. Collin, Principle of Microwave (I ) & (II ) by Huang Hongjia and so on. These pioneering works have a tremendous impact on him. However, of the later generation, he feels that he is inspired to do much of his work by studying the works of the predecessors and he wants to keep learning those works in the future.

Preface In retrospect, the writing process of the author’s works falls into two categories, one of which is conscious creation, such as the edition of Computational Microwave published in 1985. In 1982, just back from the United States, the author was eager to bring home fresh air on the other side of the Pacific – microwave in combination with modern computer and computational techniques, thus helping boost the development of this field in his own country. The author worked with all his strength to write the book. The other writing begins accidentally, but then becomes explosive and a book is composed. The book is of this kind. The author happened to read literature on the energy of the electric charge system in electrostatic field, which is thought to consist of two parts: the interaction energy among charges and the inherent energy of the charge system. It is quite clear that the inherent energy discussed here is the self-acting energy in electrostatic field. In fact, as early as 70 years ago, in the autumn of 1940, a famous American physicist, Feynman, noticed that the primary problem with electromagnetic theory was the infinite self-acting energy, a trouble that arose from the description of charges as point particles. Then Feynman made an assumption himself that electrons cannot react to themselves. Through lots of hard thinking and repeated discussions with his advisor Wheeler, he finally wrote The Classical Theory of Action at a Distance – Interaction with the Absorber as the Mechanism of Radiation, jointly published in Reviews of Modern Physics in 1945. Thus, the author proved that the self-acting energy in electrostatic field could always be zero by the mathematical limit method, which, in popular culture, was just as “it is an impossible task to leave the earth by gripping one’s hair.” The author wrote a paper and mailed it to the Journal of Electrical and Electronic Education. Thanks to the bold support of editors, the paper was accepted to be published and the author has become the journal’s columnist since 2009. It took three years till 2011 to have 18 notes published. In fact, the author somewhere crossed the fixed line and even rambled. But when viewed from another perspective, it also allowed the author to present his own difficulties and doubts in learning and research and sit down to discuss the same sincerely with readers sincerely. It must be admitted that this democratic academic environment is rare in the country at present. The great contributing factor of its emergence is editors’ breadth of mind and excellence of vision. Added are question-and-answer interviews in each note, together with one note on reflections and one note on teaching, twenty notes in total. In particular, the book covers extensive contents ranging from the self-acting energy in electrostatic field, general electromagnetic inertia to electromagnetic symmetry. The book begins with a question as in the mystery of the Fermat principle, offers an in-depth discussion of mathematical methods as in electric charge multipolar and electric current multipolar, and finally concludes with a summary of https://doi.org/10.1515/9783110527407-204

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Preface

beauty of electromagnetic theory. In addition, the ideas, concepts, methods and engineering applications of electromagnetic theory are highlighted in the book. Frankly speaking, the book has been completed, the greatest wish of the author is to see whether readers like it and whether the book is worthwhile. However, pausing and reflecting, the author finds it will be decided by two conditions: one being the author’s academic level and efforts made and another being the current environment and atmosphere. So it seems that it cannot be examined using a shortterm view. It is thought-provoking to look back at the actual situations over 30 years since 1980. Professor Wu Wanchun and the author completed Microwave Network and Its Applications in 1980, experiencing the restoration of order in 1978 just after the Cultural Revolution, when a majority of young students (some were in their middle ages) who had been depressed for many years were all desperately thirsting for learning. The longing-to-learn atmosphere at that time was seen not only in the school teaching but also in daily life. The author himself received numerous letters to request to learn and ask questions. Particularly, the motivation of the classes of 1977 and 1978 for learning still remains vivid to him. Things were very different in 1988 when the author translated the famous book Electromagnetic Wave Theory – Method of No Coordinates written by Chen Huiqing of the United States. It is often said by the author that China’s environment changes every ten years. Then academic research became a secondary interest and the book was too highbrow to be popular, so its situation was different compared with when Microwave Network and Its Applications (1980) and Computational Microwave (1985) were published. It came to 2006 and in that year Concise Microwave was published, when economic rules in the society had already been quite straightforward. When doing something, one would be asked whether he could make money out of it and how much money he would earn. Therefore, although this book was of much value as a reference book for broad professional fields, it was left out in the corner because it could not be directly used to make money. Judging from the above line of thought, apparently, the book can’t be expected highly. As Bai Yansong, a famous Chinese television host, said, “The price of a Xinhua Dictionary is as low as a kilogram of pork at all times.” The author expects that with changing time, academic research and culture will eventually become hot spots for people to pursue. In particular, Chen Xi, an outstanding young scholar, is appreciated greatly for her contributions to this book. Liang Changhong

Contents Content Summary

V

An Epigram for Electromagnetic Field Theory Teaching Series by Professor Liang Changhong VII Author’s Biography Preface

XI

List of Figures 1 1.1 1.2 1.2.1 1.2.2 1.3 1.3.1 1.3.2 1.4

2 2.1 2.2 2.2.1 2.2.2 2.2.3 2.3 2.4

IX

XXI

Self-action energy in electrostatic field 1 Introduction 1 Self-action energy in the electrostatic field 2 Total electrostatic self-Action energy of uniformly distributed charges cube 2a × 2a × 2a 3 Total electrostatic self-action energy of the charge system of any shape with volume V 4 Field calculation and source calculation 5 Field method 6 Source calculation method 7 Summary 9 Q&A 9 Recommended scholar 12 Appendix 13 Corresponding research between time-harmonic field and complex field 19 Introduction 19 The complex field representation of time-harmonic field in first-order quantity 20 The criterion for the representation of real parts 20 Calculus criterion 20 Constitutive operator 21 Second-order quantity and complex field representation of the time-harmonic field 21 Three main theorems 24

XIV

2.4.1 2.4.2 2.4.3 2.5

3 3.1 3.2 3.3 3.4

Contents

Poynting theorem 24 The Foster theorem for lossless one-port system Lorentz’s reciprocity theorem 25 Summary 26 Q&A 26 Recommended scholar 29

24

Transformation and unification of electrostatic field and constant current field 33 Introduction 33 The transformation and unification of electrostatic field and constant current field 34 * 37 Paradox of Poynting vector S Summary 38 Q&A 38 Recommended scholar 43

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Charge multipoles and current multipoles 45 Introduction 45 Generalized taylor expansion of f ðRÞ 45 Charge multipoles 46 Matrix expression of multipoles 49 Current multipoles 50 Four-dimensional current multipoles in the theory of relativity Summary 53 Q&A 53 Recommended scholar 57

5 5.1 5.2 5.3 5.4 5.5

Polarization of electromagnetic wave and its applications 59 Introduction 59 Elliptically polarized wave 59 Polarization conversion and matrix representation 65 Engineering applications of polarization 68 Summary 68 Q&A 69 Recommended scholar 71

6 6.1 6.2 6.3

Conservation of charge and conservation of current Introduction 73 Conservation of charge 74 Conservation of current 76

73

53

Contents

6.4 6.5 6.6

Current–charge conservation Examples of application 77 Summary 79 Q&A 79 Recommended scholar 84 Appendix 86

77

7 7.1 7.2 7.3 7.3.1 7.3.2 7.4 7.5 7.6

Electromagnetic reciprocal symmetry and lossless symmetry 89 Introduction 89 Lorentz reciprocal symmetry 90 Hermite lossless symmetry 92 Scalar energy inner product 92 Vector energy (power) inner product 93 Network-type foster theorem, generalized inertia 95 Electromagnetic reciprocal symmetry and lossless symmetry 97 Summary 99 Q&A 99 Recommended scholar 103

8 8.1 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.3 8.3.1 8.3.2 8.3.3 8.4

Electromagnetic symmetry and symmetry operator 107 Introduction 107 Quadratic symmetry 107 Lossless Symmetry 107 Reciprocal Symmetry 109 Geometric reciprocal symmetry 110 Compound symmetry 111 Linear symmetry 112 Geometric symmetry 112 Reciprocal symmetry 113 Anti-reciprocal symmetry 113 Symplectic inner product and electromagnetic symplectic orthogonality 114 Summary 115 Q&A 116 Recommended scholar 118

8.5

9 9.1 9.2 9.1.1 9.1.2

Plane image method and active conformal mapping Introduction 121 Generalized model of the plane dielectric image 123 ε2 ! ∞ Equivalent conductor 124 ε2 ! 0 Equivalent magnet

121 121

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XVI

9.3 9.3.1 9.3.2 9.3.3 9.4 9.4.1 9.4.2 9.5

Contents

Active Conformal Mapping of the Conducting Cylinder 124 Grounded conductor cylinder 124 Ungrounded conductor cylinder 126 Dual problem of the cylinder 128 Active conformal mapping of the complex conductor 129 There exist charges at position h between infinite double plates 130 There exist line charges at the center between semi-infinite double plates 131 Summary 132 Q&A 133 Recommended scholar 138

10 Electromagnetic loss 141 10.1 Introduction 141 10.2 Three theorems 141 10.3 Loss evaluation 145 10.3.1 The condition evaluation of loss 145 10.3.2 The geometric evaluation of loss 147 10.3.3 The phase evaluation of loss 148 10.4 Characteristics of lossy system 149 10.4.1 The openness of system 149 10.4.2 The coupling of field quantities 149 10.4.3 The encroachment of information 149 10.4.4 The directivity of conversion 149 10.5 Difficulties aroused by electromagnetic loss 150 10.5.1 The complex boundary conditions 150 10.5.2 The difficult mode theory 150 10.5.3 The blurred boundary between transmission mode and cut-off mode 151 10.6 Summary 151 Q&A 151 Recommended scholar 160 11 11.1 11.2 11.3 11.4

Complex parameter and complex theorems in electromagnetic theory 163 Introduction 163 Complex frequency 164 Complex phase angle 164 Complex frequency electromagnetic theorem 167

Contents

11.4.1 11.4.2 11.5 11.6

Complex frequency Poynting theorem 167 Complex frequency foster theorem 169 Lossless transmission lines with arbitrary load Summary 174 Q&A 175 Recommended scholar 179

172

12 12.1 12.2 12.3 12.4 12.5 12.6 12.7

and 2D static field 181 Complex operator Introduction 181 182 Complex operator Complex operator integral theorem 184 Complex partial derivative 185 Complex operator form of 2D electrostatic field 186 Complex operator form of 2D steady magnetic field 188 Summary 189 Q&A 189

13

New network theory of electromagnetic waves in multilayered media 195 Introduction 195 Basic model 195 Special angle 198 198 Brewster’s angle θB of no reflection 198 Critical angle θc in the Case of Total Reflection Electromagnetic wave transmission [C ] network 199 Oblique incidence of parallelly polarized waves 200 Oblique Incidence of Perpendicularly Polarized Waves 201 Normal incidence 202 Unified [C ] network 202 202 Wave transmission section [Cl ] Engineering applications 203 Energy conservation 205 Summary 206 Q&A 206 Recommended scholar 214

13.1 13.2 13.3 13.3.1 13.3.2 13.4 13.4.1 13.4.2 13.4.3 13.4.4 13.4.5 13.5 13.6 13.7

14 14.1 14.2

Matrix transformation in electromagnetic theory 215 Introduction 215 Two-dimensional coordinate rotation and transformation to polar coordinate matrix 216

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Contents

14.3 14.3.1

Three-dimensional coordinate matrix transformation 218 Coordinate plane rotation and its matrix transformation to cylindrical coordinate 218 Spherical Coordinate Rotation and Its Matrix Transformation to Spherical Coordinate 219 Matrix Transformation of Orthogonal Curvilinear Coordinates 220 Matrix transformation of operator ∇ 222 Matrix transformation from longitudinal field component to transverse field component 223 Summary 225 Q&A 225 Recommended scholar 229

14.3.2 14.3.3 14.3.4 14.4 14.5

15 15.1 15.2 15.3 15.4 15.5

Minimum directivity challenge of electromagnetic radiation 231 Introduction 231 Maxwell’s equations do not have isotropic solutions 231 Minimum directivity antenna 233 Minimum directivity array 235 Summary 239 Q&A 239 Recommended scholar 245 * 246 Appendix A y-direction dipole parameters and electric field E θ′ Appendix B The phases of two-dimensional eight-element 248

16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8

Mysteries of Fermat’s principle 251 Introduction 251 Expression of the law of nature 252 Minimum or maximum 253 255 Phase velocity ν p or group velocity ν g Energy E and actuating quantity s 258 Physical quantity or space 259 Least action principle 260 Summary 261 Q&A 262 Recommended scholar 272

17 17.1 17.2

Electromagnetic inertia 275 Introduction 275 Electromagnetic inertia 277

Contents

17.3 17.3.1 17.3.2 17.4 17.5 17.6

XIX

Electrostatic inertia 278 Potential φ distribution inertia 278 Electrostatic charge Σ distribution inertia 279 Green function and electrostatic inertia 280 Any antenna element cannot constitute an ideal wavelet source 281 Summary 282 Q&A 282 Recommended scholar 286

18 Beauty of electromagnetic theory 289 18.1 Introduction 289 18.2 The simple beauty of electromagnetic theory 289 18.3 Symmetry beauty of electromagnetic theory 291 18.4 The beautiful transformation of electromagnetic theory 293 18.4.1 The mutual conversion of electric field and magnetic field 293 18.4.2 The mutual transformation between space-varying and time-varying 293 18.5 The beautiful unity of electromagnetic theory 294 18.6 The duality of beauty 295 18.7 Summary 296 Q&A 296 Recommended scholar 300 19 Some thoughts on the electromagnetic theory 301 19.1 Introduction 301 19.2 Symmetry and asymmetry 301 19.2.1 The “first handshake” between electricity and magnetism 302 19.2.2 Faraday’s symmetry concept 302 19.2.3 The “second handshake” between electricity and magnetism 303 19.2.4 The research of symmetry never ends 305 19.3 Lossless and lossy 305 19.3.1 The concept of electromagnetic loss 305 19.3.2 Lossless symmetry 309 19.3.3 Uniqueness theorem of losslessness 315 19.3.4 Loss and attenuation 316 19.3.5 Ambiguity of loss 318 19.3.6 Stability of loss 318 19.3.7 Inequality of loss 319

XX

Contents

Paradox of loss 320 Lossy and lossless characteristic quantities 321 Two symmetries 322 Least action principle of lossy variation 323 325 Four-dimensional Minkovski space and
L6 Static fields and alternating fields 329 Essence of discipline development lies in innovation 331 Emphasis on the feature study of electromagnetic theory 332 Long-term study and constant practice 333 Seeking difference and innovation 334 Seeking common ground and unification 334 Developing and recreating 335 Summary 336

19.3.8 19.3.9 19.3.10 19.3.11 19.4 19.5 19.6 19.6.1 19.6.2 19.6.3 19.6.4 19.6.5 19.7

20 Teaching notes 337 20.1 Knowledge is foundation 337 20.2 The key is “Heart” 340 20.2.1 Dedication 340 20.2.2 Responsibility 342 20.2.3 Commitment 342 20.2.4 All boils down to love 343 20.3 Innovation is highlighted 343 20.4 Comprehension is the key 348 20.5 The ultimate goal is “The Way” 350 References Index

355

351

List of Figures Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8 Fig. 1.9 Fig. 1.10 Fig. 1.11 Fig. 1.12 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6

Self-action energy of volume charges in a differential cube 3 Uniformly distributed cube 2a×2a×2a 4 Charge system of any shape, with its volume being V 4 Electrostatic system with the sphere radius being R and the uniformly distributed charge density being ρ 6 * The electric field distribution under the action of E spherical charge distribution 6 * 7 The coordinate system to determine potential ’ð r ′Þ The main arrangement of the chapter 10 Annular deflection of the magnetic needle around the * 10 current I Three main ideas of physics development 11 The action of the field 11 The two topics involved in this chapter 11 Richard Feynman 12 Two methods for time domain electromagnetic field 20 Bricks – the bases of construction 27 The idea of “bases” in CCTV’s stage design in 2012 27 The research idea and method of time-harmonic field 28 Leonhard Euler 29 The Seven Bridges Problem 30 Polyhedral equation 31 Coordinate system xOy and coordinate system x′O′y′ doing relative motion with it. 34 * Poynting vector S of the electrostatic field in the moving coordinate system 37 Two axioms of Einstein’s special theory of relativity 39 Minkowski’s structural relativistic system 41 Diagram of complex rotation of four-dimensional current-charge vector 42 German mathematician, medical scientist and AmericanGerman astronomer Hermann Minkowski 43 Multipole expansion in the small source area of V and in the far field 46 47 Point charge system fqi g in the small source area of V 51 Current system fIi^li g in the small source area of V Approximation of f ðxÞ in practice by using Taylor series 54 φ approximation of the engineering applications of the charge 55 system fqi g by using multipole expansion Charge distribution of the regular triangle 56

https://doi.org/10.1515/9783110527407-205

XXII

Fig. 4.7 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4

Fig. 5.5 Fig. 5.6

Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 6.9 Fig. 6.10 Fig. A6.1 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4

List of Figures

(a) 3q Point charge contribution ’ð0Þ and (b) electric dipole * * * 57 contribution at P 1 ; P 2 ; P 3 combined ’ð1Þ ≡ 0 A general ellipse in two sets of coordinates xOy and x′Oy′ 60 Linearly polarized wave: (a) Δ’ = π in-phase linearly polarized wave and (b) Δ’ = π out-of-phase linearly polarized wave 62 Left-hand and right-hand circularly polarized waves 64 Right-hand and left-hand elliptically polarized waves: (a) α is the first quadrant angle, right-hand elliptically polarized waves, and (b) α is the fourth quadrant angle, left-hand elliptically polarized waves 64 The elliptically polarized wave considered as the center 69 A linearly polarized wave can be decomposed into two circularly polarized waves of equal amplitude and oppositely rotating direction 70 A circularly polarized wave can be decomposed into two linearly polarized waves 70 An elliptically polarized wave can be decomposed into four circularly polarized waves of oppositely rotating direction 70 Bi Dexian 71 The volume enclosed by surface S is v 74 The external volume enclosed by surface S is v
75 * 78 The ∇  A ðx; y; 0Þ problem of infinitely long current Il k^ Close relation between “conservation” and “symmetry.” 79 Conservation of trigonometric function “symmetry” on a unit circle 80 Mechanical system without consumption “symmetry” on a straight line 80 The conservation symmetry on an ellipse; 80 ðx=aÞ2 þ ðy=bÞ2 ≡ 1 The best woman mathematician in the 20th century Emmy Noether 82 Conservation and symmetry 83 Paul Dirac 84 Determining the electric field intensity by vector magnetic potential and scalar electric potential 86 A source with g = LðuÞ at point A yields a field of u at point B; a source with h = LðvÞ at point B yields a field of v at point A 91 Multi-port network [S] 92 94 The essence of the lossless system Pin = Pout Multi-port Foster theorem 96

List of Figures

Fig. 7.5 Fig. 7.6 Fig. 7.7 Fig. 7.8 Fig. 7.9

Fig. 7.10

Fig. 7.11 Fig. 7.12 Fig. 7.13 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 9.1 Fig. 9.2 Fig. 9.3 Fig. 9.4 Fig. 9.5 Fig. 9.6 Fig. 9.7 Fig. 9.8 Fig. 9.9 Fig. 9.10 Fig. 9.11 Fig. 9.12 Fig. 9.13 Fig. 9.14

XXIII

Three main methods of modern electromagnetic computation theory: network, operator and inner product 100 The network idea – determining the characteristics of an unknown system by excitation and reaction 100 The solving process of linear operator equation L ðuÞ = g 101 The expansion of the Pulse sub-domain function 101 Geometrical meaning of inner product – the projection of function vectors over the set coordinate system fui g: (a) the projection h u1 g i; h u2 g i; . . . ; h un g i of given function vector g and (b) the projection h u1 LðuÞ i; h u2 LðuÞ i; . . . ; h un LðuÞ i of unknown function vector LðuÞ 102 Foster theorem for a single-port network – the reactance slope of the port is constantly positive: (a) one-port network and (b) ∂x=∂ω > 0 103 Two representations of the network-type Foster theorem 103 Professor Harrington (left) and the author at Syracuse University 104 Linear antenna or scatterer 104 S matrix of two-port scattering 108 n-Port S network 109 Two-port transmission matrix A 109 Two-port impedance matrix Z 110 Vector inner product and symplectic inner product 114 Lorentz’s electromagnetic reciprocity theorem 115 Chen-Ning Yang 118 Solution of active conformal mapping 122 Generalized model of the plane dielectric image 122 Dielectric image method – symmetry and region solution (the shadow is the region to be solved) 123 A basic model of plane conductor images 124 A basic model of plane magnet images 124 125 Line charge ρl outside the grounded conductor cylinder From circular inversion symmetry to the plane image model 125 127 Line charge ρl outside an ungrounded cylinder From the ungrounded cylinder to the plane image model 127 Dual problem of the cylinder 128 Plane image model for the dual problem 128 Conductor cylinder with cracks 129 Active conformal mapping of the cylinder with cracks 130 130 Line charge ρl between infinite double plates

XXIV

Fig. 9.15 Fig. 9.16 Fig. 9.17 Fig. 9.18 Fig. 9.19 Fig. 9.20 Fig. 9.21 Fig. 9.22 Fig. 9.23 Fig. 9.24 Fig. 10.1 Fig. 10.2 Fig. 10.3 Fig. 10.4 Fig. 10.5 Fig. 10.6 Fig. 10.7 Fig. 10.8

Fig. 10.9 Fig. 10.10 Fig. 10.11 Fig. 10.12 Fig. 10.13 Fig. 10.14 Fig. 10.15 Fig. 10.16 Fig. 11.1 Fig. 11.2 Fig. 11.3

List of Figures

Line charge ρl between semi-infinite double plates 131 Invariance of conformal mapping: the Laplace equation of the potential function φ remains constant 133 Solution of the conformal symmetry transformation 134 Problem of the line charge outside a grounded conductor elliptic cylinder 135 Two kinds of invariance of conformal mapping 136 Orthogonality of the corresponding curves on w-plane and z-plane 136 The coaxial capacitance of the circle transformed into the plate capacitance w = ln z 137 Elliptical coaxial line 138 D. K. Cheng (Left) and the author 139 Field and Wave Electromagnetics 139 Physics includes various branches 142 Relation of electromagnetism to other branches 142 Poynting’s theorem 142 Generalized input impedance of a single-port system 143 Foster’s theorem of a single-port reactance load 144 Foster’s theorem and the lossy system 144 The condition evaluations of loss in series and parallel circuits: (a) series circuits and (b) parallel circuits 146 The contradiction between waves and good conductors: (a) waves can hardly travel into a good conductor interface and (b) 146 once traveling into a good conductor, PL is large Geometric evaluation of quality factor Q curve on the electromagnetic loss of cavity system 147 A general two-port network 148 A general directional coupler 148 Lossy electromagnetism 152 The pseudo-symmetry of Maxwell’s equations of a lossless system and a lossy (frequency domain) system 152 Four stages of the introduction of complex numbers in the electromagnetic field 153 The phase relation of the system indirectly reflects the system loss 154 Huang Hongjia 160 The generalized microwave transmission system 164 Conversion of reflection coefficients for lossless transmission lines 165 The lossy transmission line model 166

List of Figures

Fig. 11.4 Fig. 11.5 Fig. 11.6 Fig. 11.7 Fig. 11.8 Fig. 11.9 Fig. 11.10 Fig. 11.11 Fig. 11.12 Fig. 12.1 Fig. 12.2 Fig. 12.3 Fig. 12.4 Fig. 12.5 Fig. 12.6 Fig. 12.7 Fig. 12.8 Fig. 12.9 Fig. 12.10 Fig. 12.11 Fig. 13.1

Fig. 13.2 Fig. 13.3 Fig. 13.4 Fig. 13.5 Fig. 13.6 Fig. 13.7 Fig. 13.8

XXV

Poynting theorem 167 Complex frequency Foster theorem for the lossy one-port 169 The lossless transmission line with a lossy load 172 Impedance distribution curve and negative reactance slope area for a lossy load transmission line 174 ~ and complex frequency ω ~: Complex material ~ε; μ 175 ~ and complex The Maxwell’s equations of complex materials ~ε; μ ~ are exactly the same in form frequency ω 176 ~ electromagnetic Two important complex frequency ω theorems 177 ~ is used in the transmission line (space) Complex frequency ω ~ = θ′ − jθ′′ theorem, which puts forward electrical angle θ 178 R. E. Collin 179 ^ ^ = ^t × kÞ 2D integral domain Dðn 184 Relationship between the corresponding integral theorems of and real operator ∇ 185 complex conjugation operator Uniformly filled 2D electrostatic field domain D 187 A line charge q on the origin O 188 Uniformly filled 2D steady magnetic field domain D 189 Reflection of two “truths” by conjugate action of complex operator 190 Reflection of another two “truths” by complex operator action 190 Integral of complex operator action equal to 0 constantly 190 and complex partial derivative 191 Complex operator and 2D electrostatic field or steady Complex operator magnetic field 191 Two sets of coordinate systems for included angle θ, xyz and x′y′z′. 192 Take plane wave oblique incidence on the boundary plane as the basic model: (a) parallel polarization and (b) perpendicular polarization 196 Electromagnetic wave transmission [C] network 199 203 Wave transmission section network [Cl] 203 n-networks C1 ; C2 ; . . . ; Cn are cascaded Electromagnetic wave transmission in three-layered media 203 The network concept, using input–output method to study an unknown system 207 Three characteristics of spatial network 207 [C] network of the electromagnetic wave 208

XXVI

Fig. 13.9 Fig. 13.10 Fig. 13.11 Fig. 13.12 Fig. 13.13 Fig. 14.1 Fig. 14.2 Fig. 14.3 Fig. 14.4

Fig. 14.5 Fig. 14.6 Fig. 14.7 Fig. 14.8 Fig. 14.9 Fig. 14.10 Fig. 14.11 Fig. 14.12 Fig. 14.13 Fig. 14.14 Fig. 15.1 Fig. 15.2 Fig. 15.3 Fig. 15.4 Fig. 15.5 Fig. 15.6 Fig. 15.7 Fig. 15.8 Fig. 15.9 Fig. 15.10 Fig. 15.11 Fig. 15.12

Fig. 15.13

List of Figures

Transformation between linear polarization and circular polarization 209 Longitudinally magnetized ferrite waveguide system 209 Faraday rotation of the polarized plane 210 Complex network with arbitrary cascade 211 Professor Lin Weigan 214 Two-dimensional coordinates rotation 216 217 Unit vectors e^ρ and e^’ in polar coordinates The plane rotation of Cartesian coordinates (xOy) 218 The spherical rotation of Cartesian coordinates (for visual convenience, the coordinate with “′” are not at original point and should be moved to O) 219 The unit vector of spherical coordinates 220 The general orthogonal curvilinear coordinate system 221 ðq1 ; q2 ; q3 Þ Rectangular waveguide 223 Cylindrical waveguide 224 Counterparts of vector and matrix in physics 226 Eigen-mode theory of matrix 228 Linear operator L and matrix [A] 228 Dot product and generalized dot product 229 Cross-product and generalized cross-product 229 Tsung-Dao Lee 229 The eye of a tornado 233 “Curl” of hair must exist on our head 233 * 234 Radiation of electric dipole I l * 234 Radiation patterns of electric dipole I l Radiation patterns are of cross-dipole 236 The reference plane of two-dimensional eight-element circular array is perpendicular to point A 237 Radiation pattern of circular array 238 Function relationship of directivity with kR 239 Chairman Mao wrote an inscription for the magazine Communications Warrior 240 An antenna and electromagnetic waves to be radiated 240 Wang Cheng carries the radio and the wire antenna in Shangganling Mountain (China) 241 Wire antenna and “loading”: (a) the currents of wire antenna are triangle-distributed and (b) currents of wire antenna with wing on the top are near uniformly distributed 241 Jungle antenna (the “extended application” to use the wet tree as the antenna) 242

List of Figures

Fig. 15.14 Fig. 15.15 Fig. 15.16 Fig. 15.17 Fig. 15.18 Fig. 15.19 Fig. 15.20 Fig. 15.21 Fig. 15.22 Fig. A15.1 Fig. 16.1 Fig. 16.2 Fig. 16.3 Fig. 16.4 Fig. 16.5 Fig. 16.6 Fig. 16.7 Fig. 16.8 Fig. 16.9 Fig. 16.10 Fig. 16.11 Fig. 16.12 Fig. 16.13 Fig. 16.14 Fig. 16.15 Fig. 16.16 Fig. 16.17 Fig. 16.18 Fig. 16.19 Fig. 16.20 Fig. 16.21

XXVII

Parabolic antenna 242 With the uniform amplitude, the in-phase linear array has the maximum directivity 243 Two-element broadside array of uniform amplitude and inphase 243 Two-element broadside array of uniform amplitude in direction θ 244 Yagi antenna 244 Two-element end-fire array of uniform amplitude and outphase 244 End-fire array in direction θ 245 Huang Xichun 245 On Wave Velocities, Mr. Huang’s masterpiece 245 The y-direction dipole is represented with parameters θ and ’′; r is the same parameter 246 Development process of the expression of the law of nature 252 Thomson’s principle of charge distribution on a conductor * 253 ρð r Þ Laws of reflection and refraction of light 253 Light reflection of the concave mirror of the two-dimensional elliptic cylinder 254 Two extremum principles correspond to two different velocities of light 255 Energy–momentum relations of the quantum 256 Negative refraction 257 On the boundary of medium n and −n, light has countless practicable paths 257 Physical significances of energy E and actuating quantity S 258 Principle of least action 261 Fermat 262 Proving Fermat’s reflection law 263 Proving Fermat’s refraction law 265 Geometric expression of extreme values in Fermat’s principle 267 Two mirror points A′ and B′ are symmetrical 267 Circle concave mirror 268 Ellipse concave mirror with A and B at the focuses 269 Ellipse concave mirror with A and B between focuses 269 Ellipse concave mirror with A and B outside the focus length 270 Energy and energy difference in electromagnetic field 271 271 A conductor disk with a radius of r0 :

XXVIII

Fig. 16.22 Fig. 17.1 Fig. 17.2 Fig. 17.3 Fig. 17.4 Fig. 17.5 Fig. 17.6 Fig. 17.7 Fig. 17.8 Fig. 17.9 Fig. 17.10 Fig. 17.11 Fig. 17.12 Fig. 17.13 Fig. 17.14

Fig. 17.15 Fig. 18.1 Fig. 18.2 Fig. 18.3 Fig. 18.4 Fig. 18.5 Fig. 18.6 Fig. 18.7 Fig. 18.8 Fig. 18.9 Fig. 19.1 Fig. 19.2 Fig. 19.3 Fig. 19.4 Fig. 19.5 Fig. 19.6 Fig. 19.7 Fig. 19.8

List of Figures

Ye Peida 272 Hertz’s experiment 276 Unity of light and electromagnetism by Maxwell 276 Unity of dynamic field and static field by Maxwell 276 Huygens’ principle 277 Concentric circles of ripples 278 Potential φ trial function (k > 1) when the initial assumption coincides with the square conductor boundary 279 Electrostatic inertia – potential φ distribution has the tendency of the approximation graph (k > 1) 279 The charge density σ on any conductor plate also has a circular tendency 280 Green function and introduction of generalized potential 280 The dynamic field and static field electromagnetic inertia 281 There is a polarized distribution of antenna elements: (a) electric dipole and (b) a small current loop 282 Newton’s first and second laws 283 Electromagnetic theory and Maxwell’s equations 284 Applications of electromagnetic inertia: (a) the applications of electromagnetic inertia – we can receive wave no matter where we are (isotropic), and (b) “the inverse” applications of electromagnetic inertia – highly directional antenna point target (anisotropic) 285 Professor Wu Wanchun 286 The discovery of field and force lines by Faraday 290 Comparison, our habit thinking 290 Faraday’s symmetrical idea 291 The asymmetry of Oersted and Faraday 292 Maxwell complements symmetry 292 The mutual conversion of electricity and magnetism 293 The space-varying and time-varying of waves 294 Unity of light and electromagnetic waves 295 Huang Zhixun 300 The symmetrical idea of electromagnetic relationship 303 Maxwell’s symmetrical speculation 304 (a) Metal conductor and (b) carbon film resistor 305 A series system 306 A parallel system 306 Electromagnetic wave entering into a lossy medium 307 Magneto hydrodynamic power generation 308 Electromagnetic waveguide 308

List of Figures

Fig. 19.9 Fig. 19.10 Fig. 19.11 Fig. 19.12 Fig. 19.13 Fig. 19.14 Fig. 19.15 Fig. 19.16 Fig. 19.17 Fig. 19.18

Fig. 19.19 Fig. 19.20 Fig. 19.21 Fig. 19.22 Fig. 19.23 Fig. 19.24 Fig. 19.25 Fig. 20.1 Fig. 20.2 Fig. 20.3 Fig. 20.4 Fig. 20.5 Fig. 20.6 Fig. 20.7 Fig. 20.8 Fig. 20.9 Fig. 20.10 Fig. 20.11 Fig. 20.12

XXIX

Full absorbing load 308 Two-port network constraint 314 Three-port lossless network amplitude constraint 314 Uniqueness theorem of the time and the frequency domain 315 Uniqueness for a lossless region 315 Frequency spectrum 316 High-dimensional lossy ellipsoid 317 Generalized matching of a three-port network 318 320 The range of Φ12 Statistical distribution of proper phase Φ12 in the network. Moe Wind and Harold Rapaport’s experiment Pieterse and Versnel’s experiment 320 Electromagnetic open cavity 322 Relationship among the medium, operator and inner product 323 Motion boundary 327 329 Four-dimensional Minkovski space and
L6 Maxwell’s equations and the law of static field 330 Precondition of the two individual equations in Maxwell’s equations 330 * * * 331 Paradox of S = E × H in the static field Camping around Mount Everest 338 Great scientist Qian Xuesen 338 Knowledge is foundation 339 Lu Yao 341 Picture of the book Journey to the West and the monkey king 342 Teaching with all one’s heart 343 Mathematician Descartes 344 Motivation for setting up analytic geometry: putting geometric figures into coordinate systems 344 Core system innovation of MOM 345 Method innovation for Green function 346 * There are countless b i with which the same cross-product value * * * 347 can be obtained; a × b ¼ d Teaching method lies in comprehension 348

1 Self-action energy in electrostatic field The electrostatic field energy We is a very important basic concept. Most of the current literature touches upon We from the work done by point charge system fqi g. Obviously, due to the infinity problem in the self-action of point charge, self-action energy is not involved in its We . Furthermore, it can be extended to distribution charge ÐÐÐ *  *  system. Now, the stored energy can be expressed as We = ð1=2Þ v ρ r ′ ’ r ′ dv′. However, there are different opinions toward its understanding. For example, literature [3] believes that in this case, “the stored energy includes not only the interaction energy among charges, but the intrinsic energy of charge system.” In this chapter, we will prove that the self-action energy in electrostatic field in We is always zero. It is worth mentioning that the self-action energy issue can also be discussed philosophically. American physician Feynman has done great efforts to eliminate the infinity problem in point charge model. It is also mentioned that sextuple integral should be ÐÐÐ *  *  utilized to calculate stored energy by use of We = ð1=2Þ v ρ r ′ ’ r ′ dv′. Otherwise, calculation error will appear.

1.1 Introduction This is the first chapter of Electromagnetic Field Theory Teaching Series. Work connects energies in different fields such as mechanics, electromagnetics, thermotics and even chemistry. Therefore, in terms of the stored energy We in electrostatic field, most articles available start from work in the point charge system fqi g ði = 1, 2,    , N Þ [1,2]. Here, we consider the case in which the point charge system composed of q1 , q2 , ... qN is distributed in a finite space. First, we imagine charge q1 to be moved to the infinity slowly (its kinetic energy ignored). At this time, the work done by electric field force is   q1 q2 q3 ::: qN 0+ + + + W1 = q1 ’1 = 4πε0 r12 r13 r1N Then, remove q2 and we have   q2 q3 ::: qN W2 = q2 ’2 = 0+0+ + + 4πε0 r23 r2N Finally, remove qN and we have W N = 0. The work done by electric field force equals the electrostatic energy storage of the original charge system fqi g.

https://doi.org/10.1515/9783110527407-001

2

1 Self-action energy in electrostatic field

We = W1 + W2 + ::: + WN =

N 1X qi ’i 2 i=1

(1:1)

In this equation, ’i =

N X

qj 4πε0 rij j=i+1

(1:2)

We can see in the above derivation that the electrostatic energy storage We of charge system fqi g only includes interaction energy. The electric potential of charge qi at this point is “infinitely” divergent, which is the intrinsic difficulty in the point charge model. Now, if we extend eq. (1.1) to the distributed charge system, namely *  dq ¼ ρ r ′ dv′, then we have We =

1 2

ððð

*  *  ρ r ′ ’ r ′ dv′

(1:3)

v

Now, different perceptions occur. For instance, literature [3] believes eq. (1.3) “includes not only the interaction energy among charges but also the self-action energy of the charge system.” In this chapter, we will prove that in the equation of stored energy We , the selfaction energy of electrostatic field is always zero. Either the self-action energy can not be calculated due to the intrinsic problem in the model under the condition of charges or linear charges or the self-action energy is still zero under the conditions of distributed charges. In addition, sextuple integral should be used to calculate ÐÐÐ *  *  ρ r ′ ’ r ′ dv′ when applying the source method. Or else, calculation We ¼ ð1=2Þ v

error should occur.

1.2 Self-action energy in the electrostatic field Now, we will study the self-action energy of electrostatic field in the distributed charge system, namely the self-action energy of We in eq. (1.3). First, we should be clear what self-action energy is, that is, the intrinsic energy in a system. The energy of * * the charge at r ′, defined as any point r ′ in the distribution systems, can be accumulated to be the self-action energy of this system. We can convert the definition above into calculating the self-action energy of a differential cube of length 2Δ × 2Δ × 2Δ. Volume charges density ρ is distributed uniformly in it. Then, we set lim as shown in Fig 1.1. Therefore, we only need to calculate Δ!0 the potential ’ in the center of cube with the length of 2Δ. Meantime, we propose that local coordinates are used for the differential cube, which are irrelevant to coordinates.

3

1.2 Self-action energy in the electrostatic field

ρ

2Δ

2Δ 2Δ

Fig. 1.1: Self-action energy of volume charges in a differential cube.

According to Appendix, the indefinite integral of a three-dimensional cube is ððð dxdydz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = xy lnðz + rÞ + yz lnðx + rÞ + xz lnðy + rÞ x2 + y2 + z2 (1:4)     1   1 2 1 2 − 1 yz 2 − 1 xz − 1 xy − z tan − x tan − y tan 2 xr 2 yr 2 zr If we set interval ½ − Δ, Δ, then we have ðΔ ðΔ ðΔ −Δ −Δ −Δ

pffiffiffi  pffiffiffi

dxdydz 3+1 3 −1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 12Δ 2 ln pffiffiffi = 9.52031069Δ2 − tan 2 2 2 3 3 − 1 x +y +z

(1:5)

Therefore, the central potential of the differential cube is ’ðΔÞ =

ρ × 9.52031069Δ2 4πε0

(1:6)

Its electrostatic energy storage is 1 9.52031069 2 5 We ðΔÞ = ’ðΔÞρΔν = ρΔ 2 πε0

(1:7)

Equation (1.7) represents the electrostatic energy storage of the differential cube (strictly speaking, we should set lim ). Next, we will add up the whole system in Δ!0 two cases.

1.2.1 Total electrostatic self-Action energy of uniformly distributed charges cube 2a × 2a × 2a First, we discuss the simplest scenario, that is, the total self-action energy of uniformly distributed charges cube 2a × 2a × 2a, as shown in Fig. 1.2. As we can see, this system can be obtained by adding up differential cubes, namely Wes = We ðΔÞ × N

(1:8)

4

1 Self-action energy in electrostatic field

2a ρ

2a 2a

Fig. 1.2: Uniformly distributed cube 2a × 2a × 2a.

where Wes is the self-action energy of the system. 

  3 2b 3 b = N= 2Δ Δ

(1:9)

Substituting eq. (1.9) into eq. (1.8), we have Wes =

9.52031069 2 3 2 ρbΔ πε0

(1:10)

Now let lim in the differential cube, and then we obtain Δ!0

lim Wes = 0

Δ!0

(1:11)

We know that the total electrostatic self-action energy of the uniformly distributed cube is zero.

1.2.2 Total electrostatic self-action energy of the charge system of any shape with volume V Figure 1.3 shows a charge system of any shape, volume being V. We introduce ρmax * to let the charge density at any point r ′ within V satisfy the following equation: *  (1:12) ρ r ′ ≤ ρmax *′

r 2v

V ρ ( r′) Fig. 1.3: Charge system of any shape, with its volume being V.

5

1.3 Field calculation and source calculation

When Δ is small enough, volume V includes N differential cubes and we know N=

V

(1:13)

ð2 ΔÞ3

Similar to Section 1.2.1, the total self-action energy of system V can be derived: Wes ≤

9.52031069 2 ρmax VΔ2 8πε0

(1:14)

Again, let lim , and we have Δ!0

lim Wes ≡ 0

(1:15)

Δ!0

Equation (1.15) proves that the self-action energy of the electrostatic field in the distribution charge (surface or volume distribution) system constantly equals zero. Stored energy We only reflects the interaction energy among charges.

1.3 Field calculation and source calculation There are two methods to solve the electrostatic energy We in continuous charge distribution systems: field method and source method, as shown in Table 1.1. Table 1.1: Two methods for calculating electrostatic energy.

Solution formula Advantages Disadvantages

Field method ÐÐÐ We = ð1=2Þ v εE 2 dν

Source method ÐÐÐ ÐÐÐ * * * We = ð1=2Þ v′ v ρð r ′Þ’ð r − r ′Þdvdv′

Only one triple integral is calculated

Only the charge area with source ρð r ′Þ is required Sextuple integral should be calculated

Integrals should be calculated in whole space. Stored energy density * exist wherever there is field E

*

If we compare the source calculation method in eq. (1.1) with eq. (1.3), then we have ððð *  * *  ’ r − r ′ dν (1:16) ’ r′ = ν *

*

where r and r ′ represent the position vectors of field point and source point, respectively. Therefore, we need to find the sextuple integral. Here, we take as a typical example the system with sphere radius R and the uniformly distributed charge density ρ. Two methods for solving the electrostatic energy are shown in Fig. 1.4.

6

1 Self-action energy in electrostatic field

ρ

o

R

Fig. 1.4: Electrostatic system with the sphere radius being R and the uniformly distributed charge density being ρ.

1.3.1 Field method *

For a spherically symmetrical problem, it is easy to find the electric fields E within and outside the sphere according to Gauss’s law. We have 8 ρr > > < 3ε0 ^r ð0 ≤ r ≤ RÞ * (1:17) E= > ρR3 > : ^ r ð r > R Þ 3ε0 r2 *

where ^r represents the unit vector in the r direction, as shown in Fig. 1.5. The total static energy of the system is calculated by the field energy density formula 21 ε0 E2 . E ρa 3ε0

Fig. 1.5: The electric field distribution under the action of

o

a

*

r

1 We = 2

E spherical charge distribution.

ððð ε0 E2 dν = We1 + We2

(1:18)

ν

where the static energy in the sphere is ðR We1 = 2π

ρ2 r 4 2πρ2 5 dr = R 9ε0 45ε0

0

And, the electrostatic energy of spherical charge system outside the sphere is

(1:19)

1.3 Field calculation and source calculation

ð∞ We2 = 2π R

ρ2 R 6 10πρ2 5 dr = R 2 9ε0 r 45ε0

7

(1:20)

Finally, the total energy is We =

4πρ2 5 R 15ε0

(1:21)

In terms of total charge Q, we get Q = ρν =

4 πρR3 3

(1:22)

and We =

3Q2 20πε0 R

(1:23)

It is necessary to emphasize that this method shows where there is an electric field, there is energy. Thus, the integral interval includes the whole space.

1.3.2 Source calculation method *  *  Through the interaction of ρ r ′ and potential ’ r ′ , the source calculation method is derived from eq. (1.3). Therefore, we only need to integrate the contribution of an *  element of charge over the charge region. The potential ’ r ′ is determined by the coordinate system shown in Fig. 1.6.

z

r′

R r

o y

Fig. 1.6: The coordinate system to determine * potential ’ð r ′Þ.

x

*

By using symmetry, it is assumed that r ′ is in the same direction as the z axis and without loss of generality. Therefore, we can write

8

1 Self-action energy in electrostatic field

*  ’ r′ ¼

ððð

* *  ’ r  r ′ dν =

ððð

ρr2 dr sin θdθd’ * *  4πε0  r − r ′ ν   ð π d r2 + r′2  2rr′ cos θ ð ρ R r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr = 2ε0 0 2r′ 2 0 r2 + r′ − 2rr′ cos θ ð    ρ R r  r + r′ + r − r′ dr = 2ε0 0 r′

ν

(1:24)

Note (

 jr − r′ = r − r′ r ≥ r′     r − r′ = − r − r′ r < r′

(1:25)

We can obtain *  ρ ’ r′ = 2ε0

(ð

r′ 0

2r2 dr + r′

ðR r′

) 2rdr =

  ρ 1 2 R2 − r ′ 2ε0 3

Electrostatic energy of spherical charge system is given as follows:  ðR 2 ððð  *  *  1 πρ 1 4 4πρ2 5 2 We = ρ r ′ ’ r ′ dν′ = R2 r′ − r′ dr′ = R 15ε0 2 3 0 ε0

(1:26)

(1:27)

v

Of course, we can also write We =

3Q2 20πε0 R

(1:28)

Obviously, eqs. (1.21) and (1.27) are same, so are eqs. (1.23) and (1.28). In the discussion of the above example, the electrostatic energy eqs. (1.21) and (1.23) are determined again by changing the perspective of the problem. Let lim ; R!0 therefore, eq. (1.21) corresponds to 4πρ2 5 R =0 R ≥ 0 15ε0

(1:29)

3Q2 !∞ R!0 20πε0 R

(1:30)

We = lim and, eq. (1.23) is We = lim

The above results also reflect that the self-acting energy is zero in the continuous distribution of charge ρ, which is the OðR5 Þ magnitude. Being the Oð1=RÞ magnitude, the result of point charge model eq. (1.30) is difficult to diverge.

Q&A

9

1.4 Summary The problem of self-function discussed in this chapter is not only widely concerned in the physical field but also can be further extended to the philosophical level. Way back in the fall of 1940, the famous American physicist Richard Phillips Feynman learned that the main problem lies in the infinity of electron self-acting energy in the electromagnetic theory due to the description of electrons as dot particles [5,6]. Feynman made a hypothesis: An electron cannot act on itself. After repeated discussions with his mentor John Archibald Wheeler, he finally wrote The Classical Theory of Action at Distance – the Counteraction of the Absorbent Objects Damping Resistance, which was published in Modern Physics Review in 1945. In the electromagnetics theory proposed by Feynman, the radiation damping is regarded as a counteraction on the source in advanced wave form. The most important feature of this new theory is that neither does there exist the electromagnetic field nor the action of the charge on itself, thus perfectly solving the problem. So what about the electrostatic field? According to relativity, because static should be relative, it is bound to put forward Lorenz four-dimensional invariant of the advanced and delayed wave. What are they? All kinds of problems around the self-acting energy (i.e., the inherent energy) are worth further consideration.

Q&A Q: Electromagnetic theory has already been a mature discipline. Its content, method and conclusion are relatively stable, so why do you still write this chapter? A: Electromagnetic theory is indeed a very mature and stable field and hundreds of Chinese and foreign works have been published. Even so, there is still a need to do further research and explore more. This chapter is to (1) Supplement new ideas and methods. Due to a variety of constraints, it is impossible for the formal teaching and work research to discuss the ideas and methods behind the problem freely, which is exactly the things a lot of learners want to know. (2) Supplement questions. Needless to say, there is a big misunderstanding in today’s ideological trend. So long as the writing is used as type, it will be the undoubted truth. On the contrary, a lot of new discoveries and new developments start from the difficulties and problems of existing theory and problems. This shows that the core point of any study is independent thinking. (3) Supplement recommended scholars. Recently, the Qian Xuesen’s question has caused great shock and anxiety in the intellectual community. Where are the Chinese first-class talents? How to develop? It should not be seen as a rush but a relay run in history after a deep thought. It is the contribution from Einstein,

10

1 Self-action energy in electrostatic field

Godel and Weyl that makes USA what it is today. Although our generation did not complete mission for various reasons of history, we can push the young people from generation to generation to the peak through the teaching and recommended characters. That is why I write the book, as shown in Fig. 1.7.

New ideas and methods

Difficulties and problems

Recommended scholars

Fig. 1.7: The main arrangement of the chapter.

Q: What is the special significance by putting self-action energy in the electrostatic field as the first chapter? A: There are two key words in this chapter – field and action, which are exactly the important idea and concept in electromagnetic theory. Field is one of the cores in the development of physics. Material and force are the initial understanding of physics for people, and Newton’s second law is a typical representation. *

*

F ¼ ma

The gravity of a piece of iron is actually felt in the human palm. The electromagnetic and Ampere law break through the above thought. Briefly, * the annular deflection of the magnetic needle occurs around the current I , as shown in Fig. 1.8.

I

*

Fig. 1.8: Annular deflection of the magnetic needle around the current I .

The young Faraday replaces the magnetic needle with magnetic powder subtly and * tries to partition the relation between current I and magnetic particle by hard paper, * only to find the field is pervasive, and a magnetic field is generated around the current I .

Q&A

11

The third stage of human understanding of physics is symmetry. The earliest study on symmetry was presented as a research method, and it gradually developed into the core idea of physics. A detailed discussion of this point will be mentioned later. Figure 1.9 represents three main ideas of physics development. Material and force

Field

Symmetry

Fig. 1.9: Three main ideas of physics development.

The field is exactly the core concept of electromagnetics. The second thought is action. Field is generated from source and in turn acts on * other sources. The charge q in the presence of field E will be affected by the electric * * * field force F , expressed in F ¼ qE , which is called first-order action of field. Similarly, when we study any field system, its secondary action embodies the energy in the system. Figure 1.10 vividly shows the image representation of the action.

First-order action of electrostatic field

Second-order action of electrostatic field

1 Field method We = 2

F = qE

Source method We = 1 2

2

ε E dν v

ρ ( r ′ )φ (r − r ′)dvdv′ v′

v

Fig. 1.10: The action of the field.

To sum up, the first chapter deals with two topics: field and action, as shown in Fig. 1.11.

Field

Action Fig. 1.11: The two topics involved in this chapter.

Q: What kind of difficult problems does this chapter deal with? A: It is well known that the action of the field system can be divided into self-action and interaction. This chapter focuses on the point – the self-action energy of electrostatic field always is zero. * Briefly speaking, the field E produced by charge q does not produce energy at that point. Generally speaking, it does not act on itself, or as the people say: “It’s

12

1 Self-action energy in electrostatic field

impossible to leave the earth when you grab your hair with great force!” By extension, it represents a profound philosophy of physics: Only by its interaction on other object can the value of things be reflected. The Nobel Prize winner Feynman was fully aware of the divergence (infinity) difficulty of electronic self-action energy, and he stated that an electron cannot act on itself.

Recommended scholar

Fig. 1.12: Richard Feynman.

On October 13, 1999, Reference News specially published an inspirational figure recalled by Nicholas Buss – a great physicist in the 20th century, Richard Feynman (1918–1988). James Gleick, a well-known American biographer, wrote his biography – Genius. Feynman’s research work has two major characteristics.

1. Not to chase for reputation but the practical research Since he was a child, Feynman had remembered a truth taught by his father that knowing the name of a thing doesn’t mean you really understand it. This was almost his lifetime motto. Once he was teased when playing with his friends in his childhood, for he didn’t know the name of a bird in the forest. His father told him that he could know what birds are called in every language. However, knowing the name of the bird definitely doesn’t mean you understand the knowledge of birds. Feynman kept it in mind. Not to chase for reputation but for practical research.

Recommended scholar

13

2. To think independently Feynman was a genius physicist with an innate desire for innovation as well as an outstanding talent. Feynman always remembered to innovate every field and not believe any old theory. He said he didn’t study physics textbook carefully during his school days. Feynman’s job was to analyze and summarize what people believed correct. This is what we call independent thinking. Here, we briefly introduce the third form of quantum mechanics created by Feynman – quantum least action principle – a process of path integration. His physics teacher Abram Bader told Feynman the concept and thought of least action – among all paths, only the action S in the true path is least, where Lagrange is Ðb both L = T − V and S = min a Ldt. In 1941, Feynman (23 years old) thought that the microscopic world in quantum mechanics should also meet the quantum least action principle, but there has been no significant progress. Feynman was thinking about that question at a beer banquet held in Princeton, where he was looking at the ceiling in a daze by some chance. At this point, Herbert Jehle, a European physics professor who escaped from concentration camp in Germany, approached him and asked: “What are you doing?” Feynman directly asked him, “Do you know if there is a theory using action as quantum mechanics?” He answered, “Not really, but it seems that in 1932, Dirac introduced the Lagrange L into the quantum mechanics in an article, and I can find it for you tomorrow.” Feynman was eager to solve the question on that day. But unfortunately, the library was closed. The next day, the two went to the library and soon found the related paper published by Dirac, which inspired Feynman to write ðX eiðS=hÞ ’ðxa , ta Þdxa dta ’ðxb , tb Þ = where ’ represents the wave function of the particle in quantum mechanics, and S is the action. In 1942, Feynman completed his doctoral dissertation titled The Principle of Least Action in Quantum Mechanics, which also created the third form in quantum mechanics – the path integral form. It should be noted that the key propagation factor for the minimum action in quantum mechanics is the complex exponential, and the action only appears on the exponential. Due to the dimensionless exponential, we are surprised to find the Planck constant h – which represents the minimum quantum action (i is used as the imaginary number in physics while in engineering j is often used). It does widen our horizons. Feynman’s growth process and his way of thinking inspires us a lot.

14

1 Self-action energy in electrostatic field

Appendix Cubic triple integral

ða ða ða I0 = −a −a −a

dxdydz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + y2 + z 2

We can divide the above problem into two parts: indefinite and definite triple integral. Case 1: Indefinite triple integral Indefinite triple integral is ððð f ðx; y; zÞdxdydz = F ðx; y; zÞ + Gðx; yÞ + H ðy; zÞ + I ðz; xÞ + J ðxÞ + K ðyÞ + Lð zÞ + C (A1:1) Specifically, in cubic (or cuboid) triple integral, Gðx, yÞ, Hðy, zÞ, Iðz, xÞ, JðxÞ, KðyÞ, LðzÞ and C can be omitted. It is easy to obtain ððð f ðx; y; zÞdxdydz = F ðx; y; zÞ (A1:2) Now, we study specifically

ððð I¼

dxdydz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + y2 + z2

(A1:3)

(1) First, we integrate x, that is, ð Ix = Assume

dx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + y2 + z 2

8 pffiffiffiffiffiffiffiffiffiffiffiffiffi > x = y2 + z2 shu > < pffiffiffiffiffiffiffiffiffiffiffiffiffi dx = y2 + z2 chudu > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi > : x2 + ðy2 + z2 Þ = y2 + z2 chu

(A1:4)

(A1:5)

After the substitution into (A1.4) and omission Hðy, zÞ related to y and z, we obtain ( Ix = lnðx + rÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (A1:6) r = x2 + y2 + z 2 (2) The integration of y divides the following integral into two parts. ð Iy = lnðx + rÞdy = Iy1 + Iy2 The adoption of partial integration leads to

(A1:7)

Appendix

8

> > pffiffiffiffiffiffiffiffiffiffiffiffiffi > < dy = x2 + z2 sec2 νdν pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 > ν > > x + y þ z = x + z sec >  : 2 2 2 1 2 y + z = cos2 ν x sin ν + z2 It is easy to obtain

ð −z x 2

xy dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = − z tan − 1 zr ðy2 + z2 Þ x2 + y2 + z2

Therefore, we obtain Iy = y lnðx + rÞ + x lnðy + rÞ − z tan − 1 (3) Finally, we integrate z

xy zr

(A1:11)

(A1:12)

ð I=

where

(A1:10)

Iy dz = I1 + I2 + I3

R 8 > < I1 = R x lnðy + rÞdz I2 = y lnðx + rÞdz >   R : I3 = − z tan − 1 xy zr dz

For the first item I1 , use partial integration again. ð xz2 dz I1 = xz lnðy + rÞ −  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 x + y2 + z2 y+ x +y +z For the second item in eq. (A1.15), we have

(A1:13)

(A1:14)

(A1:15)

16

1 Self-action energy in electrostatic field

−

Ð

xz2 dz  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  × pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 y+

x +y +z

Ð

dz − x3 y = xy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 x +y +z

= xy lnðz + rÞ − x3 y

Ð

Ð

x +y +z

ðx2 + z2 Þ

ðx2 + z2 Þ

pdzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + y2 + z2

(A1:16)

pdzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + y2 + z2

Similarly, we assume 8 pffiffiffiffiffiffiffiffiffiffiffiffiffi > z = x2 + y2 tan w > > > pffiffiffiffiffiffiffiffiffiffiffiffiffi > x2 + y2 + z2 = sec w × x2 + y2 > > > > : x2 + z2 = 1 x2 + y2 sin2 w

(A1:17)

cos2 w

We can obtain I1 = xz lnðy + rÞ + xy lnðz + rÞ − x2 tan − 1

yz xr

(A1:18)

According to the symmetry of x , y, the integration of the second part I2 can be written directly:   xz (A1:19) I2 = yz lnðx + rÞ + xy lnðz + rÞ − y2 tan − 1 yr The I3 third part of integral is ð ð xy xy   1 I3 = − z tan − 1 dz = − tan − 1 d z2 zr 2 zr

(A1:20)

We adopt partial integration again:   1ð h xyi 1 2 − 1 xy + z2 d tan − 1 I3 = − z tan 2 zr 2 zr

(A1:21)

The second item in eq. (A1.21) is ð ð xyi 1 2 h 1 ðzrÞ2 + z4 i dz z d tan − 1 = − xy h 2 zr 2 r ðzrÞ2 + ðxyÞ2

(A1:22)

  z4 = ðzrÞ2 − z2 x2 + y2

(A1:23)

Due to

Furthermore, we have

Appendix

ð ð ð 2 2 2 xyi 1 2 h dz 1 z ðx + y Þ + 2ðxyÞ2 dz z d tan − 1 = − xy + xy 2 zr r 2 r½ðzrÞ2 + ðxyÞ2 

  yz 1 1 xz = − xy lnðz + rÞ + x2 tan − 1 + y2 tan − 1 2 xr 2 yr

17

(A1:24)

Finally, we obtain   yz 1 xy 1 xz 1 − z2 tan − 1 I3 = − xy lnðz + rÞ + x2 tan − 1 + y2 tan − 1 2 xr 2 yr 2 zr It is concluded that the indefinite triple integral of a cube or cuboid is ððð dxdydz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = xy lnðz + rÞ + yz lnðx + rÞ + xz lnðy + rÞ I= x2 + y2 + z2   yz 1 xy 1 xz 1 − z2 tan − 1 − x2 tan − 1 − y2 tan − 1 2 xr 2 yr 2 zr

(A1:25)

(A1:26)

It can be seen that the result is perfectly symmetric for x, y and z. Case 2: Definite triple integral Considering symmetry, we have ða ða ða I0 = −a −a −a

 yz a  a  a dxdydz 1    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 3 yz lnðx + rÞ − x2 tan − 1 (A1:27) 2 xr  − a  − a  − a x2 + y2 + z2

Finally, we obtain  pffiffiffi  pffiffiffi 3+1 3 = 9.52031069a2 − tan − 1 I0 = 12a2 ln pffiffiffi 3 3−1

(A1:28)

2 Corresponding research between time-harmonic field and complex field The basis of the complex representation theory of the electromagnetic time-harmonic field lies in correspondence. In other words, the real part (or imaginary part) of the first-order quantity of electromagnetic corresponds to the time-harmonic field. This method is simple and clear, so it has been widely used. However, it must be pointed out that the complex representation of an electromagnetic second-order quantity can not be derived from the general form of the energy theorem. Therefore, many literatures avoid giving the complex Poynting theorem. This chapter gives a corresponding theorem of the complex representation of the quadric field and discusses the relevance of each of them, thus laying a solid foundation for the complex Poynting theorem, the Foster theorem and the Lorentz theorem.

2.1 Introduction This is the second chapter of Electromagnetic Field Theory Teaching Series. For such a mature electromagnetic field, its basic content does not change a lot with time. All kinds of teaching materials, along with numerous textbooks, are generally similar in content. However, we can find that there are still many problems worthy of our study and discussion. Here, we will focus on the relationship between the time-harmonic field and the complex field. The electromagnetic theory is based on the time domain Maxwell’s equation. Due to the strictly matched time domain and the frequency domain, Fourier transform should be adopted when time domain is transformed to frequency domain. However, the quadratic (such as power and energy) processing in the frequency domain will be very complicated. Thus, the general time domain electromagnetic field is transformed into a single frequency (ω) time-harmonic field in almost all of the electromagnetic theory books. we have *

Acosðωt + ’a Þ Furthermore, the use of complex field representation leads to *  * _ Acosðωt + ’a Þ = Re Aejωt

(2:1)

(2:2)

where * _

*

A = Ae j’a

https://doi.org/10.1515/9783110527407-002

(2:3)

20

2 Corresponding research between time-harmonic field and complex field

* _ Usually, ejωt is omitted and A is called the complex field. Obviously, the complex field is not the frequency domain field. Its entire fundamentation is only equivalent to the time-harmonic field in eq. (2.2), as shown in Fig. 2.1.

Time-domain electromagnetics

Time-harmonic electromagnetics

Complex expression

Fourier transformation

Frequency-domain electromagnetics

Fig. 2.1: Two methods for time domain electromagnetic field.

2.2 The complex field representation of time-harmonic field in first-order quantity The first-order quantity of time-harmonic field mainly refers to the field quantity (mainly vector), calculus operation and constitutive operator. It can be attributed to the following three criteria.

2.2.1 The criterion for the representation of real parts+ It is the basic expression of the complex field, eq. (2.2). It is easy to obtain *  *  _ _ Re Ae jωt = Re A* e − jωt

(2:4)

where * represents complex conjugation. Equation (2.4) shows that the conjugate complex field and the complex field represent the same time-harmonic field.

2.2.2 Calculus criterion In the complex field, the time-dependent function has all been attributed to ejωt . Thus,

+ Some references adop imaginary parts as the criterion. As longe as the criterion is uniformed in one reference,the two representations are equivalent.

2.3 Second-order quantity and complex field representation of the time-harmonic field

(

∂ ∂t

Ð

21

! jω

dt ! jω1

(2:5)

That is, the calculus operation is transformed into algebraic operation, which is also the fundamental reason for wide application and briefness of complex field.

2.2.3 Constitutive operator ^ are all operators. We have ^ and σ According to literature [7], in the most general case, ^ε, μ ^ε = ε1 + ε2

∂ ∂2 + ε3 2 +    ∂t ∂t

(2:6)

^ have similar representation. From eq. (2.2), we can see ^ and σ μ ^ε_ = ε′ − jε′′

(2:7)

where ε′ and ε′′ are positive real numbers (here, we do not discuss the left hand material), and the minus sign in the imaginary part is necessary condition to keep the energy nonnegative. Thus, it is easy to write the complex field representation of the time-harmonic Maxwell’s equations. The exaction of the real part leads to 8 * * _ _ *_ > > ∇ × H = J + jωD > > > * > _ _

_ > > ∇  B=0 > > > * : _ ∇  D = ρ_ For the isotropic lossless medium, we further have 8 * * _ *_ _ > > ∇ × H = J + jωε E > > > * > _ _

_ > ∇  B=0 > > > > * : _ ∇  D = ρ_

(2:9)

2.3 Second-order quantity and complex field representation of the time-harmonic field A great many scholars have been concerned about the difficulties in expressing the complex field of the second-order quantity in time-harmonic field. Literature [8]

22

2 Corresponding research between time-harmonic field and complex field

indicates that the law of time-harmonic field is only applicable to the addition, differentiation and integration of sine (cosine) current or voltage, and the linear combination of these operations. Thus, when current or voltage is used for high-order operations, such as multiplication or high power, this rule does not apply. As the most important second-order quantity energy Poynting theorem in the electromag* _ netic field, literature [1,2,4] only lists Poynting vector S in the complex form and does not give the Poynting theorem. Literature [9] clearly indicates that the complex energy theorem can not be derived from the general form of energy theorem (1.46)++. In this chapter, we study the correspondence between the second-order quantity and the complex field of the harmonic field from the following theorem. Theorem 2.1: There are two pairs of vector fields. 8 * < ~ ðiÞ = Ai cosðωt + ’ai Þ A : ~ ðiÞ * B = Bi cosðωt + ’bi Þ ði = 1; 2Þ Their corresponding complex fields are 8 * _ * < Ai = Ai ej’ai _ * :* Bi = Bi ej’bi ði = 1; 2Þ

(2:10)

(2:11)

If it satisfies ~ ð1Þ × B ~ ð1Þ = A ~ ð2Þ × B ~ ð2Þ+ A we have

8 * *  * *  _ _ _ _ > < 21 Re A1 × B1 * = 21 Re A2 × B2 * * *  * *  > : 1 Re A_ 1 × B_ 1 = 1 Re A_ 2 × B_ 2 2 2

(2:12)

(2:13)

Proof: It can be seen from the first-order quantity complex number in time-harmonic field *  *  * *  * *  ~ ×B ~ = Re A_ ejωt × Re B_ ejωt = 1 Re A_ × B_ * + 1 Re A_ × B_ e2jωt A 2 2

(2:14)

Integrate time t on both sides of eq. (2.14). [0, T] is the upper and lower limits, and T = ð2π=ωÞ is the time period. We have

++ Theorem (1.46) was misprined as (1-47) in the original book, and it is revised here.

2.3 Second-order quantity and complex field representation of the time-harmonic field

ðT 1 ~ ~ 1 *_ *_  A × Bdt = Re A × B* T 2

23

(2:15)

0

It can be seen that the right-hand side of eq. (2.15) represents the average of second quantity in a time field over a fundamental period T. It can also be written as   * 1 _ *_ * ~ ~ (2:16) A × B = Re A × B 2 In eq. (2.16),

−

represents the time-average value in one cycle, and due to 1 *_ *_ 2jωt  1 * * = A × B cos½2ωt + ð’a + ’b Þ Re A × B e 2 2 1* * = A × Bfcos 2ωt cosð’a + ’b Þ − sin 2ωt sinð’a + ’b Þg 2

(2:17)

ÐT Multiply cos 2ωt and integrate in one cycle dt on both sides of eq. (2.14). We have 0

ðT

1 ~ ~ 1  * * 1 *_ *_  A × B cos 2ωtdt = A × B cosð’a + ’b Þ = Re A × B T 4 4

(2:18)

0

It can be seen that we can get the conclusion of eq. (2.13) as long as the condition of eq. (2.12) is satisfied. In order to further study the essence of the theorem, we write eq. (2.14) again as *

*

*

~ ×B ~ = f 0 + f 1 cos 2ωt + f 2 sin 2ωt A where

8 > > > > >
>  * * > * > > : f 2 =  1 A × B sinð’a + ’b Þ 2

(2:19)

*

(2:20)

So, it can be said that the theorem indicates that their corresponding Fourier component will be unique when the vector function is determined. In the sense of real part representation, eq. (2.13) can be written as 8 _ *_  *_ *_

>

* > ∂H : ∇×E=  μ ∂t It is easy to obtain

8* * * * * * < E  ð∇ × H Þ = E  J + εE  ∂ E :

*

− ð∇ × E Þ 

*

*

H = μ ∂H ∂t

*

∂t

H

(2:22)

(2:23)

The use of the vector formula and complex representation of first-order quantity leads to n *  * o _ _ − ∇  Re E ejωt × Re H * e − jωt *  *  *    * _ _ _ _ = Re J * e − jωt  Re E ejωt + εRe E ejωt  Re − jω E * e − jωt

(2:24)

 *  *  _ _ þ μRe jωH ejωt  Re H * e − jωt According to Theorem 2.1, we have

  2  * 1  *_  1 *_ 2 _ 1 *_ *_ − ∇  S = J *  E + 2jω μH  − ε E  2 4 4

Its corresponding integral form is  ðð ððð ððð   2 * 1 *_ * *_ 1  *_  1 *_ 2 _ − ∇  S = J  E dv + 2jω μH  − ε E  dv 2 4 4 s

v

(2:25)

(2:26)

v

Equations (2.25) and (2.26) hold in the sense of the real part, representing the principle of conservation of average electromagnetic energy.

2.4.2 The Foster theorem for lossless one-port system The use of vector formula and complex representation of first-order quantity leads to

25

2.4 Three main theorems

   ∂ *  ∂ *   *  * _ jωt _ * − jωt _ * − jωt _ − ∇  Re E e × × Re H ejωt + Re H e Re E e ∂ω ∂ω   *  ∂ωμ *  *  ∂ωε

* _ _ _ _ + Re E ejωt  Re E * ejωt  = j Re H ejωt  Re H * e − jωt  ∂ω ∂ω

(2:27)

According to Theorem 2.1, it is easy to obtain !  * * _* _  * 2 ∂ωμ *2 ∂ωε * ∂ E * *_ _ _ _ ∂H + × H = j H   + E   −∇  E× ∂ω ∂ω ∂ω ∂ω

(2:28)

Its corresponding integral form is ! * * ðð _* _ * ∂ E * *_ * _ ∂H + × H  d s =  4jðwm + we Þ  E× ∂ω ∂ω

(2:29)

s

where 8 *2 _ > < we = 41 ε E   * 2 > _ : wm = 41 μH 

(2:30)

express the time-average of the stored electric and magnetic energies, respectively. From this, it follows that ∂X 4ðwm + we Þ >0 = ∂ω II*

(2:31)

That is, the reactance of one-port lossless network is always constant positive.

2.4.3 Lorentz’s reciprocity theorem The use of vector formula and complex representation of first-order quantity once again leads to n *  *  *  * o _ _ _ _ ∇  Re E 1 ejωt × Re H 2 ejωt − Re E 2 ejωt × Re H 1 ejωt (2:32) *  *  *  *  _ _ _ _ = Re J 1 ejωt  Re E 2 ejωt − Re J 2 ejωt  Re E 1 ejωt That is to say, n * *  * * o * *  * *  _ _ _ _ _ _ _ _ ∇  Re E 1 × H 2 e2jωt − Re E 2 × H 1 e2jωt = Re J 1  E 2 e2jωt − Re J 2  E 1 e2jωt (2:33) According to Theorem 2.1, it is easy to obtain

26

2 Corresponding research between time-harmonic field and complex field

n* * * * o * * * * _ _ _ _ _ _ _ _ ∇  E1 × H2 − E2 × H1 = J 1  E2 − J 2  E1

(2:34)

Its corresponding integral form is ððð  ðð    * * * _ *_ _ *_ *_ *_ _ *_ ^ds = E 2  J 1 − E 1  J 2 dv  E1 × H2 − E2 × H1  n s

(2:35)

v

According to literature [10], we know that eq. (2.35) is the complex form of Lorentz reciprocity theorem. Note that this is a second formula of eq. (2.21), which is not the conjugation field product, but the direct field product – the component of Fouriercos 2ωt. This point was not pointed out previously since it could be easily proved from Theorem 2.1.

2.5 Summary It is precisely because there are two formulas of eq. (2.21) in the complex expression of second-order quantity, so in the future, we can safely use the complex form of Maxwell’s equations for vector operations – be it conjugation product or direct product. This chapter proves even for a second-order quantity; there still exists the correspondence principle between time-harmonic field and complex field. All complex form theorems of electromagnetic second-order quantity hold.

Q&A Q: Could you talk about the goal of this chapter? A: This chapter still focuses on field – time-varying field instead of electrostatic field. * * * * It can be expressed as E = E ð r ; tÞ. In the expression, radius vector r represents space variation and t represents time variation. Q: Are the research ideas and methods of time-varying fields included in this chapter? * *

A: This chapter did not seem to deal with the time-varying field E ð r ; tÞ directly. In fact, be it time-harmonic field or complex field, the common purpose is to solve this problem. The most difficult thing in time-varying research is the randomicity of its variation. As there is also space variation in the static field, the focus of our research goes further to the randomicity of time variation. The solution to random variation is to use a function system composed of bases. This seems difficult, but in fact, it is pretty simple. We see high buildings in the city and houses in rural areas. Their sizes and patterns may be totally different, but after all, they are all made of bricks or plates. The bricks are just the bases, as shown in Fig. 2.2.

Q&A

27

Fig. 2.2: Bricks – the bases of construction.

Although the idea of bases seems simple, its development and extension are profound and deep. For example, all performances require a stage, but its scenery changes randomly. In 2012, CCTV made a revolutionary change on the stage artistic design of Spring Festival Gala – chief art designer Chenyan divided the whole stage into 304 “bases” which could be raised and lowered freely. He also added 6600 m2 LED to make the stage more colorful and ever-changing. When Zhang Mingmin was singing the song “My Chinese Heart,” “The Great Wall” made up of “bases” went up slowly. And when Li Yugang was on the stage, blooming peonies made glorious scenery (see Fig. 2.3).

Fig. 2.3: The idea of “bases” in CCTV’s stage design in 2012.

Of course, from the perspective of engineering and physics, the development and application of the bases idea and concept should have strict theoretical support. Simply put, if we have a base function system ffi ðxÞg, i = 1, 2, . . ., its linear combination approaches F(x). First, the object of our study is the linear world. In general cases, both Newton’s mechanical system and Maxwell’s electromagnetic system satisfy the condition. Second, the base system must be complete. Orthogonality (even normalization) defines inner product < , > in the research domain. It can be integration or superposition. ∞ P ai fi ðxÞ, that is, any function The complete condition can be expressed as F ðxÞ = F(x) can be approached by the superposition of bases. i = 1 The orthogonality condition is hfi ðxÞ, fj ðxÞi = 0 ði ≠ jÞ. Combined with the normalization condition, then we have hfi ðxÞ, fj ðxÞi = 1.

28

2 Corresponding research between time-harmonic field and complex field

In general, the base and base system enable us to study randomly changed functions easily. Q: Is time-harmonic field the base system idea of time-varying electromagnetic fields? A: Yes. Time-harmonic field is the Fourier variable single-frequency ω expression of   * * randomly changed time-varying field. We have E = E a cos ωt + ϕa . Its research and deduction is shown in Fig. 2.4.

Randomly changed time-varying field

Fourier transform of the frequency domain

Calculation and research of time harmonic field

Fourier transform of the frequency domain

Fig. 2.4: The research idea and method of time-harmonic field.

Note that this kind of transform and inverse transform can be various. For example, we can also apply Laplace transform and inverse transform. Q: Now that the time-harmonic field is a perfect solution to problems, why bother to propose the idea and method of complex field? A: Till now, we get to discuss the core of this chapter – the idea and method of complex field. Let us answer the first question: why do we propose complex number field when the time-harmonic filed is already perfect in solving problems? In the theory of engineering, the object of electromagnetic research can be divided into first-order quantity and second-order quantity (quantities with a higher degree are rarely involved). * * First-order quantity refers to force F and field E ; and second-order quantity refers to energy W and power P. Even in a linear system, second-order quantity is nonlinear. It is very difficult to use Fourier transform and time-harmonic field. This is why we introduce the complex field. Q: How does complex field correspond to time-harmonic field? A: The very foundation of correspondence is Euler equation, that is, ejx = cos x +   j sin x.  * * * * So, we have E a cosðωt + ’a Þ = Re E a ejωω , where E a = E a ej’a .

Recommended scholar

29



*

In general, we ignore ejωt and call E a the corresponding complex field. This chapter thoroughly discusses the corresponding expression of second-order quantity in time-harmonic field and complex field. In this way, the Poynting theorem represents the conservation of average electromagnetic energy when the second-order quantity takes the real part. Foster theorem for lossless network reflects that the slope of reactance is proportional to the average total stored energy, while the Loren field reciprocity theorem represents the cosð2ωtÞ component in Fourier frequency domain. So, the problems related to second-order quantity of electromagnetism are properly solved.

Recommended scholar There is no doubt that Euler should be recommended here (Fig. 2.5).

Fig. 2.5: Leonhard Euler.

Swiss mathematician Leonhard Euler was not only the most productive mathematician in the 18th century but also the most productive one throughout the history of mathematics. Euler’s complete work include 886 works and theses. The great mathematician Laplace told young scholars: “Read Euler, as he is a teacher to all.” There are so many things that we should learn from Euler. So, we just mention four aspects here.

1. Perseverance and diligence Those who are not familiar with mathematical history may mistakenly think that Euler accomplished all those works in good conditions. In fact, it was far from the truth.

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2 Corresponding research between time-harmonic field and complex field

Euler lost one of his eyes at the age of 28 and developed total blindness at 59. Even so, he overcame extreme difficulties with great perseverance and persisted with high-level mathematical research. He asked others to record his research findings by using his extraordinary memory.

2. Bold thinking He discovered the equation ejx = cos x + j sin x, which was later called Euler’s equation. Since the beginning of mathematical history, no one has done anything like Euler’s equation to unify trigonometric function and exponential function in complex domain. It was unprecedented. How did Euler find the equation? It has always been an attractive mystery. Here, we can make a bold speculation here. When Euler was studying complex number, ( cos x = 1 − 2!1 x2 + 4!1 x4 −    sin x = x −

1 3 3! x

+

1 5 5! x

− 

and ejx = 1 + ðjxÞ +

1 1 ðjxÞ2 + ðjxÞ3 +    2! 3!

he found the infinite amazing relation between them, that is, Euler’s equation. It is worth pointing out that Euler made great theoretical contributions to complex numbers. He also found that ðjÞ − j is a real number. ð jÞ − j = e − ðπ=2Þ = 0.2078795763

3. Continuous extraction From Seven Bridges Problem, he extracted a new subject – topology. Many practical problems are well known like the Seven Bridges Problem, as shown in Fig. 2.6. But Euler took this job and extracted a brand new subject from it – topology.

Island

Island

Fig. 2.6: The Seven Bridges Problem.

Recommended scholar

31

4. Super induction ability Euler obtained the polyhedral equation V + F − E ≡ 2, where V is the number of the vertexes, F is the number of faces, and E is the number of edges, as shown in Fig. 2.7.

Fig. 2.7: Polyhedral equation.

The famous American mathematician Polya commented: “Euler is unique. He made every effort to write down all evidence of induction carefully and in detail. His explanation was special and appealing, and led him to his discoveries.” We should learn from Euler, a teacher to all of us forever.

3 Transformation and unification of electrostatic field and constant current field This chapter deals with electrostatic field and constant current field. In the existing teaching materials, these two parts are completely independent and irrelevant. It is Einstein’s relativity that unifies them as there is no absolute motion or absolute rest. * In this chapter, current J and charge ρ are regarded as unified four-dimensional * vectors. Research shows that in the moving coordinate system x′Oy′, current J can exist in dielectric medium, and charge ρ can exist in conductive medium (though very small). This is the transformation and unification of electrostatic field and * constant current field. The paradox of Poynting vector S is also covered in this chapter.

3.1 Introduction This is the third chapter of Electromagnetic Field Theory Teaching Series. The topics are electrostatic field and constant current field. It is known to all that Faraday and Maxwell, two outstanding scientists, connected electricity with magnetism for the first time and created the famous Maxwell’s equations. However, the electrostatic field and constant current field are independent of each other till now. The field produced by a stationary charge is electrostatic field, and the field produced by constant current caused by the uniform motion of electric charges is constant current (electric or magnetic) field [11]. According to Einstein’s special theory of relativity, there is no absolute motion or absolute rest [12]. Therefore, these two kinds of field can be transformed to each other and possess uniformity at the same time. The present teaching materials divide the medium environment in research into dielectric medium and conductive medium. The dielectric medium is the environment that electrostatic field is located in, while the conductive medium is the * environment of constant current field. Specifically, in dielectric medium, J = 0; while in conductive medium, ρ = 0; as shown in Table 3.1 [3]. Using the Lorenz transformation in special theory of relativity, this chapter shows that the electrostatic field and constant current field in the moving coordinate system x′Oy′ are no longer isolated from each other. There exists transformation and * unification of them in the idea of relativity. The paradox of Poynting vector S is also discussed in this chapter.

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3 Transformation and unification of electrostatic field and constant current field

Table 3.1: Electrostatic field and constant current field. Electrostatic field (environment: dielectric medium) *

Constant current field (environment: conductive medium)

J =0

ρ=0

∇  D=ρ

∇×H= J

D = εE

J = σE

*

*

*

*

*

*

*

3.2 The transformation and unification of electrostatic field and constant current field As mentioned above, electrostatic field and constant current field are two independent subjects in the present study of electromagnetics. The medium environments of them are different. So, we connect them by using Maxwell’s equation. *

J ∇  D =~ *

∇  D=ρ

(3:1) (3:2)

We will discuss the specific medium environment in different situations. In this chapter, we use a simple example as an assumption to grasp the essence of the problem. Assuming that there are current J and charge ρ in the z-direction of the coordinate system xOy (in fact, only one of the two can appear, which we will discuss later). In addition, there is coordinate system x′Oy′, in which current moves in the * opposite direction of z-direction with constant velocity of v ; as shown in Fig. 3.1.

zz′ J or ρ x

O y

x′

O′ y′

v

Fig. 3.1: Coordinate system xOy and coordinate system x′O′y′ doing relative motion with it.

For convenience, we simplify the four-dimensional problems into two-dimensional problems without changing its essence. So, we can write out 

*

∇×H

z

=J

(3:3)

3.2 The transformation and unification of electrostatic field and constant current field

 * jc ∇  D = jcρ

35

(3:4)

*

If we take the z component of ∇ × H in the left side of eq. (3.3), then J in the right side is a scalar, and c in eq. (3.4) represents the speed of light. All we need to do is multiply two-dimensional Lorentz transformation factor to transform the coordinate system xOy into coordinate system x′O′y′; that is, [12] " 1 1 L = qffiffiffiffiffiffiffiffiffiffiffi 2 jβ 1−β

− jβ

# (3:5)

1

where β = ðv=cÞ: Since the direction of v in eq. (3.5) is opposite to z direction, we can transform the right side of eqs. (3.3) and (3.4) into "

#

" 1 1 = qffiffiffiffiffiffiffiffiffiffiffi ′ 2 jβ jcρ 1−β J′

− jβ 1

#"

J

#

jcρ

(3:6)

Consider the practical condition of β ≪ 1, and we have J ′ = J + vρ

(3:7)

v ρ′ = ρ + c2 J

(3:8)

Now, Maxwell’s equation of the moving coordinate system x′O′y′ is  *  ∇′× H ′ ′ = J + vρ 

z

* v ∇′  D′ = ρ + 2 J c

(3:9) (3:10)

Note that 

*

∇′ × H ′



 z

  * * = ∇ × H + v ∇  D ′ z

*  * * v ∇′  D′ = ∇  D + 2 ∇ × H z c

(3:11) (3:12)

Keep in mind that the medium environments of these two fields are different. J and ρ cannot appear in eqs. (3.7) and (3.8) at the same time. So, we discuss them separately. Case 1: Dielectric medium In this medium environment, σ = 0; namely, J = σE = 0. So, J ′ = vρ

(3:13)

ρ′ = ρ

(3:14)

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3 Transformation and unification of electrostatic field and constant current field

The electric charge density ρ′ is unchanged in the moving coordinate system. But the equivalent current density J ′ = vρ appears in dielectric medium. The moving of charges can change an electrostatic field into constant current field, though the medium corresponds to dielectric medium. Now, we take a further step and discuss the relation between scalar electric * potential ’ and vector magnetic potential A in general cases. As ððð

ρdv′ r

(3:15)

ððð * ′ J dv r

(3:16)

1 ’= 4πε

v

μ A= 4π

*

v *

*

Using general relation J = ρv ; it is easy to obtain *

*

β ’ = cA

(3:17)

where *

*

v c

β=

(3:18) *

In addition, there is a relation between electric field intensity E and magnetic flux * density B; that is, ððð * * 1 ρr ′ E= dv (3:19) r3 4πε v

ððð * * * μ J×r ′ dv B= r3 4π

(3:20)

v

Similarly, we can obtain *

*

*

β × E = cB

(3:21) *

*

*

*

Based on the deduction above, it is clear that ρ and J ; ’ and A; and E and B are all mutually transformable. In this sense, the static electric field and constant current field are unified. Case 2: Conductive medium In this case, ρ = 0: So, we have J′ = J ρ′ =

v J c2

(3:22) (3:23)

*

3.3 Paradox of Poynting vector S

37

In other words, the current density J ′ in the moving coordinate system is unchanged. But the equivalent density ρ′ = ðv=c2 ÞJ appears in the conductive medium. It must be pointed out that in engineering v=c2  1; so ρ′ 0

(3:24)

That is to say, the transformation of these two fields is asymmetrical. In dielectric medium, the transformation from charge to current is obvious, while in conductive medium, the transformation from current to charge is scarcely evident. *

3.3 Paradox of Poynting vector S

Here, we discuss the specific concept again in depth. In the dielectric medium, if * * there are J ′ and ρ′ in the moving coordinate system, there must be E and H : In the case of 1, cylindrical coordinate system can be used and we have *

E=

*

H=

ρ ^r 2πεR

(3:25) *

ρv ρv × ^r ^=  ’ 2πR 2πR

(3:26) *

^ represent the unit vectors in the r and ’ direction, respectively, and v is where ^r and ’ in − z direction, as shown in Fig. 3.2. z′

S O′

H R

y′ E v *

Fig. 3.2: Poynting vector S of the electrostatic field in the moving coordinate system.

x′

So, the Poynting vector is given. *

*

*

S =E ×H= 

*

ρ2 ð^r × ^v × ^rÞ ρ2 v =  4π2 εR2 4π2 εR2 *

(3:27)

Equation (3.27) shows that the Poynting vector S is in the z′ direction. So, we * encounter a strange proposition: How can there be the Poynting vector S in the electrostatic field?

38

3 Transformation and unification of electrostatic field and constant current field

*

After further consideration, the Poynting vector S is found to be an indirect physical quantity defined by humans. What really makes sense should be the * ÐÐ * ^dS of S : closed surface integral S  n * s Obviously, S is perpendicular to the normal line of cylindrical side, so its contribution to the integral is zero. On the other hand, the curved area it forms with the top and bottom surfaces is 0, so there is no contribution to the integral. Finally, we have ðð * ^dS = 0 (3:28) Sn s *

In short, as long as the closed integral eq. (3.28) is 0, S has no real meaning even if it * exists. This is the answer to the paradox of Poynting vector S :

3.4 Summary This chapter has dealt with the electrostatic field and constant current field. It is shown that there may be a current in the case of a dielectric in a moving coordinate system. There may be a charge (although extremely small) in the conducting medium. This is the mutual transformation and unification between the two fields. The chapter has also discussed the paradox of Poynting vector.

Q&A Q: Is this chapter a discussion of the field again? A: Exactly, it is still about the field. However, Chapter 2 focuses on the new ideas and new methods brought about by changing from time-invariance to time-variance. This chapter discusses the changes brought about by changing from the “rest” to “motion” (of course, rest is relative). The focus of the discussion is the source. As we all know, there are two sources, * that is, charge ρ and current J – the rest is charge ρ, and the charges in motion * constitute current flow J : The ideology and method of the discussion is Einstein’s special theory of relativity. In this view, the electrostatic field and constant current field can be transformed to each other and therefore are unified.

Q&A

39

Q: Could you please introduce Einstein’s special theory of relativity in the simplest language? A: Einstein’s study on the special theory of relativity resulted from one thought when he was 16 – What would one see if he rode on a beam of “light”? Then, a further study derived the two “axioms,” as shown in Fig. 3.3.

The two axioms of Einstein’s special theory of relativity (1) The axiom of general relativity The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity) (2) The axiom of constant speed of light The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source.

Fig. 3.3: Two axioms of Einstein’s special theory of relativity.

Viewing in the perspective of philosophical logic, the system composed of these two axioms is pretty interesting. The first axiom is relative. People cannot judge whether the coordinate system where they stay is static or moving. In other word, there is no absolute rest or absolute motion. And, this is the axiom of general relativity in physics. The second axiom is absolute. The speed of light remains unchanged in every coordinate system moving relatively at a constant velocity. And, this is the axiom of constant speed of light. This idea inspired us a lot in our research work. Einstein advocates the relativity of things. In addition, he proposed and grasped an absolute standard. The theory of relativity is built on the basis of invariability of light speed. Q: Einstein’s special theory of relativity summarized above is profound, rigorous and perfect. To me, it seems that Einstein had done all the work in this field. Is there anything we can add? A: This is a good point. We know that all problems “fear” professionals. After Einstein proposed the theory of relativity, a professional stood out, his teacher Minkowski. He got one key point. That is, in Einstein’s theory of relativity, space ð x y z Þ and time ðtÞ are no longer mutually independent. On the contrary, they are closely connected with each other. At the time when Einstein enjoyed a high reputation, Minkowski proposed another structure of the theory of relativity in 1907 (43 years old). The main contents of his theory are as follows: (1) Add factor jc to time coordinate t, that is,

40

3 Transformation and unification of electrostatic field and constant current field

(

t ! jct t′ ! jct′

(2) Minkowski proposed the four-dimensional world. 8 2 3 x > > > 6 7 > > > 6 y 7 > > 7 ξ =6 > > 6 z 7 > > 4 5 > > > < jct 3 2 > x′ > > > 6 ′ 7 > > 6 y 7 > > 7 > ′ ξ =6 > > 6 ′ 7 > z > 5 4 > > : ′ jct It is clear that ct and ct′ are covered in the expression and have the same dimension with length. In addition, there is also the speed of light c in the four-dimensional world. (3) Minkowski wrote the Lorentz transformation further in the form of matrix. When the coordinate system moves along the x-axis, 2 3 1 pffiffiffiffiffiffiffiffi 0 x′ 6 1 − β2 6 ′ 7 6 6 y 7 6 0 1 7 6 6 6 ′ 7=6 0 0 4 z 5 6 4 jβ ′ p ffiffiffiffiffiffiffiffi 0 jct 2 2

1−β

0 0 1 0

3 jβ 2 3 p−ffiffiffiffiffiffiffiffi x 2 1−β 7 76 7 0 76 y 7 76 7 6 7 0 7 74 z 5 5 1 pffiffiffiffiffiffiffiffi jct 2 1−β

where β = ðv=cÞ and 2 6 6 6 L=6 6 6 4

1 pffiffiffiffiffiffiffiffi 1 − β2

0 0 jβ pffiffiffiffiffiffiffiffi 1 − β2

0 0

jβ p−ffiffiffiffiffiffiffiffi 1 − β2

1

0

0

0

1

0 1 pffiffiffiffiffiffiffiffi

0 0

3 7 7 7 7 7 7 5

1 − β2

thus forming the Lorentz complex matrix. (4) Minkowski seized the point that system of special theory of relativity is linear and space–time unified. The most amazing thing is that Minkowski introduced complex number into relativity.

Q&A

41

In Minkowski’s four-dimensional world, 8 2 3 x > > > 6 7 > > > 6 y 7 2 2 2 22 > > 7 ξ T ξ = ½ x y z jct 6 > > 6 z 7=x +y +z −c t > > 5 4 > > > < jct 3 2 > x′ > > > 7 6 > > >  6 y′ 7 > 7 = x′2 + y′2 + z′2 − c2 t′2 > ′T ξ ′ = x′ y′ z′ jct′ 6 ξ > > 7 6 > > 4 z′ 5 > > : jct′ ξTξ = ξ′ ξ′ T

where four-dimensional length is kept constant, and T represents the matrix transpose. It is easy to derive that Lorentz transformation satisfies LT = L − 1 where −1 represents a matrix inversion. In Minkowski’s theoretical system, the Lorentz complex transformational matrix is the core. In the text of this chapter, for convenience, the non-motion y and z coordinates are omitted, so that the L matrix is simplified as " # 1 − jβ 1 L = qffiffiffiffiffiffiffiffiffiffiffi 1 1 − β2 jβ Q: Minkowski transformed Einstein’s special theory of relativity to the matrix form by using complex numbers, which is a big leap in mathematics. A: This view is a little bit formal. In fact, Minkowski proposed a “structural relativistic system,” as shown in Fig. 3.4. The ideas of this system and those of Einstein’s “axiomatic relativistic system” are independent of each other. Minkowski’s structural relativistic system (1) The world is four-dimensional. (2) The world is complex Lorentz unchanged. Fig. 3.4: Minkowski’s structural relativistic system.

As far as the author knows, Mr. Shu Xingbei was the first to put forward this explicitly. In his works [12], the theory of relativity consists of two parts: Einstein’s theory of relativity and Minkowski’s theory of relativity.

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3 Transformation and unification of electrostatic field and constant current field

Shu Xingbei’s tragic experience is regrettable after returning to his home country (see Shu Xingbei Archives). Q: Could you discuss the specific applications of Minkowski’s structural relativity deeply? *

A: Looking back, we find the electrostatic field source ρ and the motion source J in Minkowski’s four-dimensional space constitutes a four-dimensional current vector and charge vector. " # " # J J′ and jcρ jcρ′ They are transformed to each other by complex Lorentz transformation. If we further introduce, 8 1 ~ > < cos θ = pffiffiffiffiffiffiffiffi 1 − β2 jβ > : sin ~θ = pffiffiffiffiffiffiffiffi 2 1−β

where ~θ is the argument. Obviously, the above equation satisfies cos2 ~θ + sin2 ~θ ≡ 1 Now, we write

"

cos ~θ

− sin ~θ

sin θ

cos θ

cos ~θ = jcρ′ sin ~θ

− sin ~θ cos ~θ

L=

#

Then, "

J′

# "

#"

J

#

jcρ

That is, in two different uniform coordinate systems, the four-dimensional currentcharge vector only has a complex rotation, as shown in Fig. 3.5. jcρ ′

jcρ

~ θ

J

~ θ

J′

Fig. 3.5: Diagram of complex rotation of four-dimensional current-charge vector.

Recommended scholar

43

It’s easy to write (

J ′ = cos ~θðJÞ − sin ~θðjcρÞ jcρ′ = sin ~θðJÞ + cos ~θðjcρÞ

Among the four-dimensional vectors, J and jcρ are unified and so are J ′ and jcρ′. Their differences are caused by the complex rotation, as shown in Fig. 3.5.

Recommended scholar Einstein has been introduced in plenty of literature, so here we recommend Minkowski (Figure 3.6), who is really worthy of an in-depth discussion.

Fig. 3.6: German mathematician Hermann Minkowski.

1. The way Minkowski saw frontier physics mathematically is worth learning Minkowski attached much weight to Einstein’s theory of relativity after its publication in 1905. He organized a special research session at Göttingen University to discuss new developments after the emergence of relativity. Minkowski made his study by combining Einstein’s work and Poincare’s work and was surprised to find that Poincare had proposed the idea of taking time t as the fourth dimension in 1906. (Minkowski was lucky that Poincare did not go further to consider visualized geometry and AQ the fourth dimension jct) So, he began to study seriously the papers of his student, Einstein, whom he had called “lazy dog” in linear algebra class. Minkowski, a genius, reacted immediately and strongly to it, “I have never thought that Einstein would be so capable!” This makes the main style of Minkowski and also the traditional style of Göttingen.

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3 Transformation and unification of electrostatic field and constant current field

2. Minkowski showed remarkable endurance in research A prominent example was with Hilbert. When speaking at the mathematics conference in 1900, Hilbert did not feel ashamed to consult his subordinate Minkowski as to what theme would be promising. Minkowski unequivocally pointed out that some mathematical issues worthy of studying in the 20th century should be addressed at the meeting, hence the historic report Twenty Unresolved Mathematical Problems. It should be noted that a report of such a magnificent kind did not recur at the conference of mathematics of 2000. 3. Minkowski, introducing complex number into relativity, achieved stunning success In this chapter, we realize once again the great significance of complex numbers. Particularly noteworthy is that in the system of relativistic complex numbers, the squared second-order quantity length, a quadratic, is not the Hermite (i.e., ½ T or   T ½ T ) but directly the squared transposed complex length, ξ T ξ or ξ ′ ξ ′. The system’s deeper function is worth investigating since it has magically overcome the weaknesses of the complex two-dimensional space and introduced the real three- or even four-dimensional space, which is unprecedented in the world of numbers. Meanwhile, the reason for which time jct is taken as imaginary has a profound philosophical meaning as well as mathematical significance. Unfortunately, in the prime of his life, Minkowski died suddenly at the age of 45 on the operating table for appendicitis. Certainly, all these also bring us opportunities and new challenges.

4 Charge multipoles and current multipoles This chapter focuses on the source (charge and current) multipole expansion in * the small source area of V and in the far field of R0 . The use of the unified three-dimensional generalized Taylor expansion together with the operator theory of vector ∇ leads to a series of conclusions of charge multipoles and current multipoles. Furthermore, the specific matrix expression of multipoles and the multipole expansion of four-dimensional current vector in the theory of relativity are discussed in this chapter. With clear concepts, standard deduction and concise conclusion, this chapter is a new attempt to study the multipole expansion theory.

4.1 Introduction This is the fourth chapter of Electromagnetic Field Theory Teaching Series. Multipole expansion is one of the recognized difficulties of electromagnetic teaching. However, it is very important in practical applications. Therefore, researchers have always been concerned about it. This chapter adopts the unified three-dimensional generalized Taylor expansion and the operator theory of vector ∇, leading to a series of conclusions of charge and current multipoles. Furthermore, this chapter discusses the specific matrix expression of multipoles and the multipole expansion of four-dimensional current vector in the theory of relativity. With clear concepts, standard deduction and concise conclusion, this chapter is a new attempt to study the multipole expansion theory.

4.2 Generalized taylor expansion of f (R) The multipole expansion in electromagnetic theory applies specially to the situation where the source area V is small and the desired field is far, as shown in Fig. 4.1. 9 * * * R = R0 − r ′ > = * (4:1) R0 = x^i + y^j + z^k > ; * r ′ = x′^i + y′^j + z′k^

R=

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðx − x′Þ + ðy − y′Þ + ðz − z′Þ

*

*

(4:2)

where R0 is called the field vector and r the source vector, so the most important constraint for the multipole expansion is

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4 Charge multipoles and current multipoles

z

(x,y,z) R0

o

x

V

R

y

r′ Fig. 4.1: Multipole expansion in the small source area of V and in the far field.

*      R0   
r′

(4:3)

The general scalar three-dimensional function f ðRÞ can be transformed into generalized Taylor expansion   1 **  *  f ðRÞ = f ðR0 Þ + r ′  ∇′f ðRÞ 0 + ð r ′ r ′Þ : ∇′∇′f ðRÞ 0 +    2!

(4:4)

where ∇′ is the operator which reacts to the source. ∂ ^ ∂ ^ ∂ +j +k ∇′ = ^i ∂x′ ∂y′ ∂z′

(4:5)

lim ; the origin of the problem model. This section sets up a “stage” 0 represents *′ ri ! 0 for the multipole expansion, and then we can study charge and current multipoles efficiently.

4.3 Charge multipoles Assume that there is a group of point charge system fqi ; i = 1; 2; . . . ; ng in the small source area of V and that the origin of the coordinate is set within V, as shown in Fig. 4.2. The electric scalar potential of point ðx; y; zÞ in the far field is n   1 X qi (4:6) ’= 4πε i = 1 Ri where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðx − xi′Þ + ðy − yi′Þ + ðz − zi′Þ 1 By using the generalized Taylor expansion and f ðRÞ = , we obtain [13]: R Ri =

(4:7)

4.3 Charge multipoles

z

47

(x,y,z) R0

Ri

qn q1 q2

y

o q3

ri

qi

x

Fig. 4.2: Point charge system fqi g in the small source area of V.

 

1 1 *′ 1 1 *′ *′ 1 ′ ′ ′ +  = + ri  ∇ + ri ri : ∇ ∇ Ri R0 R 0 2 R

(4:8)

The combination of the operator theory of vector ∇ leads to [14] *   ^0 1 R0 R ∇′ = 3 = 2 R 0 R0 R0 " ! * #  

  * 1 R 1 1 h *i = ∇′ 3 = R0 ∇′ 3 + 3 ∇′ðRÞ ∇′∇′ 0 R R 0 R R0 0

(4:9)

(4:10)

0

where *  

1 3R0 ′ ∇ = R3 0 R50

(4:11)

and h

*

∇′ðRÞ

i

   i     ∂ ^ ∂ ^ ∂ h ^ +j +k x − x′ ^i + y − y′ ^j + z − z′ ^k = i 0 ∂x′ ∂y′ ∂z′   * * =  ^i^i + ^j^j + ^k^k =  I * *

^ 0 is the unit vector, and I is the unit tensor. Finally, we have In the above derivation, R 

1 ∇′∇′ R



* *

^0R ^0 − I 3R = R30 0

(4:13)

The substitution of this equation into the electric scalar potential expression (4.6) gives       n 1 X 1 *′ 1 1 *′ *′ 1 ′ ′ ′ qi + ri  ∇ + +  ’= ri ri : ∇ ∇ 4πε i = 1 R0 R 0 2 R 0 (4:14) = ’ð0Þ + ’ð1Þ + ’ð2Þ +   

48

4 Charge multipoles and current multipoles

Clearly, the zero-order term of the generalized Taylor expansion is q 9 ’ð0Þ = 4πεR = 0 n P ; q= qi

(4:15)

i=1

q represents the total charge of the system in the source area of V, equivalent to the concentration of the sum of all point charges in the small area of V on O, the origin of the coordinate. Furthermore, the first-order term is 9   * n X ^0 > 1 1 pR * ð1Þ > ′ ′ = = ϕ = qi r i  ∇ 2 4πε R 4πεR 0 0 i=1 (4:16) n > P * * > ; p= qi r i′ i=1 *

p represents the total electric dipole moment of the system in the source area of V, and ’ is contributed by electric dipoles. Finally, the second-order term of ’ is * *!    n  * ^0R ^0 − I * 1 X 1 *′ *′ 1 1 3 R ð2Þ = (4:17) qi r i r i : ∇′∇′ M0 : ’ = 4πε i = 1 2 R 0 4πε R30

where * *

M0 =

n 1X * * qi r i′ r i′ 2 i=1

(4:18)

From literature [14], it can be known that   * *  2 1 ′ =  4πδ R0 − r ′ ∇ R

(4:19)

*

*

In view of r being in the source area of V while R0 being in the far field, namely  * * * * R0 ≠ r while δ R0 − r ≡ 0; we have   2 1 ≡0 (4:20) ∇′ R We can obtain * *

I :



   1 2 1 = ∇′ =0 ∇′∇′ R R

(4:21)

So, ’ð2Þ in eq. (4.17) can be rewritten as ð2Þ

’ * *

And for the new M; we have

* 1 * = M: 24πε

* *!

^0R ^0 − I 3R R30

(4:22)

4.4 Matrix expression of multipoles

n  ** * X * qi 3 r i′ r i′ − ri2 I

* *

M=

i=1

49

(4:23) * *

This is called the quadrupole moment of the system. Tensor M has only five independent components, and its trace (or the sum of the diagonal elements) is zero, that is, Mxx + Myy + Mzz ≡ 0

(4:24)

For the continuous distribution of charges in the source area, it corresponds to ÐÐÐ 9 ρðx′; y′; z′Þdv′ q= > > > v > = ÐÐÐ * * ′ ′ ρ= ρ r dv (4:25) v > >   * * > ÐÐÐ * 2* > ** ρ 3 r ′ r ′ − r′ I dv′ ; M= v

4.4 Matrix expression of multipoles In the multipole expansion, suppose that ðl; m; nÞ is the direction cosine of the   position vector of ðx; y; zÞ; and that x′i ; y′i ; z′i is the coordinate of the corresponding point charge qi ; which is *

*

R0  ri′ ^ cos θi′ = = R0  R0 ri′

*  ri′ lxi′ + myi′ + nzi′ = ri′ ri′

(4:26)

By literature [15], we list the Legendre polynomials, 9 > =

P0 ðcos θÞ = 1 P1 ðcos θÞ = cos θ P2 ðcos θÞ =

1 2 2 ð3cos

θ − 1Þ

> ;

and the first kind of associated Legendre polynomials, 9 > P10 ðcos θÞ = cos θ > > > > 1 > P1 ðcos θÞ = sin θ > = 2 0 3 P2 ðcos θÞ = 1 − 2 sin θ > > > P21 ðcos θÞ = 3 sin θ cos θ > > > > ; 2 P2 ðcos θÞ = 3 sin θ

(4:27)

(4:28)

It can be written as ’ð0Þ =

qp0 ðcos θÞ 4πεR0

(4:29)

50

4 Charge multipoles and current multipoles

" n P 1 ð1Þ qi x′i ’ = 2 4πεR0 i = 1

n P

qi y′i

i=1

n P i=1

2 3 # P11 ðcos θÞ cos ’ 7 qi z′i 6 4 P11 ðcos θÞ sin ’ 5

(4:30)

P10 ðcos θÞ

and ’ð2Þ = where

 2  n 2P 2 qi 3x′i − r′i 6 i=1 6 n 6 P qi 3x′i y′i ½ξ  = 6 6 i=1 6 4 n P qi 3x′i z′i

n P

6 ½η¼6 4

2

i=1

i=1 n P i=1

P22 ðcos θÞ cos 2’−2P20 ðcos θÞ 1 2 2 P2 ðcos θÞ sin 2’



qi 3x′i y′i

 2  n P 2 qi 3y′i − r′i

i=1

2 1

1 ½ξ  : ½η 24πεR30

qi 3y′i z′i

(4:31)

n P

qi 3x′i z′i

3

7 7 7 7 qi 3y′i z′i 7 i=1 7 5   n P 2 2 ′ ′ qi 3z i − r i i=1 n P

(4:32)

i=1

3 P21 ðcos θÞ cos ’   7 − 21 P22 ðcos θÞ cos 2’þ2P20 ðcos θÞ P21 ðcos θÞ sin ’7 5

P21 ðcos θÞ cos ’

pt21 P22 ðcos θÞ sin 2’ P21 ðcos θÞ sin ’

2P20 ðcos θÞ (4:33)

½ξ : ½η represents the trace operation of two 3 × 3 matrices, that is, the sum of location elements of ½ξ multiplied by corresponding elements of ½η:

4.5 Current multipoles Completely symmetrically, assume that there is a group of current element system n o Ii^li ; i = 1; 2; . . . ; n in the small source area of V, where ^li is the unit vector, as shown in Fig. 4.3. The vector magnetic potential at the point of ðx; y; zÞ is ! n * μ X Ii^li A= 4π i = 1 Ri Likewise, the use of the generalized Taylor expansion leads to       n * μ X 1 *′ 1 1 *′ *′ 1 ^ ′ ′ ′ Ii l i + ri  ∇ + +  A= ri ri : ∇ ∇ 4π i = 1 R0 R 0 2 R 0 *ð0Þ

=A

What is different is

*ð1Þ

+A

*ð2Þ

+A

+ 

(4:34)

(4:35)

51

4.5 Current multipoles

z

(x,y,z) R0

I1 lˆ1

I 2lˆ2

Ri

y

o

ri′ Ii lˆi

x

n o Fig. 4.3: Current system Ii^li in the small source area of V.

9 ^ > μI L = = 4πR 0 n P ^= ; IL Ii^li > *ð0Þ

A

(4:36)

i=1

^ represents the vector sum of the total equivalent current elements in the origin of IL ^ is the unit vector of the overall direction. It is the vector coordinates, and L  ^ * n  *ð1Þ ^ 9 * R μ P μ *R A = 4π Ii^li ri′  R02 = 4π H R02 > = 0 0 i=1 (4:37) * n * P * > ; H = Ii^li ri′ i=1

*

*ð1Þ

*ð2Þ

Obviously, H is a second-order tensor in the first-order term of A : Finally for A ; we obtain    n  *ð2Þ μ X 1 ^ *′*′ 1 (4:38) A = Ii li ri ri : ∇′∇′ 4π i = 1 2 R 0 Using exactly the same method as electric scalar potential ’, let * * *

n X

N=

 ** * * Ii^li 3 ri′ ri′ − ri2 I

(4:39)

i=1

and *ð2Þ

A

* * *

=

*  * * * μ * ^0R ^0 − I N : 3 R 24πR30

(4:40)

Similarly, N is a third-order tensor. For the continuous distribution of currents in the small source area of V, we have  RRR * 9 ^= J x′; y′; z′ dv′ IL > > > v > > * = RRR ** * J r ′dv′ H= (4:41) v > > * >   * * > RRR * * * * 2* > N= J 3 r ′ r ′ − r′ I dv′ ; v

52

4 Charge multipoles and current multipoles

*ð1Þ

For the current multipole, a detailed study should be made of A : For the continuous distribution of currents, eq. (4.37) is ððð  * *ð1Þ μ ^0  * r ′ J dv′ (4:42) R A = 2 4πR0 v

*ð1Þ

Divide A 

into two terms, namely decomposing the integrand in eq. (4.42) yields [3]

* 1 h *  * i 1 h ^ *′*  ^ **′i ^0  J * ^0  * ^0  * r′ J = r′ J − R r′ + R R R0  r J + R0  J r 2 2

(4:43)

Therefore, * ð1Þ

A1

=

μ 4πR20

ððð h ððð  1 ^ *′*  ^ **′i ′ μ 1 *′ * ^ R0  r J − R0  J r dv = r × J × R0 dv′ 2 2 2 4πR0 v

(4:44)

v

We introduce the magnetic dipole moment ððð  1 * ′ * ′ * m= r × J dv 2

(4:45)

v

And we obtain *ð1Þ

A1 =

* ð1Þ

Then, A2

μ * ^  m × R0 4πR20

(4:46)

is studied. *ð1Þ

A2 =

* *

μ 4πR20

ððð h 1 ^ * ′ *  ^ *  * ′ i ′ R0  r J + R0  J r dv 2

(4:47)

v

* Note that ∇′ r ′ = I , and we obtain

* * * * * J = J  ∇′ r ′ = J  I

(4:48)

  ^0 ^0  * r′ =R ∇′ R

(4:49)

*

and

Therefore, the integrand in eq. (4.47) is * i h   *  h *  *i * * ^0  J * ^0  * ^0  * ^0  * r′ J + R r ′ J  ∇′ r ′ + ∇′ R r′  J r′ r′= R R * i * n h h * *io *   * ^0  * ^0  * ^0  * r ′ J  ∇′ r ′ + r ′ ∇′  R r′ J − r′ R r ′ ∇′  J = R *

(4:50)

There are three terms on the right side of eq. (4.50). Since ∇′  J = 0 under the condition of steady electric current, the third term equals zero. The integral of the * second term is zero in that J will not surpass the small source area of V. And, by [3]

53

Q&A

ðð  i* *   ððð nh* * o * * * ′ ′ J  ∇′ r ′ + r ′ ∇′  J dv′  r J  ds = s

(4:51)

v

it can be seen that the first term is also zero. So, finally we get *ð1Þ *ð1Þ μ * ^  A = A1 = m × R0 4πR20

(4:52) *ð1Þ

Now, we can say that if ’ð1Þ is contributed by the electric dipole moment, then A influenced by the magnetic dipole moment.

is

4.6 Four-dimensional current multipoles in the theory of relativity *

In Einstein’s theory of special relativity, J and jcρ constitute a four-dimensional current. So, it is easy to obtain "* # ððð " * # ′ A J μ dv (4:53) = ’ 4π j jcρ R c

v

Similarly, using the generalized Taylor expansion, we can get "* #      ððð " * # A J μ 1 *′ 1 1  *′ *′  1 ′ ′ ′ +r  ∇ + +    dv′ = r r : ∇∇ 4π R 0 2 R 0 j’ jcρ R0 c

(4:54)

v

All the rest are the same.

4.7 Summary The charge and current multipoles described in this chapter are at static electric and steady electric current state, respectively. For the radiation of high-frequency ω, the concepts and principles are exactly the same, with no repetition here.

Q&A Q: With all due respect, my impression of this chapter is that the mathematical derivations are extremely tedious, and that it is difficult for beginners to follow. Could you please introduce the method of reading this chapter? A: This is exactly one of the many difficulties in writing modern scientific papers, which are usually concise and comprehensive, without any “nonsense.” Many beginners find it hard to get the background of this chapter; therefore, they often do not fully understand the book after reading.

54

4 Charge multipoles and current multipoles

In a sense, Q&A is a way by which the author tells the reader the “nonsense” in his mind, in order to have better communication. This chapter explains two points: (1) The series expansion of a function serves as an important bridge between mathematics and applications. After calculus, one will learn Taylor series. Typically, 8 1 1 1 1 > > ð − ∞ < x < ∞Þ ex = 1 + x + x 2 + x 2 + x 3 +    + x n +    > > 2! 2! 3! n! > > > > 1 1 1 1 > > > lnð1 + xÞ = x − x2 + x3 − x4 +    + ð − 1Þn − 1 xn +    ð − 1 < x ≤ 1Þ > < 2 3 4 n x x3 x5 x7 x2k − 1 > > sin x = − + − +    ð − 1Þk +  ð − ∞ < x < ∞Þ > > ð2k − 1Þ! 1! 3! 5! 7! > > > > > > x2 x4 x6 x2k > > + − +    ð − 1Þk +  ð − ∞ < x < ∞Þ : cos x = 1 − 2! 4! 6! ð2kÞ! They mostly consist of infinite sums. But in fact, we can make an approximation when a certain condition is satisfied. For example, for jxj < 1, 8 x e 1+x > > > > 1 > > > lnð1 + xÞ x − x2 > > 2 > > > > x3 < sin x x − 6 > > > > x2 > > cos x 1 − > > 2 > > > ffiffiffiffiffiffiffiffiffi > x >p n : 1+x 1+ n The key point is that for jxj < 1; all the terms in the series correspond to constantly improved precision; one more step means the precision will increase exponentially. Therefore, we can take the corresponding number of terms in the series based on actual needs. In this way, mathematics is closely connected to the reality, as shown in Fig. 4.4.

The function f (x)

Taylor series expansion f ′( ) f ′′( ) f (n)(0) n f ( x ) = f (0)+ 0 x + 0 x2 + + x + n! 1! 2!

When the condition of x < 1 is satisfied.

Bold approximation is made. f ′( ) f ( x) ≈ f (0) + 0 x x < 1 1!

Fig. 4.4: Approximation of f ðxÞ in practice by using Taylor series.

Especially, f ðxÞ f ð0Þ +

f ′ð0Þ x 1!

called the linear approximation of a function. It can be extended as follows:

Q&A

55

For jx − x0 j < 1, f ðxÞ f ðx0 Þ +

f ′ðx0 Þ f ′′ðx0 Þ ðx − x0 Þ + ðx − x0 Þ2 1! 2!

(2) The multipole expansion of the charge (current) system is an important bridge between electromagnetic theories and engineering. On the basis of the series expansion of the function, it is much easier to discuss the multipole expansion of the charge (or current). Object: The point charge system of any q1 , q2 , . . . , qn within the source area of V.   *  Objective: Scalar electric potential of R0   
r′ in the far field. n   1 X qi ’= 4πε i = 1 Ri where

Object z R0

(x,y,z)

Ri q1 x

qn o ri q i

y

a set group of point charge system { qi }

Based on the origin of the coordinate, the general scalar three-dimensional function can be transformed into generalized Taylor expansion

f (R ) = f (R 0) + r ′⋅[ ′f (R )] 0 + 1 ( r ′ r ′ ) : [ 2! where f ( R ) = 1 R

′ ′f ( R ) ]

0+

Specificly

1 1 = +r ′ Ri R 0 i

′(

1 ) + 1 ( ′r ′ ): R 0 0 2! i i

′ ′

1 R

+ 0

Likewise in engineering application, the bold approximation is as follows.

1 1 n φ= + r ′⋅( ∑ q 4πεi=1 i R 0 i

′

1 ) = φ (0 ) + φ ( 1 ) R 0

Fig. 4.5: φ approximation of the engineering applications of the charge system fqi g by using multipole expansion.

56

4 Charge multipoles and current multipoles

*

R0 = x^i + y^j + z^k

r ′ = x′^i + y′^j + z′^k

*

Similarly, we draw Fig. 4.5. If we take       n 1 X 1 *′ 1 1* * 1 ′ ′ ′ ’ = ’ð0Þ + ’ð1Þ + ’ð2Þ qi + ri  ∇ + ri ri : ∇ ∇ 4πε i = 1 R0 R 0 2 R 0 then the precision will reach the quadrupole level. Obviously, it serves as an important bridge between electromagnetic theories and engineering applications. Q: What is the difference between the multipole approximation and the Taylor series function approximation? A: Although both are based on the Taylor expansion, the latter can be specifically simplified due to its strong physical meaning. For example, for Fig. 4.6, there are three points of the charge system q in the xOy plane (the three points form a regular triangle). z R3

q

R2

R1 o

q

y

q x

Fig. 4.6: Charge distribution of the regular triangle.

Since pffiffiffi 8 1^ 3^ * > > ′ > = j i − r 1 > > 6 2 > > p ffiffiffi < 1 3^ * ^ j r2′ =  i − > 6 2 > > p ffiffi ffi > > > 3^ > :* j r3′ = 0^i − 3 *

We take R0 in the far field of the z-axis, that is, *

R0 = 0^i + 0^j + R0 ^k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Then, it is easy to obtain R1 = R2 = R3 = R = ð1=3Þ + R20

57

Recommended scholar



Therefore,

1 ∇′ R

! − ^i − ^j R

 = 0

8 pffiffiffi    > 1 1 1 3 * > ′ ′ > r  ∇ = − + > 1 > 6 R R 2 > 1 0 > > pffiffiffi >    < 1 1 1 3 * ′ ′ r2  ∇ = + > 6 R R 2 2 0 > > >  pffiffiffi   > > > 1 1 3 * > ′ ′ > : = − : r3  ∇ 6 R3 0 R

Finally, we have       1 1 1 * * * ’ð1Þ = r1′  ∇′ + r3′  ∇′ + r2′  ∇′ ≡0 R1 0 R2 R3 0 In this case, the selection of point O in the charge system makes the contribution of the electric dipole term to the electric potential ’ð1Þ ≡ 0. Therefore, ’ = ’ð0Þ + ’ð2Þ Conceptually, the origin can be regarded as the superposition of 3q and −3q, as shown in Fig. 4.7. y

y o

q

3q o o

x o

q

P3 −q –q o –q P2 P1

x q *

*

*

Fig. 4.7: (a) 3q Point charge contribution ’ð0Þ and (b) electric dipole contribution at P 1 ; P 2 ; P 3 combined ’ð1Þ ≡ 0.

In many cases, a good selection of point o can simplify the problem and improve precision.

Recommended scholar Concerning multipole expansion, I often think of Professor Cao Changqi. Actually, I am not “qualified” to introduce him, as I am neither from Peking University nor a student of Professor Cao, and I do not even find a photo of him.

58

4 Charge multipoles and current multipoles

Nevertheless, I have always wanted to introduce Professor Cao, a well-known scholar in China’s electromagnetic theory field, and his masterpiece Electrodynamics [17]. I am a “second-generation” student of Professor Cao, as Professor Wang Yiping, who taught me “electrodynamics” course in 1963, attended Professor Cao’s lectures. My first contact with this course really delighted me. As is known to all, the Chinese academia in the early 1960s was required to “completely learn from the Soviet Union.” Thus, Professor Cao’s book, with distinctive characteristics, brought some new ideas to the Chinese academia. (1) The book introduces the δ generalized function theory, unifies the discrete and * the continuous and establishes the position vector r and the operator ∇ theory systematically. It points out clearly that the operator has the duality of function and operation. * * (2) The book presents the source and the field, corresponding to r , ∇′ and r , ∇, respectively, which can be easily used for discussion. (3) The book is clear and concise, characteristic of masterpieces. It is said that back then, Cao was promoted professor mainly on account of this book.

5 Polarization of electromagnetic wave and its applications There are three important conditions for the effective reception of electromagnetic waves: the same frequency (or frequency band), the collineation in the larger * direction and the transmission and reception of electric field E having a commonorientation part – the last one is exactly the core concept of this chapter: polarization. In essence, polarization of the electromagnetic wave is the fundamental cause of the spatial anisotropy of the wave. This chapter starts with the most general elliptically polarized wave to conduct an in-depth analysis, so as to obtain the quantitative relationship among elliptic parameters α, a and b. This chapter indicates that linear polarization and circular polarization are both the degeneration of elliptic polarization. Furthermore, this chapter discusses the conversion and decomposition of polarized waves and stresses the engineering applications of polarization.

5.1 Introduction This is the fifth chapter of Electromagnetic Field Theory Teaching Series. One major event of the electromagnetic development is the experiment that the electromagnetic wave created first by Hertz was able to transmit signals through space. From this, it is gradually realized that the effective reception of electromagnetic waves is based on three conditions: the same frequency (or frequency band), the collineation in the larger direction and the transmission and reception of electric * field E having a common-orientation part, among which the last one is the core of this chapter – polarization. The polarization describes the time-varying behavior of the electric field intensity vector at a given point in the space. Thus, it can be divided into linearly polarized wave, circularly polarized wave and elliptically polarized wave.

5.2 Elliptically polarized wave The elliptically polarized wave is the most general case. When the amplitude and phase of Ex and Ey are not specifically constrained, we have ) Ex = Exm cosðωt − kz + ’x Þ   (5:1) Ey = Eym cos ωt − kz + ’y Let us suppose https://doi.org/10.1515/9783110527407-005

60

5 Polarization of electromagnetic wave and its applications

Ex ; Exm

x0 =

y0 =

Ey ; Eym

(5:2)

and u = ωt − kz + ’x

) (5:3)

Δ’ = ’x − ’y Therefore, eq. (5.1) can be written in the matrix form as # #" " # " cos u 1 0 x0 = sin u cosðΔ’Þ + sinðΔ’Þ y0 and its inverse relation " # cos u

1 = sinðΔ’Þ sin u

"

sinðΔ’Þ

0

− cosðΔ’Þ

+1

#"

x0

(5:4)

# (5:5)

y0

Its quadratic form is constituted as " ½ cos u

sin u 

Using eq. (5.5), we find ( " sinðΔ’Þ 1 ½ x0 y0  2 sin ðΔ’Þ 0

cos u sin u

# = cos2 u + sin2 u = 1

− cosðΔ’Þ +1

#"

(5:6)

sinðΔ’Þ

0

− cosðΔ’Þ

+1

#)"

x0 y0

# =1

(5:7)

Specifically,  2  1 x0 − 2 cosðΔ’Þx0 y0 + y20 = 1 sin ðΔ’Þ 2

(5:8)

In addition, Fig. 5.1 shows a general ellipse with two sets of coordinates xOy and x′Oy′ rotating around each other.

y

y′

a

b α

o

x′

x

Fig. 5.1: A general ellipse in two sets of coordinates xOy and x′Oy′.

5.2 Elliptically polarized wave

61

For coordinate x′Oy′; we have 2 2 x′ y′ + =1 a2 b2

And its corresponding matrix is 

x′

y′

"1  a2

0

0

1 b2

#"

(5:9) # x′ =1 y′

(5:10)

where a and b are the two semi-axes of the ellipse, and the angle that x′-axis makes with x-axis is α. We have #" # " # " x cos α sin α x′ (5:11) = − sin α cos α y y′ Equation (5.10) is rewritten as " #" 1 cos α − sin α a2 ½x y sin α cos α 0

0

#"

cos α

sin α

#" # x

− sin α cos α

1 b2

y

=1

(5:12)

which expands as 2 ½x

6 y 4 

1 a2

cos2 α a2

−

1 b2



+



sin2 α b2

sin α cos α

1 a2

−

1 b2



sin2 α a2

sin α cos α

+

cos2 α b2

3

" # 7 x =1 5 y

(5:13)

or 

    2  cos2 α sin2 α 2 1 1 sin α cos2 α 2 x − 2 2 − 2 sin α cos αxy + y =1 + + a2 b2 b a a2 b2

(5:14)

In eq. (5.14), let x = Ex and y = Ey : By eq. (5.8), we obtain Ey2 1 Ex2 2 cosðΔ’Þ Ex Ey 1    − + =1 2 2 sin2 ðΔ’Þ Exm sin2 ðΔ’Þ Exm Eym sin2 ðΔ’Þ Eym

(5:15)

Equation (5.15) is exactly the equation of the general elliptically polarized wave. A comparison of eq. (5.14) with eq. (5.15) leads to tanð2αÞ =

2Exm Eym cosðΔ’Þ 2 − E2 Exm ym

The angle α is 2Exm Eym 1 α = tan − 1 2  cosðΔ’Þ 2 Exm − Eym 2 and the two semi-axes of the ellipse are

(5:16) ! (5:17)

62

5 Polarization of electromagnetic wave and its applications

  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9    1 > a = 2cos Δ’ > > 2  > 2 + 1=E2 1=Exm = ym >   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    1 > > b = 2sin Δ’ > > 2  2 2 ; 1=Exm + 1=Eym >

(5:18)

Based on the above analysis, we can discuss two important degenerated cases. Case 1: Linearly polarized wave The linearly polarized wave is a degeneration of the elliptically polarized wave, and the constraint condition is that Ex and Ey are in-phase or out-of-phase, that is, Δ’ = ’x − ’y = 0 or π

(5:19)

b=0

(5:20)

we find sinðΔ’Þ = 0 and b = 0.

Note that tanð2αÞ = ±

  2 Eym =Exm  2 1 − Eym =Exm

It is easy to obtain in the case of linear polarization that   Eym tanðαÞ = ± Exm

(5:21)

(5:22)

In fact, we have Ex = Exm cosðωt − kz + ’x Þ

) (5:23)

Ey = Eym cosðωt − kz + ’x Þ *

For a fixed point z in the space, the tip of the vector E will traverse a straight line, called linearly polarized wave, as shown in Fig. 5.2.

y y Ey

E Ex

α

o

(a)

Ex

o

x

(b)

Ey

x

α

E

Fig. 5.2: Linearly polarized wave: (a) Δ’ = 0 in-phase linearly polarized wave and (b) Δ’ = π out-ofphase linearly polarized wave.

5.2 Elliptically polarized wave

63

Case 2: Circularly polarized wave An elliptically polarized wave reduces to the circularly polarized wave. The constraint condition is that Ex and Ey are equal in amplitude and differ in ± π=2 (rad) phase differences, namely, ) Em = Exm = Eym (5:24) Δ’ = ’x − ’y = ± π2 In this case, the elliptic eq. (5.15) reduces to 2 Ex2 + Ey2 = Em

(5:25)

which is a circular equation. And, the equation of the two semi-axes (5.18) becomes (5:26)

a = b = Em which is called circle radius. Then, we have Ex = Em cosðωt − kz + ’x Þ

) (5:27)

Ey = ± Em sinðωt − kz + ’x Þ Their angle measured from the x-axis is   − 1 Ey = ± ðωt − kz + ’x Þ α = tan Ex

(5:28) *

Obviously in this case, the angle α of the endpoint of electric field vector E varies with the time t. For α = þ ðωt − kz + ’x Þ, then dα = þω dt

(5:29)

*

E rotates at a uniform rate with an angular velocity ω in a counterclockwise direction. * When the fingers of the right hand follow the direction of rotation of E , the thumb points to the direction of propagation of the wave, as that of a right-handed screw. Therefore, it is called right-hand circularly polarized wave. On the contrary, for α =  ðωt − kz + ’x Þ; then dα = ω dt

(5:30)

*

In this case, E will rotate with an angular velocity ω in a clockwise direction; this is a left-hand circularly polarized wave, as shown in Fig. 5.3. Case 3: Left-hand and right-hand elliptically polarized waves Now, let’s restudy the two types of rotating direction of the elliptically polarized waves. The rotation angle α′ is

64

5 Polarization of electromagnetic wave and its applications

+ω Right-hand

y

E

Ey

−ω Left-hand α

o

x

Ex

Em

Fig. 5.3: Left-hand and right-hand circularly polarized waves.

     − 1 Ey − 1 Eym cos ωt − kz + ’y ′ = tan α = tan Ex Exm cosðωt − kz + ’x Þ

(5:31) *

In this case, the generalized rotational angular velocity ω′ of electric field E is Exm Eym ω sinðΔ’Þ dα′   = ω′ = 2 2 cos2 ωt − kz + ’ dt Exm cos2 ðωt − kz + ’x Þ + Eym y

(5:32)

Obviously, in eq. (5.32), for − π > Δ’ > 0 or sinðΔ’Þ > 0, then dα′=dt = ω′ > 0, in which case it can be called the right-hand elliptically polarized wave. On the contrary, for − π < Δ’ < 0 or sinðΔ’Þ < 0, then dα′=dt = ω′ < 0, in which case it can be called the left-hand elliptically polarized wave. It should be noted that α′, the rotation angle of the elliptically polarized wave expressed in eq. (5.31), has a completely different physical definition from elliptic angle α in eq. (5.17), the latter being a constant independent of time t. But strangely enough, α is closely related to dα′=dt = ω′. A further study of eq. (5.17) shows that for α in the first quadrant, sinðΔ’Þ > 0 is the right-hand elliptically polarized wave and that for α in the fourth quadrant, sinðΔ’Þ < 0 is the left-hand elliptically polarized wave, as shown in Fig. 5.4. y

y′

y

Positive, Right-hand ω′

ω′

a

b

α o

Negative, Left-hand

x′

y′ b x

o

x

α

a (a)

(b)

x′

Fig. 5.4: Right-hand and left-hand elliptically polarized waves: (a) α is the first quadrant angle, righthand elliptically polarized waves, and (b) α is the fourth quadrant angle, left-hand elliptically polarized waves.

5.3 Polarization conversion and matrix representation

65

5.3 Polarization conversion and matrix representation The conversion of polarized waves from one to another is not only a theoretical problem but also a practical issue. We start again with the general case of the elliptically polarized wave. Suppose that the parameters are as follows:    9 β = ωt − kz + 21 ’x + ’y > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2 + E2 E0 = Exm (5:33) ym > > > E ; ym cos θ = EExm ; sin θ = E 0

0

Then, the most basic eq. (5.1) in this chapter is written as  ) Ex = E0 cos θ cos β + 21 Δ’   Ey = E0 sin θ cos β − 21 Δ’

(5:34)

By eq. (5.34), we obtain      1  Ex = Re E0 cos θejðβ + ð1=2ÞΔ’Þ = Re E0 ejθ + e − jθ ejðβ + ð1=2ÞΔ’Þ 2      1 1 1 ¼ E0 Re cos Δ’ + j sin ðΔ’Þ ðcosðβ + θÞ + j sinðβ + θÞÞ 2 2 2       1 1 þ cos Δ’ + j sin Δ’ ðcosðβ − θÞ + j sinðβ − θÞÞ 2 2

(5.35)

Thus, we get     1 1 1 1 Ex = E0 cos Δ’ ½cosðβ + θÞ + cosðβ − θÞ − E0 sin Δ’ ½sinðβ + θÞ + sinðβ − θÞ 2 2 2 2 (5:36) Similarly, we have     1 1 1 1 Ey = E0 cos Δ’ ½sinðβ + θÞ − sinðβ − θÞ − E0 sin Δ’ ½cosðβ + θÞ − cosðβ − θÞ 2 2 2 2 (5:37) Equations (5.36) and (5.37) are the basis of polarization conversion. Case 1: Linear polarization The linear polarization can be divided into two cases: (1) ’x = ’y , or the in-phase case of Δ’ = 0. Therefore, Ex = 21 E0 cosðβ + θÞ + 21 E0 cosðβ − θÞ Ey = 21 E0 sinðβ + θÞ − 21 E0 sinðβ − θÞ

) (5:38)

66

5 Polarization of electromagnetic wave and its applications

Obviously, eq. (5.38) can be regarded as the superposition of two parts, one of which is ) Ex1 = 21 E0 cosðβ + θÞ (5:39) Ey1 = 21 E0 sinðβ + θÞ which is a right-hand circularly polarized wave, and the other is ) Ex2 = 21 E0 cosðβ − θÞ Ey2 =  21 E0 sinðβ − θÞ

(5:40)

which is a left-hand circularly polarized wave. (2) ’x − ’y = π, or the out-of-phase case of Δ’ = π. Here, we have ) Ex =  21 E0 sinðβ + θÞ − 21 E0 sinðβ − θÞ Ey =  21 E0 cosðβ + θÞ + 21 E0 cosðβ − θÞ

(5:41)

Similarly, one part is Ex3 =  21 E0 sinðβ + θÞ

)

Ey3 =  21 E0 cosðβ + θÞ which is the left-hand circularly polarized wave, and the other is ) Ex4 =  21 E0 sinðβ − θÞ Ey4 = 21 E0 cosðβ − θÞ

(5:42)

(5:43)

which is the right-hand circularly polarized wave. Both the in-phase and the out-of-phase cases show that a linearly polarized wave can be resolved into a right-hand circularly polarized wave and a left-hand circularly polarized wave of equal amplitude. Case 2: Circular polarization As Em = Exm = Eym and Δ’ = ’x − ’y = ± π=2, it is easy to write ) Ex = Em cosðωt − kz + ’x Þ   Ey = Em cos ωt − kz + ’y where ± is corresponding to the right-hand and the left-hand circularly polarized waves, respectively. We rewrite     ) Ex = Em cos β cos 21 Δ’ − Em sin β sin 21 Δ’     (5:44) Ey = ± Em cos β sin 21 Δ’ + Em sin β cos 21 Δ’ Then, a general conclusion is drawn that a circularly polarized wave can be resolved into two linearly polarized waves.

5.3 Polarization conversion and matrix representation

67

Case 3: Elliptic polarization Equations (5.36) and (5.37) are the superposition of two parts, which can be written as )     Ex1 = 21 E0 cos 21 Δ’ ½cosðβ + θÞ + cosðβ − θÞ = E0 cos θ cos 21 Δ’ cos β     (5:45) Ey1 = 21 E0 cos 21 Δ’ ½sinðβ + θÞ − sinðβ − θÞ = E0 sin θ cos 21 Δ’ cos β and )     Ex2 = − 21 E0 sin 21 Δ’ ½sinðβ + θÞ + sinðβ − θÞ = −E0 cos θ sin 21 Δ’ sin β     Ey2 = 21 E0 sin 21 Δ’ ½ − cosðβ + θÞ + cosðβ − θÞ = −E0 sin θ sin 21 Δ’ sin β

(5:46)

Both eqs. (5.45) and eq. (5.46) constitute a linearly polarized wave. Therefore, an elliptically polarized wave can be resolved into two linearly polarized waves, each of which is the sum of two oppositely rotating circularly polarized waves. A natural conclusion is that an elliptically polarized wave can be resolved into four circularly polarized waves. Thus, literatures [2,16] give a wrong description. Various polarized waves can be represented in the form of matrix. Using E + and − E to represent the most basic right-hand and left-hand circularly polarized waves, we can write #" # " # " 1 1 1 E+ (5:47) Em = − j 1 −1 E In the form of complex amplitude, it can be expressed as " #" # " #  1 1 1 ejðβ + θÞ 1 E0 Em = 2 1 − 1 e − jðβ + θÞ j Therefore, we have "

E+ E−

# " =

#"

1

1

1

−1

1

1

1

−1

#"

ejðβ + θÞ e − jðβ + θÞ

#

1 E0 2

(5:48)

 (5:49)

And we obtain "

E+ E−

# " =

ejðβ + θÞ e − jðβ + θÞ

#

1 E0 2



For the in-phase case of Δ’ = 0 in linear polarization, we have 2 jðβ + θÞ 3 e #6 " # " 7  Ex1 1 1 1 1 6 e − jðβ + θÞ 7 1 7 6 = 7 E0 jEy1 1 −1 −1 1 6 4 ejðβ − θÞ 5 2 e − jðβ − θÞ

(5:50)

(5:51)

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5 Polarization of electromagnetic wave and its applications

And for the out-of-phase case of Δ’ = π; we have 2 jðβ + θÞ 3 e #6 " # " 7  Ex2 1 − 1 1 − 1 6 e − jðβ + θÞ 7 1 7 6 =j 7 E0 jEy2 −1 −1 1 1 6 4 ejðβ − θÞ 5 2

(5:52)

e − jðβ − θÞ Finally, for the general case of elliptically polarized wave, we have         32 3 2 3 2 Ex1 cos 21 Δ’ cos 21 Δ’ cos 21 Δ’ cos 21 Δ’ ejðβ + θÞ         76 7 6 7 6  6 Ex2 7 6 j sin 21 Δ’ − j sin 21 Δ’ j sin 21 Δ’ − j sin 21 Δ’ 76 e − jðβ + θÞ 7 1 76 7=6 7 E0 6         7 6 jE 7 6 cos 1 Δ’ 6 − cos 21 Δ’ − cos 21 Δ’ cos 21 Δ’ 7 54 ejðβ − θÞ 5 2 4 y1 5 4 2 1  1  1  1  jEy2 − j sin 2 Δ’ − j sin 2 Δ’ j sin 2 Δ’ j sin 2 Δ’ e − jðβ − θÞ (5:53) where Ex = Ex1 + Ex2

)

Ey = Ey1 + Ey2

(5:54)

5.4 Engineering applications of polarization In essence, polarization of the electromagnetic wave is the fundamental cause of the spatial anisotropy of the wave. The electromagnetic waves radiated by the AM broadcast station in the far field are linearly (vertically) polarized wave with electric field perpendicular to the ground, while television signals are linearly polarized wave in the horizontal direction. For the waves to travel through the rain, circularly polarized waves are often employed. Likewise, circularly polarized waves are generally used in all motion systems, from remote control rocket to satellite – because they can be resolved into two linearly polarized waves. As a result, no matter what kind of linear polarization they become, there is always a part of waves which can be received. As electromagnetism is connected more to various aspects of the human beings, the polarization will gain more and more applications.

5.5 Summary This chapter presents an in-depth discussion of the concept of electromagnetic polarization. Starting with the most general elliptic polarization, this chapter gives

Q&A

69

important parameters, specifically angle α and two semi-axes a and b. The conversion and decomposition of various polarizations have been studied, with emphasis on their engineering applications.

Q&A Q: In the discussion of electromagnetic wave polarization, what is the difference between this chapter and other books or textbooks? A: This chapter deals with electromagnetic polarization, a domain combining theory and practice. * Polarization – simply put, is the tip of electric field vector E at fixed point in the space to change with time. As is known to all, in electromagnetic communications, the fields for receiving and transmitting signals should have shared orientation, which is the most important application in the study of polarization. This chapter has two major features: (1) The discussion is centered on elliptic polarization, with other forms of polarization considered as exceptions, as shown in Fig. 5.5. The parameters of the elliptically polarized wave are derived by using the quadratic form of matrix, which is simple and clear. (2) The decomposition and combination of polarized waves are discussed in depth, as shown in Figs. 5.6, 5.7 and 5.8, respectively.

y Elliptically polarized wave

Ex = Exm cos(ωt – kz + φx )

x

0

Ey = Eym cos(ωt – kz + φy )

Linearly polarized wave

Circularly polarized waves

y

when

when φx = φy = φ Ex = Exm cos(ωt – kz + φ) Ey = Eym cos(ωt – kz + φ)

0

x

y

Exm = Eym = Em φy = φx ± π 2

Ex = Exm cos(ωt – kz + φ) Ey = ± Eym sin(ωt – kz + φ)

0

x

Fig. 5.5: The elliptically polarized wave considered as the center.

Q: If one side for transmitting or receiving in communications is a motion system, what else should be noted? A: Obviously, the xOy planes discussed above are all perpendicular to the plane of propagation direction. If one side for transmitting or receiving in communications is

70

5 Polarization of electromagnetic wave and its applications

Linearly polarized wave

Two circularly polarized waves of equal amplitude and oppositely rotating direction

y

y

y

o

x

o

x

+

o

x

Fig. 5.6: A linearly polarized wave can be decomposed into two circularly polarized waves of equal amplitude and oppositely rotating direction.

Circularly polarized wave

Two linearly polarized waves

y

y

y

x

o

x

x

o

Fig. 5.7: A circularly polarized wave can be decomposed into two linearly polarized waves.

Two linearly polarized waves

y

Elliptically polarized wave

o

Two circularly polarized waves of equal amplitude and oppositely rotating direction

y

y x

o

x

+

o

x

o

x

y

o

x

y

o

y

y x

o

x +

Fig. 5.8: An elliptically polarized wave can be decomposed into four circularly polarized waves of oppositely rotating direction.

a motion system, then the xOy plane, called the polarization plane, changes as well and is far more important. As mentioned in this chapter, the antenna of the motion system generally uses the circularly polarized wave. So, no matter what position, there is always a part of waves which can be received.

Recommended scholar

71

Recommended scholar

Fig. 5.9: Bi Dexian.

The man recommended here, Bi Dexian (Fig. 5.9), is not only a scholar but also a great teacher. After returning from the United States, Bi devoted himself to the Xi’an Institute of Military Telecommunications Engineering (the predecessor of Xidian University). He rarely engaged in foreign affairs due to confidentiality. (1) He dedicated himself to the establishment of major teaching system in Xidian University. Mr. Bi is an “encyclopedia.” He wrote Foundations of Electromagnetic Field in 1964, translated UHF Antenna (by Ai Jinbao) from Chinese into English and spared no effort to promote the development of Pulse Technology. (2) At school, he was called “the teacher of teachers.” Whenever there was a new trend or development of technology, he would attend to it personally. He also gave lectures to teachers. (3) Mr. Bi is a kind elderly man (he was advanced in years when I met him). I often think that apart from aiming high in science and technology, today’s China also needs revered and dedicated teachers to promote Chinese science and technology to a higher level.

6 Conservation of charge and conservation of current The law of conservation is one of the most important backbones of electromagnetic theory. In this chapter, the concept of extended total space is introduced, from which follows the general forms of conservation of charge and conservation of current. The former corresponds to the time symmetry and the latter the space symmetry. In different inertial systems, current and charge constitute four dimensional vectors, which possess the Lorentz time–space symmetry. Application examples are also given in this chapter.

6.1 Introduction This is the sixth chapter of Electromagnetic Field Theory Teaching Series. The law of conservation is one of the most important backbones of electromagnetic theory, but quite a few textbooks do not pay adequate attention to it. When the Nobel Prize winner of physics Richard Phillips Feynman was speaking in the famous Messenger Lecture The Characters of Physical Law of Cornell University in 1965, he specially made “The Great Conservation Law” an important chapter [6]. He explained, “When learning physics laws, you can find many complex specific laws such as the law of gravitation, the law of electromagnetism and the law of nuclear reaction. But some general principles will emerge on top of all these diverse specific laws.” And, that is the law of conservation that Feynman intended to discuss in depth. Then, there arises a new question: How do general principles like the law of conservation come into being? To answer that, we must go back to 1916, when a prominent woman scientist Emmy Noether was invited by mathematics leading scholar Hilbert to teach at the University of Göttingen. As Hilbert put it, “I invited Noether to solve problems in the theory of energy conservation.” As expected, Noether came up to expectations of all. Within only 2 years – in 1918, she achieved her greatest contribution in physics “Noether’s theorem.” She proposed the idea of “symmetrical correspondence conservation.” The symmetry generated in rectilinear motions corresponds to the conservation of momentum, the symmetry of rotation corresponds to the conservation of angular momentum and the symmetry of time corresponds to the conservation of energy. In other words, the reason why all kinds of motion conform to laws of conservation is the inherent symmetry of everything [17]. The fields of symmetry and conservation are connected due to the outstanding work of Noether. This chapter deals with the conservation of charge and conservation of current in the electromagnetic field and their close connection with symmetry. https://doi.org/10.1515/9783110527407-006

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6 Conservation of charge and conservation of current

6.2 Conservation of charge Over the history of scientific development, there has been an obvious feature: ideas and concepts precede mathematics after experiments are conducted. Great scientists usually get to the essence of problems from the most basic concepts. Feynman had a profound understanding of the concept of charge: “I would start with a conservation law that can be easily understood—the conservation of charge. No matter what happens, the amount of charges in the world remains unchanged. If you lose charges in one place, you can find them in another place. And the total amount of charges does not change” [6]. The famous scholar Bi Dexian (academician of the Chinese Academy of Sciences) made a more specific description: “There are only two kinds of charge in the natural world: positive and negative. Charges can not be created or destroyed; they can only be transferred from one object to another, or from one part of an object to another. In other words, the algebraic sum of charges is in conservation in all physical processes. And this is the law of conservation of charge that has been verified by experiments” [1]. From the view of mathematics, Maxwell’s equations completely illustrate the macroscopic law of electromagnetism. And, the conservation of charge is compatible with the equations. We can start with the Gauss theorem. Given ððð ðð * ^1 ds ¼ ρdv (6:1) D  n v

S

^1 is the outer normal unit vector of the closed surface S; and the volume it encloses is n v, as shown in Fig. 6.1.

nˆ1 Q v S Fig. 6.1: The volume enclosed by surface S is v.

Rewrite eq. (6.1), and we get ðð * ^1 ds ¼ Q D n

(6:2)

S

where

ððð Q¼

ρ dv v

(6:3)

6.2 Conservation of charge

75

^2 as the inner normal unit vector of the closed surface S: Then, the Now, we use n external volume enclosed by surface S is v, as shown in Fig. 6.2. We introduce the concept of extended total space v∞ : v∞ ¼ v ∪
v

(6:4)

Q nˆ 2

S v Fig. 6.2: The external volume enclosed by surface S is v
.

v∞ includes point ∞; so we can write ðð *
^2 ds ¼ Q D  n

(6:5)

s

where
¼ Q

ððð ρ dv

(6:6)


v

As ^1 ^2 ¼ n n

(6:7)


¼0 QþQ

(6:8)

it can be easily deduced that

We take one more step to extend total space v∞ and rewrite v∞ ¼ V ∪
v

(6:9)


includes point ∞. Let’s define Q∞ as V includes all charges in the limited space, and V the equivalent charge at ∞, and then ððð ρ dv (6:10) Q∞ ¼ 
v

Finally

ððð ρ dvþQ∞ ≡ 0
v

(6:11)

76

6 Conservation of charge and conservation of current

The above equation, or the conservation of charge provided, shows that the sum of charge in the extended total space (including point ∞) is zero. Then, take the derivative ∂=∂t on both sides of eq. (6.11). ððð ∂ρ ∂ (6:12) dvþ Q∞ ≡ 0 ∂t ∂t
v

The conservation of charge also corresponds to time symmetry or time invariance.

6.3 Conservation of current Very few textbooks available have dealt with the conservation of current. In fact, current represents the directional flow of charges. So, the conservation of charge inevitably reflects the concept of current to some extent. First, we write the theorem of current continuity. *

∇ J þ

∂ρ ¼0 ∂t

(6:13)

Similar to what we have done with the charge, an integral form of eq. (6.13) can be deduced by creating closed surface S and volume V: ððð ðð * ∂ρ ^ ds þ dv ¼ 0 (6:14)  J n ∂t s

v

Define I∞ as the (equivalent) current at point ∞; and I∞ ¼ 

∂Q∞ ∂t

(6:15)

Then, we can write ðð * ^ ds þ I∞ ≡ 0  J n

(6:16)

s

The above equation shows the conservation of current. It should be noted that for steady current that does not change with time t; eq. (6.16) can be simplified as ðð * ^ ds ¼ 0 (6:17)  J n s *

In other words, the integral of the closed surface of any steady current J is invariably zero. Similarly, when

6.5 Examples of application

77

ððð ρ dv ¼ 0

(6:18)

v

Equation (6.17) still holds. Once again, the conservation of current corresponds to space symmetry or space invariance. More specifically, eq. (6.16) stays unchanged in different translation space coordinates.

6.4 Current–charge conservation From the above analysis of the conservation of charge and the conservation of current, we know that it is necessary to introduce the extended total space with point ∞. This is similar to the extended complex plane and generalized residue theorem in complex functions. The electric field lines originating or terminating from point ∞ are the physical essence which should be further studied. This section further studies the coordinate system of motion. According to * Einstein’s special theory of relativity, current J and charge ρ can constitute a four dimensional vector in different inertial systems. " * # J (6:19) ic ρ where c is the speed of light in vacuum and i refers to an imaginary number. In this case, the current–charge vector has the property of Lorentz invariance. Assuming the inertial system moves at the speed of v^i in the x-direction without loss of generality, we can write 2 3 iβ 1 ffi pffiffiffiffiffiffiffi ffi 0 0 pffiffiffiffiffiffiffi 2 2 1β 1β 6 7 7" * # " * # 6 6 7 J ′ 0 1 0 0 J 6 7 ¼6 (6:20) 7 6 0 7 ic ρ ic ρ′ 0 1 0 6 7 4 iβ 5 1 ffi pffiffiffiffiffiffiffiffi 0 0 pffiffiffiffiffiffiffi 2 2 1β

1β

*

where β ¼ ðv=cÞ. It is obvious that J and ρ are not in conservation independently in * movement. But the combination of current J and charge ρ satisfies the Lorentz space–time invariance, or the Lorentz conservation.

6.5 Examples of application The concept of extended total space points out that in the most general case, charge and current do not distribute in the finite field. Otherwise, unnecessary mistakes may occur.

78

6 Conservation of charge and conservation of current

Literature [13] is a classic in the electromagnetic field theory in China. In the derivation from Biot–Savart law to Ampere’s law, it is crucial to prove that the * divergence of vector magnetic potential A is zero. ðð **′ ^ds′ * *  J r n μ0 (6:21) ∇ A r ¼  r 4π s

The writer explained, “The volume integral should include the whole area with * current present. So on the integral surface, J should be zero.” Then, we have * *

∇  Að r Þ ¼ 0

(6:22)

But in fact, this is not true in many cases (such as the example in below). *

Example 6.1: The infinitely long current Il k^ is without loss of generality. Find ∇  A at point ðx; y; 0Þ; as shown in Fig. 6.3

z

Il kˆ

y

o

. A(x,y,0)

x

*

Fig. 6.3: The ∇  A ðx; y; 0Þ problem of infinitely long ^ current Il k.

Solution. This is a typical example of electromagnetic theory. We know *

∇  A ðx; y; 0Þ ¼

μ0 4π

∞ ð

Il z′ dz′   2 þ y 2 þ z′2 3=2 x ∞

It is easy to see that the integrand is an odd function, and the symmetrical finite integral is zero. So, *

∇  A ðx; y; 0Þ ≡ 0

(6:23)

In this example, no matter how large a closed S surface is, it cannot include all currents. But they still * satisfy ∇  A ¼ 0.

In conclusion, it seems that we can explain eq. (6.22) in this way: “When enclosing a * surface S with large enough current source J , we can take the center of sphere as the field point to be solved and take a sphere with radius R as the surface S without losing generality.” Then, ðð **  * *  μ ^ds′ ¼ 0 (6:24) ∇  A r ¼  0  J r′  n 4πR s

Q&A

79

It can also be pointed out that in double potential (scalar potential and magnetic potential) questions, eq. (6.22) serves as the Columb Gauge. When it is time dependent, the equation acts as the Lorentz Gauge.

6.6 Summary This chapter introduces the concept of extended total space and discusses the conservation of charge and the conservation of current. The conservation of charge shows time invariance, and the conservation of current indicates space invariance. In inertial systems of motion, their combination satisfies the Lorentz time–space invariance. It is also pointed out that in real applications, we must consider the likelihood that there might be charges or current at point ∞.

Q&A Q: It is quite difficult to fully understand this reading note. “Conservation” and “symmetry” are two concepts that we know well, but it is not easy to connect them closely, as shown in Fig. 6.4.

Conservation

Symmetry

Fig. 6.4: Close relation between “conservation” and “symmetry.”

In particular, conservation is generally believed to be a concept in physics (or chemistry), while symmetry seems to belong to the field of mathematics. A: First, we should understand the concepts in a broad way. Never limit the concepts in a specific or some specific fields. I would start the discussion of conservation and symmetry from some simple questions. 1. In mathematics In middle school, we knew the following trigonometric identity. cos2 θ þ sin2 θ ≡ 1 The identity is actually the normalized form of the Pythagorean theorem found in ancient China. The unchanged 1 represents “conservation.” Expressed geometrically, the two parts are “symmetrical” on a unit circle, as shown in Fig. 6.5.

80

6 Conservation of charge and conservation of current

sinθ

cosθ

o

Fig. 6.5: Conservation of trigonometric function “symmetry” on a unit circle.

2. In physics Also in middle school, we learned that the total energy W remains unchanged when there is no consumption in the mechanical system. So, W ≡T þ V where T is the kinetic energy of the system and V is the potential energy. Expressed geometrically, they are “symmetrical” on a straight line, as shown in Fig. 6.6. V w T o

w

Fig. 6.6: Mechanical system without consumption “symmetry” on a straight line.

A short comparison will convince us that symmetry is not restricted to a circle, and that it also exists in a straight line. Now, consider the following equation:  x 2 a

þ

 y 2 b

≡1

where x and y are symmetrical on an ellipse, as shown in Fig. 6.7. y b o

a

x

Fig. 6.7: The conservation symmetry on an ellipse; ðx=aÞ2 þ ðy=bÞ2 ≡ 1.

Some may doubt the symmetry mentioned above: straight lines and ellipses have nothing to do with symmetry! In fact, they can be “transformed” into circles easily. Let 8 x >

:Y ¼ y b

Q&A

81

and (

X2 ¼ T Y2 ¼ V

Then, we can write the same unit circle symmetry X2 þ Y 2 ≡ 1. From the simple discussion above, we find conservation and symmetry are concepts that have been “hiding” in our mind. 3. Extending to the complex domain Here, we will briefly mention the complex domain. The introduction of complex numbers enables us to see symmetry in a more generalized way. For a set of curvilinear equations, ch2 u  sh2 u ≡ 1 All we need to do is let (

z1 ¼ chu z2 ¼ ishu

It is obvious that z 1 2 þ z2 2 ≡ 1 is a complex unit circle. Q: The fields of conservation and symmetry are connected by German woman scientist Noether. Could you tell us more about her thoughts? A: The thoughts of Noether are really worth further discussion. Here, we should pay attention to three key words: Germany, University of Göttingen, and Noether. According to quantitative research by historians of science, if the scientific achievements of a country accounts for over one-fourth of the world’s total, then the country becomes the “world center of science,” and the corresponding period is called the prosperous period of science. This is the famous Yuasa phenomenon. Based on this definition, Germany was the world’s center of science during the 90 years from 1830 to 1920. The University of Göttingen is Germany’s cultural, scientific and academic center. The university, with its own style and features, was established and developed by great academic masters including Gauss, Hilbert, and Max Born. Vivid descriptions of the University of Göttingen are readily available in literature. For example, in those days teachers and students gathered in the cafe near the university. The tablecloth on the coffee tables was full of mathematical formulas and

82

6 Conservation of charge and conservation of current

physical equations. There was an unwritten rule that after using coffee, anyone could take the tablecloth away for further research and innovation. Emmy Noether (1882–1935) was the best woman mathematician in the 20th century. She did not ponder and solve the problem of conservation by accident. In fact, her academic activities were closely related to a place in the University of Göttingen – the mathematics center. Noether enrolled in the university in 1903 and obtained her doctorate in 1908, as shown in Fig. 6.8.

Fig. 6.8: The best woman mathematician in the 20th century Emmy Noether.

However, in those days in Germany, the University of Göttingen was full of feudalism: female teachers were not allowed to teach. Hilbert was the only one who strongly supported female teachers. He said, “Gentlemen, I do not think gender should be a standard for choosing a qualified teacher. After all, the university council is not a public bathhouse.” However, his efforts turned out to be fruitless. In the end, Noether had to teach the course of Invariant Theory “in the name of” Hilbert. In addition, Germany and the University of Göttingen both have an excellent tradition of integrating mathematics with physics. Note that the Algebraic invariant Theory Noether taught was opened under “a large topic,” Lectures of Mathematics and Physics. Hilbert himself had studied Einstein’s special relativity in depth. So, the close relation between mathematics and physics is obvious. Why do we introduce so much background information? Because without this tradition, in 1918, 36-year-old Noether would not have studied the universal relation between conservation and symmetry in physics. In a word, “Noether knows both mathematics and physics. The Noether theorem she developed reveals the essential relationship between physical conservation and mathematical symmetry. For any

Q&A

83

symmetry, there is a corresponding conservation law, and there is a corresponding symmetry for any conservation law” (see Fig. 6.9).

Mathematical field

Physical field

Symmetry

Conservation law

It is closely connected one-to-one corresponding relations which reflects the invariant of transformation. Fig. 6.9: Conservation and symmetry.

In particular, Einstein was also in Germany at that time. He read the paper Noether sent to him and wrote a letter of commendation to Hilbert immediately: “I am so surprised that she can explain in such a generalized way.” Q: The topic of this chapter is the conservation of charge and the conservation of current. Most of the existing textbooks do not take this topic so seriously. Why do you choose this as a special topic? Is there any other thoughts? A: Nowadays, it is generally agreed that the electromagnetic theory has developed into a mature field. In other words, there is nothing to talk about. However, every field can be divided into concepts, methods, computation and applications – and all these reflect a key word: ideas. As other subjects develop rapidly, we can find something different when we look back on the so called “mature field.” Things that we did not notice or did not pay adequate attention to now suddenly become important and profound. It is like clearing the clouds to see the sky – the conservation of charge and the conservation of current are case in point. Here, we further explain the concept of conservation in the hope that readers can feel the gap between a superficial idea and a profound idea. When Feynman was delivering a series of lectures on The Essence of Physical Laws, he sharply pointed out (1) There are two cases after analyzing the conservation of charge in depth. Case 1: Charges move from one place to another in the box. Case 2: Charges disappear in one place and appear in another place at the same time. These two things are instantly connected with each other, so the total amount of charges remains unchanged. According to Feynman, the two cases mentioned above are different. We call Case 2 the localized conservation of charge, which has much more profound implications

84

6 Conservation of charge and conservation of current

than the simple statement that the total amount of charges remains unchanged. Einstein had proved that any conservation is localized conservation. (2) The conservation of charge is actually quantum discrete conservation. Feynman said, “There is an interesting thing, a very strange thing about charges, and so far we have not had a real explanation about it. It is completely independent of the law of conservation. That is, the amount of charges is always the multiples of a basic unit.” In our words, charges are discrete quantization. Q: Are there any other profound ideas that you would like to discuss about conservation? A: Sure. When Chen-Ning Yang, a renowned Nobel Prize winner of physics, was making a speech titled The Major Developments of Theoretical Physics in the 20th Century, he pointed out that according to the conservation of charge, if a positive electron is destroyed, its charge goes to another positive electron, so the charge will not change from 1 to 0 suddenly. Based on this, the idea of electromagnetic field came into being, and then Maxwell’s equations. Similarly, the conservation of energy led to the idea of gravitational field. “The concept of conservation is closely related to phase invariance.” Note that when Mr. Yang proposed the idea, he clearly stated that “I can give a clear explanation about this.” More in-depth investigations and researches show that conservation is connected with phase invariance.

Recommended scholar The prominent physicist Paul Dirac (Fig. 6.10) will be introduced here.

Fig. 6.10: Paul Dirac.

Recommended scholar

85

1. “A stroke of genius” The ideas of Dirac were ahead of most people at that time. When Heisenberg and Schrodinger were studying quantum mechanics, Dirac created an equation in his own name, the Dirac Equation. Before then, the spinning of electrons had been deliberately added into discussion. However, after the Dirac Equation came out, a spinning could be naturally deduced from ±1 and ±j, and the magnetic moment is perfectly right. The discussion of magnetic monopoles is also an ingenious stroke of Dirac. 2. Dirac’s papers are “as clear as the autumn water” This is the high-test comment on Dirac. Chen-Ning Yang once said, “When you read Dirac’s papers, you can feel the clearness, as clear as the autumn water. His papers, without anything unnecessary, are amazing, especially the ending which is often beyond imagination.” 3. Dirac’s “large number (quasi-conservation) hypothesis” The reason why Dirac is introduced here is largely due to the amazing “large number hypothesis” he proposed. The hypothesis is a clever reflection of Dirac’s conservation idea. He studied the smallest microscopic particles and the biggest universe, and gave the specific ratio of the smallest constant to the largest one. In 1937, Dirac proposed the large number hypothesis. He studied various constants (see Table 6.1) and constructed several quantities (see Table 6.2). Table 6.1: Various constants in physics. Constant

Letter

Speed of light Charge of electron Mass of proton Mass of electron Newton’s gravitational constant Hubble’s constant Cosmic average density

c e mp me G H ρ

Table 6.2: Dirac’s large number hypothesis. 2

Ratio of minimum force to maximum force (ratio of atomic static force to gravitation)

a1 ¼ Gmee mp ¼ 2:3 × 1039

Ratio of minimum force to maximum force (ratio of atomic static force to gravitation)

a2 ¼ mee2cH ¼ 7 × 1039

Ratio of minimum force to maximum force (ratio of atomic static force to gravitation)

3

 2 8πρe3 a3 ¼ 3m ¼ 1:2 × 1039 H3 p

86

6 Conservation of charge and conservation of current

Dirac put forward a question: Why are all the dimensionless large numbers in nature related to 1039 ? This quasi-conservation hypothesis attracted the attention of a large number of physicists, along with many different opinions. To some degree, it provoked waves of controversy in the field of physics. Dirac still held to the belief that it was a powerful tool for the development of cosmology and atomic theory. In his speech A Look at the Unity of Four Interactions from a Historical Perspective, Chen-Ning Yang included the following (see Table 6.3). Table 6.3: The four interactions. Four interactions

Intensity

Strong Electromagnetic Weak Gravitation

 − − −

The ratio of gravitation to strong force is 1038 , which is worth noting.

Appendix *

As to steady field ∇  A ¼ 0, the above part has mentioned the vector magnetic potential *

A, the Lorenz Gauge and the Columb Gauge. Here, a detailed study will be made. We know that the determination of the two potentials in the electromagnetic theory is based on (* * B ¼ ∇×A * (A6:1) * E ¼ ∇ϕ  ∂∂tA *

where A is called the vector magnetic potential and ’ is the scalar electric potential. * * Hence, the previous question of determining the field intensity ðE ; BÞ is now trans* formed into the one of determining the potential ðA; ’Þ as shown in Fig. A6.1.

Determining the field

B,E

is transformed into

Determining the potential

A,φ

Fig. A6.1: Determining the electric field intensity by vector magnetic potential and scalar electric potential.

87

Appendix

All works have the following description or something similar. At this time, if we choose properly, *

∇  A ¼ με

∂’ ∂t

the potential wave equation can be simplified significantly. 8 * * * < ∇2 A  με ∂2 A2 ¼ μ J ∂t :

2

∇2 ϕ  με ∂∂tϕ2 ¼  ρε

(A6:2)

(A6:3)

Equation (A6.2) is called the Lorentz Gauge. It is noteworthy that few works mention why eq. (A6.2) is given. In fact, the essence of gauge selection is: The definition of the vector magnetic potential is given * by the rotation of vector field A (see Fig. A6.1). However, according to the famous vector field theorem, it is necessary to know both its rotation and divergence in order to determine a vector field, thus producing the freedom of selection. The essence of the * * Lorentz Gauge is to determine ∇  A through appropriate selection, thus determining A. As a corresponding quantity, the definition equation (A6.1) of scalar electric potential ’ makes it possible that it may lack one constant function (the standard point of potential is different), but people tend to ignore it. When we study the special case in the steady field, ∂’ ≡0 ∂t

(A6:4)

The Lorentz Gauge becomes *

∇A¼0

(A6:5) *

We call eq. (A6.5) the Columb Gauge, which is also a reasonable choice of ∇  A.

7 Electromagnetic reciprocal symmetry and lossless symmetry This chapter deals with two types of electromagnetic symmetry in detail: Lorentz reciprocal symmetry and Hermite lossless symmetry. Its expression in the multi-port network system is given. The paper points out that the impedance matrix of the nonreciprocal lossless network can have real part, that is, Z = R + jX and RT = − R, which satisfies the antisymmetry. The network-based Foster theorem reveals that the frequency derivative ∂S + =∂ðjωÞS of the S parameter has the property of generalized inertia, which means a + ∂S + =∂ðjωÞSa should not be too large. Finally, under the condition of the symplectic sense, the electromagnetic reciprocal symmetry is the symplectic orthogonal inner product, and the second-order normalized resistance R is a typical symplectic matrix.

7.1 Introduction This is the seventh chapter of Electromagnetic Field Theory Teaching Series. When Chairman Mao met Dr. Tsung-Dao Lee on May 30th, 1974, the first question he asked was – “Why is symmetry important?” [18]. Of course, Chairman Mao was not restricted to the physical field, but he was thinking about the question on the philosophical and social level. The simplest answer was given by Chen-Ning Yang in his paper Field and Symmetry, “Today it is generally agreed that all the fundamental forces in nature are produced by symmetry. That is to say, symmetry dominates interaction” [19]. With this point of view, it can be said that electromagnetic symmetry reveals the intrinsic nature of electricity and magnetism. To make things simple, electromagnetic problems boil down to the following operator equation [20]: LðuÞ = g

(7:1)

where L represents the linear scalar or vector operator and includes the complex actions of the medium. Equation (7.1) does not mention the boundary condition (or boundary operator) and does not loss its generality. For eq. (7.1), there are two broad understandings at the same time. One is that * given excitation g (such as charge ρ, current J , or reference magnetic charge ρm * * * magnetic current M), find the yielded generalized field u (potential ’ or field E ; H ). * * The other is that given generalized field g (such as E or H ), find another field u (such * * as H or E ). Electromagnetic symmetry falls into two major types: the symmetry of electromagnetic media and the symmetry of equations or physical quantities. This chapter will focus on the symmetry that the medium shows via operator L. https://doi.org/10.1515/9783110527407-007

90

7 Electromagnetic reciprocal symmetry and lossless symmetry

7.2 Lorentz reciprocal symmetry Lorentz reciprocal symmetry is a very important symmetry of electromagnetic media, and most of the media in engineering belong to this type, known as reciprocal media. Research shows that the properties of media are reflected in operator L, and the symmetry is represented by quadratic quantization, or the inner product method. We introduce the Dirac inner product sign h , i to reflect the electromagnetic quadratic function. Its scalar inner product is defined as ððð abdτ′ (7:2) ha, bi = v

Now, let us study two different operators L and L a , whose corresponding equations are ( L ðuÞ = g (7:3) L a ðvÞ = h Let their mutual inner products equal to each other, and we have hvT , gi = huT , hi

(7:4)

hvT , LðuÞi = huT , La ðvÞi

(7:5)

that is,

Superscript T in eqs. (7.4) and (7.5) represents transpose, which might be used for future expansion into a matrix. When eq. (7.5) is satisfied, L a is called the adjoint operator of L. And particularly, when La = L

(7:6)

L is called the self-adjoint operator, which satisfies hvT , LðuÞi = huT , LðvÞi

(7:7)

Operator L that satisfies eq. (7.7) has Lorentz reciprocal symmetry. Conceptually, it reveals the magic symmetry between the source and the field under this medium condition, as shown in Fig.7.1. As an example, let L = − ε∇2 , and we have  ððð ðð  ∂Φ ∂ψ ds′ (7:8) ðψ∇2 Φ − Φ∇2 ψÞdτ′ =  ψ −Φ ∂n ∂n v

s

When the volume integral is extended to total space, the right side of eq. (7.8) tends to be zero, that is,

7.2 Lorentz reciprocal symmetry

91

u

v

Fig. 7.1: A source with g = L ðuÞ at point A yields a field of u at point B; a source with h = LðvÞ at point B yields a field of v at point A.

B

A

h = L(u)

g = L(u)

〈uT, L(u)〉

〈vT, L(u)〉

ððð

ψ∇ Φ dτ′ =

ððð

2

v

Φ∇2 ψ dτ′

(7:9)

v

or written in the form of inner product hψ, LðΦÞi = hΦ, LðψÞi

(7:10)

Equation (7.10) shows the reciprocal symmetry (or operator self-adjoint) of the L = − ε∇2 operator. An important point is that it reflects the reciprocity of medium ε. For the general electromagnetic form, we give ððð

*

*

*

*

ðE 2  J 1 + H 2  M1 Þdτ′ =

v

ððð

*

*

*

*

ðE 1  J 2 + H 1  M2 Þdτ0

(7:11)

v

or written in the form of inner product *

*

*

*

*

*

*

*

hE 2 , J 1 i + hH 2 , M1 i = hE 1 , J 2 i + hH 1 , M2 i

(7:12)

Unlike eq. (7.10) which is the scalar inner product, eq. (7.12) is the vector inner product. eqs. (7.11) and (7.12) are called the general form of the electromagnetic Lorentz reciprocity theorem. Judging from the perspective of medium alone, the reciprocal medium satisfies the symmetry ( εT = ε (7:13) μT = μ where both ε and μ are the most general forms of tensor. In recent years, the research of electromagnetic computations tends to transform a complex system to a multi-port network, as shown in Fig. 7.2. In this case, the relationship between the incident wave parameter a and the scattering wave parameter b is b = Sa (7:14) And, Lorentz reciprocal symmetry is shown as ST = S If the network is taken as a generalized two-port block matrix, that is,

(7:15)

92

7 Electromagnetic reciprocal symmetry and lossless symmetry

1 a1 b1 2

bn n an

a2 Multi-port network [S]

b2

bj

a3 3

j aj

b3 ai

i

bi Fig. 7.2: Multi-port network [S].

" S=

SII

SIΠ

SΠI

SΠΠ

# (7:16)

Then, we have STIΠ = SIΠ

(7:17)

7.3 Hermite lossless symmetry The energy characteristics of the medium have always been a concern for people. Generally, engineers pursue the lossless system. And for convenience, small loss is often neglected and lossless approximation is used in design. In this section, energy inner product is introduced first, followed by a detailed study of Hermite lossless symmetry.

7.3.1 Scalar energy inner product We define hu + , Lu ðuÞi > 0

(7:18)

as the scalar energy inner product, which reflects the static field energy. In the equation, + is the Hermite symbol, specifically ð Þ+ = ð * ÞT = ½ð ÞT * , or transposed conjugation, and T is the transition to the matrix type. For example, if Lu = − ε∇2 u = ’ and g = ρ, then the operator equation can be written as − ε∇2 ’ = ρ

(7:19)

7.3 Hermite lossless symmetry

93

Equation (7.19) is the electrostatic potential equation, whose corresponding energy inner product is energy storage we . ððð + , L ðuÞ = εj’j2 dτ′ = we (7:20) u h i u v

On the other hand, let Lv = − μ∇ , v = ϕ and g = ρm , then 2

− μ∇2 ϕ = ρm

(7:21)

Equation (7.21) is the static magnetic potential equation, whose corresponding energy inner product is energy storage wm . ððð + μjϕj2 dl′ = wm (7:22) hv , Lv ðvÞi = v

Note particularly that under the static field condition, u and v represent the inner product of electric energy and the inner product of magnetic energy, respectively. They are independent of each other. Also, it should be pointed out that eigenvalue λ of the energy operator is positive and has a stable Rayleigh quotient equation.

7.3.2 Vector energy (power) inner product Vector inner product is defined as * *

ðð

ha ; b i =

*

*

^dS′ a×bn

(7:23)

S

It represents the inner product of cross-section S pointing at the port, while energy product is essentially the power inner product (directional) through cross-section S. We have *

*

hu + , Lu ðu Þi > 0

(7:24)

*

When expressed by electric field E , we have *+

*

Pe = hE ; Le ðE Þi

(7:25)

where Le =

1 ∇× η2 ε

(7:26)

pffiffiffiffiffiffiffiffi * and η = μ=ε is the wave impedance. Similarly, when expressed by magnetic field H , we have

94

7 Electromagnetic reciprocal symmetry and lossless symmetry

*+

*

Pm = hH , Lm ðH Þi

(7:27)

where Lm =

η2 ∇× μ

(7:28)

Hermite lossless symmetry is *+

*

*+

*

hE , Le ðE Þi = hH , Lm ðH Þi It is written in the integral form as ðð ðð h h *+ *i *+ *i ′ ^ ^dS′ E × − ε∇ × E  n dS = H × − μ∇ × H  n S

(7:29)

(7:30)

S

Conceptually, it can be understood that the inner product of the electric vector is equal to that of the magnetic vector. Essentially, the total input power of the port is equal to the total output power, as shown in Fig. 7.3. Pin Lossless System

Pout

Fig. 7.3: The essence of the lossless system Pin = Pout .

Note that the operators of the static field and the alternating field have been unified when eq. (7.29) is transformed into eq. (7.30). According to multi-port network, we know that eq. (7.30) is equal to I + ZI = − I + Z + I

(7:31)

And, the power relationship is (

Pin = 21 aþ a Pout = 21 bþ b

(7:32)

The lossless condition is Pin = Pout , or a + fS + S − I ga = 0

(7:33)

where I is the unit matrix. Since eq. (7.33) is independent of excitation a, we have S+ S = I

(7:34)

Equation (7.34) is the network-type matrix representation of Hermite lossless symmetry. Similarly, if the multi-port network is considered as a generalized twoport block network, its lossless conditions will be

7.4 Network-type foster theorem, generalized inertia

95

jdet SII j = jdet SΠΠ j

(7:35)

detðS + IΠ SIΠ Þ = detðSΠI S + ΠI Þ

(7:36)

Equations (7.35) and (7.36) are generalized module symmetry and generalized qusi-reciprocity, respectively. Especially for m = ð1=2Þn, that is, SIΠ and SΠI are both square matrices, we have [21] jdet SIΠ j = jdet SΠI j

(7:37)

8 det SII = jdet SII jejΦII > > > > < det S = jdet S jejΦΠΠ ΠΠ ΠΠ > det SIΠ = jdet SIΠ jejΦIΠ > > > : det SΠI = jdet SΠI jejΦΠI

(7:38)

Let the block matrix

We can also give the determinant of scattering matrix S det S = exp½jðΦII + ΦΠΠ Þ

(7:39)

and the generalized characteristic phase Φ Φ = ðΦIΠ + ΦΠI Þ − ðΦII + ΦΠΠ Þ = ± π From the perspective of medium alone, Hermite lossless symmetry satisfies ( ε+ = ε μ+ = μ

(7:40)

(7:41)

where both ε and μ are the general (matrix) forms of tensor.

7.4 Network-type foster theorem, generalized inertia First, we extend the Foster theorem to a multi-port lossless system, as shown in Fig. 7.4. Then, we write Maxwell’s equations in matrix form for n external ports ( * * ∇ × E =  jωμH (7:42) * * ∇ × H = jωεE In the next section, we will prove that for a multi-port lossless network, the impedance matrix Z has real part R. That is, Z = R + jX

(7:43)

RT = − R

(7:44)

And

96

7 Electromagnetic reciprocal symmetry and lossless symmetry

V–2 , I–2

V+2 , I+2 V–n , I–n V–n , I+n

V+1 , I+1

Lossless system V–1 , I–1

Fig. 7.4: Multi-port Foster theorem.

The nonreciprocal lossless R satisfies the anti-symmetry. Then, the Foster theorem of n-port lossless network is I+

∂Z I = 4ðwm + we Þ ∂ðjωÞ

(7:45)

Due to the antisymmetric property of R, we have I+

∂R I≡0 ∂ω

(7:46)

Not yielding active power, R in the lossless matrix Z still follows Foster reactance theorem. That is, I+

∂X I = 4ðwm + we Þ ∂ω

(7:47)

The right part of eq. (7.47) is related to energy storage, and we know that ∂X=∂ω is a positive definite matrix. When scattering matrix is used, we have  +  ∂S + (7:48) S a = 2ðwm + we Þ a ∂ðjωÞ Then, ð∂S + =∂ðjωÞÞS is also a positive definite matrix. Equations. (7.47) and (7.48) both indicate the rate of change in frequency but differ slightly. For currents with different frequencies, there might be zero, so ∂X=∂ω can tend to be infinity. And incident wave a can only be nonzero, so the Foster theorem given by eq. (7.48) reveals that the operator frequency is related to energy storage. It reflects the generalized inertia: with the incident power a + a being constant, a + ðð∂S + =∂ðjωÞÞSÞa cannot be very large, for it is limited by ðwm + we Þ. Note that the Foster theorem can be extended to the Q value of small loss systems. We have ωQ =

a + ð∂S + =∂ðjωÞSÞa a + ðI − S + SÞa

(7:49)

97

7.5 Electromagnetic reciprocal symmetry and lossless symmetry

eq. (7.49) is a typical Rayleigh quotient equation, with its corresponding maximum (or minimum) maxωQ = λ  det

∂S + ∂ðjωÞ

(7:50)

 + S − λðI − S SÞ = 0

(7:51)

The corresponding a is the best excitation vector, and quality factor Q is a complex multi-port system, which can be extended to open systems.

7.5 Electromagnetic reciprocal symmetry and lossless symmetry We have already discussed that the reciprocal property of operator L comes from the medium: εT = ε and μT = μ, and that its lossless property comes from the medium: ε + = ε and μ + = μ. These are two completely different and independent channels. But on the other hand, the fact that lossless media show certain generalized qusireciprocity indicates that there exists a magic “kinship” between these two electromagnetic symmetries, as shown in Table 7.1.

Table 7.1: Network reciprocity and lossless characteristics. Network parameters

Reciprocal property

Scattering matrix [S]

½S = ½ST Sij = Sjt

Transmission matrix [A]

det½A = 1

Impedance matrix [Z] Admittance matrix [Y]

½ZT = ½Z T

½Y = ½Y

Lossless property ½S − 1 = ½S + We write [S] as block matrix, that is,

SII SII I ½S ¼ SI II SII II All the block matrix are square matrices jdet½SII j = jdet½SII II j Generalized module symmetry jdet½S II I j = jdet½SI II j Generalized qusi-reciprocity qusi-reciprocity in 2n-port network jdet½Aj = 1 Anti-symmetry ½RT = − ½R, ½GT = − ½G qusi-reciprocity ½XT = ½X, ½BT = ½B

What’s more, some even proved [22] directly that the lossless property absolutely leads to reciprocal property. According to network losslessness S + S = I, we have

98

7 Electromagnetic reciprocal symmetry and lossless symmetry

Z+ +Z =0

(7:52)

For a lossless system, the impedance matrix Z should not have a real part, namely Z = jX. In the meanwhile, from eq. (7.52) follows XT = X

(7:53)

ST = S

(7:54)

and

This is the so-called proving network reciprocity by network losslessness. The fundamental error here is that they mistakenly believe the impedance matrix Z of a lossless system has no real part. In fact, for the simplest dual-port lossless network, we have Z 12 S12 = Z 21 S21

(7:55)

When the medium is nonreciprocal, S12 ≠ S21 but jS12 j = jS21 j, or Z 12 = e jθ Z 21

ðθ ≠ 0Þ

(7:56)

It is proved that Z 12 and Z 21 have real parts, namely Z = R + jX. Together with the lossless condition eq. (7.52), we prove thoroughly ( RT = − R (7:57) XT = X That is to say, impedance matrix R is antisymmetric while reactance matrix X is symmetric. As an example, the ideal lossless nonreciprocal transmission line is discussed here. Its scattering matrix is " # 0 e − jθ ðθ ≠ ’Þ (7:58) S= e − j’ 0 It is easy to derive resistance matrix R. " 0 1 R= 1 − cosðθ + ’Þ − ðcos θ − cos ’Þ

ðcos θ − cos ’Þ 0

# (7:59)

It can be written as " R = R 0 J = R0

0

1

−1

0

# (7:60)

where R0 = ðcos θ − cos ’Þ=ð1 − cosðθ + ’ÞÞ, and J = ized resistance matrix.

0 1 −1 0

is called the normal-

Q&A

99

7.6 Summary The two electromagnetic symmetries discussed in this chapter – Lorentz reciprocal symmetry and Hermite lossless symmetry – both come from the medium. For completely different fields, Hamilton transformed analytical mechanics into inherently magic symmetry in the canonical system. Based on this, the famous scientist Weyl at Princeton Institute established symplectic symmetry and symplectic theory. He defines the symplectic orthogonal inner product as [23] vT1 Jv2 = 0 where

" J=

0

I

−I

0

(7:61) # (7:62)

Surprisingly, the electromagnetic reciprocity theorem is the symplectic orthogonal inner product under the condition of symplectic sense. E2x E2y [E1x , E1y , E1z , J1x , J1y , J1z ]

I

E2z

–I 0

J2x

0

⇀ ⇀

⇀ ⇀

= E1 . J2 – E2 . J2 ≡ 0

(7.63)

J2y J2z

Similarly, the second-order normalized resistance matrix R in a nonreciprocal lossless system satisfies the definition of the symplectic matrix, that is, [23] RT JR = J

(7:64)

The combination of the electromagnetic symmetry with the symplectic symmetry in analytical mechanics is a topic worthy of further research and discussion.

Q&A Q: Could you talk about the ideas and characteristics of this chapter? A: Symmetry is one of the distinct themes in the whole book. As you remember, Chapter 6 tells us that conservation and symmetry are closely linked to each other. Actually, symmetry will be further discussed in the next chapter. Here, we will first discuss the three main methods of modern electromagnetic computation theory: network, operator and inner product, as shown in Fig. 7.5.

100

7 Electromagnetic reciprocal symmetry and lossless symmetry

Network Operator Network

L(u) = g

Inner product

a,b

Fig. 7.5: Three main methods of modern electromagnetic computation theory: network, operator and inner product.

1. Network method The network method originates from the network idea. The network idea is supposed to solve an unknown or to explore a complex system. When solving a problem, we “wrap up” the inner part of the unknown and impose an excitation to see its reaction. In this way, its characteristics can be determined, as shown in Fig. 7.6.

Exciting Network Reaction

Fig. 7.6: The network idea – determining the characteristics of an unknown system by excitation and reaction.

The specific way of handling the network idea is to use the network method, which becomes matrix equations for most linear systems. ½S½a = ½b 2. Operator method In the operator equation LðuÞ = g g is the generalized source – excitation, and u is the generalized field – reaction. It is exactly the algebraic representation of the network idea. The operator equation clearly indicates the process of determining field u when given source g. In the equation, L represents any complex process from the source to the field and shows the linear operator in most cases. It is easy to find u = L − 1 ðgÞ where L− 1 ð Þ is the corresponding inverse operator. In fact, most specific electromagnetic problems do not have analytical results.

Q&A

101

The method we employ is shown in Fig. 7.7, which has two key new concepts: (quasi) complete function system fui g and inner product.

Complete function systems

Decomposition of u

{ui}

u = ∑ aiui

Operator equations

n i=1

n

∑ aiL (ui) = g

i=1

Inner product uj

Matrix equation

∑ ai uj , L (ui) = uj, g

[S][a] = [b] [a] = [ ] [b] = [ ]

n

i=1

j = 1,2...n

Fig. 7.7: The solving process of linear operator equation L ðuÞ = g.

It seems a tough problem to be both (quasi)complete and easy to compute. But do not worry. We take the one-dimensional case as an example. If the computational domain is ½ x1 x2 , fui g can be a global base or a sub-entire-domain function. As a typical example, we take the Pulse function, as shown in Fig. 7.8. For accuracy reasons, n should be made sufficiently large.

x1

x2

x1

x2

Fig. 7.8: The expansion of the Pulse sub-domain function.

The pulse expansion is simple and computable. Only when n is very large can we use the computer to solve equation sets. 3. Inner product method To solve the equation L ðuÞ = g after establishing function system fui g, the key method is the inner product one. Physically, uj is used to excite both sides of the equation to generate reactions h uj L ðui Þ i and h uj g i; so that fai g could be determined. Geometrically, inner product is the projection of the unknown vector over each * coordinate. In the three-dimensional case, vector radius r is r = x^i + y^j + z^k

* *

r is expressed by three projection components (x, y, z), but currently the most general situation is that the function vector g is known but the function vector L ðuÞ is unknown. In this case, the so-called function system fui g, (i = 1, 2, . . ., n) is an n-dimensional coordinate system. Their projections are shown in Fig. 7.9. Set an unknown function u = a1 u1 + a2 u2 +    + an un . Note that this is exactly the * same as r = x^i þ y^j þ z^k for the three-dimensional case, as ðu1 , u2 , . . . , un Þ corresponds   to ^i ^j ^k and ða1 , a2 , . . . , an Þ corresponds to ð x y z Þ; which are the projections of the unknown function we are to solve.

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7 Electromagnetic reciprocal symmetry and lossless symmetry

Based on the above explanation, we can write LðuÞ = a1 Lðu1 Þ + a2 Lðu2 Þ +    + an Lðun Þ =

n X

ai Lðui Þ

i=1

where Lðui Þ is thought to be known since fui g is given. Then, the projection of Fig. 7.9 (b) is h uj

LðuÞ i =

n X

Lðui Þ i

a i h uj

i¼1

un

un, g

un

un, L (u)

g

L (u)

u2, g u1 , g

o

u2,L (u)

u2

u1 ,L (u)

u1

u2

o

u1

Fig. 7.9: Geometrical meaning of inner product – the projection of function vectors over the set coordinate system fui g: (a) the projection h u1 g i; h u2 g i; . . . ; h un g i of given function vector g and (b) the projection h u1 LðuÞ i; h u2 LðuÞ i; . . . ; h un LðuÞ i of unknown function vector LðuÞ.

The corresponding linear equation set is n X

h uj

Lðui Þ iai = h uj

gi

i=1

We can find ai , or (within a certain precision) the unknown function u. The network, operator and inner product discussed above are the major basis of the new method of electromagnetic computation – method of the moment. Q: This chapter discusses the network-type Foster theorem, which is quite new. Would you please explain it further? A: The Foster theorem for a lossless single-port network system is familiar to all. Simply put, it means that the reactance slope ∂x=∂ω of the port is constantly positive, as shown in Fig. 7.10. ∂x >0 ∂ω In this chapter, a single-port network system is extended to a multi-port system, as shown in Fig. 7.4.

Recommended scholar

103

x

One-port network

Zin = jx

o

ω

Fig. 7.10: Foster theorem for a single-port network – the reactance slope of the port is constantly positive: (a) one-port network and (b) ∂x=∂ω > 0.

In this case, two formulas are derived, which are equivalent to each other, as shown in Fig. 7.11.

Matrix expression of current I I+

дX I = 4 ( wm + we ) > 0 дω

Matrix expression of incident wave a

a+

дS + д( jω )

S a = 2 ( wm + w e ) > 0

Fig. 7.11: Two representations of the network-type Foster theorem.

It is obvious that ∂X=∂ω is a positive definite matrix according to the representation of current matrix I, and ð∂S + =∂ðjωÞÞS is also a positive definite matrix according to the representation of incident wave matrix a. It seems that there is no difference between them. But this is not the truth. In the representation of current matrix I, for currents I with different frequencies ω, there might be 0 point, so ∂X=∂ω can tend to be infinity. However, in the representation of incident wave matrix a, incident wave a can only be nonzero, so a + ðð∂S + =∂ðjωÞÞSÞa cannot be very large, for it is limited by ðwm + we Þ. This is what we call generalized inertia.

Recommended scholar In this chapter, the recommended scholar will be R. F. Harrington (Fig. 7.12), the famous contemporary electromagnetic theory expert. I studied overseas at Syracuse University from 1980 to 1982. Although he was not my direct adviser (my adviser Jun

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7 Electromagnetic reciprocal symmetry and lossless symmetry

Zheng is also a very famous expert in the electromagnetic field and will be introduced later), I worked with him all the time and was influenced a lot by Harrington.

Fig. 7.12: Professor Harrington (left) and the author at Syracuse University.

1. Harrington emphasizes ideas and methods From Harrington I came to know the difference between scientific and blind ways of doing things. Over years, we have thought that the purpose of doing things is to complete a task. However, what I learn from Harrington and his books is that we need to do things in a scientific way. For example, prior to Harrington, people had started to perform electromagnetic computations by computer, and functional analysis was not uncommon. However, what I sincerely admire is network, operator and inner product proposed by Harrington and his method of the moment.

dl

n S Fig. 7.13: Linear antenna or scatterer.

In his book Method of the Moment, the methods of network, operator and inner product are applied to linear antennas and scatterers (Fig. 7.13). The beginning is particularly impressive, as Harrington stated clearly that linear antennas and scatterers are the same thing.

Recommended scholar

105

Harrington said, “The basic difference between an antenna and a scatterer lies in the position of source. If the source point is far away from an object, it can be seen as a scatterer.” As a Chinese poem goes, “For a grander view, one must mount for a greater height.” Harrington is at a greater height than us and therefore he has more profound views. Let’s return to the linear antenna and scatterer. By segmentation and simple deduction, we have ½Z½I = ½V Surprisingly, this is the same as ZI = V, the Ohm’s law of extended matrix. At this moment, I was completely shocked. Ohm’s law also applies to the linear antenna – except that it is a matrix expression. I remember the comments of Weil on Loo-Keng Hua, “Hua plays matrices the way he does numbers.” I finally understand that matrix is the further extension of numbers. Any law of numbers can be found applicable in matrices. Now, we can say without any exaggeration that the matrix is a linear world, and vice versa” (see page 222 of Symmetry, Liang Changhong. China Science Press, 2010). As the saying goes, learning from an academic master is better than studying for 10 years. And Harrington is no doubt such a master. 2. Harrington taught me group and seminar Professors meet students on a regular basis to discuss work and study in an equal and free way. This is called group and seminar in American universities. Harrington invited me to join his group and report my work many times, which delighted me a lot. We should learn the advanced educational system of America, from which I gained endless benefits. Back then, Harrington had an assistant named R. Martz. He was 6 feet 3 inches tall, thin, wearing a coat that might haven’t been washed for 10 years (Syracuse is very cold, known as the City of Snow). Martz was quite an expert in computer. We said in private conversations that Harrington was lucky enough to have his mind and Martz’s hands. 3. Harrington’s great personality Harrington, as a scholar, influenced me with his personality. Once I went into his office at noon, only to find that he twisted in agony on his canvas bed beside his office table. He suffered from serious stomachache, a scene that shocked me completely. Seeing this, I said, “I am sorry” and went out quietly. After that, I wrote a lot about him in my diary. Harrington was not awarded Model Worker or Outstanding Worker like we do in China. But why was he working so hard? More than 30 years have passed but his personality has been so impressive with me. Whenever I want to slack off, Harrington’s personality charm would flash across my mind, and I would push myself even harder. This, I think, is his strong personal charisma.

8 Electromagnetic symmetry and symmetry operator In electromagnetic research, we often find such topics as lossless symmetry in energy relationship, reciprocal symmetry between source and field points and mirror symmetry in geometry. In this chapter, the symmetric operator will be introduced to describe various electromagnetic symmetries. It should be noted that electromagnetic reciprocal symmetry is the famous symplectic symmetry proposed by Weyl in mechanics. Quadratic and linear symmetry forms are also discussed in detail.

8.1 Introduction This is the eighth chapter of Electromagnetic Field Theory Teaching Series. In electromagnetic research, we often encounter an important topic – symmetry, such as lossless symmetry in energy relationship, reciprocal symmetry between source and field points and mirror symmetry in geometry. How to describe these symmetries in a unified way is a worth in-depth study. First of all, corresponding electromagnetic matrix E and symmetric operator O need to be introduced for specific problems. Then, quadratic or linear symmetry should be used to unify various standard electromagnetic symmetries. It should be noted that electromagnetic reciprocal symmetry is the famous symplectic symmetry proposed by mathematician Weyl in mechanics. In addition, electromagnetic symplectic inner product and symplectic orthogonality are also discussed in this chapter.

8.2 Quadratic symmetry Here is the general expression of quadratic symmetry. ET OE = O

(8:1)

where T is matrix transpose.

8.2.1 Lossless Symmetry The familiar electromagnetic scattering matrix S is defined as Sa = b

https://doi.org/10.1515/9783110527407-008

(8:2)

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8 Electromagnetic symmetry and symmetry operator

where b is the scattering wave matrix and a is the incident wave matrix. The two-port scattering matrix here is shown in Fig. 8.1 a2

b1 [S]

1

2

a1

b2

"

Fig. 8.1: S matrix of two-port scattering.

S11

S12

S21

S22

#"

a1

# "

a2

=

b1

#

b2

It is easy to get [24] incident wave power 1 Pin = a + a 2

(8:3)

1 PSC = b + b r 2

(8:4)

and scattering wave power

where

+

= ½* T = ðT Þ* is called Hermite sign. Thus, we can obtain lossless symmetry S + IS = I

(8:5)

Compared with the general expression (8.1), we can know E = S. The symmetric operator O = I, so it is the identity operator. Note that the coordinate rotation matrix is " # cosθ sinθ (8:6) R= - sinθ cosθ Then, it can be seen that identity matrix is the rotation matrix for θ = 0 , namely " # 1 0 = Rðθ = 0 Þ I= (8:7) 0 1 Changing both sides of eq. (8.5) into determinants, we have jdet ðSÞj = 1

(8:8)

det ðSÞ = ejð’11 + ’22 Þ

(8:9)

Specifically,

where S11 = jS11 jej’11 and S22 = jS22 jej’22 .

109

8.2 Quadratic symmetry

Note that for the S scattering matrix, the lossless symmetric eq. (8.5) can be further extended to n-port network (n ≥ 2), as shown in Fig. 8.2. In this case, we obtain 2 3 S11 S12    S1n 6 7 6 S21 S22    S2n 7 6 7 S=6 . (8:10) .. 7 .. 6 .. . 7 . 4 5 Sn1 Sn2    Snn 2

1

0

60 6 I =6 6 .. 4.



1 .. .



0

0



0

3

07 7 7 ... 7 5

(8:11)

1

b1

1

a1

bn

b2 2

n

S

a2

ai

an

bi

Fig. 8.2: n-Port S network.

i

8.2.2 Reciprocal Symmetry For two-port transmission matrix A, a quadratic can be used to express the reciprocal symmetry. Figure 8.3 shows the basic model of matrix A. I2

I1 1

V1

[A]

V2

2

Fig. 8.3: Two-port transmission matrix A.

Its basic definition is "

V2 I2

# " =

A11

A12

A21

A22

#"

V1

#

I1

When a system is reciprocal, matrix A has the following property [24]:

(8:12)

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8 Electromagnetic symmetry and symmetry operator

det ½A = 1

(8:13)

A11 A22 − A12 A21 = 1

(8:14)

Specifically,

Note that #" " 0 A11 A21 A12

A22

1

#"

1 0

A11

A12

A21

A22

# "

0

   A11 A22  A12 A21 " # 0 1 = detðAÞ 1 0 =

ðA11 A22  A12 A21 Þ

#

0 (8:15)

It can be written in the form of quadratic symmetry: AT JA = J

(8:16)

where " J=

#

0

1

1

0

(8:17)

In terms of the rotation matrix, J is equivalent to 90 rotation, namely J = Rðθ = 90 Þ

(8:18)

Here, it should be noted that eq. (8.16) is the symplectic symmetry [23] pointed out by Weyl in mechanics. Therefore, the symplectic symmetry in mechanics is the reciprocal symmetry [25] in electromagnetics. As A can be a complex number in electromagnetics, the symmetry is even more extendable than mechanics. In addition, reciprocal symmetry cannot be extended to n-port network, which is different from scattering matrix S. 8.2.3 Geometric reciprocal symmetry Here is the two-port impedance matrix Z, as shown in Fig. 8.4. I2

I1 1

V1

Z

V2

2

Fig. 8.4: Two-port impedance matrix Z.

"

Z11

Z12

Z21

Z22

#"

I1 I2

We introduce mirror symmetry operator M.

# " =

V1 V2

#

8.2 Quadratic symmetry

" M=

1

0

0

1

111

# (8:19)

If impedance matrix Z satisfies, Z T MZ = M

(8:20)

Specifically, "

Z11

Z21

Z12

Z22

#"

1

0

0

1

#"

Z11

Z12

Z21

Z22

# " =

1

0

0

1

#

That is, "

2 2  Z21 Z11

Z11 Z12  Z21 Z22

Z11 Z12  Z21 Z22

2 2 Z12  Z22

# " =

1

0

# (8:21)

0 1

We can obtain Z11 = Z22

(8:22)

Z12 = Z21

(8:23)

A set of negative solutions is ignored. Z11 = Z22 shows geometric symmetry, and Z12 = Z21 indicates reciprocal symmetry. So, it is called geometric reciprocal symmetry.

8.2.4 Compound symmetry If we introduce compound symmetric operator L for transmission matrix A, then " # 1 1 1 L = pffiffiffi (8:24) 2 1 1 The quadratic of compound symmetry is expressed as AT LA = L

(8:25)

According to coordinate rotation, L is 45° rotation, namely L = Rðθ = 45 Þ

(8:26)

Specifically, " 1 A11 pffiffiffi 2 A12 That is,

A21 A22

#"

1

1

1

1

#"

A11

A12

A21

A22

#

" 1 1 = pffiffiffi 2 1

1 1

# (8:27)

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8 Electromagnetic symmetry and symmetry operator

"

A211 þ A221

ðA11 A22  A12 A21 Þ þ ðA11 A12 þ A21 A22 Þ

ðA11 A22  A12 A21 Þ þ ðA11 A12 þ A21 A22 Þ " # 1 1 = 1 1

#

A212 þ A222

(8:28) It is easy to get 8 > > > >
A11 A22  A12 A21 = 1 > > > : A11 A12 þ A21 A22 = 0

(8:29)

Finally, 8 > < A11 = A22 A12 =  A21 > : det A = 1

(8:30)

We call eq. (8.30) compound symmetry. It deserves further study as to which real network it corresponds to. It should also be pointed out that as long as the symmetric operator satisfies jdetðOÞj = 1

(8:31)

we can always get the electromagnetic matrix E jdet Ej = 1

(8:32)

In most cases, det E = 1.

8.3 Linear symmetry Here is the general expression of linear symmetry. ET O = OE

(8:33)

8.3.1 Geometric symmetry We introduce geometric symmetry factor G. " G=

0

1

1

0

# (8:34)

8.3 Linear symmetry

113

If the two-port impedance matrix satisfies, Z T G = GZ

(8:35)

Specifically, 8" # " #" > Z21 0 1 Z11 Z21 > > = > > < Z12 Z22 1 0 Z22 " #" # " > > Z11 Z12 0 1 Z21 > > = > : 1 0 Z Z Z11 21 22

Z11 Z12 Z22

# #

(8:36)

Z12

Then, we have Z11 = Z22

(8:37)

which satisfies the conditions of geometric symmetry.

8.3.2 Reciprocal symmetry Apply the identity operator I to the impedance matrix Z, and we get Z T I = IZ

(8:38)

Z 12 = Z 21

(8:39)

This is the reciprocal symmetry.

8.3.3 Anti-reciprocal symmetry Similarly, if the mirror symmetric operator M and impedance matrix Z are introduced, we have Z T M = MZ

(8:40)

Specifically, 8" # " #" # > Z11 Z21 1 0 Z11 Z21 > > = > > < Z12 Z22 0 1 Z12 Z22 " #" # " # > > Z11 Z12 1 0 Z11 Z12 > > = > : 0 1 Z Z21 Z22 21 Z22

(8:41)

Finally, Z12 = − Z21

(8:42)

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8 Electromagnetic symmetry and symmetry operator

We call it the anti-reciprocity condition. The realization of this type of electromagnetic network is worthy of further study. Here a problem is left: Why do we discuss the quadratic symmetry first? In fact, the linear symmetric eq. (8.33) is also another form of quadratic symmetry. ET ½OE − 1 = ½O

(8:43)

In this sense, all we need is quadratic symmetry (definition might be different). Similarly, the quadratic symmetry (8.1) can also be written as ET ½O = ½OE − 1

(8:44)

In fact, it becomes a linear symmetry with a different definition.

8.4 Symplectic inner product and electromagnetic symplectic orthogonality Compared with the common vector inner product, matrix J is the core of symplectic inner product. Definition 8.1: Given two state vectors v1 and v2 , then we call v1 T Jv2

(8:45)

v1 T Jv2 = 0

(8:46)

symplectic inner product, and

symplectic orthogonality. Specifically, for the 2 × 2 dimension, " #" # v1 0 1 ½ v 1 v2  =0 1 0 v2

(8:47)

Figure 8.5 shows the general symplectic inner product and the comparison of vector orthogonality with symplectic orthogonality. vector inner product

symplectic inner product

aTIb

V1TJV2

vector orthogonality

symplectic orthogonality

a

TIb = 0

V1TJV2 = 0

Fig. 8.5: Vector inner product and symplectic inner product.

As we know, there is a famous law called Lorentz reciprocal theorem in the electromagnetic reciprocity system. It defines that the field E1 produced by source J 1 at point B has the same inner product as the field E2 produced by source J 2 at point A. That is,

8.5 Summary

hE1 , J 2 i = hE2 , J 1 i

115

(8:48)

where E1 is the electric field generated by the source current J 1 , E2 is the electric field generated by another source current J 2 and h, i is the generalized inner product symbol (e.g., representing a volume integral). In other words, Lorentz’s reciprocity theorem reflects the symmetry between field and source, as shown in Fig. 8.6.

E1

E2

A

B J1

J2

Fig. 8.6: Lorentz’s electromagnetic reciprocity theorem.

If we introduce a state vector, 8 " # > E1 > > v = > > < 1 J1 " # > > E2 > > > v2 = : J2 then the reciprocity theorem can be rewritten as [25] " # " #+ * 0 I E2 ½E1 ; J 1  =0 I 0 J2

(8:49)

where 2

1

6 I =40 0

0

0

3

1

7 05

0

1

A simpler form of the equation above is v1 T Jv2 = 0

(8:50)

Here again, we can see that one of the physical meanings of symplectic orthogonality is electromagnetic reciprocity.

8.5 Summary It is of universal significance to study the electromagnetic symmetry through different symmetric operators, which can provide typical characteristics such as lossless symmetry, reciprocal symmetry and geometric symmetry. It should be noted that

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8 Electromagnetic symmetry and symmetry operator

electromagnetic reciprocal symmetry is the symplectic symmetry proposed by famous mathematician Weyl in the field of mechanics. Symplectic orthogonality is the electromagnetic reciprocity theorem.

Q&A Q: What is the relationship between symmetry and symmetry operators in electromagnetics? A: This question is the very topic of this chapter, that is using a variety of symmetric operators to define symmetry. As aforementioned, there are generally two kinds of symmetric operators: linear symmetric operator and quadratic symmetric operator. Figure 8.1 lists several typical linear symmetric operators, and Fig. 8.2 lists several typical quadratic symmetric operators. Most people believe that symmetry is mainly used in mathematics, mechanics and electromagnetics. But it is not true. When choosing a life partner, an important standard is the symmetric facial features. Modern medical research has found the biological basis for this – it is beneficial to future generations. It is a new approach to study symmetry by means of operator because it is irrelevant to the object of study and the field of study. Therefore, it is discovered that the symplectic symmetry proposed by Weyl in mechanics is the reciprocal symmetry in electromagnetics Tables 8.1 and 8.2.

Table 8.1: Linear symmetric operator. Linear symmetric operator O satisfies OE = E T O

Geometric symmetry

GZ = Z T G; G =

Reciprocal symmetry

IZ = Z T I; I =

Anti-reciprocal symmetry

MZ = Z T M; M =



0 1 ; Z 11 = Z 22 1 0

1 0 ; Z 12 = Z 21 0 1

1 0 ; Z 12 = − Z 21 0 1

Q: Could you explain more about symplectic symmetry, symplectic inner product and symplectic orthogonality? A: As we know, Weyl is a distinguished mathematician in the 20th century as well as Hilbert’s versatile successor. Hitler’s Nazism “sent” him to the Institute for Advanced Study in Princeton, where he made great contributions to the world.

Q&A

117

Table 8.2: Quadratic symmetric operator.  T E OE = 0 Quadratic symmetric operator O satisfies þ E OE = 0 S 6 S + 6 S IS = I; ½S = 4  S

Reciprocal symmetry

AT JA = J; A =

Geometric reciprocal symmetry

Z T MZ = M; ½Z =

Compound symmetry

AT LA = L; ½A =







A12 0 1 ; ½J = A22 1 0

A11 A21



3  0  7 7  5  1

3 2 1  S 6  S 7 7; ½I = 6    4  5 0  S

2

Lossless symmetry

Z 11 Z 21

A11 A21



Z 12 1 ; ½M = Z 22 0

0 1



A12 1 ; ½L ¼ 21 A22 1

1 1





Weyl’s ideas are not only clear but also normative. He used symmetric operator J to define symplectic symmetry: ST JS = J where S is called symplectic matrix. Table 8.3 gives four main properties of symplec

0 1 is called entanglement operator. tic matrix and J = 1 0 J 2 = − I,

J T = − J,

J −1 = − J

detðJÞ = 1 (det is the abbreviation of determinant)

Table 8.3: Four main properties of symplectic matrix S. det S = ± 1 (det is the determinant of S) ()

It is obvious from the definition that det ST det J det S = det J Generally, det S = 1

()

ST is also symplectic matrix

()

If S1 and S2 are symplectic matrices, then S = S1 S2 is also symplectic matrix

  ST JS = ðS1 S2 ÞT JðS1 S2 Þ = ST2 ST1 JS1 S2 T = S2 JS2 = J

()

Symplectic matrix has its inverse matrix S − 1

It is shown in property ()

By definition, we inverse it S − 1 JS − T = J (multiplied by S on the left ST on the right)  T SJST = ST JST = J

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8 Electromagnetic symmetry and symmetry operator

Table 8.4: Symplectic inner product, symplectic orthogonality and Lorentz’s electromagnetic reciprocal theorem. Symplectic inner product

State vectors v and v are defined as symplectic inner products

Symplectic orthogonality

v and v are symplectic orthogonal

Lorentz’s electromagnetic reciprocity theorem



E E Let v 1 = 1 ; v 2 = 2 J1 J2



0 1 E2 ≡0 then the reciprocity theorem is ½E 1 J1  1 0 J2 Symplectic orthogonality is electromagnetic reciprocity

Table 8.4 gives symplectic inner product, symplectic orthogonality and Lorentz electromagnetic reciprocity theorem.

Recommended scholar The most influential researcher in the field of symmetry is Chen-Ning Yang (Fig. 8.7).

Fig. 8.7: Chen-Ning Yang.

Yang is a master in science, much greater than merely an electromagnetic theory expert. Here, we show two points on how to set goals and how to do scientific research. 1. Yang has been at the peak of modern physics for more than half a century At the turn of the year 2000, Chen-Ning Yang delivered a very important lecture The Major Theme of Physics in the 20th Century. Physics in the 20th century has undergone or is undergoing tremendous changes. Yang summarized three most important themes: (1) quantum; (2) symmetry; (3) phase factor.

Recommended scholar

119

The world is surprised that Mr. Yang’s research is mainly around these three themes, and in certain aspects, he has done some pioneering work. Mr. Yang’s vision is unique and sets a good example to us. What should we focus on in a day, a year or a period of time? This is probably one of the things that young scholars are most concerned about. Two points should be noted: the importance of research objectives and the researcher’s academic foundation and adaptability. To sum up, one should strike a balance between his goal and adaptability when doing scientific research. 2. Forty years of teaching On March 2, 1983, Chen-Ning Yang made an important speech on the 20th anniversary of the Chinese University of Hong Kong – Forty Years of Studying and Teaching. I would like to point out a few things. The first thing is the speech title. People tend to think that Yang must have been studying and doing scientific research for 40 years. But they are wrong! He has been teaching for 40 years. It can be seen that teaching and research are not contradictory, but they promote each other. This can be shocking to those who look down on teaching in China and even think teaching is inferior to research. In addition, I would like to list a few subtitles of each part of his speech. – Solid foundation: Southwest Associated University – Scientific research and styles Yang said, “I particularly admire three people: Einstein, Fermi and Dirac, who are all great physicists in the 20th century. Their styles are different, but what they have in common is that they could develop new ideas from complex physical phenomena and express them in a mathematical way.” Their research papers are straightforward and keep to the point. In fact, this is exactly the style of Chen-Ning Yang. – Vivid and Alive Physics: University of Chicago. – Science, especially engineering, must be put into practice. – Experience in experiments. This is the master and recommended scholar in Chapter Eight – Chen-Ning Yang.

9 Plane image method and active conformal mapping This chapter combines the image method with complex conformal mapping. A large class of problems in practice can be solved by using plane image as a generalized model. By active conformal mapping, a complex shape can be transformed into the plane image model. Interestingly, the solution region after mapping may have image charges at infinity, which is a breakthrough for the common image method.

9.1 Introduction This is the ninth chapter of Electromagnetic Field Theory Teaching Series. The analytical methods of solving problems in a static field play a crucial role in electromagnetic theory. In the book Electromagnetic Field Theory, Lin Weigan pointed out, “In modern times, few problems in electromagnetics can be solved accurately. So electrostatic methods are often used to find approximate solutions. It is also the basis of solving electrodynamic problems” [11]. The image method and conformal mapping are two basic methods to solve electromagnetic problems. Table 9.1 lists their main features. In this chapter, plane dielectric images are treated as a generalized model. That is, conductor problems are exceptional cases when ε2 ! ∞, while magnet models correspond to ε2 ! 0. There are two types of conformal mapping. One is passive conformal mapping without singular line charge ρl , which can be used to determine the capacitance between conductors and the characteristic impedance Z0 of transmission lines. The other is active conformal mapping with singular line charge ρl , which can be used to find the potential distribution function ’ around the source. Active conformal mapping is rarely discussed in books [1,4,26], and is therefore the focus of this chapter. For two-dimensional conductors, a large class of problems can be solved by treating the plane image as a basic model, since any complex geometric shape can be transformed into the plane image after active conformal mapping. The solution is shown in Fig. 9.1. In research, we find that by active conformal mapping the solution region after mapping may have image charges at infinity. Because of this, we can see the original charge and the image charge appear in pairs, which is a breakthrough for the common image method.

https://doi.org/10.1515/9783110527407-009

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9 Plane image method and active conformal mapping

Table 9.1: The image method and conformal mapping method. Image method

Conformal mapping method

. Symmetry in solving problems (namely, the same medium is applied to all regions). . Add image charges (they must not appear in solution regions). . Solution by area and boundary conditions splicing. . The basis is the uniqueness theorem. Namely, the solution satisfying governing equations and boundary conditions is the only correct solution.

. The physical quantity φ, such as potential, does not change after mapping. . The source ρ does not change after mapping. . The potential and force line correspond to themselves after mapping. . The potential and force line remain orthogonal after mapping. . The boundary of the medium must be equipotential. . The approach is to solve the potential function φ by changing from asymmetry to symmetry.

Applies to two- and three-dimensional problems.

Applies only to two-dimensional problems.

y

v

Region I

ε0 O

Active conformal mapping

ρ

ρ

w= f ( z)

l

ε0

u

φ=

l

2πε0

ln

r2 r1

O

ε0

z-plane

−ρl

Line charge outside complex geometric shape

ρ

r2

l

x

φ

r1

w-plane

Fig. 9.1: Solution of active conformal mapping.

9.2 Generalized model of the plane dielectric image As mentioned earlier, we treat the plane dielectric image as a generalized model for a class of problems, as shown in Fig. 9.2. It has been pointed out in literature [26] that symmetry can be formed by making both regions ε1 : The solution is shown in Fig. 9.3. y

Region I d

ρ

l

ε1

O

x ε2

Region II Fig. 9.2: Generalized model of the plane dielectric image.

9.2 Generalized model of the plane dielectric image

y

d

y

φΙ

r1

ε1

ε1 d

ρ

l

Region I

ρ

l

Region I

r2

x O

ε1

x

O

Region II −d

123

−ρ′l

Region II ε1

r φΙI

Fig. 9.3: Dielectric image method – symmetry and region solution (the shadow is the region to be solved).

Its image charge is − ρl′ = 

  ε2 − ε1 ρ ε2 + ε1 l

(9:1)

Obviously, we have ρl − ρl′ =



 2ε1 ρ ε2 + ε1 l

The potential functions ’ of the two regions are 9 8 > >   = < ρl 1 ε2 − ε1 1 ’I = ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − > 2πε1 > ε2 + ε1 : x2 + ðy − dÞ2 x + ðy + dÞ2 ; 9 8 > >   = < ρ 2ε1 1 ’Π = l ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2πε1 > : ε2 + ε1 x2 + ðy − dÞ2 ;

(9:2)

(9:3)

(9:4)

They fulfill the governing equations and boundary conditions. In general, the medium interface for y = 0 is not equipotential. Here, two important equivalence cases should be pointed out.

9.1.1 ε2 ! ∞ Equivalent conductor For ε2 ! ∞, a uniform plane medium can be seen as the equivalent of a conductor, as shown in Fig. 9.4. Its image charge is   ε2 − ε1 ρ =  ρl (9:5) − ρ′l = lim − ε2 !∞ ε2 + ε1 l

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9 Plane image method and active conformal mapping

y d ε1

φ

r1

ρ l

r2 x

O − d −ρl ′

ε1

Fig. 9.4: A basic model of plane conductor images.

9.1.2 ε2 !0 Equivalent magnet For ε2 ! 0, a uniform plane medium can be seen as the equivalent of a magnet, as shown in Fig. 9.5. Its image charge is   ε2 − ε1 ρ =ρ (9:6) − ρ′l = lim − ε2 !0 ε2 + ε1 l l

y ε1

φ

r1

d ρ l

r2

x

O ε1

−d −ρl′ Fig. 9.5: A basic model of plane magnet images.

9.3 Active Conformal Mapping of the Conducting Cylinder As mentioned earlier, all we need to do is use the conductor plane image as the basic model, and other problems can be simplified into the plane image by active conformal mapping. As one of the most typical problems, the line charge outside a cylinder is divided into three cases for discussion.

9.3.1 Grounded conductor cylinder In the previous electromagnetic theory works, the conductor cylinder is often dealt with by the image method, and real line charge ρl and image line charge − ρl satisfy the inversion relation, as shown in Fig. 9.6.

9.3 Active Conformal Mapping of the Conducting Cylinder

125

y

φ

Region I

Region II −ρl

O

−R D

ρ

R C

B

l

x

A

R ε0 Fig. 9.6: Line charge ρl outside the grounded conductor cylinder.

z-plane

(

OA = r OB = R2 =r

(9:7)

Therefore, we have OA  OB = R2

(9:8)

which satisfies the inversion symmetry. This chapter employs active conformal mapping and maps it to the plane image model, which is specifically   z−R (9:9) w=j z+R As shown in Fig. 9.7, it is easy to find the parameter k: k=

ε0

r1 jk

A ρ

l

r2

C

D

(9:10)

φ

v

Region I

r−R r+R

D

u

O ε0

Region II

−jk B

−ρl Fig. 9.7: From circular inversion symmetry to the plane image model.

126

9 Plane image method and active conformal mapping

And by w-plane, we can directly obtain the potential function ’ produced in Region I by the cylinder line charges of z-plane:  !      r z − R2   r2   ρl ρ w + jk ρ  r   = l ln ln  = l ln ’= (9:11)  2πε0 r1  2πε0 w − jk 2πε0 R z − r  Let ’0 =

r ρl ln 2πε0 R

(9:12)

then  !  z − R2  ρl  r  ln ’=  + ’0 2πε0  z − r 

(9:13)

It seems to be different from the original problem by a constant. In fact, it is easy to prove that the constant is exactly the constraint condition of the cylinder being grounded ð’jr = R = 0Þ: Again we write    rz − R2  ρl   ln ’= (9:14) 2πε0 Rz − rR Suppose z = Rej’ , which means it is at the boundary of the circumference, and we obtain     Re − j’ − r ρ  ≡0 ’ = l ln  − ej’  j’ (9:15) 2πε0 Re − r  hence the result.

9.3.2 Ungrounded conductor cylinder Line charge ρl outside an ungrounded conductor cylinder is shown in Fig. 9.8. This case corresponds to the one where charge ρl is added in Region II. Being ungrounded means: (1) the surface of the cylinder conductor remains equipotential and (2) the total charge in Region II is 0. Again, we employ the active conformal mapping equation and map it to the plane image model of w-plane, as shown in Fig. 9.9. [New principle for active conformal mapping] When using active conformal mapping to find the line charge, the solution region (w-plane) after mapping may contain the line charge at point ∞ of z-plane, which is the mirror image for z = 0: As a matter of fact, only in this case can we realize the principle that charges and image charges have to appear in pairs. In our problem, image charge − ρl at w=j(z=∞) is added in Region I. Therefore,

9.3 Active Conformal Mapping of the Conducting Cylinder

127

y Region I ε0

Region II ρ

−ρ

l

D

ρ

−ρ

O

A

x

l

l

l

C

B

R

z−plane

v

Fig. 9.8: Line charge ρl outside an ungrounded cylinder.

φ

Region I r4 ε0

j −ρ

l

jk ρ

l

r1 r2 r3

u

O −jk − ρl ε0

−j ρ

l

Region II w-plane

Fig. 9.9: From the ungrounded cylinder to the plane image model.

      w + jk w − j r2 r4  ρl ρl ρl       ’= ln  ln ln + = 2πε0 r1 r3  2πε0 w − jk 2πε0 w + j

(9:16)

We further derive  !  r z − R2  z ρl ρ    r  ln ’=  + l ln  2πε0 R z − r  2πε0 R It is written in the form of the original cylinder problem as   z  r2  ρ ρ   ’ = l ln  + l ln  + ’0 2πε0 r1 2πε0 R which is exactly the same as the original solution.

(9:17)

(9:18)

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9 Plane image method and active conformal mapping

9.3.3 Dual problem of the cylinder The dual problem of cylinder is shown in Fig. 9.10.

y Region II

φ

Region I

ε0

D −R

ρ l

A

C

r

O

R

r2

r1

B

R

R r

−ρl

x

2

z-plane Fig. 9.10: Dual problem of the cylinder.

The active conformal mapping is w=  j

v Region I ε0

~ jk

ρ

(9:19)

φ

r1

l

r2 C

D

  z−R z+R

O

D u

ε0

~ − j k −ρ l Region II w-plane

Fig. 9.11: Plane image model for the dual problem.

As shown in Fig. 9.11, we map it to the plane image basic model, where ~k = R − r R+r Again, we have

(9:20)

129

9.4 Active conformal mapping of the complex conductor

  z − R2  ρl  r ’= ln  + ’0 2πε0  z − r 

(9:21)

and ’0 =

r ρl ln 2πε0 R

(9:22)

which is the same as the original problem in form.

9.4 Active conformal mapping of the complex conductor When there are line charges outside a complex conductor, the method is to map them to the plane image model. Two typical cases will be discussed here. Case 1: Problem of the cylinder with cracks Figure 9.12 shows a conductor cylinder with cracks, and line charge ρl is at the origin of coordinates. y Region II

Region I

ε0 ρ

A

l

C

D

O R

Δ

B

x

z-plane Fig. 9.12: Conductor cylinder with cracks.

We employ the linear fractional transformation 8   z1 =  j zz +− RR > > > >

z = z > 3 2 + K2 > > : pffiffiffiffi w = z3 As shown in Fig. 9.13, we have Δ = jBCj = jCDj and

ð9:23Þ ð9:24Þ ð9:25Þ ð9:26Þ

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9 Plane image method and active conformal mapping

C B D

A

y3

y2

y1 Region I ε0 A x1 z1-plane

−K2 C

Region I ε0 A D x B A 2 z2-plane

Region I ε0 A D

C

B K2 A z3-plane

x3

v Region I ρ r ε0 jw l 1

φ

r2

0

A

B C D −K K −jw0 − ρ l w-plane

A u

Figure 9.13: Active conformal mapping of the cylinder with cracks.

 K=

Δ 2R − Δ

 (9:27)

Finally, we obtain ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  z−R 2 w=j − K2 z+R

(9:28)

As z=0 corresponds to the position of line charge ρl , we map it to point jw0 on w-plane, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi pffiffiffiffiffiffiffiffiffiffiffiffi Δ w0 = 1 − K 2 = 1 − (9:29) 2R − Δ Finally, we obtain the potential function   w + jw0  ρl   ln ’= 2πε0 w − jw0 

(9:30)

Case 2: Problem of the line charge ρl between double plates This case can be discussed from two aspects.

9.4.1 There exist charges at position h between infinite double plates The infinite double plates are 2a wide, and there is line charge ρl at position h: Suppose h ≤ a without loss of generality, as shown in Fig. 9.14. y

y1

ε0 −a

ρl

O a h z-plane

x

y2

ε0

h O ρl

jsin 2a x1

z1-plane

Fig. 9.14: Line charge ρl between infinite double plates.

πh 2a

v

ρl

O x cos π h 2 z2-plane 2a

h jsin π ρ 2a l O

r1

w-plane

φ

r2 u

9.4 Active conformal mapping of the complex conductor

131

The employment of active conformal mapping leads to z1 = z + a

(9:31)

πz1

z2 = ej 2a

(9:32)

  πh w = z2 − cos 2a

(9:33)

It is easy to obtain the potential distribution function ’:  πz    jejð2aÞ − e − jðπh w + j sinπh  2a Þ  ρl ρl   2a  πh  = ’= ln ln  πh 2πε0  w − j 2a  2πε0  jejðπz 2aÞ − ejð 2a Þ 

(9:34)

For the line charge at the center when h = a, we further obtain  πz     ejð2aÞ + 1  ρl ρl πz    ln lntan ’0 = =    Þ 2πε0 ejðπz 2πε 4a 0 2a − 1

(9:35)

9.4.2 There exist line charges at the center between semi-infinite double plates There is line charge ρl at the center between semi-infinite double plates, that is, ρl is at position (0, 0) and the double plates are short-circuited at the position of y =  jd, as shown in Fig. 9.15.

v r1 φ π d ρl jsh r2 ε0 2a

y

y1 ε0

–a

ρ

ρ

l

O

z-plane

a

x

jd l πd ε0 jsh 2a a −a O z1-plane

u x1

O ε0 πd −ρ −jsh 2a

l

w-plane

Fig. 9.15: Line charge ρl between semi-infinite double plates.

The employment of active conformal mapping leads to ( z1 = z + jd   1 w = sin πz 2a hence basic model of the plane image model. It can be seen that

ð9:36Þ ð9:37Þ

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9 Plane image method and active conformal mapping

  πz πd πz πd πz πd = sin + j cos w = sin ch sh +j 2a 2a 2a 2a 2a 2a Note that line charge ρl is at the position of z = 0, that is,     πd πd = jsh w0 = sin j 2a 2a

(9:38)

(9:39)

It is easy to obtain the potential distribution function ’ :            sin πz ch πd + j 1 + cosπzshπd r2    ρl ρ w + w ρ  0 2a 2a   = l ln  2a  2a  πz  πd  ’= ln  = l ln πd  2πε0 r1  2πε0 w − w0  2πε0 sin πz ch − j 1 − cos ch 2a 2a 2a 2a (9:40) Finally, it is simplified to          sin πz ch πd + j cosπzshπd   ρl πz ρ   4a 2a 4a 2a l + πz πd lntan ln πz  πd ’=  2πε0  4a  2πε0 cos 4a ch 2a − j sin 4a sh 2a 

(9:41)

Especially for d ! ∞, that is, when the situation changes from semi-infinite plates to infinite double plates, we have     πd πd 1 πd sh e 2a ! ∞ (9:42) ch 2a 2a 2 It is easy to give       sin πz ch πd + j cosπz shπd  4a 2a 4a 2a      lim      d!∞cos πz ch πd − j sin πz sh πd  4a 2a 4a 2a

(9:43)

Finally, we obtain lim ’ = ’0 = 

d!∞

    ρl πz  lntan 2πε0 4a 

(9:44)

which is exactly the same as eq. (9.35).

9.5 Summary With plane image as the basic model, this chapter provides an in-depth discussion about applications of active conformal mapping. It is pointed out that after mapping, the region to be solved allows existence of the image charge at infinity in the original problem, thus breaking the principles for the general image method. It is found that the image charge and the original charge have to appear in pairs.

Q&A

133

Q&A Q: The title of Chapter 9 comes straight to the point that it discusses two methods: plane image method and active conformal mapping. What is the connection between these two methods? What is the theme of this chapter? A: Probably surprising to most people, Chapter 9 still discusses symmetry. It is about complex symmetry and symmetry transformation. The image method belongs to complex symmetry, while active conformal mapping involves symmetry transformation. First, let us look at the image method. This chapter uses a generalized model of plane medium images. If the structure is symmetrized, that is, the whole space is filled with ω1 , there will be an additional image charge in the problem, with its position pðdÞ =  d and its quantity is LðρÞ = 

  ε2 − ε1 ρ ε2 + ε1

where ρ and L represent the position symmetry operator and the charge symmetry operator, respectively. This is a typical composite symmetry in electromagnetics. Note that the use of the unified model for solutions needs to be conducted in various regions, as shown in Fig. 9.3. Now let us look at conformal mapping that this chapter employs to solve the complex electromagnetic problem with active line charges. Conformal mapping reflects the most important invariance of the transformation. Specifically, the problems before and after mapping are the same. The fundamental basis is that the Laplace equation of the potential function ’ remains unchanged, as shown in Fig. 9.16. Chapter 9 further applies the theory to Poisson’s equation. That is, the problem to be solved allows existence of the source, hence the invariance of active conformal

Before

After

y

φ

v ρ

ρ x

O

z-plane 2

2

дφ дφ + =0 дx 2 дy2

w = f (z) Invariance of conformal transformation

u O

−ρ 2

w-plane 2

дφ дφ + =0 дu2 дv 2

Fig. 9.16: Invariance of conformal mapping: the Laplace equation of the potential function φ remains constant.

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9 Plane image method and active conformal mapping

mapping. To meet this application, the boundary must be at the equipotential surface. Therefore, what is discussed are all problems of the conductor (or magnet) and the image method of conductors to handle complex regions. Q: Probably due to the in-depth mathematics involved in this chapter, it seems pretty hard to understand the key points. A: Many people feel the same way, for it is a fresh new idea to employ the invariance of conformal mapping to deal with symmetry transformation. Before and after mapping, potential and source ρe remain the same. Take the problem of the line charge outside a grounded conductor cylinder as an example, as shown in Fig. 9.17.

To solve the original problem of region I Region ε0 I

Conformal symmetry transformation

ρ

R

l

A r

z −R w= j z+R

The transformation from the original problem to the plane conductor v Region I jK

ε0 o

u

image solution r v 1 Region jK ρl I

ε0

ε0 − jK

φ

r2

u

O − ρl w-plane

φ the invariance of image method is satisfied to the original problem

Fig. 9.17: Solution of the conformal symmetry transformation.

First, let us solve the problem of Region I, the complex (or the line charge outside a grounded conductor cylinder) region. The most important thing is that we do not need to know the position or the magnitude of image charge ρ′ inside the cylinder. What we need is the conformal symmetry transformation:   z−R w=j z+R which changes the original problem into one of the plane conductor. From this, it is clear that the complex boundary of the original problem must be equipotential. The original problem is the same as the plane conductor problem, which can be solved by the image method. Therefore, the potential of the latter is the one of the original problem. This is the solution by using conformal symmetry transformation. Q: Apart from the problem of the line charge outside a grounded conductor cylinder, can it solve other kinds of complex boundary problems? A: Sure. Take the problem of the line charge at position r1 ðr1 > aÞ outside a grounded conductor elliptic cylinder with its semimajor axis a and semiminor axis b, as shown in Fig. 9.18.

Q&A

135

y1 b1 O

z-plane

ρl a1 r1

z1 = x1 + jy1

x1 Fig. 9.18: Problem of the line charge outside a grounded conductor elliptic cylinder.

The use of conformal mapping transforms it into the problem of line charge ρe outside a grounded conductor cylinder, as shown in Fig. 9.6. We have   1 h2 z1 = z+ z 2 It is easy to obtain

(

R=a+b h2 = a 2 − b 2

z = z1 +

qffiffiffiffiffiffiffiffiffiffiffiffiffi z12 − h2 ðTake + Þ

The corresponding position of the line charge outside the conductor cylinder is qffiffiffiffiffiffiffiffiffiffiffiffiffi r = r1 + r12 − h2 Similarly, we transform the problem of the cylinder into the plane conductor image. Given  2  8 < ’ = ρl ln z − R =r + ’0 2πε0 z−r : ’ = ρl ln r  0 2πε R 0

We finally obtain 8 0 1 pffiffiffiffiffiffiffiffiffiffi ða + bÞ2 > pffiffiffiffiffiffiffiffiffi z1 + z12 − h2 − > > 2 2 r1 + r −h > ρ B C > 1 > ’ = 2πεl ln@ pffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi > ffiA + ’0 > 0 2 − h2 − r + 2 − h2 > z + z r > 1 1 1 1 > > < ffi  pffiffiffiffiffiffiffiffiffi r1 + r12 − h2 ρ > ’0 = 2πεl ln > a + b > 0 > > > ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p > > > h = a2 − b2 > > > : The elliptic parameter a;b and the position of r1 are given: The example can be called double conformal symmetry mapping. In particular, the cleverest part of employing the conformal mapping is that the problem can be solved by the plane image method without knowing the position or magnitude of the image charge inside the conductor elliptic cylinder.

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9 Plane image method and active conformal mapping

Q: It can be seen from the aforementioned examples that conformal mapping is indeed useful for symmetry transformation. A: True. Let us further extend the applications of complex conformal mapping in the electromagnetic field. Conformal mapping has two kinds of invariance, as shown in Fig. 9.19. Based on conformal transformation, Poisson's equation remains the same. After mapping, both potential φ and source ρl keep unchanged. 2

дφ дx

2

дu

2

v = q1

2

+

дφ дy

2

= ρlδ ( r )

+

дφ дv

2

u1 u = u2

u = u1 C v = q2

2

2

дφ

Invariance of capacitance C of conformal transformation .

= ρ lδ ( r )

(a) Before

q1

C

q2

u2 (b) After

Fig. 9.19: Two kinds of invariance of conformal mapping.

The second kind of invariance of conformal transformation can be used to determine the two-dimensional capacitance C in complex regions. Suppose ( u = Γ1 v = Γ2 If Γ 1 is orthogonal to Γ 2 , their corresponding curves C1 and C2 on z-plane are also orthogonal to each other based on the conformal mapping of w = f ðzÞ, as shown in Fig. 9.20. With u as the equipotential line and v as the electric field line, twodimensional capacitance C has

v

y

Γ2

C2

Γ1

o

C1

u o

w-plane

x z-plane

Fig. 9.20: Orthogonality of the corresponding curves on wplane and z-plane.

C=

q2 − q1 v2 − v1

The problem can be solved by transforming the complex region into the parallel-plate capacitor.

Q&A

137

Example 9.1: Determine the cylindrical coaxial capacitance C of unit length. Suppose that the coaxial line is on z-plane, and the use of conformal mapping leads to w = ln z = ln r + jθ = u + jv Therefore, we have (

u = ln r v =θ

If u represents the equipotential line transformation and v represents the electric field line transformation, the parallel-plate capacitance is formed on w-plane. The equipotential line ranges from ln a to ln b, and the electric field line ranges from 0 to 2π, as shown in Fig. 9.21.

y

v 2π

b

a o

x o

z-plane

u

ln a ln b

Fig. 9.21: The coaxial capacitance of the circle transformed into the plate capacitancew = ln z:

w-plane

We are given the capacitance of parallel-plate capacitor C=

εS d

where (

S = 2πl d = ln b − ln a = ln

b a

Unit length means l = 1: Based on the invariance of capacitance C before and after conformal mapping, we know that the parallel-plate capacitance is

C=

2πε  ln ba

or the cylindrical capacitance of the circle.

Example 9.2: Determine the elliptical coaxial capacitance C with the semimajor axis and the semiminor axis a1 , b1 and a2 , b2 , respectively, as shown in Fig. 9.22. The Joukowski transformation transforms the elliptical coaxial line into one of the cylindrical, and we have w=

  1 k2 = u + jv z+ z 2

138

9 Plane image method and active conformal mapping

y

y

b2 b1

O

a1 a2

x

x

O

w-plane

z-plane

Fig. 9.22: Elliptical coaxial line.

Suppose z = rejθ , and it is easy to obtain 8   2 > < u = 21 r + kr cos θ   > : v = 1 r − k2 sin θ 2 r We find r=a+b Based on the invariance of capacitance C, we derive 2πε  

C= ln

a2 + b2 a1 + b1

Recommended scholar In this chapter, I will introduce David K. Cheng, a well-known electromagnetics scholar and my advisor in America. As shown in Fig. 9.23. Mr. Cheng is an eminent professor at Syracuse University School of Engineering, as famous as Harrington. Entering his office, you can see only one certificate hung on the white wall, the doctoral diploma from Harvard University. Mr. Cheng values this certificate a lot, and he has been working hard to live up to the Harvard certificate. While studying in America, I worked with Mr. Cheng (Fig. 9.23) for long hours, and was deeply impressed by his knowledge and expertise. Mr. Cheng has two extraordinary traits: (1) Mr. Cheng used to teach us that “it is nothing to present difficult problems in a difficult way, but that it is a real thing to present difficult problems in a simple way.” His one significant trait is “explaining difficult problems in simple ways,” and his masterpiece Field and Wave Electromagnetics (Fig. 9.24) is a perfect example. (2) Mr. Cheng is a man of few words, but he does things with extreme preciseness and seriousness. Compared with him, I often feel ashamed.

Recommended scholar

139

Fig. 9.23: D. K. Cheng (Left) and the author.

Fig. 9.24: Field and Wave Electromagnetics.

Here I cite only two examples. (1) Mr. Cheng mentioned several times that before the publication of Field and Wave Electromagnetics, the publisher had sent five first drafts to different scholars for review. Finally, the collected drafts that had been reviewed surprised them all – for not a single word had been revised. (2) After arriving in America in 1980, I handed in my first research work to Mr. Cheng. He returned it to me in less than a week. I felt ashamed when seeing the manuscript full of suggestions. From then on, I have been more strict with myself. The biggest gain of studying in America is that I see what a top scholar is like in America and across the world. One can hardly understand unless he experiences by himself. Thus, I know the real difference between me and a top scholar and the difference between my country and the developed countries.

10 Electromagnetic loss This chapter focuses on the discussion of electromagnetic loss, with an emphasis on situations where electromagnetism changes from a loss state to an open state. The condition evaluation, geometry evaluation and phase evaluations of loss are described in detail. It is found that loss leads to significant changes of a general electromagnetic system, such as the openness of the system, the coupling of the field quantity, the encroaching of information and the directivity of the evolution. By overcoming the difficulties, the lossy electromagnetic theory is worth to be established.

10.1 Introduction This is the 10th chapter of Electromagnetic Field Theory Teaching Series. As is known to all, electromagnetism is a part of such fields as physics, chemistry and materials science. Take physics as an example, which includes mechanics, electromagnetism, optics, acoustics, thermology and quantum mechanics. In teaching, these parts are independent of each other and are taught as different courses, as shown in Fig. 10.1. However, branches in physics (or even chemistry) are actually related to each other in two ways: conversion and loss. For example, electromagnetism can be converted to mechanical motions (such as the electric motor) and to acoustic vibrations (such as the loudspeaker). Besides, electromagnetic loss is converted to heat – a directional degradation, as shown in Fig. 10.2. This chapter focuses on the situation where the connection between electromagnetism and loss leads to an open field and a series of dramatic effects.

10.2 Three theorems The space of electromagnetism with no loss is closed and complete, with all modes independent of each other (or orthogonality) [7,9,10,27]. Essentially, electromagnetic loss is energy which is certainly shown in the theorem of second-order quantity.

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10 Electromagnetic loss

Mechanics Electromagnetism Optics

Physics

Acoustics Thermology Quantum mechanics

Fig. 10.1: Physics includes various branches.

Convert

Convert Electromagnetism

Mechanics

Acoustics loudspeaker

electric motor Loss

degradation

Thermology

Fig. 10.2: Relation of electromagnetism to other branches.

Theorem 10.1: Poynting’s theorem A theorem of energy conservation and conversion in the electromagnetic field, Poynting’s theorem has a time domain form of  ðð  ððð ððð  * * * * * ∂ 1 2 1 2 J  E dV (10:1) − εE + μH dV =  E × H  dS + ∂t 2 2 V

S

V

Figure 10.3 shows a corresponding physical model. From eq. (10.1), we know that ð − ∂=∂tÞ; the time rate of decrease of electric and magnetic energies stored into volume V, equals the Poynting power which flows out of the boundary of volume V (area S) plus the power dissipated in volume V (converted to thermal energy ~ J = σ~ E). dS

V S Fig. 10.3: Poynting’s theorem.

It is this theorem that ends the closed electromagnetic space. A part of electromagnetic energy is dissipated and (directionally) degraded into heat.

10.2 Three theorems

143

T

Zin



Fig. 10.4: Generalized input impedance of a single-port system.

T

Theorem 10.2: Kirchhoff’s theorem Kirchhoff’s theorem is a theorem of secondorder quantity in the frequency domain. By eq. (10.1), it can be further derived the complex Poynting’s theorem. ððð  ðð ððð * * 1  * *    * 1 1 * *  * *  E  J dV + j2ω (10:2)  E × H  − dS = B  H − E  D dV 2 2 4 s

V

V

We introduce the complex Poynting vector, *

S=

and we have

1 * *   E×H 2

8 ÐÐÐ * *  1 > B  H dV < Wm = 4 V * > W = 1 ÐÐÐ * : E  D dV e 4

(10:3)

ð10:4Þ ð10:5Þ

V

which represent magnetic energy storage and electrical energy storage, respectively. And, ððð * * 1 PL = E  J dV (10:6) 2 V

represents the power dissipation of the system. Equation (10.2) is rewritten as ðð * ^dS = PL + j2ωðWm − We Þ (10:7) S n In Kirchhoff’s theorem, as shown in Fig.10.4, generalized input impedance is defined as zin =

PL + j2ωðWm − We Þ = R + jX ð1=2ÞII 

(10:8)

Especially for Wm = We ; the reactance of the system X ≡ 0; suggesting that system resonance occurs at frequency ω: Generalized impedance R is positive, which

144

10 Electromagnetic loss

shows that the electromagnetic space turns toward openness. Electromagnetic loss connects electromagnetism and heat, hence some features and problems. Theorem 10.3: Foster’s theorem Foster’s theorem, being rather special, is not a theorem of loss but a theorem of second-order quantity of a lossless system in the frequency domain. Its physical model is shown in Fig. 10.5. T V +, I+ 

Reactance load

V –, I –

Fig. 10.5: Foster’s theorem of a single-port reactance load.

T

V and I are defined as the equivalent voltage and the equivalent current on end face T–T, which satisfy ! * * ðð * * ∂H ∂E * ∂I  ∂V  ^ dS = V + ×H  n + I (10:9)  E× ∂ω ∂ω ∂ω ∂ω T

By Foster’s theorem, it is derived that ∂X 4ðWm + We Þ = ∂ω II 

(10:10)

It indicates that reactance slope ∂X=∂ω in a lossless system, directly proportional to the total energy of the terminal, will never be negative. ∂X >0 ∂ω

(10:11)

Actually, it agrees to the fact that the system is certainly lossy as long as reactance slope ∂X=∂ω is less than 0, typically as shown in Fig. 10.6. And, this is the reason why Foster‘s theorem is applied to the lossy system. X Foster theorem дX

дω O

X

>0

ω

O

Fig. 10.6: Foster’s theorem and the lossy system.

ω

10.3 Loss evaluation

145

10.3 Loss evaluation One main issue in the study of electromagnetic loss is how to evaluate the magnitude of loss. In essence, loss is a typical amplitude quantity directly related to the electromagnetic secondary quantity – energy. This section has a focus on three issues which are often neglected: the condition evaluation, geometric evaluation and phase evaluations of loss.

10.3.1 The condition evaluation of loss Electromagnetic loss is divided into electrical loss and magnetic loss. Given its intrinsic symmetry, here we only talk about electrical loss. Maxwell’s first equation is rewritten in the frequency domain as *

*

*

*

*

*

∇ × H = jωεE + J = jωεE + σE = jω~ε E

(10:12)

where ~ε = εð1 − jðσ=ωεÞÞ is called complex dielectric constant. Generally, we have 8 σ > > > 100 The material is a conductor: > > ωε > > < σ < 100 The material is a semiconductor: 0:01 < (10:13) > ωε > > > > > : σ < 0:01 The material is an insulator or medium: ωε We can see that it is related to three factors: conductivity of the material σ; dielectric constant ε and working (angular) frequency ω: For electromagnetic loss, one common misunderstanding is the material evaluation. That materials are categorized into large-loss type and small-loss type, which is actually incomplete. First, look at the contradictions revealed from the following problem. On the one hand, wires (with large σ) produce nearly no electromagnetic loss, while resistance R (with small σ) produces large loss. On the other hand, by eq. (10.6), the definition equation of electromagnetic loss power PL ; we obtain ððð ððð  2 * * 1 1 *  E  J dV = σ E  dV (10:14) PL = 2 2 V

V

Therefore, the larger the σ, the greater the electromagnetic loss. Then, what are the materials with large loss? We can further discuss from two aspects. Case 1: From the aspect of circuits [28] Typical circuits can roughly be divided into two main types: series circuits and parallel circuits. For series circuits, see Fig. 10.7(a), where circuit I (or ~ J) is the invariant in the circuit. In this case, the condition evaluation is that the larger the σ; the lower the loss power PL :

146

10 Electromagnetic loss

R

Wire

V

V

V

R

Wire

I

(a)

(b)

Fig. 10.7: The condition evaluations of loss in series and parallel circuits: (a) series circuits and (b) parallel circuits.

1 PL = 2

 2  ððð * J σ

V

dV

(10:15)

For parallel circuits, see Fig. 10.7(b), where voltage V (or ~ E) is the invariant in the circuit. Obviously, the condition evaluation turns into the situation where the larger the σ; the greater the loss power PL : Case 2: From the aspect of waves [29] Correspondingly, the electromagnetic loss of waves also reveals a contradiction. On the one hand, when traveling into a good conductor interface with large σ, waves will form a significant reflection ðΓ − 1Þ; and the tangential component of the total electric field on the interface is close to 0. In other words, waves cannot travel into the conductor, and the larger the σ; the fewer waves travel into the conductor, the lower the loss power PL ; as shown in Fig. 10.8(a). Ei

Et

Hi

Sr

Si

PL small

σ

Hr

Si

Ht

σ

PL large

Er Γ ≈ –1

(a)

(b)

Fig. 10.8: The contradiction between waves and good conductors: (a) waves can hardly travel into a good conductor interface and (b) once traveling into a good conductor, PL is large.

On the other hand, once traveling into a good conductor, waves will dissipate soon, and the larger the σ; the greater the loss power PL ; as shown in Fig. 10.8(b). From this point of view, the use of waveguide as the transmission line for transmitting waves, with a large waveguide wall σ; has the possibility of causing a

10.3 Loss evaluation

147

large loss after waves travel into the waveguide wall. However in practice, waves can hardly travel into the waveguide wall ð~ E 0Þ: Waves are just like beautiful seagulls, flying across the sea with their wings hardly wet. So far, we can make a brief summary. The study of electromagnetic loss is not only about the material but also about the application environment and conditions. This is the condition evaluation of loss.

10.3.2 The geometric evaluation of loss Electromagnetic loss is a physical mechanism; however, it can be evaluated by geometric curves. (1) The first type of geometric evaluation The reactance of generalized transmission lines has a negative slope in the lossy condition. According to the Theorem 10.3 and Fig. 10.6, the reactance curve slope ∂X=∂ω remains positive in a lossless system. It is easy to reason out that electromagnetic loss is closely related to negative slope, as shown in Fig. 10.6(b). The smaller the negative slope ∂X=∂ω; the larger the system loss will be. Therefore, we can make geometric evaluations about the system. (2) The second type of geometric evaluation Enclosed or to-be enclosed conductor cavities have geometric curve of quality factor Q. The width Δω where frequency is 3 dB and the central frequency ω0 constitute Q=

ω Δω

(10:16)

as shown in Fig. 10.9. Therefore, we can make geometric evaluations on the loss of the cavity system based on Q curve. P P0

1 P 2 0 ∆ ω ω0

ω

Fig. 10.9: Geometric evaluation of quality factor Q curve on the electromagnetic loss of cavity system.

Exploring the physical essence of electromagnetic loss in the geometric way has profound significance and is worth further research.

148

10 Electromagnetic loss

10.3.3 The phase evaluation of loss From the perspective of electromagnetism, loss represents the square amplitude of a signal and therefore is totally different from the phase of a signal. Strangely, in many cases, we can evaluate the loss of a system by observing its phase. (1) Characteristic phase evaluation on the loss of two-port network According to literature [24], a lossless two-port network, as shown in Fig. 10.10, has a characteristic phase Φ that meets Φ = ð’11 + ’22 Þ − ð’12 + ’21 Þ = ± π

b1

a2

Two-port network

1

(10:17)

2

a1

b2

Fig. 10.10: A general two-port network.

where s11 = js11 jej’11 , s22 = js22 jej’22 , s12 = js12 jej’12 , s21 = js21 jej’21 : If the network has loss, then its characteristic phase ’ is no longer ± π: We can evaluate the loss of a two-port network according to the difference between characteristic phase and ± π: (2) Characteristic phase evaluation [30] on the loss of directional coupler The directional coupler is a typical four-port network, as shown in Fig. 10.11.

4

S31

3

1

S21

2 Fig. 10.11: A general directional coupler.

where ①→② is the master port, ①→③ is the coupling port and port ④ is isolated. When there is no loss in the directional coupler, it can be proved that ’31 − ’21 =

π 2

(10:18)

So, it is also called a 90° coupler. However, when there is loss in the system, eq. (10.18) is not equal to π/2. Based on the difference, we can evaluate the system loss of the coupler. The loss (amplitude) has close relationship with the phase, another interesting question that is worth exploring.

10.4 Characteristics of lossy system

149

10.4 Characteristics of lossy system Loss brings qualitative changes to a general electromagnetic system.

10.4.1 The openness of system Loss makes the electromagnetic space more open. The previous situation in which electromagnetism and other physical quantities do not convert to each other is broken. On the one hand, the openness of system brings a variety of changes and promotes innovation. On the other hand, the openness also brings great challenges to research ideas, methods and tools. It constitutes the two sides of one issue.

10.4.2 The coupling of field quantities Loss constitutes an important electromagnetic medium. It generates an amazing intercoupling between field quantities (E and H) that were originally isolated. And, these originally isolated waves now have energy exchanges among each other.

10.4.3 The encroachment of information Nowadays, electromagnetic exploration is a very active field. Conceptually, it utilizes the reflection information of waves to acquire the features of unknown targets. However, once the underground loss is taken into consideration, the electromagnetic waves entering into the ground have obvious attenuation, and the reflection information will be encroached. In this way, the results of the target will be influenced. From another point of view, if the target is invisible, the encroachment of information will have positive effects.

10.4.4 The directivity of conversion As discussed before, there are two kinds of correlation between electromagnetism and other fields: conversion and loss. Broadly speaking, loss is also a type of conversion, a conversion from electromagnetism to heat. But it should be noted that this is a special conversion. Electromagnetic loss is a degraded conversion: electromagnetism can be converted to heat, but heat can never be converted to electromagnetism completely. This is the directivity of electromagnetic loss conversion. It is necessary to mention that sometimes we deliberately convert electromagnetism to heat in daily life, such as the electric kettle, the electric stove for heating

150

10 Electromagnetic loss

and the microwave oven for cooking. In this case, the electromagnetic loss is converted to what we desire.

10.5 Difficulties aroused by electromagnetic loss There is no doubt that the existing electromagnetic theory is built on the basis of a “fully-enclosed” lossless network. Once loss occurs, the research work will face various difficulties.

10.5.1 The complex boundary conditions In a typical example, the corresponding boundary condition of an ideal conductor is *

^×E=0 n

(10:19)

If it is in the nonideal condition where there is loss, the Jieotobny boundary condition should be taken into account. *

*

^ × E = zm H n

(10:20)

where zm represents the surface impedance of a good conductor. It can be deduced that rffiffiffiffiffiffiffi ωμ zm = ð1 + jÞ (10:21) 2σ ~ thus forming a complex situation of At this time, there is coupling between ~ E and H; the boundary condition.

10.5.2 The difficult mode theory One important research method in the electromagnetic theory is to expand any complex field and wave in a complete mode system, such as the Eigen-mode expansion of an ideal conductor waveguide or an ideal conductor resonant cavity. However, there is a lethal problem when there is loss in the waveguide wall or cavity wall – the orthogonality between different modes disappear and completeness is difficult to achieve. Strictly speaking, the basis of the whole mode system is struck by a strong “earthquake” at this point.

Q&A

151

10.5.3 The blurred boundary between transmission mode and cut-off mode An important feature of lossless transmission theory lies in its modes: transmission mode and cutoff mode [31]. But the boundary between these two modes becomes obscure when there is loss. The characteristic impedance in the transmission mode has a reactance component, while it has a resistance component in the cutoff mode. Also, the coupling between these two components brings great difficulty to research work. With loss in the system, we are faced with far more problems than have been mentioned above. Although electromagnetism has developed over a hundred years, the real lossy electromagnetism has not emerged. Currently what we do in engineering are merely some compromising methods like perturbation.

10.6 Summary The topic of this chapter is electromagnetic loss, with an emphasis on situations where electromagnetism changes from a loss state to an open state. It should be pointed out that energy conversion observes the rule of energy conservation. This chapter has discussed the condition evaluation, geometric evaluation and phase evaluation in depth. At the same time, we should overcome various difficulties and try to establish the real lossy electromagnetism.

Q&A Q: What is the purpose of discussing electromagnetic loss separately in this chapter? A: This is a good question. Many people have a misunderstanding about electromagnetic theory that it is merely a theoretical course. This is wrong. One of the major targets of electromagnetic theory is that theory must be applied to practice. Generally speaking, all science and engineering courses have the same target. There are two aspects regarding to the core of electromagnetic theory. On the one hand, some reasonable simplifications like the lossless, orthogonal and complete mode system must be done at the beginning in order to establish a theoretical system. In this way, we grasp the main problem of engineering design and applications. And, the whole system reflects strong symmetry. On the other hand, we must take electromagnetic loss into account in practice. At this point, we are not rebuilding the system (to be frank, a real lossy system is hard to build at present) but regard loss as perturbation (which is quite large sometimes) in a lossless system. Thus, the lossy electromagnetism is established, as shown in Fig. 10.12.

152

10 Electromagnetic loss

Considering the loss of medium ⎡ ⎛ σ ⎞⎤ ε~ = ε ⎢1 − j ⎜ ⎟⎥ ⎝ ωε ⎠ ⎦ ⎣ ~ μ

Lossless electromagnetic theory {Ei } {Hi } is lossless, orthogonal and complete mode system considering the loss of medium

The lossy electromagnetism is established (original system is kept unchanged)

Fig. 10.12: Lossy electromagnetism.

Q: Is there anything special about the research method of electromagnetic loss in this chapter? A: I am not sure whether you have noticed where this chapter is in the whole book. We discussed electromagnetic symmetry in several previous chapters, and the following chapters will discuss the application of complex numbers in electromagnetism. It is obvious that electromagnetic loss here acts like a joint that connects symmetry and complex numbers. The lossless electromagnetism is symmetrical. The introduction of complex number materials in the frequency domain to represent loss breaks the electromagnetic symmetry. However, it is absolutely the same as the lossless condition in form. In this case, the lossless condition and the lossy condition constitute an amazing pseudo-symmetry, as shown in Fig. 10.13.

Maxwell's equations of a lossless system

Maxwell's equations of a lossy system

Δ

× E = − jωμH

Δ

Δ

× H = − jωε E

× E = − jωμ~H × H = − jωε~E

Δ

Fig. 10.13: The pseudo-symmetry of Maxwell’s equations of a lossless system and a lossy (frequency domain) system.

Q: The introduction of complex numbers in electromagnetism is really a useful method. A: This is a widely accepted opinion. In fact, what complex numbers do in electromagnetism is more like proposing a new idea, establishing a new system and most of all revealing the essence. Generally speaking, there are four stages, as shown in Fig. 10.14. Some of the contents mentioned above fall into later chapters and will be discussed thoroughly later. Complex numbers are introduced step by step. The author has made a clear description in the preface to Complex Variables Functions Reading Note.

153

Q&A

The introduction of complex numbers in frequency domain jω

The introduction of complex numbers in medium

The introduction of complex numbers in frequency

The introduction of complex numbers in operator

Corresponding stage

Similar stage

Developing stage

Exploring stage

ε~ ↔ε μ~ ↔ μ

ω~ → ω′ − jω″

R e e j(ωt−kz) = cos(ωt−kz) Euler's equation

=

д д +j дx дy

Fig. 10.14: Four stages of the introduction of complex numbers in the electromagnetic field.

The original application of complex numbers is completely based on correspondence principle. As long as the complex numbers establish a correspondence relationship with z = x + jy surface, we can acquire various engineering applications in time domain. For the time-harmonic field, we have *

*

E = E 0 cosðωt − kzÞ

By using the Euler equation ejx = cos x + j sin x; we can enter into the complex field again and obtain h* i * E = Re E 0 ejðωt − kzÞ However, the past 20th century was full of innovations. Things are undergoing essential changes without being noticed – complex numbers and complex functions become more practical in a real sense. Now, it is safe to say that the complex function belongs to the real physical world. As the exploration of the laws of microscopic particles continues and the Heisenberg matrix mechanics and Schrodinger’s wave mechanics emerge, the complex number has become a fundamental concept in physics. Their basic equations are both in the form of complex numbers. ( pq − qρ = − j
h j
h ∂’ ∂t = H’ If j is canceled and the above equation is written as a real part and an imaginary part, then the equation will lose its real meaning. Feynman proposed the third quantum mechanics, with its path integral form being * *  ð * * .* * * * * ’ r ′; t ′ = K r ′; t ′ r ; t ’ r ; t d r dt   P jðs=hÞ  ; and the actuating quantity is where the integral kernel K ~ r ′;~ t′ ~ r;~ t = e placed on the phase factor of the negative exponent. Weyl intended to establish a unified field theory, and after repeated deductions, he finally got the beautiful melody of complex exponent gauge theory.

154

10 Electromagnetic loss

0

ðq

1

e exp@ − j Au dxu A
hc p

In the 1970s, the great scientist Chen-Ning Yang pushed forward related research continuously. His achievements mainly include the following discoveries: (1) all interactions are some kind of gauge field, (2) the gauge field has close connections with the mathematical fiber bundle concept, and (3) every single fiber is an integral kernel position or a more generalized phase. Finally, a fundamental principle of modern physics was formed – all fundamental forces are phase fields. It is obvious that complex numbers are truly significant for the development of physics. Q: This chapter mentioned that phase can be used to evaluate the system loss. How does it work? A: It is obvious that loss shows the amplitude relation of the system field and is reflected in phase, as shown in Fig. 10.15. The phase relation of the system indirectly reflects the system loss.

The amplitude relation is reflected by loss.

Fig. 10.15: The phase relation of the system indirectly reflects the system loss.

We divide it into two situations and discuss them in detail, as shown in Fig. 10.10. Case 1: Two-port network (

½b = ½s½a ½b + = ½a + ½s +

We can write   ½b + ½b = ½a + ½s + ½s ½a For the lossless situation, ½b + ½b = ½a + ½a = ½a + ½I ½a

1 0 represents the unit matrix. where I = 0 1 It is easy to write ½s + ½s = I; that is, " #" # " s11 s12 s11 s12 1 =   s21 s22 0 s21 s22

0 1

#

Q&A

155

Specifically, "

js11 j2

s11 s12 + s21 s22

s12 s11 + s22 s21

js22 j2

# " =

1

0

0

1

#

It is very clear s11 s12 + s21 s22 = 0 So, the characteristic phase Φ of two-port network satisfies Φ = ð’11 + ’22 Þ − ð’12 + ’21 Þ = ± π To find out the reason, we explore furthermore and have   ðjs11 j2 + js21 j2 Þja1 j2 + 2Re ðs11 s12 + s21 s22 Þa1 a2 + ðjs12 j2 + js22 j2 Þja2 j2 = ja1 j2 + ja2 j2 Finally, it is found that the true reason of s11 s12 + s21 s22 = 0 is the coupling power that prevents a1 and a2 : Now, we start to discuss the two-port lossy network. The only difference between a two-port lossy network and a two-port lossless network lies in the right part of the above equation, that is,    ðjs11 j2 + js21 j2 Þja1 j2 + 2Re s11 s12 + s21 s22 a1 a2 + ðjs12 j2 + js22 j2 Þja2 j2 = jb1 j2 + jb2 j2 If we let 8 2 2 2 2 >

: 2 js21 j + js22 j2 = 1 − Δ2 where δ1 ; δ2 ; Δ1 ; Δ2 are real numbers less than 1. Specifically,    2Re s11 s12 + s21 s22 a1 a2 = ðΔ1 − δ1 Þja1 j2 + ðΔ2 − δ2 Þja2 j2 Let s = s11 s12 + s21 s22 : We have 8 h i < ReðsÞRea a  − ImðsÞIma a  = 1 ðΔ − δ Þja j2 + ðΔ − δ Þja j2 1 2 1 2 1 1 1 2 2 2 2     : ReðsÞIm a1 a2 + ImðsÞRe a1 a2 = 0 Based on the second fraction above, we can deduce   Im a1 a2   ReðsÞ ImðsÞ = − Re a1 a2 Substituting the result into the first fraction, we get

156

10 Electromagnetic loss

"



 

Re a1 a2

 # i Im2 a1 a2 1h    ReðsÞ = ðΔ1 − δ1 Þja1 j2 + ðΔ2 − δ2 Þja2 j2 + 2 Re a1 a2

That is, i 1h ja1 a2 j2    ReðsÞ = ðΔ1 − δ1 Þja1 j2 + ðΔ2 − δ2 Þja2 j2 2 Re a1 a2 For the reciprocal network, we have js12 j = js21 j: It can be deduced that k  j Re a1 a2 ðΔ1 − δ1 Þja1 j2 = ðΔ2 − δ2 Þja2 j2 js12 j½js11 j cosð’11 -’12 Þ + js22 j cosð’22 − ’21 Þ = 2ja1 a2 j2 We come to the conclusion that the S parameter phase in a two-port lossy network has close connections with amplitude, especially the coupling power of a1 ; a2 : At this point, we need to look back on what was mentioned in the text part. “The loss constitutes an important electromagnetic medium. It generates amazing inter~ and the originally coupling between the originally isolated field quantities (~ E and H), isolated waves now have energy exchanges among each other.” These sentences really have profound meaning. Case 2: Directional coupler network For a general directional coupler in Fig. 10.11, in an ideal situation, we take symmetry into account. s24 = s13 s34 = s12 For the reciprocal network of an ideal directional coupler, we have s14 = s23 = s32 = s41 = 0 We can write 2

0

6 6 s12 ½s = 6 6s 4 13 0

s12

s13

0

0

0

0

s13

s12

It is easy to get s12 s13 + s13 s12 = 0 That is,

0

3

7 s13 7 7 s12 7 5 0

Q&A

157

js12 jjs13 jcosð’13 − ’12 Þ = 0 Finally, we have ’31 − ’21 = ’13 − ’12 =

π 2

This equation shows the phase characteristics of a lossless directional coupler. The lossy electromagnetism can be developed in two aspects: lossy field theory and lossy network. Theorem 10.4: In a two-port network, the restriction of characteristic phase Φ will disappear if s11 = s22 = 0; that is, " # 0 s12 ½ s = s21 0 " # js12 j2 0 + ½s ½s = 0 js12 j2   ½a + ½s + ½s ½a = ½b + ½b Specifically, js21 j2 ja1 j2 + js12 j2 ja2 j2 = jb1 j2 + jb2 j2 without the restriction of Φ: The result remains the same even in a lossy network. Theorem 10.5: In a symmetrical lossless directional coupler, both s11 and s14 are 0, or neither of them are 0. Otherwise, contradictions will occur. Proof: As an example, we discuss the situation when s11 ≠ 0; s14 = 0: Specifically, 2  32 3 2 3 s11 s12 s13 0 s11 s12 s13 0 1 0 0 0 6  76 7 6 7 6 s12 s11 0 s13 76 s12 s11 0 s13 7 6 0 1 0 0 7 6 76 7 6 7 + ½s ½s = 6 76 7=6 7   76 6 s 7 6 7 0 s s s 0 s s 0 0 1 0 13 11 12 11 12 54 4 13 5 4 5    0 s13 s12 s11 0 s13 s12 s11 0 0 0 1 We can get the following three equations. 8  s s12 + s12 s11 = 0 > > < 11 s11 s13 + s13 s11 = 0 > > :  s12 s13 + s13 s12 = 0

158

10 Electromagnetic loss

That is, 8 > < js11 jjs12 jcosð’11 − ’12 Þ = 0 js11 jjs13 jcosð’11 − ’13 Þ = 0 > : js12 jjs13 jcosð’12 − ’13 Þ = 0 From the equations above, we have 8 ’ − ’12 = π2 > < 11 ’11 − ’13 = π2 : > : ’12 − ’13 = π2 The second fraction minus the first fraction leaves ’12 − ’13 = 0; which is apparently contradictory with the third fraction. Now, we can start to discuss the symmetrical lossy directional coupler. 2 32 3 a1 0 0 s12 s13 + s13 s12 js12 j2 + js13 j2 6 76 7 2 2 6 6 7 7 0 ζ 0 js12 j + js13 j 6 76 a 2 7 ½ a1 a2 a3 a4 6 76 7 6 76 a 3 7 0 ζ 0 js12 j2 + js13 j2 4 54 5 2 2 a4 ζ 0 0 js12 j + js13 j 2 ½ a1 η + a4 ζ

a2 η + a3 ζ

a3 η + a2 ζ

a4 η + a1 ζ

a1

3

6 7 6 a2 7 6 7 6 7 6 a3 7 4 5

a4      = ja1 j2 + ja2 j2 + ja3 j2 + ja4 j2 η + 2 Re a1 a4 + Re a2 a3 ζ 



= jb1 j2 + jb2 j2 + jb3 j2 + jb4 j2 Let jb1 j2 + jb2 j2 + jb3 j2 + jb4 j2 = ja1 j2 ð1 − Δ1 Þ + ja2 j2 ð1 − Δ2 Þ + ja3 j2 ð1 − Δ3 Þ + ja4 j2 ð1 − Δ4 Þ;        η = 1 − δ2 Re a1 a4 + Re a2 a3 s12 s13 + s12 s13      = 4js12 jjs13 j Re a1 a4 + Re a2 a3 cosð’12 − ’13 Þ = ja1 j2 ðδ − Δ1 Þ + ja2 j2 ðδ − Δ2 Þ + ja3 j2 ðδ − Δ3 Þ + ja4 j2 ðδ − Δ4 Þ

159

Q&A

Finally, we have ja1 j2 ðδ − Δ1 Þ + ja2 j2 ðδ − Δ2 Þ + ja3 j2 ðδ − Δ3 Þ + ja4 j2 ðδ − Δ4 Þ    4js12 jjs13 j Reða1 a4 Þ + Re a2 a3

cosð’12 − ’13 Þ =

It is obvious that ð’1 − ’2 Þ is closely related to the inputs a1 ; a2 ; a3 and a4 in the lossy situation. It is, of course, also related to the loss factors Δ1 ; Δ2 ; Δ3 ; Δ4 and δ: Let 8 η = js11 j2 + js12 j2 + js13 j2 + js14 j2 > > > > < ζ = s s + s s + s s + s s 12

11 12

12 11

13 14

14 13

> ζ 13 = s11 s13 + s12 s14 + s13 s11 + s14 s12 > > > : ζ 14 = s11 s14 + s12 s13 + s13 s12 + s14 s11 then, 2

s11

s12

s13

6  6 s12 6 ½ s ½ s = 6 6 s 4 13

s11

s14

s14

s11

s14

s13

s12

+

½ a1

" =

a2

a3

s14

32

s11

76 6 s13 7 76 s12 6 7 6 s12 7 54 s13 s11 2

s14

η 6 6 ζ 12 6 a4 6 6ζ 4 13 ζ 14

s12

s13

s11

s14

s14

s11

s13

s12

ζ 12

ζ 13

η

ζ 14

ζ 14

η

ζ 13

ζ 12

a1 η + a2 ζ 12 + a3 ζ 13 + a4 ζ 14 a1 ζ 13 + a2 ζ 14 + a3 η + a4 ζ 12

ζ 14

s14

3 2

η

ζ 12

ζ 13

7 6 6 s13 7 7 6 ζ 12 7=6 6 s12 7 5 4 ζ 13

η

ζ 14

ζ 14

η

ζ 14

ζ 13

ζ 12

s11 32

3

ζ 14

3

7 ζ 13 7 7 7 ζ 12 7 5 η

a1 76 7 6 7 ζ 13 7 7 6 a2 7 76 7 6 7 ζ 12 7 5 4 a3 5 η

a4

2

a1

3

#6 7 7 a1 ζ 12 + a2 η + a3 ζ 14 + a4 ζ 13 6 6 a2 7 6 7 7 a1 ζ 14 + a2 ζ 13 + a3 ζ 12 + a4 η 6 4 a3 5

a4  = ja1 j2 + ja2 j2 + ja3 j2 + ja4 j2 η + a1 a2 ζ 12 + a1 a3 ζ 13 + a1 a4 ζ 14 + a2 a1 ζ 12 + a2 a3 ζ 14 

+ a2 a4 ζ 13 + a3 a1 ζ 13 + a3 a2 ζ 14 + a3 a4 ζ 12 + a4 a1 ζ 14 + a4 a2 ζ 13 + a4 a3 ζ 12        = ja1 j2 + ja2 j2 + ja3 j2 + ja4 j2 η + 2 Re a1 a2 + Re a3 a4 ζ 12           + 2 Re a1 a3 + Re a2 a4 ζ 13 + 2 Re a1 a4 + Re a2 a3 ζ 14

160

10 Electromagnetic loss

As 8 > < ζ 12 = 2½js11 jjs12 jcosð’11 − ’12 Þ + js13 jjs14 jcosð’13 − ’14 Þ ζ 13 = 2½js11 jjs13 jcosð’11 − ’13 Þ + js12 jjs14 jcosð’12 − ’14 Þ > : ζ 14 = 2½js11 jjs14 jcosð’11 − ’14 Þ + js12 jjs13 jcosð’12 − ’13 Þ jb1 j2 + jb2 j2 + jb3 j2 + jb4 j2 = ja1 j2 ð1 − Δ1 Þ + ja2 j2 ð1 − Δ2 Þ + ja3 j2 ð1 − Δ3 Þ + ja4 j2 ð1 − Δ4 Þ; η = 1 − δ It can be simplified as      4 Re a1 a2 + Re a3 a4 ½js11 jjs12 j cosð’11 − ’12 Þ + js13 jjs14 j cosð’13 − ’14 Þ      þ Re a1 a3 + Re a2 a4 ½js11 jjs13 j cosð’11 − ’13 Þ + js12 jjs14 j cosð’12 − ’14 Þ

     + Re a1 a4 + Re a2 a3 ½js11 jjs14 j cosð’11 − ’14 Þ + js12 jjs13 j cosð’12 − ’13 Þ ¼ ja1 j2 ðδ − Δ1 Þ + ja2 j2 ðδ − Δ2 Þ + ja3 j2 ðδ − Δ3 Þ + ja4 j2 ðδ − Δ4 Þ Similarly, the phase has close and complicated relations with the inputs a1 ; a2 ; a3 and a4 : Of course, it is also related to the loss factors Δ1 ; Δ2 ; Δ3 ; Δ4 and δ:

Recommended scholar Here, we recommend Mr. Hongjia Huang (Fig. 10.16), a leading scholar in the electromagnetic microwave field in China.

Fig. 10.16: Huang Hongjia.

1. Compared with surroundings, one’s personality is more important Hongjia Huang graduated from Southwest Associated University majoring in telecommunication. He studied there from 1940 to 1944. It is well known that Southwest Associated University was established due to the anti-Japanese war. It experienced twists and turns, and the study and working environment there was terrible. But it

Recommended scholar

161

was this university that cultivated many great masters, among whom was the famous physicist Chen-Ning Yang. Mr. Huang overcame various difficulties and achieved excellent grades at school. In particular, he enlisted into the army after graduation. He worked as a major translator in expeditionary forces and served in Burma Road until the war was over. In 1949, he gave up the opportunity of studying for a doctorate in America and returned to serve his own country in October. 2. Compared with superior conditions, one’s effort is more important From 1963 to 1964, the Science Press specially published Huang’s masterpiece, the first and second volumes of Microwave Theory, which had nearly a million words. Since then, the development of this field in China was brought to international frontier. He was also awarded Significant Contribution Award at the National Science Conference in 1978. However, few people know the story behind Microwave Theory. From 1959 to 1961, the economy of our country faced difficulties temporarily. People had little to eat, and there was prejudice against professional work. Despite all these adverse conditions, Mr. Hongjia Huang kept writing. He said to me personally: “To seize the time, I take every chance to prepare materials and outlines for my book.” Every time when I think of the scene of Mr. Huang writing his book, I have great respect for him. Surely, he was more hardworking than most of us. 3. Continuous innovation, continuous progress Hongjia Huang is a great role model who combines teaching and research very well. For decades, he has been teaching and doing research. In 1965, he published the famous thesis From Microwave to Light, the earliest literature in our country that discussed the possibility of applying optical fiber in communication. In 1981, he published Coupled Mode Theory at New York Institute of Technology. In 1998, he published Microwave Methods in Highly Irregular Fiber Optics with John Wiley & Sons. He also published Light Waveguide Science, creating the concept of “supermode.” Mr. Hongjia Huang is indeed innovating and keeping forward!

11 Complex parameter and complex theorems in electromagnetic theory This chapter discusses the complex parameter and complex theorem in the electromagnetic theory. As the complex number is introduced into the electromagnetic field, two physical concepts in the time domain – transmission and loss, resistance and reactance, and resonant frequency and quality factor are unified on a higher level. According to the complex frequency Poynting theorem given in this chapter, there exists a generalized quality factor Q = ω′W=PL in both open space and enclosed space. ~ = 0: As is pointed out, for And, the complex frequency Foster theorem indicates ∂~z* =∂ω a lossy system, the reactance xin in the input impedance has a negative slope area.

11.1 Introduction This is the 11th chapter of Electromagnetic Field Theory Teaching Series. The introduction of complex numbers is a revolutionary change for mathematics, and the application of complex numbers in engineering has brought a brand new appearance. The famous French mathematician Jacques Hadamard made a brilliant explanation: The shortest way between two truths in the real domain is through complex domain [32]. In fact, it was by using complex numbers that Euler magically connected the trigonometric function with the exponential function, namely the famous Euler equation. ejx = cos x + j sin x

(11:1)

The Nobel Prize winner in physics, Dr. Chen-Ning Yang also has profound feelings about complex numbers. In the fall of 1940, he was studying Differential Geometry taught by Shing-Shen Chen. He was puzzled by a difficult problem: how to prove conformal transformation exists in every two-dimensional surface and plane. Yang knew how to transform the measurement and tensor into the form of A2 du2 + B2 dv2 ; but he could not go further on to solve the problem. One day, Chen advised him to use complex variables and wrote down. C dz = A du + jB dv

(11:2)

From this equation, Chen-Ning Yang learned a clever way to solve problems. He said that it was the most unforgettable experience in his whole life [33]. This chapter introduces the complex parameter and complex theorem into the electromagnetic theory. In this way, “two truths” in the real domain like transmission loss, resonant frequency ω and quality factor Q are connected in essence. And, we can observe and study related questions from a higher level. https://doi.org/10.1515/9783110527407-011

164

11 Complex parameter and complex theorems inelectromagnetic theory

11.2 Complex frequency Till now, the electromagnetic theory has developed to a high level of maturity. Most mathematical tools have been applied to this field successfully. Take complex variables functions as an example. In the time-harmonic field, we introduced the ejωt factor [7,10], electromagnetic network complex extension s = σ + jω and conformal mapping in the electrostatic field. The complex frequency ω′ð1 − jð1=2QÞÞ was introduced in the study of closed conductor cavity [7,10,34], which has not been further extended and developed, though. In this chapter, we extend this idea to the most general electromagnetic system ~ We can write so that the lossy part of the system would go to complex frequency ω: complex frequency Maxwell’s equations in the most general form. 8 * * < ∇ × H = jωε ~ E (11:3) * * : ~ H ∇ × E = − jωμ Note that ε and μ are both real parameters in this case (only, the isotropic medium is studied here). The complex frequency Maxwell’s equation in eq. (11.3) reverts to the lossless form. The generalized quality factor is defined as Q=

ω′ 2ω′′

(11:4)

It can be used in the closed resonant cavity. What’s more, it can be further extended to open systems and any case of nonresonance.

11.3 Complex phase angle ~ belongs to the generalized resonance system, then ~θ belongs to the If the frequency ω generalized transmission system, as shown in Fig. 11.1.

z0 ~ θ

Fig. 11.1: The generalized microwave transmission system.

Theorem 11.1: For the small loss generalized microwave transmission system, we have   θ′′ ω′′ λg 2 = (11:5) θ′ ω′ λ0 and

11.3 Complex phase angle

 2 # 1 λg ~θ = θ′ 1 − j 2Q λ0

165

"

(11:6)

Proof: As both ω′′ and θ′′ are very small, we write ω′ as ω; and θ′ as θ; so θ′′ Δθ ω′′ Δω = = ; θ ω θ′ ω′   θ = 2π=λg l is given, where λg is the waveguide wavelength of the system and l is the system length. We obtain Δθ 2πl = 2 Δλg λg Considering 1=λ2g = 1=λ20 − 1=λ2c ; where λc is the cutoff wavelength of the transmission mode, we have  3 Δλg λg = Δλ0 λ0 λ0 = c=f = 2πc=ω is also given, we obtain Δλ0 =Δω =  λ0 =ω: Finally, we have   Δθ Δθ Δλg Δλ0 2π λg l = = Δω Δλg Δλ0 Δω ω λ0 2 That is to say,   Δθ Δω λg 2 = θ ω λ0 We obtain the two complex phase angle expressions eqs. (11.5) and (11.6). Particularly in the case of the transverse electromagnetic (TEM) wave (i.e., λc = ∞), since λg = λ0 ; we obtain   ~θ = θ′ 1 − j 1 (11:7) 2Q The application of the complex phase angle begins with the transmission line. For Γ1 load connected in the lossless transmission line shown in Fig. 11.2, we have

Γin

z0 = 1 ~ θ

Γl Fig. 11.2: Conversion of reflection coefficients for lossless transmission lines.

166

11 Complex parameter and complex theorems inelectromagnetic theory

Γin = Γl e − j2θ

(11:8)

It represents a complex transformation of the load reflection coefficient Γ1 to the input reflection coefficient Γin . When Γ1 =  1; we further obtain Γin =  e − j2θ

(11:9)

Now, we introduce the complex phase angle ~θ = θ′ − jθ′′; as shown in Fig. 11.3.

Γin

z0 = 1

Γl = –1 Fig. 11.3: The lossy transmission line model.

~ θ

It is easy to write ~

′′

Γin = e − j2θ = e − 2θ e − j2θ

′

(11:10)

On the other hand, due to the arbitrary reflection Γ1 = jΓ1 jej’1 ; for the lossless transmission line, we have Γin =  jΓl je − jð2θ + Δ’l Þ

(11:11)

A comparison of eqs. (11.10) and (11.11) leads to 1 θ′ = θ + Δ’l 2

(11:12)

Δ’l = π − ’l

(11:13)

and ′′

e − 2θ = jΓl j

(11:14)

8 ~ Γin =  e − j2θ* > > > > > < ′ θ = θ + 21 Δ’l >   > > > > : θ′′ = 21 ln 1 jΓl j

(11:15)

The final result is

11.4 Complex frequency electromagnetic theorem

167

In the transmission line model of complex phase angle ~θ; it is normalized to be lossless in form and Γ1 is always taken as short circuit (equal to −1). The total loss across the load is normalized to complex ~θ and the input impedance is further normalized
zin =

1 + Γin = j tan~θ 1 − Γin

(11:16)

which can be seen as the transmission line characteristic impedance z0 = 1 but without the loss of generality, as shown in Fig. 11.3. Note     1 1 1 ρ+1 1 = th − 1 (11:17) θ′′ = ln = ln 2 jΓl j 2 ρ−1 ρ where ρ is the standing wave ratio of the system, in complete agreement with the results of literature [29].

11.4 Complex frequency electromagnetic theorem This section will discuss two of the most important complex frequency electromagnetic theorems.

11.4.1 Complex frequency Poynting theorem The Poynting theorem of a general electromagnetic system is to study the relationship between the volume internal energy storage and loss within the enclosed volume and the electromagnetic power flowing into the closed surface S, as shown in Fig. 11.4. S=

1 (E×H *) 2 nˆ

V S

Fig. 11.4: Poynting theorem.

It is easy to write ðð * ^ dS = PL + j2ω′ðWm − We Þ −  S n s

where

(11:18)

168

11 Complex parameter and complex theorems inelectromagnetic theory

* 1  * **  S= E ×H 2

(11:19)

which is called Poynting vector. In the above equation, the unit normal vector is ^ points outward. The negative sign represents that the electromagnetic defined by n power is injected from outside to inside. PL is the total power loss in V; while Wm and We represent the stored electric and magnetic energies within V; respectively. We have 8 * * < Wm = 1 μ H  H * 4 (11:20) ** : W = 1 ε* E  E e 4 Now, let us consider the general complex frequency ω = ωi of electromagnetic system represented by eq. (11.3). A completely similar derivation leads to ðð * ^ ds = 2jðωW ~ m−ω ~ * We Þ (11:21) − S n s

Equation (11.21) is the complex frequency Poynting theorem. Given ~ = ω′ − jω′′ ω it is easy to know another form of eq. (11.21) ðð * ^ ds = 2ω′′ðWm + We Þ + j2ω′ðWm − We Þ − S n

(11:22)

s

Compare eqs. (11.18) and (11.22), and we introduce W = Wm + We which represents the total energy storage in V. Then, we have   ω′ PL ω′ ′′ = ω = 2 ω′W 2Q

(11:23)

(11:24)

where Q=

ω′W PL

(11:25)

which is called the generalized quality factor. It shows that generalized Q can also be used to reflect the extent of its loss in the open space within any volume (not necessarily a closed conductor surface S), both resonant and non-resonant. Thus, we further write   1 ~ = ω′ 1 − j (11:26) ω 2Q In form, eq. (11.26) is similar to the complex frequency [7,10] introduced by the conductor cavity, but its practical significance has been extended.

169

11.4 Complex frequency electromagnetic theorem

11.4.2 Complex frequency foster theorem Foster theorem is another famous electromagnetic theorem, also known as the Foster reactance theorem. When the one-port in Fig. 11.5 is a reactive termination, it can be proved that [10]

Pin

Generalized system Fig. 11.5: Complex frequency Foster theorem for the lossy one-port.

Pout

∂X >0 ∂ω

(11:27)

That is, the reactance slope of the lossless system is always positive. Now, let us study complex frequency Foster theorem for the general lossy oneport from electromagnetic complex frequency theory. By ( * * ~ *ε E * ∇ × H * =  jω (11:28) * ~ *μ H * ∇ ×~ E * = jω we obtain * *! ðð * ∂H * * ∂ E * ^ ds +H× n  E× ~ ~ ∂ω ∂ω s

ððð 

= j v

 ððð * ** ∂ω ~ * μ * ** ∂ω ~ *ε dv + 2ω′′ HH +E  E ~ ~ ∂ω ∂ω v

* * *∂ H * *∂ E * μH +εE Þdv ~ ~ ∂ω ∂ω (11:29)

Define mode voltage V and mode current I [10] at the port, and the left side of eq. (11.29) becomes ∂I * ∂V * (11:30) +I ~ V ~ ∂ω ∂ω and V = ZI

(11:31)

Z = R + jX

(11:32)

Thus, the left side can be further written as

170

11 Complex parameter and complex theorems inelectromagnetic theory

2RI

∂I * ∂Z * + II * ~ ~ ∂ω ∂ω

(11:33)

Complex frequency Foster theorem The Foster theorem in the general electromagnetic system can be expressed as ∂Z * =0 ~ ∂ω

(11:34)

Proof: The general loaded impedance Z of one-port electromagnetic system is ~ It satisfies the Cauchy–Riemann condianalytic and is an analytic function of jω: tion, namely ∂X ∂R = ′ ∂ω ∂ω′′

(11:35)

∂X ∂R = ∂ω′′ ∂ω′

(11:36)

∂ ∂ω′ ∂ ∂ω′′ ∂ + = ′ ~ ~ ∂ω′′ ~ ∂ω ∂ω ∂ω ∂ω

(11:37)

and

As 8 < ω′ = 21 ðω ~ +ω ~ *Þ : ω′′ = 1 jðω ~ −ω ~ *Þ 2 we have

  ∂ 1 ∂ ∂ + j = ~ 2 ∂ω′ ∂ω ∂ω′′

(11:38)

(11:39)

Note that ∂Z * ∂ 1 = ðR − jXÞ = ~ ∂ω ~ ∂ω 2



   ∂R ∂X ∂R ∂X + − +j ∂ω′ ∂ω′′ ∂ω′′ ∂ω′

(11:40)

Considering the Cauchy–Riemann conditional eqs. (11.35) and (11.36), we thus prove eq. (11.34). Theorem 11.2: For the general one-port system, we have 1 ∂I * 1 ∂W 1 = +j ~ W ∂ω ~ I * ∂ω 4W

ððð ϑm v

* *! * ∂E* * ∂H * +εE  dv μH  ~ ~ ∂ω ∂ω

(11:41)

11.4 Complex frequency electromagnetic theorem

171

where ϑm ðÞ represents the imaginary part in the parentheses, and W is the total energy storage. ~ we obtain Proof: According to the meaning of the complex derivative ∂=∂ω; ~ *μ ∂ω =0 ~ ∂ω

~ *ε ∂ω =0 ~ ∂ω

Reconsidering eqs. (11.29) and (11.33), we can write * *!   ððð * ∂H * * ∂E* 1 ∂I * ′′ =ω μH  +εE  dv RII * ~ ~ ~ ∂ω ∂ω I * ∂ω

(11:42)

(11:43)

v

Substituting the following equation (

ω′′ =

PL 2W

(11:44)

RII * = 2 PL into eq. (11.44), and we obtain eq. (11.41). ~ Equivalently, the It is worth noting that the Foster theorem above is a derivation of ω. ~ * can also be written. derivative of ω * *! ððð * ∂H* * ∂E* ∂I * ∂Z * 2RI + II * =  j4W + 2ω′′ μH  +εE  dv (11:45) ~* ~* ~* ~* ∂ω ∂ω ∂ω ∂ω v

where       ∂Z * 1 ∂Z * ∂Z * 1 ∂R ∂X ∂X ∂R −j = − + −j = ~ * 2 ∂ω ∂ω 2 ~′ ~ ′′ ∂ω ∂ω′ ∂ω′′ ∂ω′ ∂ω′′

(11:46)

Reconsidering the Cauchy–Riemann condition, we have ∂Z * ∂R ∂X −j = ~ * ∂ω′ ∂ω ∂ω′

(11:47)

Finally, we obtain   ððð 1 ∂I * 1 ∂R 1 ∂X 1 ω′′ −j + = j + ~* I * ∂ω R ∂ω′ R ∂ω′ ω′′ PL v

* *! * ∂H * * ∂E* μH  +εE  dv ~* ~* ∂ω ∂ω

(11:48)

In particular, if the lossless condition is adopted in the very beginning, then ∂X 4W = >0 ∂ω′ II *

(11:49)

172

11 Complex parameter and complex theorems inelectromagnetic theory

This is the theorem that the reactive slope in the lossless system is always positive, which is also equivalent to lim

∂R

R!0 ∂ω′′

>0

(11:50)

11.5 Lossless transmission lines with arbitrary load As an example, we discuss a TEM wave lossless transmission line with a length of l: The characteristic impedance Z0 = 1 and does not lose generality. The corresponding phase angle of l is ω′ θ′ = l c

(11:51)

Now, the normalized resistance
rl = 1=ρ is connected to an end, and ρ represents the voltage standing wave ratio (VSWR) of the system, as shown in Fig. 11.6.

zin

1 rl = ρ

z0 = 1

Fig. 11.6: The lossless transmission line with a lossy load.

θ'

In this case, the input impedance
zin can be written as
zin =
rin + jx
in = It is easy to write

1 + j ρ tanθ′ ρ + j tanθ′

(11:52)

  ρ 1 + tan2 θ′ rin = ρ2 + tan2 θ′

(11:53)

ðρ2 − 1Þtanθ′ ρ2 + tan2 θ′

(11:54)

xin =

The complex ~θ transmission line shown in Fig. 11.3 is used again here.   ~θ = l ω′ − jω′′ = θ′ − jθ′′ c

(11:55)

Specifically, 8 < θ′ =

ω′ c l

: θ′′ =

θ′ 2Q

(11:56)

11.5 Lossless transmission lines with arbitrary load

173

Because this model load is always short-circuit
zL =  1, we have
in
zin = j tan ~θ* =
rin + jx

(11:57)

 ′′  cthðωc lÞ 1 + tan2 θ′ rin = ′′ cth2 ðω lÞ + tan2 θ′

(11:58)

We also have

c

xin =

  ′′ cth2 ðωc lÞ − 1 tanθ′ ′′

cth2 ðωc lÞ + tan2 θ′

(11:59)

A comparison of eqs. (11.58) and (11.59) with eqs. (11.53) and (11.54) leads to   1 th ω′′=c l = ρ

(11:60)

Thus, the transmission line quality factor Q can be obtained. Q=

θ′ 2th ð1=ρÞ −1

(11:61)

Particularly, eq. (11.59) shows that the series and parallel resonant points in the transmission line do not change regardless of the value of ρ: ~ to the microHere, we further apply the Foster theorem of complex frequency ω wave transmission line with a lossy load. It is easy to get         2 ∂
rin ∂x
in sech ω′′=c l sec2 θ 1 − tan2 θ th ω′′=c l l = = (11:62)     2 ′′ ′ c 2 2 ∂ω ∂ω 1 + tan θ th ω′′=c l which does satisfy the Cauchy–Riemann conditional eq. (11.1). Note that the lossy load itself is an antenna radiation model. Equation. (11.62) shows when    ′   tan2 θ′th Q  > 1 (11:63)  2Q  there must be a negative slope in the system. In fact, the impedance distribution curve of any load clearly shows the existence of a negative reactance slope area. Also, its physical concept corresponds to the inevitable result of continuous curve conditions from the series resonant to the parallel resonant, as shown in Fig. 11.7. Note the difference between the conclusion here and literature [35]. Besides, we can obtain      2  ∂
rin ∂
xin 2tanθ′th ω′′=c l sec2 θ′sech2 ω′′=c l l = = (11:64)    2 c ∂ω′ ∂ω′′ 1 + tan2 θ′th2 ω′′=c l

174

11 Complex parameter and complex theorems inelectromagnetic theory

zin xin rin 1 ρ l

O

Negative reactance slope area Fig. 11.7: Impedance distribution curve and negative reactance slope area for a lossy load transmission line.

which satisfies Cauchy–Riemann conditional eq. (11.2). Thus, the correctness of the ~ = 0 in any system is proved again. From eq. (11.64), it is clear Foster theorem ∂z* =∂ω that whether ∂
rin =∂ω′ is positive or negative depends on the sign of tan θ′; which can also be seen in Fig. 11.7.

11.6 Summary This chapter focuses on complex parameter and complex theorem in the electromagnetic theory. It should be noted that the complex variable function theory still has great potential in the applications of electromagnetism. Thus, great attention should be paid to it. Chen-Ning Yang made an incisive analysis in his famous report The Main Themes of the Development of Theoretical Physics in the 20th Century [36]: We know that a symphony has some main melodies, and the different but related melodies interact with each other and develop to form the whole symphony. With this in our analysis of the development of physics in the 20th century, we find that there are also three main melodies: quantization, symmetry, and phase factor. You might be surprised with phase factor, which was originally put forward by Weyl in 1918. Phase factor, also called stretch factor, is a real exponential factor in order to reflect the micro-local transformation of fields including general relativity. Unfortunately, Einstein immediately pointed out its serious error that time clocks are different for different paths. After repeated studies, Schrödinger suggested in around 1925 that it be replaced by the imaginary exponential factor – the complex phase. It was one of the foundations of the famous Young Milles gauge later. The Einstein contradiction was overcome as a

Q&A

175

result of the introduction of complex phase factor. The canonical symmetry really ascended the stage of theoretical physics. The review about this part of history shows that the introduction of complex numbers into physics is not only a methodology problem but also a philosophical issue. The applications of complex numbers in the electromagnetic theory are worth further research and exploration.

Q&A Q: The previous chapter discusses frequency domain electromagnetism of complex ~ ~; while this chapter further discusses complex frequency ω dielectric materials ~ε and μ electromagnetism. What on earth are the differences between them? A: What you ask is important but often ignored in books and literature of various types. Figure 11.8 shows their relationship and differences. Generally speaking, * ~ reflect the attenuation of space ð r Þ of electromagnetic complex materials ~ε and μ ~ reflects the attenuation of time (t) of electromagwaves while complex frequency ω netic waves. Complex materials ε~, μ~ ~ ε = ε' – jε", μ~ = μ' – jμ" Space discussion r

Complex frequency

e–k"z reflects attenuation decayed with space r

e–ω"z reflects attenuation decayed with time

~ = ω' – jω" ω

Time discussion t

~

~ t) propagation factor (k r – ω contains both attenuation factors

~ k is probable local. Different r may have different values.

~ must be global which ω reflects common variation.

~: ~ and complex frequency ω Fig. 11.8: Complex material ~ ε; μ

The differences above show that they are used in different situations and have their own characteristics. Surprisingly, complex frequency is also suitable in space situation by transformation in some cases (such as the transmission line), which will be further discussed later.

176

11 Complex parameter and complex theorems inelectromagnetic theory

Meanwhile, the Maxwell’s equations given by them are the same in form, as shown in Fig. 11.9.

Δ

~H × E = –jωμ

× H = jωε~E

Δ

Δ

× H = jωε~E

~ Complex frequency ω Maxwell’s equations Δ

~ Complex materials ε~, μ Maxwell’s equations

~H × E = –jωμ

~ are exactly the ~ and complex frequency ω Fig. 11.9: The Maxwell’s equations of complex materials ~ ε; μ same in form.

~ a new idea to deal with problems? Q: Is complex frequency ω A: This question can be discussed from two perspectives. On the one hand, complex frequencies are already present in earlier electrical network synthesis, generally called S surface network synthesis, s = σ + jω: Because jω is generally considered to represent only the imaginary axis, and s = σ + jω a complex plane, the S-plane can be expressed by conformal mapping geometrically in synthesis. ~ to s here, we can write If we relate complex frequency ω   ~ = j ω′ − jω′′ = ω′′ + jω′ = σ + jω′ s = jω A comparison of them leads to σ = ω′′: ~ are both complex frequencies, they have On the other hand, although s and jω ~ is used for the real loss of totally different uses – s is used for synthesis while jω electromagnetism. ~ electroQ: Could you please introduce another two important complex frequency ω magnetic theorems? A: This chapter has described two most important theorems, namely complex frequency Poynting theorem and complex frequency Foster theorem, as shown in Fig. 11.10. Poynting theorem is the law of conservation of electromagnetic energy. The wave energy of the input system is divided into two parts: power loss and energy storage. It is easy to derive the quality factor Q, which is represented by ω′ and ω′′ in the complex frequency Poynting theorem. Q=

ω′ 2ω′′

Here are three benefits. (1) The above quality factor reflects its attenuation with time as a result of the relationship between the frequency and time t. (2) Since ω′ and ω′′

Q&A

Complex frequency Poynting theorem – S . nˆ dS s = 2ω"(wm + we) + j2ω' (wm – we)

Compared with complex and realfrequency Poynting theorem, – S . nˆ dS = PL + j2ω' (wm – we) s we obtain PL = j2ω" (wm – we)

Quality factor Q = ω' ω" (the transmission line) Q=

θ'

λg

2θ" λ0

2

177

Complex frequency Foster theorem дZ * 0 ~ = дω (where Z = R + jX is the input impedance of one-port network) Cauchy–Riemann дX дR = дω' дω" дR дX =– дω' дω"

The lossless system дX >0 дω'

~ electromagnetic theorems. Fig. 11.10: Two important complex frequency ω

are not directly related to space, the definition of Q is more generalized. Both closed and open systems are not influenced by the “shadow” of Q. Specifically, ω′ refers to the general operating frequency rather than merely the resonant frequency. (3) An important conclusion that comes along is that complex frequency can also be represented by Q. Thus, we have   1 ~ = ω′ 1 − j ω 2Q Foster theorem is also known as the reactance slope theorem in the lossless system. That is to say, the reactance slope is always positive under the lossless condition. Thus, we have ∂x >0 ∂ω′ However, in the general complex frequency system, the complex frequency slope of conjugate impedance z = R + jx is zero. We have ∂z* =0 ~ ∂ω The above equation shows a magic complex slope, or z is an analytic function of jω; which must satisfy the Cauchy–Riemann condition. Here, we further prove that it is also an equivalent equation. ∂z =0 ~* ∂ω

178

11 Complex parameter and complex theorems inelectromagnetic theory

As ∂ ∂ω′ ∂ ∂ω′′ ∂ + = * * ′ ~ ~ * ∂ω′′ ~ ∂ ω ∂ω ∂ω ∂ω Based on eq. (11.36), it is easy to write   ∂ 1 ∂ ∂ −j = ~ * 2 ∂ω′ ∂ω ∂ω′′      

∂z 1 ∂ ∂ 1 ∂R ∂x ∂R ∂x −j + − ðR + jxÞ = −j = ~ * 2 ∂ω′ ∂ω 2 ∂ω′ ∂ω′′ ∂ω′′ ∂ω′′ ∂ω′ From Cauchy–Riemann condition eqs. (11.33) and (11.34), we also have ~ * ≡ 0: ∂z=∂ω In particular, it is worth noting that we can obtain a duality of complex derivatives of analytic functions in Functions of Complex Variables. Theorem 11.3: Complex function w = u + jv is the analytic function of z = x + jy, which satisfies the Cauchy–Riemann condition. ∂u ∂v ∂u ∂v = , = ∂x ∂y ∂y ∂x We obtain the duality of complex derivative. 8 ∂w > > =0 < ∂z* > > ∂w* = 0 : ∂z Complex electrical angle of transmission lines 1 ~ θ = θ' – jθ" = (ω' – jω" ) c θ" ω" λg = θ' ω' λ0

Lossy transmission lines 1 ~ θ = θ' 1 – j 2Q

2

Lossless transmission lines with a lossy load 1 ~ θ = θ' –jth ρ

~ is used in the transmission line (space) theorem, which puts forward Fig. 11.11: Complex frequency ω ~ = θ′ − jθ′′: electrical angle θ

Recommended scholar

179

~ = ω′ − jω′′ theory be applied to solve space problems? Q: Can the complex frequency ω ~ theory is essentially A: In the beginning, we mentioned that the complex frequency ω an attenuation theory of t, which is independent of space. However, on the other hand, we should continue to expand its applications. The theory plays an important role in dealing with lossy transmission lines or lossless transmission lines with a load. Broadly speaking, the length of the transmis  sion line can be represented by electrical angle θ, θ′ = ω′=c l: Naturally, complex electrical angle ~θ = θ′ − jθ′′ can be introduced for the lossy transmission line. However, for the lossless transmission line with a load, we have θ′′ = th − 1 ð1=ρÞ; where ρ is the VSWR, as shown in Fig. 11.11.

Recommended scholar In this chapter, we introduce the famous American microwave expert R. E. Collin (Fig. 11.12).

Fig. 11.12: R. E. Collin.

1. He is the theoretical successor and principal representative of the MIT School It is appropriate to introduce the MIT School here. Due to urgent time, many people do not take scientific research seriously enough. Or, there is always a feeling that theory can hardly be applied to practice. Let us look back on what it was like in the 1940s. In response to World War II, the United States was in great need of research on advanced equipment including radar, antenna and navigation. The electromagnetic and microwave studies were almost a brand new field back then. On the other hand, scientists felt the urgency of conducting research. The MIT Radiation Laboratory was a major unit to carry out related research. They not only accomplished the task well but also created a new theory – a truly

180

11 Complex parameter and complex theorems inelectromagnetic theory

close combination of theory and practice, and theory and engineering. The publication of Waveguide Handbook (MIT Radiation Laboratory Series, Vol. 10, McGraw-Hill, 1951), led by N. Marcuvitz, was impressive. This book, from the theoretical point of view, was unfamiliar and difficult for most people. As an important bridge, R. E. Collin published Waveguide Field Theory in time to systematically introduce the Eigen-mode orthogonality, normalization and completeness theory of the MIT School. Besides, the mode matching theory for two kinds of component interfaces and the network representation theory of modes were also dealt with in the book. Having been published for nearly 50 years, Waveguide Field Theory has not only undergone historical tests but has also been a peak in the microwave field. Needless to say, Waveguide Field Theory is profound theoretically. Looking back on the highlights of the MIT School during my graduate studies, I am still excited as I was. In order to facilitate the research of most engineering and technical personnel, Collin later published Fundamentals of Microwave Engineering, which also played an important role in promoting the understanding of theories. It is no exaggeration to say that the main ideas of the MIT School are publicized and promoted by Collin. 2. Collin is an excellent educator, famous for being strict and rigorous Collin is indeed an excellent educator, and he is most adept at anatomical analysis of the profound knowledge of the MIT School. His famous book Waveguide Field Theory itself constitutes a rigorously complete theoretical system. The most notable feature of Collin is his rigorousness. Surprisingly, he can extract definitions, characteristics and difficulties exactly from theories and is meticulous even in terms of the letters, logic, derivation and conclusions. 3. Every master has his own characteristics and style, and Collin is no exception Let us cite the example of Chinese calligraphy. For the same Chinese character, although the writing varies from calligrapher to calligrapher, people familiar with it will immediately recognize it at the sight of a piece of calligraphy. It is the same with science and technology. All masters have their own distinctive characteristics and styles. It will be totally different when they talk about the same thing. I was deeply impressed while in the United States. Even for a vector arrow, Collin wrote it above the letter, but Harrington wrote it below the letter. Their different styles make electromagnetism more interesting and colorful.

12 Complex operator

and 2D static field

Famous mathematician Jacgues Hadamard gave us a strange proposition that the shortest distance between two truths in real domain is to pass through complex and its corresponding domain. Hereby, the paper proposes complex operator theory, merging the 2D divergence ∇ and curl ∇ × in vector theory and Gauss theorem and Stokes theorem in integral theory into a unified whole. It makes study on 2D electrostatic field and steady magnetic field, unifying its governing integral equations into a residue theorem, while line charge and current are the residue source to provide integral contribution.

12.1 Introduction This is the 12th chapter of Electromagnetic Field Theory Teaching Series. Famous mathematician Jacgues Hadamard gave us a strange proposition that the shortest distance between two truths in real domain is to pass through complex domain [32], which means the best way to link the two real domains is to express with complex domain. This is a focus that needs to be discussed in this paper. First, the correspondence between complex numbers and plane vectors is given, that is to enable a one-to-one correspondence between Gauss plane and Cartesian plane. Complex number vector a = ax + jay b = bx + jby

*

a = ax^i + ay^j

*

b = bx^i + by^j

(12:1)


represents complex conjugate of a. Upon complex multiplication of a
and in which a b, we can obtain       
b = ax − jay bx + jby = ax bx + ay by + j ax by − ay bx (12:2) a Moreover, let’s write the dot product and cross-product of the corresponding plane vector, obtaining (* * ð12:3Þ a  b = ax bx + ay by  * *  ð12:4Þ a × b = ax by − ay bx k^ where ^k represents the unit vector of the third dimension in z direction. Introduce the symbol:

a × b = ax by − ay bx

https://doi.org/10.1515/9783110527407-012

(12:5)

182

12 Complex operator

and 2D static field

It further represents the cross-product of plane vector after removing the direction ^k. Take note that signs are still kept in eq. (12.5); hereby, we can write

ab = a ⋅b + j a × b

(12:6)


b on the left; the dot product of In eq. (12.6), there is complex multiplication a corresponding vectors is given in the real part and the cross-product (exclusive of direction) is given in the imaginary part on the right; the complex domain contains two truths in real domain. This correspondence was first discovered by the famous mathematician G. Polya [37].

12.2 Complex operator Know that the operator ∇ in the vector theory refers to ∂^ ∂^ i+ j ∂x ∂y

∇=

(12:7)

It possesses two properties: vector property and operator property. If introducing * vector function w ¼ u^i þ v^j; we can quickly write the divergence ∇ and curl ∇ × of * the w; being, respectively, 8   ∂u ∂v * > > ð12:8Þ + >∇ w = < ∂x ∂y   > ∂v ∂u ^ * > > k ð12:9Þ − :∇× w = ∂x ∂y In the paper, we further give play to the idea of G. Polya to define a new complex operator in the complex domain correspondingly, obtaining =

∂ ∂ +j ∂x ∂y

(12:10)

Similarly, it also has two properties: complex property and operator property. By introducing the corresponding complex function w = u + jv and remembering vector correspondence,

w

ab

a

w

b

(12.11)

12.2 Complex operator

183

we can get

w=

∂ ∂ ∂u ∂v ∂v ∂u u + jv ) = −j +j + − ( ∂x ∂y ∂x ∂y ∂x ∂y

Thereby, the important corresponding equation of the complex operator vector operator ∇ can be obtained: w=

.w +j

×w

(12:12) and (12:13)

is closely linked to the ∇ and Equation (12.13) shows that the complex operator not only reflects the physical essence of the dual ∇ × of vector operator; therefore, ∇ but also refers to the two truths in real domain. Moreover, if introducing the dual form of eq. (12.13), we can obtain

w=

∂ ∂ ∂u ∂v ∂v ∂u u + jv ) = +j +j – + ( ∂x ∂y ∂x ∂y ∂x ∂y

(12:14)

It is very interesting to point out that the Cauchy–Riemann conditions of analytic function in this case can be uniformly expressed with the following complex operator equation as w=0

(12:15)

It is equivalent to 8 ∂u ∂v > > < ∂x = ∂y > ∂u ∂v > : =− ∂y ∂x

(12:16)

Besides, the validity of operator eq. (12.15) is irrelevant to the coordinates; for analytic functions, eq. (12.15) is always true; if there are polar coordinates, we can further obtain =

∂ 1 ∂ +j ∂r r ∂θ

In this case, Cauchy–Riemann condition is 8 ∂u 1 ∂v > > < = ∂r r ∂θ > ∂v 1 ∂u > : =− ∂r r ∂θ

(12:17)

(12:18)

184

12 Complex operator

and 2D static field

12.3 Complex operator integral theorem For 2D domain D and positively oriented contour C, three unit vectors can be defin^ and z -direction unit vector ^k (see able: tangential unit vector ^t, normal unit vector n Fig. 12.1). î

y

n̂

W( x, y) k D

C

O

x

  ^ = ^t × ^ Fig. 12.1: 2D integral domain D n k .

There are two famous theorems in vector theory: 2D Gauss theorem þ ðð * * ^ dl ∇  w dS = w  n D

(12:19)

C

which can also be written in components  þ ðð  ∂u ∂v dx dy = u dy − v dx + ∂x ∂y D

(12:20)

C

and 2D Stokes theorem ðð D

∇ × w  ^kdS = *

þ

*

w  ^t dl

C

which can also be written in components similarly  þ ðð  ∂u ∂v dx dy = u dx + v dy − ∂x ∂y D

(12:21)

(12:22)

C

Thus, it is easy to derive the integral theorem of complex operator w dxdy = j w dz D

C

(12:23)

wherein d
z represents the complex conjugate of dz, that is, d
z = dx − j dy Its corresponding relationship is shown in Fig. 12.2. Similarly, the integral theorem of complex dual operator can be given

(12:24)

185

12.4 Complex partial derivative

Integral theorems of complex conjugation operator

w dxdy = j

D

Imaginary part Stockes theorem

Real part Gauss Theorem Δ

∫∫D

⋅ w ds =

∫C w dz

∫∫D

∫C w ⋅ nˆ dl

Δ

∫∫

× w ⋅ kˆ ds =

∫C w ⋅ tˆ dl

Fig. 12.2: Relationship between the corresponding integral theorems of complex conjugation and real operator ∇. operator

w dxdy = –j w dz C

D

(12:25)

Particularly, when w is analytic function, we can obtain w=0

Then, the eq. (12.25) will be þ w dz = 0

(12:26)

C

Equation (12.26) is the famous Cauchy–Goursat integral theorem. It shows that closed contour integral of analytic function is zero. The root is that complex operator w equals zero. Moreover, we can uniformly express eqs. (12.23) and (12.25) with integral operator, obtaining dxdy

–j dz

D

(12:27)

C

.

12.4 Complex partial derivative We already know that 8 1 > > < x = ðz +
zÞ 2 1 > > : y = ðz −
zÞ 2j We can further write

(12:28)

186

12 Complex operator

and 2D static field

w = wðx, yÞ = wðz,
zÞ

(12:29)

It is easy to give complex partial derivative 8   ∂ 1 ∂ ∂ > > −j > = < ∂z 2 ∂x ∂y   > ∂ 1 ∂ ∂ > > = +j : ∂
z 2 ∂x ∂y

ð12:30Þ ð12:31Þ

Thereupon, we can make the complex partial derivative correspond with complex operator , obtaining ∂ ∂z =2 ∂ ∂z

(12.32)

=2

(12.33)

and Laplace operator ∇2 2

=

=4

∂2 ∂z ∂z

(12:34)

Since we already know that analytic functions have important properties, ∂w =0 ∂
z

(12:35)

there is dw ∂w 1 = = w'( z ) = d z ∂z 2

w

(12:36)

×w }

(12:37)

finally obtaining

.

1 w′( z ) = {( . w) + j 2

12.5 Complex operator form of 2D electrostatic field For uniformly filled 2D electrostatic field studied in this paper, its differential governing equations are *

∇×E=0

(12:38)

ρ ε

(12:39)

*

∇E=

And, the corresponding integral equations are

12.5 Complex operator form of 2D electrostatic field

þ ðð

*

E  ^t dl = 0

C *

þ

∇  E ds = D

*

^ dl = En

(12:40) n X qk k=1

C

187

(12:41)

ε

Equation (12.40) is called loop integral, and eq. (12.41) refers to the famous Gauss theorem, as shown in Fig. 12.3. î

y ...

qn

q2 k q1 D

O

n̂

C ε x

Fig. 12.3: Uniformly filled 2D electrostatic field domain D.

By just considering E = Ex + jEy and using complex operator ρ E= ε

, we can quickly write (12:42)

It is very obvious that the two real differential equations are all included in eq. (12.42); if the integral theorem is written in the form of customary residue theorem, there is þ n X − jqk − jEdz = 2πj (12:43) 2πε c k=1 Equation (12.43) is the residue theorem form of complex integral of 2D electrostatic field; compared with theory of functions of a complex variable, we can know that Res½ − jE, zk  =

− jqk 2πε

ðk = 1, 2, . . . , nÞ

(12:44)

This clearly shows that the residue of electrostatic field is closely related to the 2D line charge. 2D electrostatic field is regarded as one of the physical backgrounds of complex residue theorem. In order to reveal physical essence, we give the simplest examples to make a study on the line charge q placed at origin of coordinates, as shown in Fig. 12.4. According to vector theory, in this case, there is * * qr E= 2πεr2

(12:45)

188

12 Complex operator

and 2D static field

y

ε q O

1

x

Fig. 12.4: A line charge q on the origin O.

According to the corresponding relationship between complex number and vector, we can write r2 = jzj2 = z
z

(12:46)

Hence, the complex electric field is E=

q 2πε
z

The obtained complex integral of 2D electrostatic field is þ þ − jq
− jE dz = dz 2πεz C

(12:47)

(12:48)

C

According to residue theorem,  
0 = lim z  − jq = − jq Res − jE, z!0 2πεz 2πε

(12:49)

we can obtain þ C

 
dz = 2πjRes − jE,
0 =q − jE ε

(12:50)

12.6 Complex operator form of 2D steady magnetic field As symmetry, we can absolutely use the above results of electrostatic field to 2D steady magnetic field in uniform medium; its differential governing equations are ( * * ∇× H = J ð12:51Þ * ∇  H =0 ð12:52Þ It is easy to give complex operator form H = jJ

(12:53)

189

Q&A

For the uniformly 2D steady magnetic field domain D shown in Fig. 12.5, we can write its residue integral theorem as well

î

y ...

Jn

J2 k J1

O

n̂

C ε x

Fig. 12.5: Uniformly filled 2D steady magnetic field domain D.

þ


dz = 2πj − jH

C

 n  X − Jk k=1

2π

(12:54)

By comparison, we can know that the residue is  
zk = − Jk Res − jH, 2π

ðk = 1, 2, . . . , nÞ

(12:55)

It is very clear that for the complex integral form of all the 2D static fields – residue theorem, the charge and current sources are the physical roots of the residue.

12.7 Summary The introduction of complex operator not only unifies the 2D divergence ∇ and curl ∇ × in vector theory as well as Gauss theorem and Stokes theorem into a unified whole but also makes a new intersection with complex function theory for electrostatic field and steady magnetic field. Integral theorem of 2D static field is identical with residue theorem in complex function. Physical nature of corresponding residue is the line charge source or line current source. as a theory. It is worth of deep study on complex operator

Q&A Q: You have mentioned the famous saying of Jacgues Hadamard in two of your notes. “The shortest line between two truths in real domain is through complex domain.” So what profound meaning do you intend to express? A: Yes, what Hadamard proposed represents the profound idea of complex function. It includes real part and imaginary part, both of which are not mutually isolated.

190

12 Complex operator

and 2D static field

Both real part and imaginary part are real functions and connected closely – namely the shortest line between two truths in real domain is through complex domain. This note puts forward the idea of complex operator =

∂ ∂ +j ∂x ∂y

Then, the function of it and complex function w = u + jv generate two truths, as shown in Fig. 12.6.

Divergence

Imaginary part

Rotation

Δ

w

Real part

Δ

Conjugate action of complex operator

⋅w of vector function w

×w

of vector function

w (direction neglected)

Fig. 12.6: Reflection of two “truths” by conjugate action of complex operator.

Surprisingly, another two truths are generated when complex operator action is zero – Cauchy–Riemann conditions of analytic function, as shown in Fig. 12.7.

Complex operator action

Real part Cauchy—Riemann conditions of analytic function

w =0

Imaginary part

дu дx дu дy

− +

дv

=0

дy дv дx

=0

Fig. 12.7: Reflection of another two “truths” by complex operator action.

We have proven the prediction of Hadamard in detail through discussion on conjugate action of complex operator and complex operator action. It is really amazing that differential and integral are further in inverse operation. Then, there are conjugation and complex operator integral theorem: real part corresponds to Gauss theorem and imaginary part corresponds to Stokes theorem, as shown in Fig. 12.2. Symmetrically, for analytic function w, it is indicated by Fig. 12.8, namely the famous Cauchy–Goursat theorem (see Fig. 12.8), that complex operator action

Cauchy—Riemann conditions

w dxdy D

0

дu

Real part S

дx

−

дv

дy ⎟

dxdy

0

Cauchy—Goursat conditions -j wdz = 0 C

Imaginary part

дv S

дx

+

дu

дy ⎟

dxdy

0

Fig. 12.8: Integral of complex operator action equal to 0 constantly.

191

Q&A

corresponds to Cauchy–Riemann conditions. The source for complex closed contour w = 0. integral of zero is Q: From the above conditions, the concept of complex operator attractive!

is new and

A: Your statement is not comprehensive. The concept of complex operator new and old, as shown in Fig.12.9.

is both

Complex operator Complex partial derivative

2

д

дz д 2 дz

2

д2 =4 дzдz

Fig. 12.9: Complex operator partial derivative.

and complex

Δ

regards itself as the combination of 2D operators ∇ In addition, complex operator and ∇ × and also regards itself as the combination of Cauchy–Riemann conditions. is the new From this point of view, it is innovative. Therefore, complex operator from this sense. Q: What’s the connection between complex operator

and electromagnetic theory?

in 2D electrostatic field and steady magnetic field, as A: Typically, we can apply shown in Fig. 12.10. They bear the identical idea with residue theorem.

Two-dimensional electrostatic field

Two-dimensional steady magnetic field

ρ E=ε Fig. 12.10: Complex operator

H = jJ and 2D electrostatic field or steady magnetic field.

Q: What are other innovations available in complex number and complex function fields? A: This is a good question. Investigating from the nature, we can totally say that innovation exists everywhere and the key just lies in diligence. Here, I’d like to give an example: namely combining complex operator and matrix.

192

12 Complex operator

and 2D static field

(1) Complex rotation matrix Two sets of coordinate systems for included angle θ, xyz and x′y′z′ are given in Fig. 12.11. y

yʹ

xʹ θ θ

x

O Fig. 12.11: Two sets of coordinate systems for included angle θ, xyz and x′y′z′.

Given (

x0 = x cos θ + y sin θ y0 = − x sin θ + y cos θ

we write it into matrix form: " # #" # " # " x x cos θ sin θ x0 =T = y − sin θ cos θ y y0

cos θ sin θ is called real rotation matrix, and there is det T = 1.  sin θ cos θ Now, let’s introduce the definition of complex rotation matrix [CT] " # cos θ − j sin θ ½CT = − j sin θ cos θ

where T ¼

det½CT ≡ 1. Then, we can get "

x0 jy0

# " =

cos θ

− j sin θ

− j sin θ

cos θ

#"

x jy

# :

It equals to (

x0 = x cos θ + y sin θ jy0 = jð − x sin θ + y cos θÞ

Amazingly, real rotation matrix is asymmetrical, but complex rotation matrix is symmetrical, and inverse matrix

Q&A

" ½CT

1

=

cos θ

j sin θ

j sin θ

cos θ

193

#

is also symmetrical. (2) Matrix form of complex operator h i ∂ ∂ j ∂y Further, we can also bring icomplex operator into matrix form and write ∂x : h ∂ ∂ j and conjugation ∂x ∂y ; then the complex operator and its conjugation are

=

∂ ∂x

=

∂ ∂ – j ∂x ∂y

j

∂ ∂y

1 1 1 1

Note that complex function w corresponds to matrix " # u jv ½w = jv u then w=

∂ ∂ –j ∂x ∂y

w=

∂ ∂x

j

∂ ∂y

u

jv 1

jv u u

1

jv 1

jv u

1

=0

(3) Complex operator rotation Given complex operator = ∂ ∂x + j ∂ ∂y if represented by ðx, yÞ, now represent it   by x′, y′ after rotating θ

=

∂x′ ∂ ∂y′ ∂ ∂x′ ∂ ∂y′ ∂ + +j + ∂x ∂x′ ∂x ∂y′ ∂y ∂x′ ∂y ∂y′

= cos θ

∂ ∂ ∂ ∂ − sinθ + j sinθ + cosθ ∂x′ ∂y′ ∂x′ ∂y′

Then, the matrix form can be written as follows: #

"

cos θ j sin θ ∂ ∂ ∂ ∂ ∂ = j 0 = j 0 ∂x ∂y ∂x ∂y ∂x0 j sin θ cos θ This is called coordinate rotation of

j

∂ ½CT − 1 : ∂y0

194

12 Complex operator

and 2D static field

Q: As you know, whether different complex operators are introduced somewhere else? A: Of course. An example is given in the work Electrodynamics of Mr. Cao Changqi, version of July 1962, People’s Education Press. In discussion on special relativity of Einstein, complex operator of Lorentz (P284) is introduced. 2 3 jβ 1 ffi pffiffiffiffiffiffiffi ffi 0 0 pffiffiffiffiffiffiffi 2 2 1β 7 6 1β 6 7 6 0 1 0 0 7 6 7 α=6 0 1 0 7 6 0 7 4 jβ 5 1 ffi pffiffiffiffiffiffiffiffi 0 0 pffiffiffiffiffiffiffi 2 2 1β

1β

Here introduces the famous educator and mathematician of America, G. Polya. If the saying of Hadamard is prediction, what G. Polya gives is example with strong * * persuasion. In his famous work Complex Function, he compared 2D vectors a ; b with complex numbers a = ax + jay and b = bx + jby deeply for the first time and discovered that a *b = ( axbx + ay by ) + j ( axby − ay bx ) = a b + j a × b

The seemingly simple question covers deep idea of G. Polya. Besides, he wrote the famous work of How to Solve It for Princeton Series. In teaching, we shall first learn his skill of discovering problems in simple questions and learn to contrast and compare totally different things, thus achieving extraordinary teaching in ordinary work.

13 New network theory of electromagnetic waves in multilayered media This chapter discusses the reflection and refraction of electromagnetic wave transmission in multilayered media. As a basic model, the case that the electromagnetic wave oblique incidence on the boundary plane between two kinds of media is considered. Then, we apply the model to the situations of normal incidence, conductor boundary and magnetic boundary. By combining electromagnetic wave propagation and network theories, this chapter proposes the multilayered media [C] network theory innovatively. Thus, the mode of electromagnetic wave transmission in multilayered media can be converted to the cascaded [C] matrix model with simple concepts and uniform results. Specific examples are also given in the discussion.

13.1 Introduction This is the 13th chapter of Electromagnetic Field Theory Teaching Series. Electromagnetic wave propagation, especially the incidence, reflection and refraction in multilayered media, is not only a cutting-edge theoretical problem but also an important application issue. Prospecting, geodesy and foreign material detection depend on its basis – electromagnetic wave propagation into multilayer media. This chapter, based on three principles, considers the electromagnetic wave oblique incidence on the boundary plane between two kinds of media as a basic model. Then, we apply the model to the situations of normal incidence, conductor boundary and magnetic boundary. By combining electromagnetic wave transmission and network theories, this chapter proposes the multilayered media [C] network theory innovatively. Thus, the mode of electromagnetic wave transmission in multilayered media can be converted to the cascaded [C] matrix model. A series of useful conclusions are obtained from application examples.

13.2 Basic model Incidence, reflection and refraction all involve the transmission of electromagnetic plane waves [1,11,16,26,38–45]. First of all, this chapter presents three principles for the basic model: 1. Take the plane boundary between two kinds of media as the basic unit. 2. Take the oblique incidence electromagnetic plane wave as the basic model and * divide it into two situations: the parallel polarized wave whose electric field E is parallel to the incident plane and the perpendicularly polarized wave whose * electric field E is perpendicular to the incident plane. https://doi.org/10.1515/9783110527407-013

196

3.

13 New network theory of electromagnetic waves in multilayered media

When the incident angle θi ; the reflex angle θr and the refraction angle θt are all 0 ; that is, θi = θr = θt = 0 ; the problem should be perpendicular incidence situation.

It must be noted that due to the third principle, the results of this book are somewhat different from those of most other books [1,11,16,26,38–45]. Figure 13.1 shows two cases of the basic model.

x

x Sr

Er

Sr Hr

Et θr θi

θt O

Hr Er Et

St

Ht

θr θi

y Hi

Ei

θt O

St Ht z

Si

Si Ei

Hi ε1, μ1

ε2, μ2

ε1, μ1

ε2, μ2

Fig. 13.1: Take plane wave oblique incidence on the boundary plane as the basic model: (a) parallel polarization and (b) perpendicular polarization.

Case 1: Parallelly polarized wave In this case, the matching condition of the electromagnetic field at the plane boundary is 9 E0i cos θi + E0r cos θr = E0t cos θt = (13:1) i r Et E0 E0 ; 0 η − η = η 1

1

2

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where η1 = μ1 =ε1 and η2 = μ2 =ε2 . If reflection coefficient Γ== and refraction coefficient T== are further defined, respectively, 9 Er > Γ== = 0i > E = 0 (13:2) t > E0 > ; T== = i E

0

13.2 Basic model

we have Γ== =

η2 cos θt − η1 cos θi η2 cos θt + η1 cos θi

T== =

2η2 cos θi η2 cos θt + η1 cos θi

197

9 = (13:3)

;

The Fermat principle θi = θr has been taken into account. From eq. (13.3), we can easily get  η (13:4) T== = 1 − Γ== 2 η1 Case 2: Perpendicularly polarized wave In this case, the matching condition of the electromagnetic field at the plane boundary is 9 E0i + E0r = E0t = (13:5) t i r E0 cos θi E cos θ E cos θr − 0η = 0η t; η 1

1

2

Similarly, we can get reflection coefficient Γ? and refraction coefficient T? : 9 η cos θ − η cos θ Γ? = η2 cos θi + η1 cos θt = 2

T? =

i

1

t

2η2 cos θi η2 cos θi + η1 cos θt

;

(13:6)

According to the third principle in determining the basic model, eqs. (13.3) and (13.6) can be combined in the case of perpendicular incidence (viz. θi = θr = θt = 0 ). Γ = Γ== ð0 Þ = Γ? ð0 Þ =

η2 − η1 η2 + η1

T = T== ð0 Þ = T? ð0 Þ =

2η2 η2 + η1

9 = ;

(13:7)

and T? = 1 + Γ?

(13:8)

The oblique incidence plane of electromagnetic wave is adopted as the basic model for a deep reason. It not only includes the perpendicular incidence situation but also summarizes the case when the second medium is an ideal conductor (electric wall). Now, set η2 = 0; and we have ) Γe = Γ== ð0 Þ = Γ? ð0 Þ =  1 (13:9) Te = T== ð0 Þ = T? ð0 Þ = 0 In the case of magnetic wall when the second medium is an ideal magnet, if η2 ! ∞; then ) Γm = Γ== ð0 Þ = Γ? ð0 Þ = 1 (13:10) Tm = T== ð0 Þ = T? ð0 Þ = 2

198

13 New network theory of electromagnetic waves in multilayered media

Readers may be confused with Tm = 2 in eq. (13.10). In fact, all you need to note is that Γ and T in this chapter are both reflection and refraction of the electric field. It is obvious in what follows that they satisfy the energy conservation law.

13.3 Special angle Two special angles deserve our attention when the plane electromagnetic wave arrives at an interface before reflection and refraction: Brewster’s angle θB in the case of no reflection and critical angle θc in the case of total reflection.

13.3.1 Brewster’s angle θB of no reflection For parallelly polarized waves, the no reflection condition is η1 cos θi = η2 cos θt

(13:11)

From Fermat’s theorem, cos θt =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε1 μ1 1 − sin2 θt = 1 − sin2 θi ε2 μ2

(13:12)

we have Brewster’s angle == θB

= sin

−1

"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1 − ðε1 μ2 =ε2 μ1 Þ 1 − ðε1 =ε2 Þ2

(13:13)

But for perpendicularly polarized waves, the no reflection condition is η2 cos θi = η1 cos θt The corresponding Brewster’s angle is "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1 − ðε2 μ1 =ε1 μ2 Þ ? −1 θB = sin 1 − ðμ1 =μ2 Þ2

(13:14)

(13:15)

In particular, for dielectric medium (μ1 = μ2 ), there is no Brewster’s angle for perpendicularly polarized waves. Only in the case of parallelly polarized waves, there will be Brewster’s angle of no reflection.

13.3.2 Critical angle θc in the Case of Total Reflection From eqs. (13.3) and (13.6), it is obvious that only if

13.4 Electromagnetic wave transmission [C ] network

cos θt = 0 there will be total reflection. Then, we have   Γ==  = jΓ? j = 1

199

(13:16)

(13:17)

And from eq. (13.16), we can obtain sin2 θi =

ε2 μ2 ε1 μ1

(13:18)

Then, we have θt = 90

(13:19)

It means that electromagnetic waves can only propagate/transmit along the interface and they will not transmit into a second medium. Be it parallelly polarized wave or perpendicularly polarized wave, the critical angle θc in the case of total reflection is rffiffiffiffiffiffiffiffiffi ε2 μ2 −1 θc = sin (13:20) ε1 μ1 Therefore, the condition for total reflection is ε1 μ1 > ε2 μ2

(13:21)

For dielectric medium of μ1 = μ2 ; we have ε1 > ε2

(13:22)

In other words, there will be total reflection only if waves transmit from an optically denser medium to an optically thinner medium. It was this principle that led to the emergence of optical fiber in recent years.

13.4 Electromagnetic wave transmission [C ] network In order to study complex cases of oblique incidence into multiple dielectric interfaces, this chapter combines electromagnetic wave transmission with the network theory and proposes [C] network theory, as shown in Fig. 13.2.

a2

b1 [C]

1

a1

2

b2

Fig. 13.2: Electromagnetic wave transmission [C] network.

200

13 New network theory of electromagnetic waves in multilayered media

As a typical example, we study how [C] network represents the medium’s interface in transmission. From the perspective of symmetrical ports, b1 and b2 represent the electric field reflection of port ① and port ②, respectively, while a1 and a2 represent their corresponding electric field incidence. It can be shown in the following matrix. " # " #" # a2 C11 C12 b1 = (13:23) a1 C21 C22 b2 If we study from the perspective of generalized transmission from port ① to port ②, a1 is the incidence of port ①, a1 = E0i ; b1 is the reflection of port ①; b1 = E0 r b2 is the refraction of port ②; b2 = E0 t a2 for the network with single-layer interface as indicated in Fig. 13.2, we have a2 = 0

(13:24)

For the multilayered network, a2 represents the reflection of the second layer.

13.4.1 Oblique incidence of parallelly polarized waves Based on eqs. (13.1) and (13.2), and considering the symmetric form of each layer of medium, it is easy to obtain " #" # " #" # cos θt cos θt cos θi cos θi b1 a2 = (13:25) 1 1 1 1 −η −η a1 b2 η η 1

1

2

2

Then, "

b1 a1

#

2 cos θ t η1 4 η1 + = 2 cos θi cos θt − η 1

cos θi η2

cos θt η1

−

cos θi η2

cos θi η2

cos θt η1

+

cos θi η2

3" 5

a2

# (13:26)

b2

For convenience of expansion, we introduce ( θ1 = θi

(13:27)

θ2 = θt

The electromagnetic wave matrix [C== ] of the parallelly polarized wave can be given. " 1 # 1 1 1 h i η cos θ cos θ η cos η1 cos θ1 − η2 cos θ2 θ1 + η2 cos θ2 1 2 1 == 1 C = (13:28) 1 1 1 1 2 cos θ1 η cos θ − η cos θ η cos θ + η cos θ 1

1

2

2

1

1

2

2

In particular, in the case of single-layer interface and a2 = 0; specifically we have

13.4 Electromagnetic wave transmission [C ] network

b1 =

η1 cos θi cos θt 2 cos θi

a1 =

η1 cos θi cos θt 2 cos θi

 

1 η1 cos θi

−

1 η2 cos θt

1 η1 cos θi

+

1 η2 cos θt



201

9 = b2 >

 ; b2 >

(13:29)

Then, Γ== =

b1 a1

=

η2 cos θt − η1 cos θi η2 cos θt + η1 cos θi

T== =

b2 a1

=

2η2 cos θi η2 cos θt + η1 cos θi

9 = (13:30)

;

The result is consistent with eq. (13.3).

13.4.2 Oblique Incidence of Perpendicularly Polarized Waves Similarly, from eq. (13.5) and considering symmetry, we have " #" # " #" # 1 1 1 1 b1 a2 = cos θt cos θi cos θi cos θt − η − η a1 b2 η η 1

1

2

(13:31)

2

Then, "

b1 a1

#

2 cos θ i η1 4 η1 + = 2 cos θi cos θi − η 1

cos θt η2

cos θi η1

−

cos θt η2

cos θt η2

cos θi η1

+

cos θt η2

3" 5

a2

#

b2

We have the [C? ] network of the perpendicularly polarized wave. 2 cos θ 3 cos θ2 cos θ2 cos θ1 1  ? η1 − η2 η1 4 η1 + η2 5 C = 2 cos θ1 cos θ1 − cos θ2 cos θ1 + cos θ2 η1

η2

η1

(13:32)

(13:33)

η2

In particular, in the case of single-layer interface and a2 = 0; specifically we have   9 cos θi η = b1 = 2 cos1 θ − cosη θt b2 > η 1 2 i (13:34)   cos θi η > cos θt ; a1 = 2 cos1 θ + b 2 η η 1

i

2

Finally, we get 

Γ? =

b1 a1

=

T? =

b2 a1

=

η2 cos θi − η1 cos θt η2 cos θi + η1 cos θt



2η2 cos θi η2 cos θi + η1 cos θt

Again, the result is consistent with eq. (13.6).

9 > =  > ;

(13:35)

202

13 New network theory of electromagnetic waves in multilayered media

13.4.3 Normal incidence When considering θ1 = θ2 = 0 ; Eqs. (13.28) and (13.33) both become the same normal incidence network, as shown below. η ½C = 1 2

"

1 η1

+

1 η2

1 η1

−

1 η2

1 η1

−

1 η2

1 η1

+

1 η2

# (13:36)

13.4.4 Unified [C ] network We introduce unified impedance Z, which is defined as 8 > < η -Normal incidence Z = η== -Oblique incidence of parallelly polarized waves > : ? η -Oblique incidence of perpendicularly polarized waves

(13:37)

and ==

==

η1 = η1 cos θ1 ; η2 = η2 cos θ2 η? 1 =

η1 cos θ1

; η? 2 =

Thus, we have the unified [C] network "1 AZ1 Z1 + ½C = 2 Z1 − 1

)

η2 cos θ2

1 Z2

1 Z1

−

1 Z2

1 Z2

1 Z1

+

1 Z2

(13:38)

# (13:39)

and ( A=

1 cos θ2 cos θ1

Normal incidence and perpendicular polarization oblique incidence Parallel polarization oblique incidence (13:40)

Note that in the case of oblique incidence of parallelly polarized waves, the factor A = cos θ2 = cos θ1 does not affect the expression of the reflection coefficient Γ but affect that of the refraction coefficient T.

13.4.5 Wave transmission section [Cl ] Take the wave transmission section as medium (ε2 ; μ2 ), and section l does not lose generality, as shown in Fig. 13.3.

13.5 Engineering applications

1

203

2

2

2

[Cl] l

Fig. 13.3: Wave transmission section network [Cl].

From its definition, we have "

b2 ð1Þ a2 ð1Þ

# " =

e − j’

0

0

ej’

#"

a2 ð2Þ

#

b2 ð2Þ

(13:41)

where ’ = k2 l cos θ2 pffiffiffiffiffiffiffiffiffi k 2 = ω ε2 μ 2 :

and

(13:42)

13.5 Engineering applications The greatest feature of electromagnetic wave transmission [C] network is that it satisfies cascade conditions. As shown in Fig. 13.4, where n networks C1 ; C2 ; . . .,Cn are cascaded, we can get the overall network. n

½C = Π ½Ci  i=1

[C1]

[C2]

(13:43)

[Cn]

Fig. 13.4: n-networks C1 ; C2 ; . . . ; Cn are cascaded.

It is obvious that this feature is very suitable for the research of multiple dielectric interfaces with different constitutive parameters. This part will discuss three-layered lossless medium in engineering applications, as shown in Fig. 13.5. We can use [C] matrix and write ε1, μ1

x ε2, μ2

ε3, μ3

θ3 θ2 θ1

O

z

l

Fig. 13.5: Electromagnetic wave transmission in three-layered media.

204

13 New network theory of electromagnetic waves in multilayered media

"

b1

#

= ½C

a1 We have A12 A23 Z1 Z2 ½C = 4

"

1 Z1

+

1 Z2

1 Z1

−

1 Z2

1 Z1

−

1 Z2

1 Z1

+

1 Z2

"

#"

0

# (13:44)

b3

e − j’

0

0

ej’

For parallelly polarized waves, we have A12 =

cos θ2 cos θ1

A23 =

cos θ3 cos θ2

#"

1 Z2

+

1 Z3

1 Z2

−

1 Z3

1 Z2

−

1 Z3

1 Z2

+

1 Z3

# (13:45)

9 = ;

(13:46)

Then, 8 1 > > < A13 =

> > :

A12 A23 =

Normal incidence and perpendicular polarization oblique incidence cos θ3 cos θ1

(13:47)

Parallel polarization oblique incidence

It is very easy to derive ½C = "

A13 4Z2 Z3

ðZ2 +Z1 ÞðZ3 + Z2 Þe − j’ + ðZ2 − Z1 ÞðZ3 − Z2 Þej’ ðZ2 − Z1 ÞðZ3 +Z2 Þe − j’ + ðZ2 +Z1 ÞðZ3 − Z2 Þej’

#

ðZ2 +Z1 ÞðZ3 − Z2 Þe − j’ + ðZ2 − Z1 ÞðZ3 + Z2 Þej’ ðZ2 − Z1 ÞðZ3 − Z2 Þe − j’ + ðZ2 + Z1 ÞðZ3 + Z2 Þej’ (13:48) For a3 = 0; we have

  b1 ðZ2 Z3 − Z1 Z2 Þ cos ’ + j Z2 2 − Z1 Z3 sin ’   = Γ= a1 ðZ2 Z3 + Z1 Z2 Þ cos ’ + j Z2 2 + Z1 Z3 sin ’

(13:49)

Note that whatever the case is, be it normal incidence, oblique incidence of parallelly polarized waves or perpendicularly polarized waves, the reflection coefficient Γ will remain consistent in form, except that Z corresponds to η; η== and η? ; respectively. If we further introduce the input impedance Zin ,   Z3 + jZ2 tan ’ (13:50) Zin = Z2 Z2 + jZ3 tan ’ Then, eq. (13.48) can be expressed as Γ=

Zin − Z1 Zin + Z1

It is consistent with the correct result [38]. In addition, the refraction coefficient T is

(13:51)

13.6 Energy conservation

T=

b3 2Z2 Z3 =A13   = a1 ðZ2 Z3 + Z1 Z2 Þ cos ’ + j Z2 2 + Z1 Z3 sin ’

205

(13:52)

For normal incidence, oblique incidence of parallelly polarized waves and oblique incidence of perpendicularly polarized waves, the expressions of Z and A13 are somewhat different.

13.6 Energy conservation As the cases discussed in this chapter are restricted to the lossless medium model, the energy conservation law applies among the power of incidence wave, reflection wave and refraction wave during the electromagnetic waves transmission. First, let us examine the case of single-layer lossless interface. From eqs. (13.3), (13.6) and (13.7), we have ) Γ2 + kT2 ≡1 (13:53) k = A212 ZZ1 2

Specifically for normal incidence, A12 = 1; we have k=

η1 η2

(13:54)

For the oblique incidence of perpendicularly polarized waves, as A12 = 1 still holds, we have k=

η1 = cos θ1 η2 = cos θ2

(13:55)

Finally, for the oblique incidence of parallelly polarized waves, A12 = cos θ2 = cos θ1 and Z1 =Z2 = η1 cos θ1 =ðη2 cos θ2 Þ; we have k=

η1 = cos θ1 η2 = cos θ2

(13:56)

Surprisingly, the results of the energy relationship are unique and identical. Now, let’s proceed to look at three-layered lossless medium. From eqs. (13.49) and (13.52) alone, we can get 8 < jΓj2 + kjTj2 ≡ 1   (13:57) : k = A213 Z1 Z 3

The energy conservation relation is independent of the second medium layer ’ = k2 l cos θ2 : It is conceptually natural. k can also be written as

206

13 New network theory of electromagnetic waves in multilayered media

 k = ðA12 A23 Þ

2

Z1 Z2 Z2 Z3

 (13:58)

In this way, there is reason to generalize this to n-layered lossless media, with the general conservation relation shown below. 8 < jΓj2 + kjTj2 ≡ 1   (13:59) : k = A21n Z1 Zn

13.7 Summary The innovation of this chapter, if any, is the combination of electromagnetic wave theory and network theory. It converts the problem of multiple dielectric interfaces to the matrix theory, which is clear and simple. Lossy and nonreciprocal multiple dielectric interfaces will be discussed in other chapters.

Q&A Q: Will network problems be discussed in the beginning from this chapter? A: In fact, the general electromagnetic theory consists of two parts: electromagnetic field theory and electromagnetic field network. Usually, we think the electromagnetic theory is just the electromagnetic field theory. The foundation of the network theory is the matrix theory, which is an important mathematical tool we should master. The [C] network of electromagnetic wave transmission in multiple dielectric interfaces discussed in this chapter is rarely touched upon. It has two main points: 1. Cascade model Electromagnetic waves are transmitted layer by layer in multiple dielectric interfaces. The corresponding networks are cascaded in a similar way. The overall network parameter is the product of parameters of each network by order of sequence, as shown in Fig. 13.4. 2. The system is linear A problem must be linear if matrix is applied to solve it (thus, the matrix theory corresponds to linear algebra). Coincidentally, most electromagnetic problems satisfy this condition. In our teaching practice, we find that many students, although they have studied network theory for a long time, still have little knowledge about its core

Q&A

207

concepts. By network is meant what “includes” the object being studied, by means of which we examine the output after inputting excitation constantly. This is the “network concept,” which ignores the “specific conditions” within the network and only studies the input–output relationship, as shown in Fig. 13.6.

Network

Fig. 13.6: The network concept, using input–output method to study an unknown system.

The essence of the network theory is as follows: for a specific network, we only need to study a finite number of input–output relations to get the total input–output relations. The specific number depends on the complexity of the network concerned. Q: Could you talk about the main characteristics of the wave network [C]? A: The general microwave network theory focuses on field network. Specifically, the study concentrates on the behaviors and characteristics of the field around waveguide coaxial cables, micro-strips and various components. The [C] network in this chapter is purely wave network focusing on the behaviors and characteristics of waves transmitting in space. Therefore, it can also be called spatial network. It has three characteristics as shown in Fig. 13.7: medium, angle (incidence angle, reflection angle and refraction angle) and polarization.

Medium ε, μ, σ

Angle θi, θr and θt

Polarization Perpendicular polarization Parallel polarization

Fig. 13.7: Three characteristics of spatial network.

It is obvious that the [C] network should be discussed according to the polarization type. Moreover, the parameters of the [C] network include angles, which are given in Fig. 13.8.

208

13 New network theory of electromagnetic waves in multilayered media

a2

b1 [C ]

1

2

a1

b2

Oblique incidence of parallelly polarized waves

η1cosθ1 cosθ2 [C ]= cosθ1 2 󴀂󴀂

Electric field lying in the plane of incidence Oblique incidence of perpendicularly polarized waves Electric field perpendicular to the plane of incidence

[C ⊥]=

Normal incidence (namely θ = θ = 1 2

1 1 n1 η1 + η2 [C ]= 2 1 – 1 η1 η2

0)

η1

a1 = E i0 b1 = E0r b2 =E0t θi = θ1, θt = θ2

1 1 + η1cosθ1 η2cosθ2 1 1 – η1cosθ1 η2cosθ2

cosθ1 cosθ2 + η1 η2

1 1 – η1cosθ1 η2cosθ2 1 1 + η1cosθ1 η2cosθ2

cosθ1 cosθ2 – η1 η2

2cosθ1 cosθ1 cosθ2 cosθ1 cosθ2 – + η1 η2 η1 η2 1 1 – η1 η2 1 1 + η1 η2

Fig. 13.8: [C] network of the electromagnetic wave.

Q: Could you explain further the specific applications of the network? A: It’s indeed necessary. The network theory and the field theory have distinctly different characteristics. The network theory has simple principles but flexible applications. Let’s cite two examples. Example 13.1: Transformation between linear polarization network and circular polarization network. The commonly used field is linear polarization, with waves Ex and Ey traveling in z direction. However, in actual applications, some media only responds to circular polarization. Positive circular polarized waves transmit according to θ + = β + , while negative ones according to θ − = β − ,; a famous phenomenon known as the Faraday effect. For these problems, it is convenient to adopt the network theory to illustrate the mutual transformation between linear polarization and circular polarization, as shown in Fig. 13.9. (1) L ! C network, from linear polarization network to circular polarization network. A linear polarized wave can be decomposed into two circular polarized waves with equal amplitude. It is easy to get "

E+ E

If we let

#

" 1 1 = pffiffiffi 2 1

j −j

#"

Ex Ey

#

Q&A

Ex

Ey

L

C

From linear polarization network to circular polarization network

E +, θ +

E –, θ –

C

L

209

Ex

From circular polarization network to linear polarization network

Ey

Fig. 13.9: Transformation between linear polarization and circular polarization. " 1 1 ½TLC  = pffiffiffi 2 1

#

j −j

then "

E+

#

" = ½TLC 

E

#

Ex Ey

(2) C ! L network, from circular polarization network to linear polarization network. Similarly, "

Ex Ey

#

" = ½TCL 

#

E+ E

where 1 ½TCL  = pffiffiffi 2

"

1

1

−j

j

#

and ½TCL  = ½TLC  − 1 (3) Faraday transmission network. For example, the longitudinally magnetized ferrite waveguide system is shown in Fig. 13.10.

Ex 1 Ey 1

H0

Ex2 Ey2

Fig. 13.10: Longitudinally magnetized ferrite waveguide system.

In this case, the positive and negative circular polarized waves are the Eigen waves of the propagation system, with independent transmission angles. We have "

E2+ E2

# " =

e − jθ +

0

0

e − jθ −

#"

E1+ E1

#

210

13 New network theory of electromagnetic waves in multilayered media

Let

" ½Tθ  =

#

e − jθ +

0

0

e − jθ −

we have "

Ex2

#

" = ½T 

Ey2

Ex1

#

Ey1

where " 1 1 ½T  = ½TCL ½Tθ ½TLC  = 2 j

1

#"

e − jθ +

0

0

e − jθ −

−j

#"

1

1

1

−j



# =e

−j

θ+ +θ− 2

"

cos θ

− sin θ

sin θ

cos θ

#

and 1 ðθ − − θ + Þ 2

θ= From a mathematical point of view, "

cos θ

− sin θ

sin θ

cos θ

#

is a coordinate rotation matrix, the right-hand rotation (the condition is θ − > θ + ). Let the input be a linear polarized wave in the x-direction and then we have "

# " # 1 = Ey1 0 Ex1

It does not lose generality, so "

Ex1

# = ½T 

Ey1

" # 1 0

" = e − jððθ + + θ − Þ=2Þ

cos θ

#

sin θ

That is, in forward transmission, the polarized plane is rotated by an angle of θ; known as Faraday rotation, as shown in Fig. 13.11.

Ex1 Ex2 z

z

E2 Ey2 Fig. 13.11: Faraday rotation of the polarized plane.

* Example 13.2: Study the transformation between general electric field E = Ex^i + Ey^j + Ez k^ and paral*== *? lelly polarized wave E or perpendicularly polarized wave E in the coordinate system, as shown in Fig. 13.1. Obviously, we have

3 2 cos θi Ex 6 7 6 0 4 Ey 5 = 4 2

Ez

− sin θi

3 # 0 " == 7 E 15 E? 0

Q&A

211

and its inverse transformation "

E == E?

# " =

1

0

0

1

2 3 # Ex 1 6 7 4 Ey 5 0 Ez

Note that the corresponding transformation is not matrix, and therefore, there is no inverse matrix but inverse transformation. Apparently, the network method is simple and direct.

Q: In addition to the cascade connection, are there other applications of the network theory? A: The network theory has wide applications, among which the most common is the multi-port arbitrary connection of [S] parameter, as shown in Fig. 13.12. a3

3

b3

Net ΙΙΙ

am+3

a2 2

m+3

b2 a1

m+3 Net Ι bam+5 m+5

1

b1

bm+5 an–1

bn–1 an

bm+4 m+4 am+4 bm+6

b4 4

Net II

m+6

a4 b5 5

a5

am+6

bn

Net ΙV

bn

m

am

Fig. 13.12: Complex network with arbitrary cascade.

Net I, Net II etc. of a complex network and their corresponding [S] parameters are given, and Port m+1 is connected to Port m+2, . . ., Port n − 1 connected to Port n. [S] is the matrix when the system has not been connected. Then, we have Si;i + 1 = 0

ði = m + 1; m + 3; . . . ; n − 1Þ

212

13 New network theory of electromagnetic waves in multilayered media

Thus, 2

S11

6  6 6 6 Sm1 6 ½SC = 6 6 Sm + 1;1 6 6 4  Sn1

S12



S1m

S1;m + 1













Sm2



Smm

Sm;m + 1



Sm + 1;2



Sm + 1;m

Sm + 1;m + 1













Sn2



Snm

Sn;m + 1



3

S1;n

 7 7 7 Smn 7 7 7 Sm + 1;n 7 7 7  5 Sn;n

and ½b = ½SC ½a Expressed in the block matrix form, we have "

# "

bI

#"

SI;I

SI;II

SII;I

SII;II

#

aI

= bII

aII

where 8 2b 3 2b 3 > 1 m+1 > > > 6 7 6 7 > > > 6  7 6  7 > > 6 7 6 7 > > 6 7 6 7 > > 6 7 6 7 > > 6 6 7 7 > b ½  = ½  =   b I II > 6 6 7 7 > > 6 7 6 7 > > 6 6 7 7 > > 6 6 7 7   > > 4 4 5 5 > > > > > < bm bn 2a 3 > > 1 > > > > 6 7 > > 6  7 > > 6 7 > > 6 7 > > 6 7 > > 7 > ½aI  = 6 > > 6  7 > > 6 7 > > 6 7 > > 6  7 > > 4 5 > > > : am

2a

m+1

6 6 6 6 6 ½aII  = 6 6 6 6 6 4

   an

;

8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > >
> Sm + 1;1    Sm + 1;m > > > 7 7 > >   6 7 7 > > SII;I = 6    7 6 7 > > > 5 4 7 > 7 > > > 7 >  Snm Sn1 > 7 > > 7 > 2 3 > 7 > > Sm + 1;m + 1    Sm + 1;m 7 > > 5 > > 7 > >  6 > 6 7 >          S = > 6 7 II;II > > 4 5 > > :  Snn Sn;m + 1 2

Connect the imaginary n-port network according to the actual situation, namely connecting m+1 to m+2, m+3 to m+4, n − 1 to n. The corresponding condition is

Q&A

213

8 bm + 1 = am + 2 > > > > > bm + 2 = am + 1 > > > > > > > < bm + 3 = am + 4 bm + 4 = am + 3 > > > >  > > > > > bn − 1 = an > > > : bn = an − 1 After connection, ½aII  and ½bII  can be expressed by ½ε matrix. The connection condition is ½aII  = ½ε½bII  and 2

0

6 6 1 6 6 0 6 6 ½ ε = 6 6 0 6 6 6 6 0 4 0

1

0





0

0





0

0

1



0

1

0











0





0

0





1

0

3

7 0 7 7 0 7 7 7 0 7 7 7 7 7 1 7 5 0

We can regard ½ε as a generalized negative matrix ½ΓL : Obviously, we have       − 1  ½Sm  = SI;I + SI;II ½ε ½I  − SII;II ½ε SII;I which is the matrix form of Sm = S11 +

S12 S21ΓL 1 − S22 ΓL

: Since

½ε − 1 = ½ε we can obtain the ½Sm matrix of the multi-port network with arbitrary connection.       − 1   ½Sm  = SI;I + SI;II ½ε − SII;II SII;I

214

13 New network theory of electromagnetic waves in multilayered media

Recommended scholar

Fig. 13.13: Professor Lin Weigan.

This time I will recommend Professor Lin Weigan, a famous microwave expert. Mr. Lin Weigan has two distinctive characteristics. 1. Solid mathematical foundations As a teacher, one should have wide knowledge and solid foundations of his professional field. Mr. Lin has a first-class “weapon” – his mathematical knowledge. Therefore, problems and difficulties do not stand in his way. 2. Always speaking considerably but quick in action Mr. Lin is a man of few words. It is no exaggeration to say that Lin is very focused on the microwave and electromagnetic field.

14 Matrix transformation in electromagnetic theory In this chapter, we will discuss various matrix transformations involved in the application of the electromagnetic theory, including coordinate matrix transformation, ∇ operator matrix transformation and field matrix transformation. It is pointed out that the corresponding matrix transformations of the unit vector, differential and operator in different coordinates remain unchanged. Similarly, the matrix of any cylindrical waveguide transforming from the longitudinal field component to transverse field component is also unchanged.

14.1 Introduction This is the 14th chapter of Electromagnetic Field Theory Teaching Series. Dr. Tsung-Dao Lee, a Nobel Prize winner of Physics, wrote a very special book, Mathematical Methods in Physics [46]. It has two distinct features. For one thing, this book is the lectures by Dr. Lee when he was teaching doctoral students at Columbia University. The book was not written by Lee himself; instead, it was based on the lecture notes of students who attended his classes. Therefore, what is described in this book is vivid and alive. For another, this book analyzes abstruse mathematical problems profoundly and provides important application examples. The book describes the matrix as follows. “When the relationship between a linear operator and a matrix is established, it gives us an analytical tool to study the linear operator in finite dimensional space.” Therefore, “All matrix theorems can be transformed into operator theorems. In essence, the matrix is attractive to us because it is a tool used to study operators. The application of the matrix in physics is mainly coupled with studies on linear operators.” In a word, the matrix in a linear system is the operator and also transformation. With this concept in mind, this chapter focuses on matrix transformation. If the electromagnetic theory is compared to a splendid drama, then the stage, makeups, dressing and actors are indispensable. The scenes should be built based on the plot, the characters are shaped with makeups and dressing according to the plot and actors must understand the plot to create different roles. This is like the three topics in this chapter: coordinate matrix transformation, operator matrix transformation and field matrix transformation.

https://doi.org/10.1515/9783110527407-014

216

14 Matrix transformation in electromagnetic theory

14.2 Two-dimensional coordinate rotation and transformation to polar coordinate matrix The matrix is the best mathematical expression of linear systems. We start with twodimensional coordinate rotation. Two-dimensional coordinates x′Oy′ are xOy rotated by ’ in a counterclockwise direction, as shown in Fig. 14.1. y y′ x′

φ φ

x

o

Fig. 14.1: Two-dimensional coordinates rotation.

It is easy to get "

# " cos ’ x′ = ′  sin ’ y

sin ’

#" # x

cos ’

y

(14:1)

Similarly, the unit vectors of the two coordinate systems x′Oy′ and xOy are also matrix transformation. #" # " # " ^i′ cos ’ sin ’ ^i = (14:2) ^j′  sin ’ cos ’ ^j Equations. (14.1) and (14.2) are the matrix transformation of the rotation coordinate system. It must be noted that in this case, the rotation angle ’ is a constant parameter. Now let’s look at the transformation between polar coordinates and Cartesian coordinates. We have ( x = ρ cos ’ (14:3) y = ρ sin ’ and its corresponding inverse relation. ( pffiffiffiffiffiffiffiffiffiffiffiffiffi ρ = x2 + y2  ’ = tan1 xy

(14:4)

14.2 Two-dimensional coordinate rotation and transformation

217

y êφ

êρ φ

φ

φ

o

x

Fig. 14.2: Unit vectors ^ eρ and ^ e’ in polar coordinates.

The unit vectors ^eρ and ^e’ in polar coordinates are shown in Fig. 14.2. We have [14] "

^eρ ^e’

# " =

cos ’  sin ’

#" # ^i cos ’ ^j sin ’

(14:5)

Apparently, eq. (14.5) has the same matrix transformation as eqs. (14.1) and (14.2). However, it should be noted that ’ is a variable in the polar coordinate system.  2  2 Therefore, ^eρ and ^e’ are variable unit vectors, and ^eρ  ^eρ = ^eρ  = 1, ^e’  ^e’ = ^e’  = 1 and ^eρ  ^e’ = 0: That is to say, polar coordinates are orthogonal coordinates. 8 ∂^e ^eρ < ∂ρρ = 0; ∂∂’ =  sin ’ ^i + cos ’ ^j = ^e’ (14:6) : ∂^e’ = 0; ∂^e’ =  cos ’ ^i  sin ’ ^j =  ^e ρ ∂ρ ∂’ Furthermore, 8 2 ∂ ^e > < 2ρ =  ^eρ ∂’

(14:7)

> : ∂2 ^e’ =  ^e’ ∂’2

By using eq. (14.3), we can get the differential matrix transformations of polar coordinates and Cartesian coordinates # #" " # " dρ cos ’  sin ’ dx (14:8) = ρd’ sin ’ cos ’ dy and their inverse relations. "

dρ ρd’

# " =

cos ’

sin ’

 sin ’

cos ’

#"

dx dy

# (14:9)

eqs. (14.1), (14.2), (14.5) and (14.9) derived in this chapter all show the invariance of the matrix transformation.

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14 Matrix transformation in electromagnetic theory

14.3 Three-dimensional coordinate matrix transformation We will discuss three cases here. 14.3.1 Coordinate plane rotation and its matrix transformation to cylindrical coordinate The plane rotation of Cartesian coordinates (xOy) is shown in Fig. 14.3. z

φ x

x′

o

y′

φ

y

Fig. 14.3: The plane rotation of Cartesian coordinates (xOy).

This is similar to the two-dimensional coordinates. We have 32 3 2 3 2 x cos ’ sin ’ 0 x′ 76 7 6 ′7 6 4 y 5 = 4  sin ’ cos ’ 0 54 y 5 z′

0

0

and the unit vector matrix transformation 2 3 2 ^i′ cos ’ sin ’ 6 ^′ 7 6 =  4 j 5 4 sin ’ cos ’ 0 0 k^′

1

(14:10)

z

32 3 ^i 76 ^ 7 0 54 j 5 0 1

(14:11)

^k

Similarly, the unit vector transformation from Cartesian coordinate system to cylindrical coordinate system is 32 3 2 3 2 ^i ^eρ cos ’ sin ’ 0 76 ^ 7 6^ 7 6 (14:12) 4 e’ 5 = 4  sin ’ cos ’ 0 54 j 5 ^k ^ez 0 0 1 and the differential matrix transformation is 2 3 2 cos ’ sin ’ dρ 6 7 6 4 ρd’ 5 = 4  sin ’ cos ’ dz

0

0

0

32

dx

3

76 7 0 54 dy 5 1

dz

Once again we see the invariance of the transformation matrix.

(14:13)

14.3 Three-dimensional coordinate matrix transformation

219

14.3.2 Spherical Coordinate Rotation and Its Matrix Transformation to Spherical Coordinate The spherical rotation of Cartesian coordinates is shown in Fig. 14.4. z

x′ z′ θ

o

y′

φ

y x

Fig. 14.4: The spherical rotation of Cartesian coordinates (for visual convenience, the coordinate with “′” are not at original point and should be moved to O).

It is easy to get 3 2 sin θ cos ’ x′ 6 ′7 6 4 y 5 = 4 cos θ cos ’ 2

z′

 sin ’

sin θ sin ’ cos θ sin ’ cos ’

32 3 x 76 7  sin θ 54 y 5 cos θ 0

(14:14)

z

and the unit transformation matrix 2

^i′

3 2

sin θ cos ’

6 ^′ 7 6 4 j 5 = 4 cos θ cos ’ ^k′  sin ’

sin θ sin ’ cos θ sin ’ cos ’

32 3 ^i 76 ^ 7  sin θ 54 j 5 cos θ 0

(14:15)

^k

In the rotation, θ and ’ are constant parameters based on Cartesian coordinates. Now, let’s look at the transformation from Cartesian coordinates to spherical coordinates. We have 8 > < x = r sin θ cos ’ (14:16) y = r sin θ sin ’ > : z = r cos θ and the inverse relation 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 y2 + z 2 >

 : ’ = tan1 xy

(14:17)

pffiffiffiffiffiffiffiffiffiffiffiffiffi where ρ = x2 + y2 : The unit vector of spherical coordinates is shown in Fig. 14.5. It is easy to get

220

14 Matrix transformation in electromagnetic theory

z

eˆr eˆφ θ eˆθ

o φ

y x

Fig. 14.5: The unit vector of spherical coordinates.

2

^er

3 2

sin θ cos ’

6^ 7 6 4 eθ 5 = 4 cos θ cos ’ ^e’  sin ’

sin θ sin ’ cos θ sin ’ cos ’

32 3 ^i 76 ^ 7  sin θ 54 j 5 cos θ 0

From eq. (14.16), we have the differential matrix transformation 32 3 2 3 2 dx sin θ cos ’ sin θ sin ’ cos θ dr 76 7 6 7 6 4 rdθ 5 = 4 cos θ cos ’ cos θ sin ’  sin θ 54 dy 5 r sin θd’

 sin ’

cos ’

(14:18)

^k

0

(14:19)

dz

It must be noted that in the case of spherical coordinates, 8 ∂^e ∂^er r ^ ∂^er ^ > ∂r = 0; ∂θ = eθ ; ∂’ = sin θ e’ > > < ∂^eθ ∂^eθ ∂^eθ ^ ^ ∂r = 0; ∂θ =  er ; ∂’ = cos θ e’ > > > ∂^e’ ∂^e’ :∂^e’ ^ ^ ∂r = 0; ∂θ = 0; ∂’ = ez × e’

(14:20)

Furthermore, there is 8 2 ∂ ^er > > > ∂θ2 =  ^er >

∂θ2 > > > : ∂^er  ∂^er + ∂^eθ  ∂^eθ = 1 ∂’ ∂’ ∂’ ∂’

(14:21)

Also an orthogonal coordinate system, spherical coordinates have invariance in matrix transformation.

14.3.3 Matrix Transformation of Orthogonal Curvilinear Coordinates Here, we extend the problem to the general orthogonal curvilinear coordinate system ðq1 ; q2 ; q3 Þ; as shown in Fig. 14.6.

221

14.3 Three-dimensional coordinate matrix transformation

q3 eˆ3 q1 O eˆ1

q2 eˆ2 Fig. 14.6: The general orthogonal curvilinear coordinate system ðq1 ; q2 ; q3 Þ:

8 > < q1 = q1 ðx; y; zÞ q2 = q2 ðx; y; zÞ > : q3 = q3 ðx; y; zÞ

(14:22)

We can write the matrix transformation of the unit vector. 2

^e1

3

2

1 ∂x H1 ∂q1

6 6 ^ 7 6 1 ∂x 4 e2 5 = 6 H2 ∂q2 4 ^e3 1 ∂x

H3 ∂q3

3

2 3 ^ 7 i 1 ∂z 76 ^ 7 H2 ∂q2 74 j 5 5 ^k 1 ∂z

1 ∂y H1 ∂q1

1 ∂z H1 ∂q1

1 ∂y H2 ∂q2 1 ∂y H3 ∂q3

(14:23)

H3 ∂q3

where H1, H2 and H3 are called Lamè coefficient. Specifically, 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2  2  2ffi > ∂y > ∂x ∂z > = + ∂q + ∂q H > 1 ∂q1 > 1 1 > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r <      2

2

2

∂y ∂x ∂z + ∂q + ∂q H2 = ∂q2 > 2 2 > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > 2  2  2 > > ∂y > ∂x ∂z : H3 = + ∂q + ∂q ∂q 3

3

(14:24)

3

We introduce the transformation matrix ½T which is 2

1 ∂x H1 ∂q1

6 6 ∂x ½T  = 6 H12 ∂q 2 4

1 ∂x H3 ∂q3

1 ∂y H1 ∂q1

1 ∂z H1 ∂q1

1 ∂y H2 ∂q2

1 ∂z H2 ∂q2

1 ∂y H3 ∂q3

1 ∂z H3 ∂q3

3 7 7 7 5

(14:25)

Obviously, ½T1 = ½TT where ½ T is transposed matrix and

(14:26)

222

14 Matrix transformation in electromagnetic theory

2

1 H1 ∂q ∂x

6 6 1 ½T1 = 6 H1 ∂q ∂y 4 1 H1 ∂q ∂z

∂q2 ∂x ∂q H2 ∂y2

H2

H2

∂q2 ∂z

∂q3 ∂x ∂q H3 ∂y3

H3

H3

∂q3 ∂z

3 7 7 7 5

(14:27)

As an example, we study two-dimensional polar coordinates again, in which H1 = 1 and H2 = ρ: Then, we have 2 3 " # ∂y ∂x cos ’ sin ’ ∂ρ ∂ρ 5= (14:28) ½T = 4 1 ∂x 1 ∂y  sin ’ cos ’ ρ ∂’

ρ ∂’

Similarly, the differential matrix transformation is 2 1 ∂x 3 1 ∂y 1 ∂z 2 3 2 3 H1 ∂q1 H1 ∂q1 H1 ∂q1 H1 dq1 6 7 dx 7 6 7 6 ∂x 1 ∂y 1 ∂z 76 4 H2 dq2 5 = 6 H12 ∂q H2 ∂q2 H2 ∂q2 74 dy 5 2 4 5 1 ∂x 1 ∂y 1 ∂z dz H3 dq3 H3 ∂q3

H3 ∂q3

(14:29)

H3 ∂q3

We can see clearly that there is indeed constant transformation matrix ½T in the normal orthogonal coordinate system.

14.3.4 Matrix transformation of operator ∇ Operator ∇ is so important in the electromagnetic theory that for the Maxwell’s equations, they are written in ∇ × and ∇: Its basic formula is ∂ ^∂ ^ ∂ ∇ = ^i +j +k ∂x ∂y ∂z

(14:30)

It is easy to obtain    1 ∂ + ∂q2 ∂ + ∂q3 ∂ ^ ∂q1 ∂ ∇ = ^i ∂q ∂x ∂q1 ∂x ∂q2 ∂x ∂q3 + j ∂y ∂q1 +   ∂q3 ∂ ∂q2 ∂ 1 ∂ + ^k ∂q ∂z ∂q + ∂z ∂q + ∂z ∂q 1

2

∂q2 ∂ ∂y ∂q2

+

∂q3 ∂ ∂y ∂q3

(14:31)

3

Then, we get the derivative matrix transformation of coordinate 32 3 2 ∂ 3 2 ∂q1 ∂q ∂q 1 ∂ H1 ∂x H2 ∂x2 H3 ∂x3 H1 ∂q1 ∂x 76 7 6∂7 6 7 ∂q 76 ∂q 1 6 7=6 H1 ∂q H2 ∂y2 H3 ∂y3 76 H12 ∂q∂2 7 4 ∂y 5 6 ∂y 4 54 5 ∂ 1 ∂ ∂q3 ∂q2 1 H1 ∂q H H ∂z H ∂q 2 ∂z 3 ∂z 3 3 ∂z According to eq. (14.27), we know that



(14:32)

223

14.4 Matrix transformation from longitudinal field component

2

3

4

7 7 7 = ½T 5

1 ∂ H ∂q 6 1 1 6 1 ∂ 6 H2 ∂q2 1 ∂ H3 ∂q3

3

2

∂ ∂x 6∂7 6 7 4 ∂y 5

(14:33)

∂ ∂z

From eqs. (14.23) and (14.27), we have 8  > 1^ ^e1 = H1 ∂q > > ∂x i + > > <  2^ ^e2 = H2 ∂q ∂x i + > >  > > > : ^e3 = H3 ∂q3 ^i + ∂x

 k^  ∂q2 ^ ∂q2 ^ k j + ∂y ∂z  ∂q3 ^ ∂q3 ^ ∂y j + ∂z k

∂q1 ^ ∂q1 ∂y j + ∂z

(14:34)

Finally, the operator ∇ in the normal orthogonal curvilinear coordinates is given ∇ = ^e1

1 ∂ 1 ∂ 1 ∂ + ^e2 + ^e3 H1 ∂q1 H2 ∂q2 H3 ∂q3

(14:35)

or 2 h i ∇ = ^i; ^j; ^k

3

∂ ∂x 6∂7 6 7 = ½^e1 ; ^e2 ; ^e3  4 ∂y 5 ∂ ∂z

2

3

4

7 7 7 5

1 ∂ H ∂q 6 1 1 6 1 ∂ 6 H2 ∂q2 1 ∂ H3 ∂q3

(14:36)

Identically, the matrix transformation of operator ∇ remains constant.

14.4 Matrix transformation from longitudinal field component to transverse field component The electromagnetic waveguide with constant cross-section is one of the typical applications of Maxwell’s equations in cylindrical coordinates. And, the field in waveguide is the Eigen-mode in physics. Literature [29] has discussed the typical matrix waveguide and cylindrical waveguide. Figure 14.7 shows the rectangular waveguide. The matrix transformation from longitudinal field component to transverse field component is derived: y

b z

o

a

x

Fig. 14.7: Rectangular waveguide.

224

14 Matrix transformation in electromagnetic theory

2

Ex

2

3

γ

0

jωμ

0

6 6 7 6 Ey 7 1 6 0 γ jωμ 6 7= 6 6 H 7 k2 6 0 jωε γ 4 x5 c4 jωε 0 0 Hx

0 0 γ

32 ∂Ez 3 ∂x 7 76 ∂Ez 7 6 76 ∂x 7 76 76 ∂Hz 7 54 ∂x 7 5

(14:37)

∂Hz ∂x

where kc = 2π=λc is the cutoff wavelength of the waveguide and the z direction variation of the electromagnetic field in the waveguide is eγz : The cylindrical waveguide is shown in Fig. 14.8. Literature [29] also derives the matrix transformation using longitudinal field component (z) to express transverse field component ðρ; ’Þ:

y

z

o

x

Fig. 14.8: Cylindrical waveguide.

2 3 3 ∂Ez ∂ρ Eρ γ 0 0 jωμ 6 7 76 1 ∂Ez 7 6 7 6 6 E’ 7 1 6 0 γ jωμ 0 76 ρ ∂’ 7 76 6 7 6 7 6 H 7 = k2 6 0 6 z 7 jωε γ 0 7 56 ∂H 4 ρ5 7 c4 ∂ρ 4 5 H’ jωε 0 0 γ 1 ∂Hz 2

3

2

(14:38)

ρ ∂’

This chapter will derive the curved waveguide in normal orthogonal curvilinear * coordinate (q1 ; q2 ; q3 ). Under this condition, curl of vector field A is given [14] as   ^ez   H1 ^e1 H2 ^e2   * 1  ∂ ∂ γ  ∇×A= (14:39)  ∂q1 ∂q2  H1 H2  H A  HA HA 1 e1

2 e2

3 e3

In eq. (14.39), according to the conditions of cylindrical waveguide, H3 = 1,^e3 = ^ez and ∂=∂q3 = ∂=∂z ! γ

(14:40)

From Maxwell’s equations, (

*

*

∇ × H = jωεE *

*

∇ × E =  jωμH and the curl formula (14.33), it is easy to get two sets of equation

(14:41)

Q&A

8 < jωεEe1  γHe2 =

1 ∂Hz H2 ∂q2

: γEe1 + jωμHe2 = and

225

(14:42)

1 ∂Ez H1 ∂q1

8 < jωεEe2 + γHe1 =  H1

∂Hz 1 ∂q1

(14:43)

: γEe2 + jωμHe1 =  1 H

∂Hz 2 ∂q2

We can also get the matrix transformation from the longitudinal field component to transverse field component 2 3 3 1 ∂Ez 2 3 2 γ 0 0 jωμ 6 H1 ∂q1 7 Ee1 76 1 ∂Ez 7 6 7 6 6 Ee2 7 1 6 0 γ jωμ 0 76 H2 ∂q2 7 76 7= 6 6 7 (14:44) 6 H 7 k2 6 0 6 z7 jωε γ 0 7 56 H1 ∂H 4 e1 5 7 c4 ∂q 1 1 4 5 jωε 0 0 γ He2 1 ∂Hz H2 ∂q2

Till now, we can see clearly that the matrix transformation keeps constant for all orthogonal curvilinear coordinate systems.

14.5 Summary One of the focuses in scientific research is transformation and its invariance. Invariance not only simplifies issues, but it reflects the features and essence in transformation, which deserves our further study.

Q&A Q: This chapter mainly discusses the matrix problem. However, you recommend the famous physicist Tsung-Dao Lee’s work (he is not a mathematician). Do you have deeper meaning on this point? A: Your question is crucial and it meets the key point of this chapter. Roughly we can divide mathematics into two aspects: abstract mathematics and engineering (or applied) mathematics. The former has its own system and selfdeveloped while the latter depends on the development in engineering, or we can say the practical requirements and development is the source of life of engineering mathematics. Abstract mathematics, of course, requires application and needs to be verified through practice. About Theory of the Matrix, we cite what Tsung-Dao Lee said, “The application of matrix in physics is coupled with research on various linear operators.”

226

14 Matrix transformation in electromagnetic theory

He combines theories with practice, which deepens our understanding of matrix transformation. Another similar case is Chen-Ning Yang’s studies in Complex Function Theory. He concluded three themes in the development of physics in 20th century: quantization, symmetry and phase factor. The third theme, phase factor, is a complex domain, a concept which is surprising and thoughtful in philosophy. It is Chen-Ning Yang and other physicists that contribute to the studies of this aspect. Their contributions mainly include the following aspects. They discover that all interactions are some forms of gauge field; the gauge field is closely related to the mathematical concept of fiber bundle, in which every fiber is a complex phase or a phase in general sense. Finally, they concluded a basic principle of modern physics: All basic forces are phase fields. The previous two examples show that physics and physicists play an important role in the development of mathematics. Therefore, in this chapter, we refer to Matrix Theory as living, developing and it can find great applications. Popularly speaking, it can be summed up in one sentence: Matrix Theory by Tsung-Dao Lee is different from normal mathematical books. Q: In which sense does Theory of the Matrix by Tsung-Dao Lee differ from general mathematics books? A: Now I would like to have discussions on his concept of matrix and its applications from the following aspects: (1) Tsung-Dao Lee explicitly concludes the reason to utilize matrix and matrix theories. The counterparts of vector and matrix in physical studies are illustrated clearly as shown in Fig. 14.9.

Vector

Directed physical quantity

Matrix

Transformation

Fig. 14.9: Counterparts of vector and matrix in physics.

Evidently, matrix and vector are always in pairs. It is convenient to extend the case to n dimensions after introducing matrix, thus corresponding to n-dimensional physical quantity. The utilization of matrix makes it convenient for calculation. Namely, modern matrix theory and computing matrix theory can be applied to physics.

Q&A

227

(2) It is a high conception to adopt invariance principle to define vector. Tsung-Dao Lee is such a tru master that he touches upon problems from surprising perspectives. The definition of vector is one of the typical examples. He introduces the principle of invariance. The invariant quantity when the coordinate system rotates is called scalar. Vector Theorem Suppose a principle makes it possible for every coordinate system * to provide three numbers (A1, A2, A3). If by this principle, every vector B = ðB1 ; B2 ; B3 Þ 3 * P Ai Bi a scalar (invariance property), then A = ðA1 ; A2 ; A3 Þ is a vector. makes i=1

Proof: When the coordinate system rotates, vector h ! 0 changes according to the principle. 3 X B′i = uij Bj ði = 1; 2; 3Þ j=1

  Suppose under the same condition, (A1, A2, A3) changes into A′1 ; A′2 ; A′3 : We need to prove the following equation holds. 3 X uij Aj ði = 1; 2; 3Þ A′i = j=1

We have assumed that 3 X

A ′ i B ′i =

3 X

i=1

So, we can get 3 X

A′i B′i =

i=1

3 X

A′i

Ai Bi

i=1

3 X

i=1

uij Bj =

j=1

3 X 3 X j=1

 uij A′i Bi

i=1

*

Vector B is arbitrary and therefore 3 X Aj = ukj A′k

ðj = 1; 2; 3Þ

k=1

By multiplying the above equation by uij and finding the sum of j, we have 3 X

uij Aj =

j=1

3 X 3 X j=1 k=1

uij ukj A′k =

3 X

A′k

k=1

3 X

uij ukj =

j=1

3 X

δik A′k = A′i

ði = 1; 2; 3Þ

k=1

We can get A′i =

3 X

uij Aj

j=1

This is exactly what we want to prove. Then, we can understand the importance of matrix in rotating coordinate systems (or coordinate systems in general sense).

228

14 Matrix transformation in electromagnetic theory

(3) Tsung-Dao Lee made a thorough discussion on the Eigen-mode theory of matrix. Notably, electromagnetic waveguide, coaxial line and strip line are all decomposed from Eigen-mode, as shown in Fig. 14.10. A12

...

A1n

A21 ...

A22 − λ ...

... ...

A2n ...

An1

A2n

...

Ann − λ

det

Eigenvalue equation

Eigenvalue and eigenvector

A11 − λ

=0

Eigenvalue number system {Ai } and eigenvector (namely, eigenmode) {xi } (i = 1,2, ... n) Any vector x can be obtained through addition of eigenvector

Completeness

n

{ xi } , namely x = ∑ ki xi i =1

Orthonormality

x i⋅ x j = δij =

1

i= j

0

i≠ j

Fig. 14.10: Eigen-mode theory of matrix.

(4) Tsung-Dao Lee considered matrix [A] as the finite dimensional (n) linear operator. This practice is important theoretically, as shown in Fig. 14.11. All above are Tsung-Dao Lee’s main ideas about matrix and its application in physics. Linear operator L Lx =y L ( x1 + x 2 ) = L x1 + L x 2

Matrix [A]

[ A ][ x ] = [ y ] [ A ] ( [ x1 ] + [ x2 ] ) = [ A ][ x1 ] + [ A ][ x2 ]

L (λ x) = L ( x)

[ A ][ λ x ] = λ [ A ] [ x ]

Identity operator Ix = x

Identity (unit) matrix [ I ][ x ] = [ x ]

Given the base, there is a matrix [A] that corresponds to each linear operator L. Inverse operator –1 L y=x Any operator theorems

Given the base, each matrix [A] has a linear operator L. Inverse matrix [ x ] = [ A ] −1 [ y ] Corresponding matrix theorems

Fig. 14.11: Linear operator L and matrix [A].

Q: Do you have anything to add on vector theory? A: The vector theory is well developed and it has been applied in physics and engineering extensively. But there are problems left systematically. Let’s start from comparison. Figure 14.12 shows the dot product comparison of vector, n-dimensional (n > 3) vector and matrix.

Recommended scholar

Vector

x ⋅y = x y cos θ is a quantity.

n-dimensional vector (n>3)

Cross product has been defined.

Matrix (square matrix)

229

Cross product has not been defined.

Fig. 14.12: Dot product and generalized dot product.

For n-dimensional vector, *

a = a1^i1 + a2^i2 +    + an^in

*

b = b1^i1 + b2^i2 +    + bn^in

Their generalized dot product is *

*

ab=

n X

ai bi

i=1

We further study cross-product in Fig. 14.13.

Vector

iˆ

ˆj

kˆ

x × y = x1 y1

x2 y2

x3 is a vector. y3

n-dimensional vector (n>3) No dot product definition Matrix (square matrix)

No dot product definition

Fig. 14.13: Cross-product and generalized cross-product.

The result of the research above is not satisfying, though further exploration is required.

Recommended scholar This part will introduce the Nobel Prize winner Tsung-Dao Lee (Fig. 14.14). He is called by the Chinese former premier Enlai Chou “a master in studies.” Our focus here is not on his studies but on his teaching ideas.

Fig. 14.14: Tsung-Dao Lee.

230

14 Matrix transformation in electromagnetic theory

As a saying goes, “The spectators see the chess game better than the players.” Lee spent a long period of time abroad. He, however, has insight on domestic education problems, which can be concluded as follows. (1) Think while studying Mr. Lee believes that the elementary education in China tries to lay a foundation, but “students think less while studying much.” On the contrary, the American education focuses on innovation. “Students think much while studying less.” We should reflect on our talent cultivation mode if we are to solve the problem that “China has no Nobel Prize winner.” China or even the whole Asia has some shortages in education. Shortage of creative education is one of the main ones among these. (2) “Learn creatively,” instead of studying for tests Grade is attached great importance in China. High grade is a symbol of excellent students while the creativity education has been ignored. Right now, the teaching method in China focuses on tests. Excessive quizzes and exams eliminate students’ desire for knowledge. It is even more dangerous for them to lose the interest in exploring science. Einstein said that even if it is a healthy wild beast, with whips forcing it to eat when it is not hungry may cause the loss of its nature of edacity. The forced teaching of knowledge can only deprive the desire for exploration and creativity. (3) Teachers and parents should enlighten children, instead of arranging everything for them The teacher’s role is to arouse students’ curiosity for exploration through teaching and interacting with them. The teacher’s responsibility is to help students build up right values through their own personality. (4) Tsung-Dao Lee puts forward “three conditions for talents” Personality: One should hold solid life outlook and value; live with self-esteem, independence and continuous self-renewal, bearing deep love to his nation; improve personality endlessly. Hard-working: No matter where, when and how well the material civilization develops, he should always bear hardships and stand hard work. Adaptability: He should be able to adapt to the changing world and overcome all failures and barriers. These are the thoughts and views on talents of Tsung-Dao Lee.

15 Minimum directivity challenge of electromagnetic radiation This chapter is titled Minimum Directivity Challenge of Electromagnetic Radiation. Why? The reason is that Maxwell’s equations do not have isotropic solutions. So, the electromagnetic radiation can never be without directivity. However, we hope to approximate this goal in many communication cases and practices. Therefore, the contradictory subject which is the core of this chapter must be accepted and researched.

15.1 Introduction This is the 15th chapter of Electromagnetic Field Theory Teaching Series. In the new century, there is no doubt that the electromagnetic wave is one of the vital information transmission media. It is linked with the life of average people every second. From Internet, mobile phones, televisions to vehicles with GPS, from the national information systems of each country to the Bluetooth technology in households, electromagnetic wave propagation and information carrying technologies are involved. To construct various information systems, it is highly wished to develop the antenna without directivity in practice, making it possible for electromagnetic waves to reach all users without dead zones of signal. However, for other hand, directivity is one of the most important properties of antenna radiation. A lot of papers and literature have pointed out that all antennas must be directive. Antennas which can radiate equal energy to all directions do not exist [38]. It is this contradiction that becomes the topic of this chapter. On the one hand, we should find out why nondirectional antennas do not exist. On the other hand, we should take the challenge to find an approach to minimum directivity.

15.2 Maxwell’s equations do not have isotropic solutions First, it is clearly stated in this chapter that the macroscopic electromagnetic theory systems, constructed by Maxwell’s equations, have no isotropic solution. By the use of spherical coordinates, the electric field at any point in space can be expressed as *

E ¼ Er ^er þ Eθ ^eθ þ E’ ^e’

(15:1)

If we take the method of proof by contradiction, suppose the isotropic solution exists 9 Er ¼ f1 ðrÞ > = Eθ ¼ f2 ðrÞ (15:2) > ; E’ ¼ f3 ðrÞ

https://doi.org/10.1515/9783110527407-015

232

15 Minimum directivity challenge of electromagnetic radiation

In frequency domain, the Helmholtz’s wave equation is *

*

∇2 E þ k2 E ¼ 0 pffiffiffiffiffi where k ¼ ω με is the wave number. Note that in the spherical coordinate system, the Laplace operator is   * 2Er 2ctgθ 2 ∂Eθ 2 ∂E’ ^er  2 ∇2 E ¼ ∇2 Er  2  2 Eθ  2 r r r ∂θ r sin θ ∂’   2 ∂Er Eθ 2 cos θ ∂E’ ^eθ  þ ∇2 Eθ þ 2  r ∂θ r2 sin2 θ r2 sin2 θ ∂’   2 ∂Er 1 2 cos θ ∂Eθ ^e’ E þ ∇2 E’ þ 2 þ  ’ r sin θ ∂’ r2 sin2 θ r2 sin2 θ ∂’

(15:3)

(15:4)

If we consider the isotropic property, we have 9 ∂E’ ∂Er ∂Eθ > ¼ 0; ¼ 0; ¼ 0> = ∂θ ∂θ ∂θ ∂E’ ∂Er ∂Eθ > ; ¼ 0; ¼ 0; ¼ 0> ∂’ ∂’ ∂’

(15:5)

Equation (15.4) can be simplified as       * 2Er 2ctgθ Eθ 1 2 ^e’ ^ ∇2 E ¼ ∇2 Er  2  2 Eθ ^er þ ∇2 Eθ  E þ ∇ E  e ’ ’ θ r r r2 sin2 θ r2 sin2 θ (15:6) Here, we only take the component ^eθ as an example. ∇2 Eθ þ k2 Eθ ¼ gðrÞ

(15:7)

But its term 

Eθ r2 sin2 θ

(15:8)

consists of a varying function with θ as variable, which is contradictory with the hypothesis. Therefore, we disprove Maxwell equations have isotropic solution. In addition, physically, electromagnetic radiation is closely related to two curl equations ∇ × From the concept of curl, we know that a tornado, no matter how big it is, is not isotropic but it has an eye of typhoon, shown in Fig. 15.1. Mathematically, there must be a zero vector point in tangential vector field, which is continuously differentiable on the surface of a sphere, pointed out Su Buqing in Differential Geometry [47]. “Curl” must exist on human’s head or cows’ skin, as shown in Fig. 15.2. In fact, literature [48] had made extensive studies on the anisotropism of electromagnetic radiation early in 1984. A conclusion was drawn that if all the components

15.3 Minimum directivity antenna

233

Fig. 15.1: The eye of a tornado.

Fig. 15.2: “Curl” of hair must exist on our head.

of the vector are independent of spherical coordinate variable θ and ’; then it is impossible to get the solutions to Maxwell’s equations. The anisotropism of electromagnetic radiation is determined not only by the field structure but by the topological property of the tangential vector on the sphere.

15.3 Minimum directivity antenna Here, we will touch upon two aspects, namely the minimum directivity antenna and the minimum directivity array. What is a minimum directivity antenna? There is no specific conclusion as to this * problem. Pragmatically, we first examine the familiar antenna the electric dipole I l ; as shown in Fig. 15.3. The far-zone radiation fields of electric dipole are

234

15 Minimum directivity challenge of electromagnetic radiation

eˆr

z

eˆφ eˆθ

θ

y

φ

x

*

Fig. 15.3: Radiation of electric dipole I l .

pffiffiffiffiffiffiffiffiffiffiffi 9 ðμ=εÞIl sin θ jkr > > = Eθ ¼ j e 2λr > Il sin θ jkr > ; H’ ¼ j e 2λr

(15:9)

Its radiation patterns are shown in Fig. 15.4. z

o

o

y

y

x *

Fig. 15.4: Radiation patterns of electric dipole I l :

φ = 0, φ = 30, φ = 60, φ = 90 The general equation for determining the directivity is D¼

4π R2π

d’

0

Rπ

(15:10)

F2 ðθ; ’Þ sin θdθ

0

where Fðθ; ’Þ is the amplitude of antenna radiation fields. jEj ¼ jEjm F ðθ; ’Þ

(15:11)

jEjm is the field density amplitude in the direction with most intensive radiation. For electric dipole, Fðθ; ’Þ ¼ sin θ and we have D0 ¼

4π R2π 0

Rπ

¼ 1:5

(15:12)

3

d’ sin θ dθ 0

In the principle of duality, the magnetic dipole (circular antenna) is identical and we will not touch upon it.

235

15.4 Minimum directivity array

As mentioned above, we cannot prove that electronic (or magnetic) dipole is the minimum directivity antenna. From Fig. 15.4, it can be easily seen that there is a “curl” at the origin of yOz plane, namely the anisotropism. A cross-dipole is presumed in this chapter, composed of two dipoles in the z and y directions, with an attempt to reduce the directivity. We take into consideration Appendix (A15.14) and place polarization in the θ direction. Then, we have

(15:13) F ðθ; ’Þ ¼ F0 ðsin θ  cos θ sin ’Þ^eθ þ ð cos ’Þ^e’ where F0 is normalized coefficient. Namely, the maximum amplitude is 1. The radiation patterns are drawn in Fig.15.5. And, we calculate the directivity D0 ¼

R2π 0

Rπ

4πjF ðθ; ’Þj2max 2

d’ ðsin θ  cos θ sin ’Þ sin θdθ þ 0

R2π

cos2 ’d’

Rπ

0

¼ 1:5

(15:14)

sin θdθ

0

The analysis above ignores the influence of cross-coupling between antennas.

15.4 Minimum directivity array Minimum directivity array is another problem which required extensive studies. Directivity function is F ðθ; ’Þ ¼ FA ðθ; ’ÞFD ðθ; ’Þ

(15:15)

where FA ðθ; ’Þ stands for the array factor while FD ðθ; ’Þ element factor. Equation (15.15) can be obtained after certain conditions satisfied. In array studies, we put aside element factor for a while. This is another topic without specific conclusions. To have a clear and direct understanding, we discuss the two-dimensional eight-element circular array as shown in Fig. 15.6 and draw conclusion without considering the directivity function of element factor. We derive the directivity function of the circular array in Appendix B. pffiffiffi

pffiffiffi

o n 2 2 kR cos ’ cos kR sin ’ F ð’Þ ¼ F0 cos½kR cos ’ þ cos½kR sin ’ þ 2 cos 2 2 (15:16) kR = 2πkR = 1.5π D0 jkR¼2π ¼

2π R2π 0

½Fð’Þ2 d’

¼ 2:27304

(15:17)

210

180

150

240

270

300

240

120

210

330 0.5

150

210

120

240 270

270

300 1.0

0.5

0.0

0

330

0.5

30

210

180

240 270

90

1.0

1.0

0.5

180

150

120

90

φ = 30

θ = 45

120

90 0.0

60

1.0

90 60

150

30

θ = 30

180

0

φ=0

Fig. 15.5: Radiation patterns are of cross-dipole.

1.0

0.5

0.0

0.5

1.0

1.0

0.5

0.0

0.5

1.0

300

60

300

60

0

330

30

330

0

30 150

150

210

1.0

0.5 210

0.0 180

0.5

1.0

1.0

0.5

0.0 180

0.5

1.0

240

120

240

120

270

90

θ = 60

270

90

φ = 60

300

60

300

60

330

0

30

330

0

30

1.0

0.5

0.0

0.5

1.0

1.0

0.5

0.0

0.5

1.0

210

180

150

210

180

150

240

120

240

120

270

90

θ = 90

270

90

φ = 90

300

60

300

60

330

0

30

330

0

30

236 15 Minimum directivity challenge of electromagnetic radiation

15.4 Minimum directivity array

The reference plane 2

y 3 4

A 1

φ

5 o 6

237

x

8 Fig. 15.6: The reference plane of two-dimensional eightelement circular array is perpendicular to point A.

7

2π

D0 jkR¼1:5π ¼

R2π

¼ 1:19076

(15:18)

2

½Fð’Þ d’

0

D0 jkR¼π ¼

2π R2π

¼ 1:00916

(15:19)

¼ 1:00006

(15:20)

2

½Fð’Þ d’

0

2π

D0 jkR¼π=2 ¼

R2π

2

½Fð’Þ d’

0

We draw out the radiation pattern of circular array when kR = 2π, 1.5π, π and π/2, as shown in Fig. 15.7, and get the corresponding directivity coefficient (Fig.15.8). If we consider the directivity function of a single dipole antenna, then the directivity function of two-dimensional eight-element circular array is  pffiffiffi

pffiffiffi

 2 2 kR cos ’ cos kR sin ’ F ðθ; ’Þ=F0 sin θ cos½kR cos ’ + cos½kR sin ’þ2 cos 2 2 (15:21) D ¼ 2π π R R

4π

¼

R2π

F 2 ðθ; ’Þ sin θdθd’

0 0

4π 3π ¼ 2π Rπ R F 2 ð’Þd’ sin3 θdθ F 2 ð’Þd’

0

0

(15:22)

0

When kR = 2π, 1.5π, π and π/2, the directivity coefficients of eight-element circular array are D0 jkR¼2π ¼

3π R2π

¼ 3:40956

(15:23)

¼ 1:78614

(15:24)

2

½Fð’Þ d’

0

D0 jkR¼1:5π ¼

2π R2π 0

2

½Fð’Þ d’

238

15 Minimum directivity challenge of electromagnetic radiation

kR = 2π

kR = 1.5 90

90 120

1.0

60

30

150

0.5

30

150

0

180

0.0

180

0

0.5

240

330

210

330

210

1.0

1.0

300

240 270

kR = π

kR = π/2

120

1.0

0.0

300

270

90

0.5

60

0.5

0.5

0.0

120

1.0

90 60

120

1.0

30

150

0.0

30

150

0.5

0

180

60

180

0

0.5

0.5 330

210

330

210 1.0

1.0 240

300

240

270

300 270

Fig. 15.7: Radiation pattern of circular array.

D0 jkR¼π ¼

2π R2π

¼ 1:51374

(15:25)

¼ 1:50009

(15:26)

2

½Fð’Þ d’

0

D0 jkR¼π=2 ¼

2π R2π

2

½Fð’Þ d’

0

Now we have calculated the minimum directivity of cross-dipole array to be 1.5. Therefore, we might assume that the minimum directivity of the antenna is 1.5. For strict proof, we need to think further. The directivity of antenna array is related with electrical size. The smaller the electrical size is, the more the directivity of antenna array will be 1.0; this is quite a reasonable conclusion.

Q&A

239

3.0

Directivity

2.5

2.0

1.5

1.0 π/2

π KR

1.5π

2π

Fig. 15.8: Function relationship of directivity with kR.

15.5 Summary The minimum directivity of electromagnetic wave radiation is a research topic which theory highly links with practice. We have put up challenges in many aspects. Further study is now in progress.

Q&A Q: You specially discussed electromagnetic radiation in this chapter. Could you introduce the significance of this topic? A: Electromagnetic radiation is a topic where theory is applied to practical use. As stated in the very beginning, every communication and information system requires transmission of data. The most important part of it is the antenna, which transmits and receives all information by electromagnetic wave radiation. Chairman Mao, the revolution leader in China, knew its importance and wrote “You are the clairvoyance and clairaudient of science” to encourage the researchers (Fig. 15.9). As a form of electromagnetic motion, radiation has two essential conditions. One is that the receiver must be in the “far field” (which is a specialized term in the electromagnetic theory, the ratio of distance , to wavelength λ(,=λ)). For long waves, the “far field” means several thousand meters away while for millimeter waves, several meters can be the “far field.” The other is that power (energy) must be transmitted smoothly. Based on electromagnetic studies, Eθ =H’ ¼ η is a real number and is called the real wave impedance.

240

15 Minimum directivity challenge of electromagnetic radiation

Fig. 15.9: Chairman Mao wrote an inscription for the magazine Communications Warrior.

In many science popularization activities, electromagnetic waves are always imagined as the generalized three-dimensional water waves propagating outward, as shown in Fig. 15.10.

Transmitting antenna

Donor antenna Base station coverage area

Repeater Base station coverage area

Fig. 15.10: An antenna and electromagnetic waves to be radiated.

Q: What unique features does electromagnetic wave radiation have? A: Since radiation is a form of electromagnetic wave motion, it follows the laws of electromagnetic motion, that is, Maxwell’s equations. It is pointed out clearly in this book that on the one hand, any antenna must be directional, so the directionless antenna which radiates energy uniformly in all directions of the space is unlikely to exist in practice. On the other hand, the directivity of any antenna has an “upper limit,” the maximum directivity, which actually is worth further study. Q: Can you introduce the maximum directivity of the wire antenna? A: The wire antenna is one of the simplest and the most important antennas. In the movie Heroic Sons and Daughters, there is a whip antenna on the radio Wang Cheng

Q&A

241

Fig. 15.11: Wang Cheng carries the radio and the wire antenna in Shangganling Mountain (China).

(a character in the movie) carries on his back, as shown in Fig. 15.11. Briefly speaking, the wire antenna is directionless in direction ’ (east, west, north, south) but is directional in direction θ: The biggest problem of wire antenna is that currents are triangle-distributed, thus making its gain G very small. Generally, we “load” a wing on the top of the wire antenna so that currents are near distributed uniformly, thus improving the efficiency, as shown in Fig. 15.12.

I

I

Fig. 15.12: Wire antenna and “loading”: (a) the currents of wire antenna are triangle-distributed and (b) currents of wire antenna with wing on the top are near uniformly distributed.

Various methods are available for the loading of wire antenna, but the principle is the same, to make currents distributed uniformly. Simple as it is, the wire antenna has flexible applications. For example, there are dense woods and much rain in the jungles in the south of China, the latter of which humidifies the jungles. As a result, the great electromagnetic loss prevents the communication. However, if we hang a wire antenna on the top of a higher tree we have picked, the tree becomes a higher “wire antenna” due to the conductivity of the wet tree itself and then helps transmit and receive signals, just like the jungle antenna shown in Fig. 15.13. It can indeed be applied in practice. Q: Please tell us about the radiation of the surface antenna. A: The surface antenna is one application of secondary radiation. It has one radiation source first, like a horn which is placed on the focal point of the paraboloid. Electromagnetic waves are emitted to the paraboloid by the primary radiation of the horn and to the target by the secondary radiation of the paraboloid, as shown in Fig. 15.14.

242

15 Minimum directivity challenge of electromagnetic radiation

Fig. 15.13: Jungle antenna (the “extended application” to use the wet tree as the antenna).

Fig. 15.14: Parabolic antenna.

To obtain the most efficient radiation, we must employ the primary pattern of the horn antenna to match the magnitude of the paraboloid so that the missing power is low; the paraboloid is small and the amplitude of the paraboloid is distributed uniformly. Q: Now, could you tell us about the radiation of antenna array? A: For array, the discussion here is still focused on the maximum directivity. For simplicity and convenience, we only discuss linear array, which has only two key points, amplitude and phase. The conclusion is very easy to remember: to obtain the maximum directivity of linear array, the phase must be in-phase and the amplitude must be uniform. In this case, the beam width of 3 dB.   λ 2θ3 dB ¼ 51

, It is very clear that the larger the array length ,; the smaller the θ3 dB ; hence the super large directivity, as shown in Fig. 15.15. If the phase is in-phase, and the amplitude is distributed in accordance with the cosine, we have   λ 2θ3 dB ¼ 68

, This array is called broadside array. The typical two-element broadside array of uniform amplitude is shown in Fig. 15.16.

Q&A

243

2θ 0.5

Fig. 15.15: With the uniform amplitude, the in-phase linear array has the maximum directivity.

λ /2

e− jkr

− jk ( r + d cosθ )

e

θ

θ

I1

I2 λ /2

Fig. 15.16: Two-element broadside array of uniform amplitude and in-phase.

The E-field intensity of array is     E = e − jkr e − jkd cos θ + 1 = e − jkðr + ðθ=2Þ cos θÞ e − jð1=2Þkd cos θ + e jð1=2Þkd cos θ   1 = 2e − jkðr + ðd=2Þ cos θÞ cos kd cosθ 2 Considering k = 2π=λ and d = λ=2; we have π  E = 2e − jkðr + ð1=2Þd cos θÞ cos d cos θ 2 Two-element broadside array of uniform amplitude in direction θ is shown in Fig. 15.17. Another famous array is Yagi antenna, also known as end-fire array. If the feature of broadside array is uniform amplitude and in-phase, then the biggest feature of end-fire array is out-phase. It is a traveling wave antenna, as shown in Fig. 15.18. We still take the two-element end-fire array of uniform amplitude and out-phase as an example for explanation. The distribution of currents has I2 ¼ I1 ej’ ; that is, phase I2 advances phase I1 (the difference of the two is d ¼ λ=4), as shown in Fig. 15.19.

244

15 Minimum directivity challenge of electromagnetic radiation

θ

o

Fig. 15.17: Two-element broadside array of uniform amplitude in direction θ.

θ Fig. 15.18: Yagi antenna.

e−jk(r + d cos θ)

θ

e−jkr

θ

I1 = 1 d = λ/4

I2 = e−jφ

Fig. 15.19: Two-element end-fire array of uniform amplitude and out-phase.

Here, E-field intensity can be expressed as     E = e − jkr e − jkd cos θ + e − jψ = e − jðkr + ψÞ e − jðkd cos θ − ψÞ + 1   1 ψ = 2e − jðkr − ð1=2Þkd cos θ + ðψ=2ÞÞ cos kd cos θ − 2 2 Considering d = λ=4; we take ψ = π=2 and then obtain

π E = 2e − jðkr − ð1=2Þkd cos θ + ψ=2Þ cos ð1 − cos θÞ 4 Figure 15.20 shows the end-fire array in direction θ:

Recommended scholar

245

θ

Fig. 15.20: End-fire array in direction θ.

Recommended scholar This chapter will introduce Mr. Huang Xichun (Fig. 15.21), a senior professor at Xi’an Jiaotong University.

Fig. 15.21: Huang Xichun.

First, Xi’an Jiaotong University deserves special mention. In the 1950s, there was a great gap between eastern China and western China. At the mention of Xi’an, the easterners would imagine it as a barren wasteland, so how admirable a group of pioneers abandoned favorable conditions and took roots in Xi’an! Mr. Huang is a conscientious and strict scholar who had studied in Germany. He leaves us very few books, but every word in his few books is powerful and valuable. Without any exaggeration, few of us can fully understand his book On Wave Velocities (Fig. 15.22).

Fig. 15.22: On Wave Velocities, Mr. Huang’s masterpiece.

246

15 Minimum directivity challenge of electromagnetic radiation

On the New Year’s Day of 1986, I visited Mr. Huang at his home for recommending me as professor and doctoral supervisor (Mr. Huang was the only second-ranking professor in this field in northwestern China). What was more regrettable was that he passed away on January 8. However, his personality and works are still clear in my mind until now.

Appendix A y-direction dipole parameters * and electric field E θ′ There is also a dipole in direction y, as shown in Fig. A15.1, where ^eθ and ^e’ represent * direction ^eθ ; and parameters θ and ’ represent electric field E : ^θ and e ^’ ^θ′ by e Expression of e

1

z eˆθ ′ r θ′

y Fig. A15.1: The y-direction dipole is represented with parameters θ and ’′; r is the same parameter.

x

Suppose the dipole coordinate in direction z is ^er ; ^eθ ; ^e’ – ^x, ^y, ^z, and that in direction y is ^er′ ; ^eθ′ ; ^e’′ – ^x′; ^y′; zb′: In the above coordinate system, the common unit vector is ^er ¼ ^er′

(A15:1)

Here, we first study ^eθ′ ; that is, the far-field radiation electric field direction of the dipole in direction y, represented by ^eθ and ^e’ : It is easy to obtain 32 3 2^ 3 2 er ′ ^x′ sin θ′ cos ’′ sin θ′ sin ’′ cos θ′ 7 6 7 6 ^e 7 6 4 θ′ 5 ¼ 4 cos θ′ cos ’′ cos θ′ sin ’′  sin θ′ 54 ^y′ 5 ^e’′ ^z′  sin ’′ cos ’′ 0 32 3 2 3 2 ^x 0 0 1 ^x′ 76 ^ 7 6 ′7 6 4 ^y 5 ¼ 4 1 0 0 54 y 5 ^z′

0

1

0

^z

(A15:2)

(A15:3)

Appendix A y-direction dipole parameters and electric field

2 3 2 ^x sin θ cos ’ 6^7 6 4 y 5 ¼ 4 sin θ sin ’ ^z cos θ

cos θ cos ’ cos θ sin ’  sin θ

 sin ’

32

^er

247

3

76 7 cos ’ 54 ^eθ 5 ^e’ 0

(A15:4)

Therefore, 2^ 3 2 3 er′ ^er 6 ^e 7 6^ 7 4 θ′ 5 ¼ T 4 eθ 5 ^e’′ ^e’

(A15:5)

and we have 32 0 sin θ′ cos ’′ sin θ′ sin ’′ cos θ′ 76 6 ′ ′ ′ ′ ′ T ¼ 4 cos θ cos ’ cos θ sin ’  sin θ 54 1 0  sin ’′ cos ’′ 0 2 3 sin θ cos ’ cos θ cos ’  sin ’ 4 sin θ sin ’ cos θ sin ’ cos ’ 5 cos θ  sin θ 0 2

0

1

3

0

7 05

1

0

2

sin θ′ sin ’′ sin θ cos ’ + cos θ′ sin θ sin ’ + sin θ′ cos ’′ cos θ 6 = 4 cos θ′ sin ’′ sin θ cos ’ − sin θ′ sin θ sin ’ + cos θ′ cos ’′ cos θ cos ’′ sin θ cos ’ − sin ’′ cos θ sin θ′ sin ’′ cos θ cos ’ + cos θ′ cos θ sin ’ − sin θ′ cos ’′ sin θ sin θ′ sin ’′ cos θ cos ’ + cos θ′ cos θ sin ’ − sin θ′ cos ’′ sin θ cos ’′ cos θ cos ’ + sin ’′ sin θ 3 − sin θ′ sin ’′ sin ’ + cos θ′ cos ’ 7 − cos θ′ sin ’′ sin ’ − sin θ′ cos ’ 5

(A15:6)

− cos ’′ sin ’ Based on transformation matrix T, we obtain the unit vector of ^eθ′ .   ^eθ′ = cos θ′ sin ’′ cos θ cos ’ − sin θ′ cos θ sin ’ − cos θ′ cos ’′ sin θ ^eθ (A15.7)   − cos θ′ sin ’′ sin ’ + sin θ′ cos ’ ^e’

2 Expressions of sin θ′, cos θ′, sin ’′ and cos ’′ As have been mentioned, the common part of the coordinates with and without an apostrophe is ^er : That is, ) ^er ¼ ^x sin θ cos ’ þ ^y sin θ sin ’ þ ^z cos θ (A15:8) ^e ¼ ^x′ sin θ′ cos ’′ þ ^y′ sin θ′ sin ’′ þ zb′ cos θ′ r

248

15 Minimum directivity challenge of electromagnetic radiation

A comparison of the dipoles in direction z and in direction y leads to ^x′ ¼ ^z;

^y′ ¼ ^x;

^z′ ¼ ^y

(A15:9)

We obtain 9 sin θ′ cos ’′ ¼ cos θ > = sin θ′ sin ’′ ¼ sin θ cos ’ > ; cos θ′ ¼ sin θ sin ’ And, it can be solved that (only the first quadrant is considered here) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 sin θ′ = cos2 θ + sin2 θ cos2 ’ > > > > > > > ′ = cos θ = sin θ sin ’ sin θ cos ’ ffi sin ’′ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 cos θ + sin θcos ’

cos θ ffi cos ’′ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 cos θ + sin θcos ’

> > > > > > > ;

(A15:10)

(A15:11)

*

3 Electric field E θ′ of the y-direction dipole * Electric field E θ′ is completely symmetric to the dipole in the z-direction. If the dipole in direction y is expressed by using the coordinate with an apostrophe, as shown in eq. (15.9), it is easy to obtain pffiffiffiffiffiffiffiffiffiffiffi * ðμ=εÞIl sin θ′ jkr e (A15:12) E θ′ ¼ ^eθ′ j 2λr Based on eqs. (A15.7) and (A15.11), we obtain *

*

*

E θ′ ¼ E θ þ E ’

(A15:13)

where *

pffiffiffiffiffiffiffiffiffiffiffi ðμ=εÞIl ½ cos θ sin ’ 2λr pffiffiffiffiffiffiffiffiffiffiffi * ðμ=εÞIl ð cos ’Þ E ’ ¼ ^e’ j 2λr

E θ ¼ ^eθ j

Appendix B The phases of two-dimensional eight-element The distribution of an eight-element employs A as the reference plane. The phase in element 1 is

(A15:14) (A15:15)

249

Appendix B The phases of two-dimensional eight-element

ejkRð1cos ’Þ

(B15:1)

The phase in element 2 is

’Þ

ejkR½1cosð45

(B15:2)

The phase in element 3 is ejkR½1cosð90

’Þ

(B15:3)

The phase in element 4 is

þ’Þ

ejkR½1þcosð45

(B15:4)

ejkR½1þcos ’

(B15:5)

The phase in element 5 is

The phase in element 6 is

’Þ

ejkR½1þcosð45

(B15:6)

The phase in element 7 is ejkR½1þcosð90

’Þ

(B15:7)

The phase in element 8 is

þ’Þ

ejkR½1cosð45

(B15:8)

We extract common factor ejkR ; and the eight elements all have an amplitude of I, so we have  2I cos½kR cos ’ + cos½kR cosð45 − ’Þ + cos½kR cosð45 + ’Þ + cos½kR cosð90 − ’Þ ¼ 2IfcosðkR cos ’Þ þ cosðkR sin ’Þ pffiffiffi  pffiffiffi  2 2 þ2 cos kR cos ’ cos kR sin ’ 2 2

(B15:9)

Finally, we obtain the directivity function of circular array antenna pffiffiffi

pffiffiffi

2 2 F ð’Þ = F0 fcos½kR cos ’ + cos½kR sin ’ + 2 cos kR cos ’ cos kR sin ’ 2 2 (B15:10) F0 is a normalization coefficient as well.

16 Mysteries of Fermat’s principle Pierre de Fermat, a talented scientist, proposed the famous optical extremum principle – Fermat’s principle in 1662, which brought a fresh atmosphere to science and provoked a storm of argument. However, it also caused a series of problems and mysteries. For example, the expression of the principle is algebraic while the result is geometrical; no criterion of the minimum or the maximum is mentioned in the principle; the phase condition corresponds to the equations of motion while the amplitude condition cannot be deduced from the equations of motion. This chapter uncovers all the mysteries of Fermat’s principle in the hope of arousing further attention.

16.1 Introduction This is the 16th chapter of Electromagnetic Field Theory Teaching Series. One of Maxwell’s greatest contributions is the establishment of the unified theory of light and electromagnetism. However, people have totally different understanding of these two fields as time went on. On the one hand, as early as the ancient times, humans stroke a light for warmth and then realized the presence of light, not to mention that the sun has always been a close companion with our ancestors. It can be said that light is one of the earliest phenomena for humans to understand natural physics. On the other hand, electromagnetic fields and waves are totally different. People realized the presence of electromagnetism very late since it cannot be seen, heard or felt. And, communicating electromagnetic waves (with signals) are totally “man-made.” Pierre de Fermat (1601–1665) to be discussed in this chapter is the one who made an outstanding contribution in optics. Born in France in 1601, Fermat worked initially as a councilor at the Parliament of Toulouse (one of the High Courts of Judicature in France) and was soon promoted as a grand justice at the Supreme Court, Parliament of Toulouse. However, it was surprising that he made many first-class or even superclass contributions in mathematics and physics “out of” his busy government affairs. No more interpretation is needed, for everyone knows that Fermat’s last theorem xn + yn = zn

ðn = 3; 4; 5; . . .Þ

(16:1)

has no integer solution, which was finally proved, 358 years (in 1995) after Fermat’s last theorem had been proposed, by Andrew Wiles, a mathematician from Cambridge University [49]. What this chapter discusses in depth is the tremendous innovation achievement which Fermat made for the path light takes in physical optics, a totally different field. In 1662, 61-year-old Fermat realized deeply that the nature always achieves its aim in the simplest way. Specifically, “when encountering a boundary between two https://doi.org/10.1515/9783110527407-016

252

16 Mysteries of Fermat’s principle

media, light will always take the path that requires the least time either by reflection or refraction.” This is the famous Fermat’s principle. Fermat is always a legendary figure. Though 380 years has passed since Fermat proposed the principle, its essence and highlights still have great enlightenment to the study of electromagnetism, and the mysteries leaving behind inspire our more thinking and development.

16.2 Expression of the law of nature To live in harmony with the nature, what humans need to do first is to understand the nature and reveal the law of nature [50]. How on earth the law of nature has expressed itself in the past thousands of years? People have experienced an extremely long understanding process, as shown briefly in Fig. 16.1.

Pythagoras number

Newton (1687) equation

Fermat (1662) functional extreme value (inequation)

Noether, Chen-Ning Yang symmetry

Fig. 16.1: Development process of the expression of the law of nature.

It should be noted that though Fermat’s principle was proposed earlier than Newton’s Philosophiae Naturalis Principia Mathematica (1687), Fermat’s principle is more advanced, thus not being understood and accepted by most people for long period of time. Newton was among them. The first highlight of Fermat’s principle is that it does not show the specific path light takes (or equation) while it gives the extremum principle which the motion of light follows – following the path of least time t. This is indeed an original thought, which corresponds to Maupertuis’s principle of least action and Euler’s variational method later. The second highlight is that Fermat’s principle is a coordinate-free principle with no need of the specific coordinate, which increasingly shows its universality. The first person to try tomato is always the rarest and bravest explorer. Thereafter, J. J. Thomson also proposed the similar extremum principle for charge distribution on a conductor in electrostatics, “the charge on a conductor should be distributed in the way that keeps electrostatic stored energy We minimum.” This is called Thomson’s principle, hence a new path of the extremum expression of the law of nature, as shown in Fig. 16.2. As is known to all, the law of reflection and the law of refraction had existed before Fermat’s principle was proposed, that is,

253

16.3 Minimum or maximum

S

ρ (r )

V

Fig. 16.2: Thomson’s principle of charge distribution on a * conductor ρð r Þ:

8 < θi = θr sin θi νi nt : = = sin θt νt ni

ð16:2Þ ð16:3Þ

where ν represents velocity and n represents refractive index, as shown in Fig. 16.3.

θi θr

n1

n2

θt Fig. 16.3: Laws of reflection and refraction of light.

The first mystery Fermat’s principle brings is that the expression of the principle is algebraic while the result is geometrical. Despite the strict deduction, the deeper reason why the algebraic form turns to the geometrical form deserves a further exploration.

16.3 Minimum or maximum Euler admired the extremum principle most. He once said, “Since the structure of the universe is the most perfect possible and the work of the wisest possible creator, nothing happens which has not some maximal or minimal property” [51]. Whether does the universe take the maximum or minimum? The scientific community almost leans to one side and prefers that “the nature is basically lazy, and it always strives for the minimum path.” Therefore, what is developed from Fermat’s principle is also called the principle of least action. However, the fact is not as simple as imagined. Please look at the following example. Example 16.1: Concave mirror of the elliptic cylinder For simplicity as well as illustration, let us study the light reflection of the concave mirror of the twodimensional elliptic cylinder as shown in Fig. 16.4. Please consider an exception: light is reflected by the mirror from point Aðx; yÞ where x > c and y > 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (focal length c = a2 − b2 ) to point B, with point B just at one focal point c2 of the ellipse.

254

16 Mysteries of Fermat’s principle

y A( x, y ) C2 B

C1

a

O

θ 2 θ1 C

x

b Fig. 16.4: Light reflection of the concave mirror of the twodimensional elliptic cylinder.

D

In Fig. 16.4, the semimajor axis and the semiminor axis are a and b, respectively, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi OC1 = OC2 = c = a2 − b2

(16:4)

where OC1 is the focus point. As is known to all, the ellipse has two important features: (1) The sum of distances from the two focus points ðC1 ; C2 Þ to any point on the ellipse is 2a; (2) OC divides ff BCA equally, that is, θ1 = θ2

(16:5)

and OC is the normal of the ellipse at point C. Obviously, the ray of light AC1 C ! CB satisfies the angle of incidence equal to the angle of reflection, and its path sum is AC + BC = AC1 + C1 C + CB =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx − cÞ2 + y 2 + 2a

(16:6)

For comparison, we take point D on y-axis, that is, light travels along AD ! BD: We note that BD = C1 D = a (due to BD + CD = 2a and its symmetry), so AD + BD = a +

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 + ðy + bÞ2

(16:7)

In triangle C1 DA; AC1 + C1 D > AD

(16:8)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx − cÞ2 + y 2 + a > x 2 + ðy + bÞ2

(16:9)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx − cÞ2 + y 2 + 2a > x 2 + ðy + bÞ2 + a

(16:10)

AC + BC > AD + BD

(16:11)

We have

or written as

That is,

On the condition of this concave mirror, light takes the maximum path, which is a surprising conclusion. We note that Ref. [52] shows in the footnotes that “in some examples, such as the

16.4 Phase velocity νp or group velocity νg

255

reflection of light by concave mirror, the path light takes needs very long time, the fact of which is known by Fermat and described clearly by William R. Hamilton.” Though having realized that the principle of least time in Fermat’s principle had some problems, some great scholars proposed no idea and criteria that in what conditions the minimum or the maximum appeared, hence the second mystery of Fermat’s principle.

16.4 Phase velocity ν p or group velocity ν g In the development of Fermat’s principle, one key point must be mentioned that back then from different perspectives existed two different extremum principles: Fermat’s principle and Maupertuis’s principle, one of which is inversely proportional to the velocity of light while the other of which is directly proportional to the speed of light, thus forming an obvious contradiction [53]. The times produce their heroes. It was young French doctoral student Louis de Broglie who studied quantum mechanics with very sharp eyes and revealed profoundly that in the existing extremum principles at that time, Fermat’s principle is related to the phase velocity νp while Maupertuis’s principle is associated with group velocity νg : Thus, two different expressions are unified, as shown in Fig. 16.5. In the figure, n represents relative refractive index n=

c νp

(16:12)

and P represents (the magnitude of) momentum. Then, P = mνg

Fermat’s principle

δ ndS = δ

c vp

Maupertuis’s principle

δ PdS = δ mvg dS= 0

dS = 0

Wave optics represents phase radiation

Microscopic optical represents particle trajectory

vp vg = c 2 Fig. 16.5: Two extremum principles correspond to two different velocities of light.

(16:13)

256

16 Mysteries of Fermat’s principle

By using Planck’s quantum theory (E represents energy, h represents Planck constant and v represents the frequency of the radiational electromagnetic waves) E = hν

(16:14)

and Einstein’s theory of relativity (m0 represents the mass of the object at rest, m represents the mass of the object moving at velocity v) m0 c2 E = mc2 = qffiffiffiffiffiffiffiffiffiffiffi 1 − β2

(16:15)

where β = ðν=cÞ; and c represents the velocity of light in a vacuum, Louis de Broglie derived the momentum relation totally corresponding to Planck’s energy relation, m0 ν g Eνg hν h = P = mνg = qffiffiffiffiffiffiffiffiffiffiffi = 2 = c νp λ 1 − β2

(16:16)

νλ = νp

(16:17)

which employs

as shown in Fig. 16.6.

Planck’s energy relation

E = hv

Louis de Broglie’s momentum relation

P=

h λ

Fig. 16.6: Energy–momentum relations of the quantum.

And, νλ0 = c

  c λ0 λ0 P n= = = λ h νp Since ðλ0 =hÞ is a constant, we have ð ð δ n ds = δ P ds = 0

(16:18) (16:19)

(16:20)

which indeed proves the unity of Fermat’s principle and Maupertuis’s principle in the movement of light. It can be seen clearly from the discussion above that Fermat’s principle physically represents the ray of phase wave and corresponds to phase velocity νp : Therefore, it is

16.4 Phase velocity νp or group velocity νg

257

not directly related to the conservation of energy. In other words, Fermat’s principle will not violate the conservation of energy or Poynting’s theorem. Double-negative media or left-hand materials proposed in recent years can be interpreted as negative refractive index − n or negative phase − νp (where n and νp are positive). Then, when light is refracted from the air to the medium of − n; the refracted ray and the incident ray are on the same side of the normal, as shown in Fig. 16.7.

A

θi

n0

O

−n

θt

B

Fig. 16.7: Negative refraction.

And, we have sin θi c = sin θt νp

(16:21)

Especially when medium 1 is n and medium 2 is −n, we have sin θi = 1 sin θt

(16:22)

In this case, light needs some time when moving from point A to point B at group velocity νp ; but when points A and B are vertically symmetrical, the total time (the wave phase) of the movement of light at group velocity νg is 0. Hence from point A to point B, there are countless rays. Then, it seems that “light cannot find its way,” hence the third mystery of Fermat’s principle, as shown in Fig. 16.8. A

n −n

B

Fig. 16.8: On the boundary of medium n and −n, light has countless practicable paths.

258

16 Mysteries of Fermat’s principle

16.5 Energy E and actuating quantity S In the process of the development of Fermat’s principle into the principle of least action, many scholars have made contributions, especially Joseph Lagrange and William Rowan Hamilton [51,54]. Now, the general principle of least action can be written as 8 t1

> > > * * > ∂ A ð r ;tÞ *

> > Dð r ; tÞ = εE ð r ; tÞ > > * * :* * Bð r ; tÞ = μH ð r ; tÞ

(16:32)

*

The principle can be extended as: “Provided that the correct values of A and ’ are given in the region V at the initial time t0 and final time t1 and on the closed surface S * enclosing V during the time interval ½t0 ; t1 , then A and ’ in V for which δS = 0 * becomes stationary can be determined, and those A and ’ thus determined give the true fields throughout the region V” (see Fig. 16.10). Thus, the integration with respect to the time t in (16.31) becomes the integration with respect to the frequency f as follows:

16.8 Summary

261

nˆ

J ( r , t) ρ ( r , t) V

E ( r , t) B( r , t ) D (r , t ) H ( r ,t) S

Fig. 16.10: Principle of least action.

S= ∞ ð

dω -∞

 ð h i ** * * * * * * * * 1** * * E ð r ; ωÞD* ð r; ωÞ−H ð r ; ωÞB* ð r ; ωÞ þAð r ; ωÞ J * ð r ; ωÞ−ρð r ; ωÞ’* ð r ; ωÞ dν 2 r

(16:33) and 8 ** * * > B ð r ; ωÞ = ∇ × A ð r ; ωÞ > > * * > * *

> Dð r ; ωÞ = εE ð r ; ωÞ > > * * :* * Bð r ; ωÞ = μH ð r ; ωÞ

(16:34)

In this book, we have proved that from δS = 0, all Maxwell’s equations and boundary conditions can be derived. In other words, the electromagnetic least action principle is equivalent to Maxwell theory. From the previous discussions, we can see that energy E reflects amplitude conditions while action S reflects phase conditions. Here, we have the seventh mystery: Why phase conditions can equal motion equations while amplitude conditions can not?

16.8 Summary Now, we can discuss this issue from the perspective of philosophy. Closely related to action S, the Lagrange L includes the difference between the two kinds of energy. If readers take a step further, it is obvious that these two types of energy are different. The difference between kinetic energy T and potential energy V is contradictory to that between electric energy We and magnetic energy Wm : In this sense, it is a problem of generalized resonance to find the extreme value of L and S. Further, in generalized sense, the minimum and the maximum can be analogized to serial resonance and parallel resonance of the network. This idea deserves attention.

262

16 Mysteries of Fermat’s principle

This chapter is not so much a paper as a collection of puzzling questions around Fermat principle. This fashion of questioning has been prevalent since ancient times in Chinese history. Chhu Yuan, a sage in ancient China, had raised sharp questions about the universe in his well-known work Thien Wen (Questions about the Heaven), a Chinese book of origins. “Raising a question is half way to its solution.” Here, I would like to ask far-sighted people, especially young scholars, to provide answers to these questions.

Q&A Q: You unintentionally touched upon the development history of physics, especially electromagnetics, which is enlightening indeed. It seems that the necessity is enveloped by chance: In mechanic studies, Newton gave us equations (Newton’s second * * * law F = ma and law of universal gravitation F = kðMm=r2 Þ^r). Meantime, in his optical studies, Fermat proposed inequations – the principle of extremum. Then is there a deep reason underneath? A: From this question I can see you have learned how to think by yourself step by step. As we can see, it is natural and necessary to pose equations when applying mechanics to solve problems in usual practice. Moreover, it is a reasonable process to explore the nature from low level to advanced level, from elementary layer to complex layer and from concrete to abstract. Fermat’s concept originates from this law. Then, the essence of your question is unraveled: Newton’s equations, from the start, occurred in the “particle property” visual to objects while Fermat’s principle originated from the “wave property” (regardless of the duality of light). The Fermat is shown in Fig. 16.11.

Fig. 16.11: Fermat.

Q&A

263

Q: Recently is there some progress about Fermat’s principle? A: Yes, there are new discoveries in Fermat’s principle. We proposed a brand new equivalent expression of Fermat’s principle. “The total phase of light waves keeps extreme values during light propagation.” And ϕ=

2πf , = ωt v

where ’ is optical phase; 2πf = ω; where f is frequency and ω is angular frequency; , is transmission length and ν is light velocity; ,=ν = t (time). And, v=

c n

where c is the light velocity in vacuum and n is the refractive index during corresponding distance. In the 20th century, one of the outstanding achievements in physics was the great attention attached to the concept of phase. In his paper How important were Hermann Weyl’s Contribution to Physics written in 2008, Chen-Ning Yang pointed out that a basic principle in modern physics formed thanks to the resent development – all basic forces are phase fields. In 2003, Mr. Yang concluded three themes of physic development in the 20th century in his speech: quantization, symmetry and phase factor. Now, let us review the Fermat’s principle from its beginning.

1 Fermat’s reflection law Fig. 16.12 shows the proving of Fermat’s reflection law, where h1 ; h2 and x0 are given. How to choose the path from A through reflection boundary O to B? And x is unknown.

y

A

l1

h1

B

h2

θ1 θ 2

l2

x x0 − x

x x0

Fig. 16.12: Proving Fermat’s reflection law.

264

16 Mysteries of Fermat’s principle

According to Fermat’s principle, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi L = ,v ð ,1 + ,2 Þ = ,v ðx0 − xÞ2 + h21 + x2 + h22 where ,v means to find extreme values, minimum or maximum. We can obtain dL ðx0 − xÞ x =  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + pffiffiffiffiffiffiffiffiffiffiffiffiffi2 = 0 2 dx 2 x + h2 ðx0 − xÞ + h21 That is, ðx0 − xÞ x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 + h2 2 x 2 2 ðx0 − xÞ + h1 or sinθ1 = sinθ2 θ1 = θ2 When the distance is assigned extreme value, we know that incidence angle equals reflection angle. From the perspective of wave phase, when f and v are assigned constant values, from ’ = 2πf ,=ν, we have   2πf ð,1 + ,2 Þ ,v ð,1 + ,2 Þ = ,v ð’1 + ’2 Þ = ,v ’ For the case of reflection, it is corresponding to optical phase ’ and extreme values from A to B.

2 Fermat’s refraction law Figure 16.13 shows the proving of Fermat’s refraction law, where ,1 ; ,2 > 0: According to Fermat’s principle, the time is the shortest from A ! B: We can obtain   ,1 ,2 + T = ,v ð t 1 + t 2 Þ = , v v1 v2 Note that in terms of dealing with refraction problems, ν1 and ν2 are different when light travels in two kinds of media. We have v=

c ; n1

v=

c n2

Q&A

265

y

A

l1 θ1

n1 n2

x

x0 − x O

x

l2 x0

B Fig. 16.13: Proving Fermat’s refraction law.

Further, we can divide it into 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðx0 − xÞ2 + h21 x2 + h22 A + T = ,v @ v2 v1 and dT 1 − ðx0 − xÞ 1 x = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + pffiffiffiffiffiffiffiffiffiffiffiffiffi2 = 0 dx v1 v2 x2 + h2 2 2 ðx0 − xÞ + h1 We have sinθ1 sinθ2 = v1 v2 Finally, we give sinθ1 v1 n2 = = sinθ2 v2 n1 In other words, this is the most general refraction law. Again, we move back to the perspective of the wave phase and we have  T = , v ð t 1 + t 2 Þ = ,v

 2πf ,1 2πf ,2 = ,v ð ’ 1 + ’ 2 Þ + v1 v2

Now, we see that the issue can be equivalent to finding wave phase and the extreme value of ð’1 + ’2 Þ: The use of phase and extreme values unifies this issue: be it reflection or refraction, one medium or two media, or even continuously changing media, we can get

266

16 Mysteries of Fermat’s principle

Z ,v

 ’ð,Þd,

On the other hand, amplitude was raised to a momentous position in the past because it was visible and tangible. However, phase plays a decisive role here. Actually, discussions in this chapter that light velocity ν of Fermat’s principle corresponds to phase velocity νp have left out its recent development and extension. Relevantly, it can be called “phase physics.” However, the secrets within can not be explained, which are left to be further studied. Q: It is intriguing to use extreme values in Fermat’s principle. Could you explain the mathematical expressions of maximum and minimum? A: To emphasize the concept, we focus on the reflection of light on a flat surface. There are two expressions. 1. Algebra expression To find the second order of the objective function T using the previous symbols, we have d2 T 1 ðx0 − xÞ2 1 x2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q = − − h i3=2 + pffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 2 2 2 dx x + h2 x2 + h22 ðx0 − xÞ2 + h21 ðx0 − xÞ2 + h21 =h

h21

h22 i3=2 −  3=2 > 0 x2 + h22 ðx0 − xÞ2 + h21

When the second order exceeds 0, minimum will be obtained, which is also the shortest path. 2. Geometric expression * We draw A using mirror symmetry as shown in Fig. 16.14. AO + OB = A′ OB *

By looking at E ; we have A′ O′ + O′ B > A′ OB The sum of two edges is greater than the third edge, namely, straight line is the shortest between ðA′ BÞ: It is necessary note that the mirror point A′ of A and the mirror point B′ of B are symmetrical as shown in Fig. 16.15.

Q&A

267

y

A

B

l1 l2

x o′

o

l1 Fig. 16.14: Geometric expression of extreme values in Fermat’s principle.

A′

y

A

B

l1 l2

x o B′ A′

Fig. 16.15: Two mirror points A′ and B′ are symmetrical.

Q: Should we take the maximum or minimum in Fermat’s principle? A: It is a difficult question. We will split it into two cases. For convenience, we only consider the two-dimensional xoy. 1. Circle concave mirror The circle concave mirror is shown in Fig. 16.16. To simplify the case, we presume A and B are on the same diameter of the circle concave mirror. Since AO = OB = r; the optical length is AC + CB = LO and we have pffiffiffiffiffiffiffiffiffiffiffiffiffi LO = AC + CB = 2 R2 + r2 Now, we presume ffCOC′ = δ is an angle of deflection, and then AC′ = AO + OC′ − 2AO  OC′  cosð90 + δÞ = R2 + r2 + 2Rrsinδ 2

2

268

16 Mysteries of Fermat’s principle

y

R A

r

r

C′′

B

x

o δ θ θ

C′ C

Fig. 16.16: Circle concave mirror.

C′B2 = BO + OC′ − 2BO  OC′  cosð90 − δÞ = R2 + r2 − 2Rrsinδ 2

Therefore, the optical length L to C′ is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L = AC′ + C′B = R2 + r2 + 2Rr sinδ + R2 + r2 − 2Rr sinδ (1) The quasi differential case when δ ! 0: Set x = 2Rr sin δ=ðR2 + r2 Þ and we know jxj < 1: When δ ! 0; x is very small. ( ð1 + xÞ1=2 1 + 21 x − 81 x2 ð1 − xÞ1=2 1 − 21 x − 21 x2 Thus, pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi R2 r2 sin2 δ L = R 2 + r 2 1 + x + R2 + r 2 1 − x 2 R2 + r 2 − ðR2 + r2 Þ3=2 Obviously in this case, L < LO : For light AC + BC; the incidence angle θ equals the reflection angle θ and the maximum is satisfied. (2) Comprehensive study of the circle concave mirror. If we have an overall study of the circle concave mirror, all we need is to set δ = 90 : Lp = AC′′ + C′′B = ðR + rÞ + ðR − rÞ = 2R Lp < LO is satisfied and the value is a minimum (if the obstacle between A and B is left out). Now, Incidence angle = reflection angle = 0.° Therefore, for the circle concave mirror, there exists not only the optical path of maximum but the path of minimum as well.

Q&A

269

2. Ellipse concave mirror For the ellipse concave mirror, we will split the case into three aspects. (1) When source A and reception point B are at two focuses, as shown in Fig. 16.17. According to the property of ellipse, an infinite number of radials satisfy the requirements. L0 ≡ 2a y b

A

B

o

a

Fig. 16.17: Ellipse concave mirror with A and B at the focuses.

C′

C

x

(2) When A and reception point B are within the focal length as shown in Fig. 16.18. That is, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r < a2 − b2 = c Then, pffiffiffiffiffiffiffiffiffiffiffiffiffi Li = 2 r2 + b2 < L0 which is the case when the value is the minimum. y b

r A

r o

B

a

x

θθ

C

Fig. 16.18: Ellipse concave mirror with A and B between focuses.

(3) When source A and reception point B are outside the focal length, as shown in Fig. 16.19.

270

16 Mysteries of Fermat’s principle

y b

r

r o

A θ

B

a

x

θ

C

Fig. 16.19: Ellipse concave mirror with A and B outside the focus length.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > a2 − b2 = c Then, pffiffiffiffiffiffiffiffiffiffiffiffiffi Le = 2 r2 + b2 > L0 which is the case of maximum. Even though it is the case, there is another path. minL′ = AC′ + C′B = 2a = LO .(obstacle ignored) We must point out that in the examples of this chapter, we can find another case with minimum (if only the concave mirror is intact enough). And, we can obtain (1) As a famous physicist William Rowan Hamilton pointed out, maximum exists in the concave reflection, which was also known by Fermat. Here, we propose that maximum must coexist with minimum. (2) Now, the question concerned is that what choice light will make when it faces several extreme values (if it is “human”). It deserves further study. (3) When source and reception point are at the ellipse focuses, light will be at lost. This is another key issue – extreme values are everywhere. From the discussions above, we can see that Fermat’s principle remains to be extended. Q: In this chapter, we also mention the Joseph John Thomson’s principle, known to us as the J. J. Thomson’s principle. It is described as “Charges in conductors are distributed in a way by which the electrostatic energy We remains the minimum.” It opens up a new approach for the electromagnetic principle. Is there anything that requires further discussion? A: My answer is yes. As Fig. 16.9 has shown, E = T + V and energy is relevant to R amplitude while as shown in δS = δ ðT − VÞdt; energy difference is relevant to phase, where T is kinetic energy and V is potential energy. In addition, let’s consider the electromagnetic field as shown in Fig. 16.20. Specially, note that in J. J. Thomson’s principle,

Q&A

W = We + Wm

Energy sum is relevant to amplitue

δ S = δ (We + Wm ) dt

Energy difference is relevant to phase

271

Fig. 16.20: Energy and energy difference in electromagnetic field.

Wm = 0 Here, we have W = We while Lagrange’s function is L =  We Now, we can see that the sum of energy and energy difference is nearly unified (except for a negative symbol). Since J. J. Thomson did not know the minimum action principle, it was called the minimum energy principle, precisely the maximum action principle (the smaller We is, the greater its negative number We will be). It appears that it is more exact to describe action as extremum action principle. Q: Could you take an example to show the application of the J. J. Thomson’s principle in daily use? A: Now, let’s look at the charge distribution in the infinitely thin conductor disk with a radius of r0 as shown in Fig. 16.21. z

O x

Q

r0

y Fig. 16.21: A conductor disk with a radius of r0 :

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We will discuss two distributions. One is σ = σ0 = 1 − ðr=r0 Þ2 (correct) while the other is σ = σ0 uniform distribution (incorrect). From Table 16.1, we can get Q and V and capacitance C1 and C2 from C = Q=V: Then, we can figure out the we electric energy 8 2 Q2 < We1 = 21 QC = 16εr :W = e2

1

0

1 Q2 2 C2

Q2 4πεr0

=

272

16 Mysteries of Fermat’s principle

Table 16.1: Conductor disks with a radius of r0 :

Correct distribution of charges σ0 σ = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 − ðr=r0 Þ 2 Rπ Rr0

σ0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r dr dϕ 2 0 0 1 − ðr=r0 Þ   Rr0 d 1 − ðr=r0 Þ2 = − πσ 0 r02 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 1 − ðr=r0 Þ

Q=

Uniform distribution σ 0

Q = πσ 0 r02

= 2πσ 0 r02 Potential V of the disk center V=

ÐÐ 4πε

=

σ0 r0 2ε

=

πσ 0 r0 4ε

σ 0 r dr dϕ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 − ðr=r0 Þ

r sinðr=r0 Þj00

Distribution capacitance C C1 =

Q V

= 8εr0

Potential V of the disk center V=

ÐÐ

σ 0 r dr dϕ 4πεr

=

σ0 2ε

=

σ 0 r0 2ε

r

rj00

Evenly distributed capacitance C C2 =

Q V

= 2πεr0

In other words, under the condition that charges Q are equivalent, in the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distribution, σ0 = 1 − ðr=r0 Þ2 , We1 < We2 :

Recommended scholar In the following part, I will introduce Mr. Ye Peida (Fig. 16.22) with great respect.

Fig. 16.22: Ye Peida.

Recommended scholar

273

He is a first-rate scholar and his sincerity, encouragement and support have left a deep impression on me. Mr. Ye, as a first-generation expert in electromagnetic microwave in China, introduced advanced theories developed in MIT during World War II as aforementioned. He tries his best to introduce the book Advanced Theory of Waveguides by L. Lewin, especially the eigen-mode expansion and the network matching theory. He encouraged me greatly when I wrote Computational Microwaves after I finished my studies in America. He believed that it was a direction in the future to combine microwave with calculation. In 1985 when this book was published, he wrote a preface for me. The support from a great master has filled me with warmth to date. This kind of support, I presume, will be the important condition for Chinese science to scale the heights. Even if we are not able to scale the heights ourselves, we should learn from Mr. Ye to be a “ladder” for others.

17 Electromagnetic inertia The great physicist Isaac Newton proposed the mechanical inertia law: In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force. This chapter deals with electromagnetic inertia in the electromagnetic field: under the electrostatic condition, potential φ or source σ tends to keep the state of sphere (three-dimensional) or circle (two-dimensional) unless electromagnetic motions or boundaries change this state.

17.1 Introduction This is the 17th chapter of Electromagnetic Field Theory Teaching Series. In the development history of physics, prophesies and discoveries of electromagnetic wave are earthshaking events, in which the talented James Clerk Maxwell and his successor Heinrich Rudolf Hertz both play an important role. As is known to all, before Maxwell, there had been Ampere’s law and Faraday’s law. 8 * * > ðAmpere 1825Þ

:∇×E = − ðFaraday 1831Þ ∂t However, a contradiction appears in the circuits with capacitors C (viz. current I exists in the external circuit but no such current appears in the capacitor C). After repeated consideration and attempt, Maxwell magically introduced displacement current. *

∂D JD= ∂t

*

(17:2)

Then, the previous contradiction was overcome. In this case, eq. (17.1) can be changed into * 8 * * ∂D > >

* > ∂B : ∇×E= − ∂t In this way, the electromagnetic mutual transformation was achieved – time varia* tion in the electric field ∂D=∂t could be transformed to space variation in the magnetic * * field ∇ × H : In addition, time variation in the magnetic field ∂B=∂t could also be * transformed to space variation in the electric field ∇ × E : It was the two-way transformation that held the key to the emergence of electromagnetic waves. Young Maxwell creatively derived the wave equations of electromagnetic wave. For the simple one-dimensional case, we have https://doi.org/10.1515/9783110527407-017

276

17 Electromagnetic inertia

*

*

∂2 E ∂2 E + εμ 2 = 0 2 ∂z ∂t

(17:4)

The wave velocity is light speed c, namely 1 c = pffiffiffiffiffi εμ

(17:5)

Maxwell’s prediction was verified by Hertz experimentally, altogether laying the key foundation for modern communication. Hertz’s two works, Transverse Free Space Electromagnetic Waves Traveling at a Finite Speed over a Distance published in January 1888 and Electromagnetic Radiation in December 1888, brought electromagnetic waves to people’s vision, as shown in Fig. 17.1 [56]. Sender

Receiver

Fig. 17.1: Hertz’s experiment.

All these discoveries contributed to the great success of Maxwell’s electromagnetic theories. Two forms of unity were achieved: the unity of light and electromagnetism, and the unity of dynamic field and static field as shown in Figs. 17.2 and 17.3 [57]. Hereafter, the new chapter of electromagnetic extensive study was opened.

Light waves EM waves

Fig. 17.2: Unity of light and electromagnetism by Maxwell.

Dynamic fields Static fields

ω

ω=0 Fig. 17.3: Unity of dynamic field and static field by Maxwell.

17.2 Electromagnetic inertia

277

17.2 Electromagnetic inertia Few people notice that there is another important property of electromagnetic waves, namely electromagnetic inertia. Generally speaking, electromagnetic inertia consists of the wave inertia in the dynamic field and the potential inertia and charge inertia in the static field. By inertia, we refer to the essential properties of objects without any constraint condition. As Newton pointed out, in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force [25]. The inertia mentioned is mechanical inertia, namely Newton first law. When * force F is 0, the essential state of objects is static or moving along a straight line at constant speed. Correspondingly, Christiaan Huygens put forward the original concept of electromagnetic inertia, which revealed the formation of waves and their essence. He pointed out that “the wavefront of a propagating wave of light at any instant conforms to the envelope of spherical wavelets emanating from every point on the wavefront at the prior instant (with the understanding that the wavelets have the same speed as the overall wave),” as shown in Fig. 17.4.

Old wavefront

cΔt

New wavefront

Wavelets

Old wavefront

cΔt

New wavefront

wavelets Case1

Plane wave

Case2 Spherical wave

Fig. 17.4: Huygens’ principle.

Obviously, Huygens’s principle has two highlights. (1) Concept of wavelet. Every point on the wave can be equivalent to an independent ideal source, which creates conditions for its propagation. (2) Concept of spherical wave. Every wavelet emanates circular (two-dimensional) or spherical (three-dimensional) waves, which can be seen as the free state of wave. As a Chinese proverb goes, “A tossed stone raises a thousand ripples.” If we drop a pebble in a calm pond, no matter how complicated the conditions are on the bank, a

278

17 Electromagnetic inertia

circular wave on the two-dimensional surface of the pond will emanate outward until they reach the bank, as shown in Fig. 17.5.

Fig. 17.5: Concentric circles of ripples.

This chapter discusses the Newton’s mechanical inertia and puts forward electromagnetic inertia: “under the electrostatic condition, potential φ or source σ will tend to keep the state of sphere (three-dimensional) or circle (two-dimensional) unless electromagnetic motions or boundaries change this state.” It can be said that the electromagnetic constraint is motion or the boundary, and the spherical state and the circular state are unconstrained free states. In fact, this idea can be further extended to other situations, such as water waves and sound waves, which are not discussed here.

17.3 Electrostatic inertia As mentioned earlier, Maxwell has united the dynamic field with the static field. The latter only does not change with time, so how does the dynamic field reflect the inertia in the electrostatic field when it has the characteristics of electromagnetic inertia? After research, it can be pointed out that a static object still has static inertia, which can be reflected in two aspects: potential φ distribution inertia and electrostatic charge σ distribution inertia.

17.3.1 Potential φ distribution inertia It is very coincidental that this important characteristic was discovered in a typical example occasionally [57]. It is well known that in an electric field, the conductor boundary is an equipotential surface. The study of the problem starts with the square coaxial line. According to the boundary conditions shown in Fig. 17.6, it is easy to know that the testing function of the potential is also square (see Fig. 17.6). However,

17.3 Electrostatic inertia

279

y

o

a

ka

x

Fig. 17.6: Potential φ trial function (k > 1) when the initial assumption coincides with the square conductor boundary.

it is surprising that the calculated capacitance C and the characteristic impedance Z0 based on this idea differ greatly from the exact value. The above contradiction gives us the opportunity to think deeply. Finally, we summarize the distribution inertia principle of the electrostatic potential φ. In the electrostatic field, it always has the tendency to form a circle or sphere once away from the potential φ of the conductor boundary. It is worth noting that limited by the boundary, it also struggles as much as possible between the ellipse and ellipsoid so as to obtain their own freedom in some cases. In short, the circular or spherical electrostatic φ distribution is an unconstrained free state, as shown in Fig. 17.7, which is fully verified by numerical calculations [57]. y

C1

o

a

ka

x

Fig. 17.7: Electrostatic inertia – potential φ distribution has the tendency of the approximation graph (k > 1).

17.3.2 Electrostatic charge Σ distribution inertia In the electrostatic field, the physical quantity corresponding to potential φ is the charge density σ. The charge distribution of the general shape conductor plate also

280

17 Electromagnetic inertia

has the above characteristics. The example in Fig. 17.8 shows that the charge density σ distribution also has a circular tendency.

Fig. 17.8: The charge density σ on any conductor plate also has a circular tendency.

Finally, we conclude the distribution inertia principle of electrostatic charge density σ again: “The charge density σ distribution on the electrostatic field conductor always has a tendency to form a circle.”

17.4 Green function and electrostatic inertia In the electromagnetic theory, the Green function is a generalized potential, as shown in Fig. 17.9.

L(u)=g L: linear operator

L: linear operator δ ( r − r ′ ) : δ source

g: common source

G (r

u: generalized potential required

r′

) : Green function, the

generalized potential of δ source

Fig. 17.9: Green function and introduction of generalized potential.

It is clear that the δ function can be seen as a wavelet source or subsource, and G is a wavelet. The specific process of transformation from the dynamic field to the static field electromagnetic inertia can be shown clearly in Fig. 17.10.

The dynamic field ∇2 G + k2 G = − 4πδ

The static field ∇2 G = − 4πδ

With wave characteristic * *′  * *  * *  G r = r ′ = e − jkð r − r Þ = r − r ′

Without wave characteristic * *  * *  G r = r ′ = 1= r − r ′

17.5 Any antenna element cannot constitute an ideal wavelet source

Green function Huygen’s principle

281

e–jk(r–rʹ ) |r–rʹ| G1(r/rʹ)

r 1ʹ . Wavelets

WAVE FRONT

r 2ʹ .

G2(r/rʹ)

r 3ʹ . G3(r/rʹ) Generalized Huygen’s principle

Green function

l |r–rʹ| G1 r 1ʹ . r 2ʹ .

Sourcelets

G2

r 3ʹ . Equipotential surface

G3

Fig. 17.10: The dynamic field and static field electromagnetic inertia.

17.5 Any antenna element cannot constitute an ideal wavelet source In the discussion above, there is a key point in electromagnetic inertia, namely isotropic, which indicates a spherical wave with the same amplitude in all directions, or a uniform spherical wave. This point is reflected in Green function * *′ * r e − jkj r − r j G * = * *  :  r − r ′ r′

(17:5)

However, in fact, what we encounter is completely another situation: Any antenna can not constitute an ideal subwave source that is uniformly radiated in all directions. The key of the problem lies in the polarization. Whether it is an electric dipole or a small current loop, as shown in Fig. 17.11. Their distributions are not isotropic, and their corresponding directivity coefficient is 1.5, that is, min G = 1:5

(17:6)

282

Il

θ

17 Electromagnetic inertia

r

θ

IS

r

Fig. 17.11: There is a polarized distribution of antenna elements: (a) electric dipole and (b) a small current loop.

We can speculate with certainty that this may also be the minimum directivity of any electromagnetic radiation.

17.6 Summary *

*

What Newton has brought to mankind is not only the law of gravity and F ¼ ma : From the mechanics point of view, he profoundly reveals one of the most important concepts of matter – inertia. “In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.” It essentially indicates that the inertia represents the free state of the object mechanics as well as an essential attribute of the object. The electromagnetic inertia proposed in this chapter can be said to be the corresponding characteristics of things in the electromagnetic field. Under the electrostatic condition, potential φ or source σ tends to keep the state of sphere (three-dimensional) or circle (two-dimensional) unless electromagnetic motions or boundaries change this state. It is very interesting to note that for mechanical inertia, in fact, infinite approximation can be achieved in the uniform linear motion. On the contrary, for motion electromagnetic inertia, it is impossible for the actual antenna (external source) radiation to reach isotropic spherical waves. The philosophical significance of this contrast is well worth further exploration. In contrast with Newton’s mechanics, the generalized electromagnetic inertia is discussed here.

Q&A Q: This chapter is titled “Electromagnetic Inertia.” How did you come up with this idea? A: A great many things achieve success from failure. As a saying goes, “Failure is the mother of success.” I have a deep understanding of it. Although we have given a brief introduction in the text, there are still a few words necessary to argue that the starting of the problem is very simple and clear – we wanted to study capacitance C and characteristic impedance Z0 of the square coaxial line per unit length a few years ago, as shown in Fig. 17.6. We need to assume the potential function ’ in the calculation. The most straightforward and easiest

Q&A

283

condition we can imagine is that it is the square shape from inside to outside, which is simple and satisfies the internal and external boundary conditions. However, the result should have failed! It was a great blow to me. Through a very complex intermediate process, we finally knew the interposition distribution function between the inner and outer squares that satisfy the boundary conditions has the tendency to be circle, but certainly not the standard circle. It shows clearly that electric and magnetic have the tendency to be circle (2D) and sphere (3D) once they are free. In other words: Circle and sphere reflect the electromagnetic inertia. Q: What is the relationship between the concept of electromagnetic inertia and Maxwell’s equations? A: You do catch the point. This sentence just caught the point of the problem. We still start with the mechanical analogy. Newton’s First Law and Newton’s Second Law are * * given in Fig. 17.12, F = ma : Newton’s First Law

Inertia law: In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Newton’s Second Law In an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma

Fig. 17.12: Newton’s first and second laws.

Here are three levels of analysis. First, many people have the wrong idea before the establishment of the equation by Newton. The force must exist when there is movement, an error which was first revealed by Newton. Movement exists without force imposed: uniform linear motion. Second, Newton’s First Law shows that the free state of an object is the inertia, and the inertia of mechanics is geometrically represented as a straight line (uniform straight line). Third, it is generally difficult to achieve the situation without force completely. Thus, the main inertia line is in an ideal and a free state in the Second Law equation, and the equation is the state of constraint of force. We need to further point out that Newton established two equations in his * * lifetime. Newton’s Second Law is F = ma ; where m is called inertial mass. The reason is that the conversion from the uniform straight line into the accelerated movement with the acceleration a, we must overcome the gravity of m. The greater the m is, the * greater the gravity is; the gravitation equation is F = kðMm=r2 Þ^r; where Newton calls * the m as gravitational quality again. Obviously, gravity F is proportional to m. Few people take these two kinds of definitions of quality seriously before Einstein. Only Einstein unifies the two definitions in general relativity.

17 Electromagnetic inertia

Maxwell’s equations дD ×H=J+ дt Δ

Electromagnetic inertia Under the electrostatic condition, potential φ or source σ tends to keep the state of sphere (three-dimensional) or circle (two-dimensional) unless electromagnetic motions or boundaries change this state.

Δ

284

×E=–

дB дt

Fig. 17.13: Electromagnetic theory and Maxwell’s equations.

Now, let’s look at the electromagnetic correspondence, as shown in Fig. 17.13. The comparison shows that if the free state of mechanics indicates that inertia is a straight line, so the free state of electromagnetic is circle or sphere, that is, electromagnetic inertia. We found that electromagnetic inertia tends to be isotropic after deep study. * * The mechanical equation F = ma is in constraint state, so does Maxwell’s equa * *  * * * tions ∇ × H = J + ∂D=∂t and ∇ × E = − ∂B=∂t: However, the most amazing thing is that there is no isotropic solution for Maxwell’s equations. This is also the “biggest contradiction” between the electromagnetic free state and the constrained state. The free state tends to be isotropic, while the restraining state rejects isotropic. We should make further exploration in this aspect. Here, it is necessary to mention Christiaan Huygens, who made a great contribution as the founder of wave science. The most important contribution is the idea of wavelet and spherical waves. However, when it involves in the field of electromagnetic waves, there occurs new contradiction – wave polarization. Polarization is one of the essences of electromagnetic wave source, and there is still inherent polarization even when the minimum directivity of dipole or the current loop is achieved. I think whether we can say or not that polarization is the most fundamental cause of * preventing wave isotropy. Just as the force F is most fundamental cause of preventing the object from the uniform straight movement. This makes the minimum directivity of the electromagnetic antenna less than 1. However, when it comes to the electrostatic field, both the potential ’ and the charge density σ distribution tend to be isotropic obviously, since the concept of polarization is not established in this case. In addition, for the generalized wave of Huygens, propagation of the waves, such as water waves and sound waves, depends on the medium, and Maxwell electromagnetic waves do not need any “ether.” Although this property has been proven in various experiments and relativity, the differences between the various waves are worth studying. Finally, the wave has not been found by gravity. How does it travel over such a long distance? Does wave travel take some time? We once again study the electromagnetic and force and explore the essential differences. The two curl equations of Maxwell’s equation truly complete the two-way conversion between electric and magnetic as well as the conversion between

Q&A

285

time and space in the true sense, as described in this chapter. The time variation in * the electric field ∂D=∂t can be transformed into space variation in the magnetic field *

*

∇ × H : The time variation in the magnetic field − ∂B=∂t can be transformed into the * space variation in the electric field ∇ × E : It is this harmonious two-way transformation that is the key reason for the emergence of electromagnetic waves. * In contrast, F = kðMm=r2 Þ^r; the gravity equation in Newton’s mechanics, seems to be “lonely” for lack of conversion. Perhaps, looking for transformational factors in gravitational problems is the possible research direction in the future. Q: According to the above discussion, can the generalized inertia be regarded as a general physical thought to examine the free state of various objects? A: In fact, the discussion of generalized inertia is far more than the physical aspect and can also be examined geometrically and spatially. The generalized electromagnetic inertia has the tendency of the sphere (3D), which indicates that three-dimensional space we are in is isotropic fundamentally. The inertia is closely related to the space in which we live. Q: What are the applications of electromagnetic inertia? A: The study of any problem has its two sides. The isotropic tendency of electromagnetic makes the blind side less for satellite as well as GPS so that we can receive it no matter where we are. In addition, human is able to develop a highly directional antenna that facilitates transmitting and receiving, for the polarization destroys the isotropy. On the military, missiles can even strike targets along the antenna beam. From this point of view, it can also be said that the strong directivity antenna is an important application of anti-electromagnetic inertia, as shown in Fig. 17.14.

(a)

(b)

Fig. 17.14: Applications of electromagnetic inertia: (a) the applications of electromagnetic inertia – we can receive wave no matter where we are (isotropic), and (b) “the inverse” applications of electromagnetic inertia – highly directional antenna point target (anisotropic).

Q: In this chapter, you discuss the concept of wavelet research by Green function. Could you further extend it?

286

17 Electromagnetic inertia

A: As described in this chapter, δ function can be regarded as wavelet source or subsource, and Green function G is seen as wavelet. The most important thing is that the Green function in the dynamic field and that in the static field differ only by a phase factor. * *

′ e − jkj r − r j * *   r − r ′

Here appears the phase factor again. At Princeton Institute for Advanced Study, one of the centers for physics development in the 20th century, the talented physicist Hermann Weyl had been trying to combine electromagnetics and gravitation (general relativity) together to form a unified field theory. It was Weyl who first proposed that field of local transformation needs to be given, also known as gauge transformation. He specifically wrote a transformation factor, but unfortunately, it was an exponential factor. Later, he realized it was not correct and the correct nonintegral phase factor was given after the deep research by Schrödinger Fork, London and other scientists. 0 1 ðq e Au dxu A exp@ − j
hc p

Finally, we got the correct gauge theory. Here, the comparison of the Green function will inspire us greatly in thought.

Recommended scholar

Fig. 17.15: Professor Wu Wanchun.

Here, I will introduce Wu Wanchun, professor at Xidian University and also an educator (Fig. 17.15). As I was a young teacher, he recommended me as a member of his tutor team. He had thought of me in the compilation of the Microwave Network and Its Applications. Here, I would first like to thank the generous help from the old educators.

Recommended scholar

287

As a microwave expert, Mr. Wu has always been working hard. His perfect manuscript and solid foundation have won him praise from all over the university. It was after the Cultural Revolution that Wu worked to open up some new and practical research fields at Xidian University, such as microwave network and circuit. His contributions will be remembered forever.

18 Beauty of electromagnetic theory This chapter discusses the beauty of electromagnetic theory. With simple beauty, symmetrical beauty, transformational beauty and unified beauty as the focus, we try to use the history of electromagnetic theory as a specific research object. Just as everything in the world has duality, beauty is no exception. Beauty can bring us benefits, but if you are not careful, it may also bring us problems. This is the duality of beauty in scientific research.

18.1 Introduction This is the 18th chapter of Electromagnetic Field Theory Teaching Series. At present, innovation is important in all areas of work. The source of innovation is clearly based on practice. But what puzzles us is that why some people are innovative, while others are not. A quotation from the famous sculptor Auguste Rodin may be appropriate: “Life is not lacking in beauty, but the eyes to discover beauty” [25]. It can be seen that people’s mind and taste are very important. Only when our taste is improved can we find and reveal the internal nature of things to innovate. This chapter attempts to use the history of electromagnetic theory as a specific research object, with simple beauty, symmetrical beauty, transformational beauty and unified beauty as the focus. It is worth noting that plenty of literature about the beauty of science often overemphasizes the merits of beauty. But everything in the world has duality, and beauty is no exception. It can be said that “science is beautiful, but beauty does not necessarily represent science,” or “beauty can bring us benefits and problems as well.” This is the duality of beauty. Even so, numerous scholars pursued and are pursuing beauty with all their efforts possible. In 1983, Chandra Sacca, Nobel Prize winner in physics, made a summary: “Everyone can be satisfied in the pursuit of beauty and science in our own way.”

18.2 The simple beauty of electromagnetic theory Tsung-Dao Lee, a Nobel Prize winner, has ever said many times, “The most important thing is often the simplest.” The situation is so true throughout the history of the development of electromagnetic theory. The simplest and most important thing is to set up the concept of field rather than ether. https://doi.org/10.1515/9783110527407-018

290

18 Beauty of electromagnetic theory

Chen-Ning Yang, also a Nobel Prize winner in physics, clearly pointed out in his famous speech Field and Symmetry that the field is one of the most important concepts in the 20th century. In fact, Hans Christian Oersted and Michael Faraday used small needles and ferrous powder, respectively, to prove the existence of magnetic around current I in almost the same experiment, as shown in Fig. 18.1.

I

I

Fig. 18.1: The discovery of field and force lines by Faraday.

It is this seemingly unobtrusive difference that made Faraday solemnly announce: There exists a magnetic field at any point in the space around a current, which can be represented by a force line vividly. Thus, Faraday established the space stage in the electromagnetic field, which makes the electromagnetism fundamentally different from the distance action mechanics put forward by Newton. It was the concept of field by Faraday that inspired Maxwell to think of waves, which is helpful to today’s global communications, time services and aerospace business. If it is not easy to build a simple and important concept, then it is far more difficult to abandon the simple and wrong concept. In the electromagnetic theory, the concept of ether is a case in point. The most commonly used method in science is comparison, as shown in Fig. 18.2.

The fluid in water wave

The air in sound wave

Dose electromagnetic wave needs ether?

Fig. 18.2: Comparison, our habit thinking.

The fluid in water wave and the air in sound waves are the carriers of the wave propagation. They are static. Waves propagate in the carrier, so (pay attention to people’s habitual thinking) electromagnetic waves must also have a carrier which we call ether. The discovery of the Special Theory of Relativity by Einstein is fundamentally against the ether. Electromagnetic waves do not require ether, which is the most important feature that distinguishes itself from other waves and is also the essence of

18.3 Symmetry beauty of electromagnetic theory

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the permanent principle of light velocity. The simple beauty in the colorful electromagnetic phenomena is reflected in the entire Maxwell system, and the specific details will be described below. As Einstein summarized, “The simplicity of natural laws is also a fact, and the right conceptual system must keep the subjective and objective aspects of this simplicity balanced.”

18.3 Symmetry beauty of electromagnetic theory The formation of an important field and system often tends to be accompanied by a pair of symmetrical or complementary figures, which are called perfect match by later generations, Newton, Kepler and Galileo, to name just a few. In the field of electromagnetic theory to be discussed, Faraday and Maxwell are recommended first. They are a truly perfect match. When Oersted discovered that there was magnetic field around a wire with current I, young Faraday was so excited and encouraged. He thought that according to the idea of symmetry, magnetic field should also be able to induce electric field since electric field can induce magnetic field. In 1822, Faraday stated a goal in his note for himself – turn the magnetic field into electric field [58]. It was this firm pursuit that took him 10 years (1821–1831) of his young time. Faraday (30–40 years old) was in his golden age. Initially, Faraday thought that using a strong magnet to approach the wire would * * produce a current I in the wire, or that there would be a new current I ′ in the near wire if a wire is with strong current. His symmetrical idea is shown in Fig.18.3. It is regrettable that countless experiments failed during the 10 years.

Oersted’s experiment electricity induces magnetic field

I

Faraday’s experiment magnetic field induces electric field

I

Fig. 18.3: Faraday’s symmetrical idea.

On October 17, 1831, Faraday still used wires to connect the galvanometer and the hollow coil. Thinking of thousands of failures, he was not hopeful for this test at all. But Faraday inexplicably inserted a magnet into the coil with a rush, and the current meter suddenly deflected. The first few times, he thought it was accidental. After many times of repeats, the “spring” after 10 years finally came – the law of electromagnetic

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18 Beauty of electromagnetic theory

induction was produced: not only electricity could produce magnetic field but also magnetic field (change of flux linkage) can generate electric field. Faraday was also delighted with the discovery of such an important achievement – the symmetrical idea [14,25]. And then, a great character in the “symmetry” – young Maxwell appeared on the historical stage. When he came on the stage and was determined to overcome the difficulties of the electromagnetic system, Faraday, as an eldership, was afraid that the mathematical form would “submerge” the essence of electromagnetic phenomena. On the contrary, Maxwell first discovered the asymmetry of Oersted and Faraday, as shown in Fig. 18.4.

The idea of Faraday’s experiment. Magnetic field − дB (change of дt flux linkage) could induce electric field.

The idea of Oersted’s experiment. Electricity (source) could induce magnetic field.

Fig. 18.4: The asymmetry of Oersted and Faraday.

In other words, Faraday used the symmetrical idea to find the asymmetry, and genius Maxwell boldly added Faraday’s symmetrical idea in the condition of * immature circumstances – that is the proposal of displacement current J D ; as shown in Fig. 18.5. *

JD ¼

Faraday’s experiment ×E =−

*

∂D ∂t

(18:1)

Oersted’s experiment

дB дt

×H =

*

дD +J дt

Fig. 18.5: Maxwell complements symmetry. *

Δ

It is not difficult to find that J in the right side has no symmetrical quantity J m – magnetic flow in the left frame in Fig. 18.5. In fact, owing to symmetrical idea, many scholars believe that there must be magnetic charge and magnetic current corresponding to the charge and current. The team led by Tom Fennel of the LaueLangevin Institute in France and the team led by Jonathan Morri of the German Helmholtz Center Berlin for Materials and Energy both reported particle and magnetic currents similar to magnetic monopoles observed in spin ice matter Dy2 Ti2 O7 in Science on September 14, 2009 and in Nature on October 15, respectively. The process of pursuing symmetry in the electromagnetic theory has continued.

18.4 The beautiful transformation of electromagnetic theory

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18.4 The beautiful transformation of electromagnetic theory The main goal of electromagnetic theory is to study the law of electric and magnetic changes. Here are Maxwell’s two curl equations [29]: *

*

∇×H ¼

∂D * þJ ∂t

(18:2)

*

*

∇×E ¼ 

∂B ∂t

(18:3)

18.4.1 The mutual conversion of electric field and magnetic field The left side of eq. (18.2) is magnetic field and the right side of it is electric field (the left side of eq. (18.3) is electric field and the right side of it is magnetic field), the equal sign being in the middle, which profoundly reveals the mutual conversion of electricity and magnetism. They are interdependent, mutually opposed of each other and coexisting in an electromagnetic field as a whole, as shown in Fig. 18.6. H E

H E

E

Fig. 18.6: The mutual conversion of electricity and magnetism.

18.4.2 The mutual transformation between space-varying and time-varying The left side of eqs. (18.2) and (18.3) is the curl (∇ × ) spatial operation and the right side is the derivative (∂=∂t) time operation. The equal sign in the middle profoundly further indicates that a space-varying electric field could transform into time-varying magnetic field, and vice versa, which is the source of electromagnetic waves, as shown in Fig. 18.7. In fact, there are various transformations in electromagnetic theory. In order to simplify the solution, we could transform the solving of the electric * * * field E and the magnetic field H into the solving of the magnetic vector A and the electric potential ’ (including the Lorentz constraint), turn Maxwell’s equations into the variational operation of the inequality and convert the electromagnetic scattering problem into optical calculation, and so all. All of these transformations show its strong beauty based on its magical faces.

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18 Beauty of electromagnetic theory

Space Time

Wave

Fig. 18.7: The space-varying and time-varying of waves.

18.5 The beautiful unity of electromagnetic theory Unity is the highest pursuit of many scholars in their research works. As a typical example, the great Albert Einstein boldly “wasted” (which is held by others rather than himself) more than 20 years of his life for the sake of “the unified field theory.” Throughout the history of the development of electromagnetic theory, all kinds of unity ideas prosecuted to the end. In the early days, Franklin, an American, flied a kite in the thunderstorms to prove the unity of atmospheric electricity and earth electricity. Moreover, the leader of electromagnetic theory Maxwell was even more so. In 1864, he read a famous paper A Dynamic Theory of the Electromagnetic Field to the Royal Society. Maxwell directly stated, The theory I propose may therefore be called a theory of Electromagnetic Field because it has to do with the space in the neighborhood of electric and magnetic bodies, and it may be called a Dynamical Theory, because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced. It was very courageous Maxwell who derived the wave equation from the equations. *

*

1 ∂2 E ¼0 c2 ∂t2

*

1 ∂2 H ¼0 c2 ∂t2

∇2 E  ∇2 H 

(18:4)

*

(18:5)

Maxwell boarded the academic peak – formally put forward the theory of optoelectronic unity. He said, “it is almost impossible for us to reject the following conclusion: light is made up of a medium transverse wave vibration, which is the cause of the existence of electrical and magnetic phenomena.” The most popular language is that light is electromagnetic waves with a very short wavelength, as shown in Fig. 18.8.

18.6 The duality of beauty

Electromagnetic wave

295

Light

Fig. 18.8: Unity of light and electromagnetic waves.

It is this discovery that gives us great inspiration: physics is objective, but there are subjective elements in man’s knowledge of physics. The unification of light and electromagnetic waves indicates that there is no difference between the two. However, light is visible for human beings, and electromagnetic waves cannot be seen for people. This chapter would like to put forward a sharp question: light and electromagnetic waves are unified, and light possesses wave particle duality, then where is the corpuscular property of electromagnetic waves? The pursuit of unity is beautiful, but the pursuit of unity is also extremely difficult.

18.6 The duality of beauty In fact, the concept of beauty in physics is not fixed – the meaning of beauty in physics is constantly developing. It is not only characterized by its simplicity, symmetry, unity and could be diverse, singular, noble and romantic. More importantly, beauty has duality, the same as any other things in the world. It is to say, beauty not only contributes to the rapid development of electromagnetic theory, and beauty will make researchers go astray if left unattended. In 1968, there was a climax to “invent” small antennas in the East China Sea: goldfish antenna, lotus antenna, grass antenna, flower basket antenna and so on. It is said to be varied and magnificent with “very good” performance. After all the samples were studied and tested, all the antennas were only loaded on the line antenna, and there was no any new principle in it. Thus, we cannot be spellbound by abovementioned “lively beauty.” To the late 1980s, the research on “electromagnetic missile” was heated, and it was said to able to break the law 1=r2 : It seemed beautiful in theory, but the test of time proved that there is no new principle. These lessons are worthy of our study, and especially the “nonacademic factors” which interfere with the academic should be resolutely resisted.

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18 Beauty of electromagnetic theory

18.7 Summary A few years ago, the “innovative teaching team” project of Ministry of Education brought us not so much joy as heavy burden of pressure in our hearts. How to innovate teaching and what innovation we could achieve was like a goal we could not grasp. It seemed that in the early spring, the green of life stimulated and toggled our nerves, and we thought of years of teaching experience, difficulties and doubts encountered in teaching. Suddenly, a bold challenge occurred in my mind. I am soberly aware that the desire to innovate is burning in the heart. Could we challenge the old ideas specifically? This kind of teaching idea is that teachers are condescending, preaching and even trying to pretend to be “the embodiment of truth.” And, the challengers should put forward teaching while learning, have a heart-to-heart talk and equal discussion these normal teaching ideas.

Q&A Q: I still do not understand why we should discuss beauty since the electromagnetic theory is a discipline of science and technology applied to the practice. A: This question can be considered from two aspects. In the first aspect, man’s need in the world for any practice must be sublimated. In the simplest case, human wear clothes to keep warm and conceal. But now the mainstream of most model clothes is pursuing beauty. Initially, human ate food only to fill his stomach; however, currently good food needs to be goodlooking, sweet-smelling and tasty. Science and electromagnetic theory is the same. It is constantly pursuing beauty as it is applied to practice! This is the sublimation of things. In the second aspect, we inspect great research masters’ brilliant views. –

H. Bondi (remembering Einstein). What impressed me most was that Einstein did not argue with me when I came up with an idea that I thought was justified, but only said, “Ah, how ugly!” As long as he thinks this equation is ugly, he will not have any interest in it. And he can not understand why some people would like to spend so much time on it. Einstein is convinced that beauty is a guiding principle in the search for important results in theoretical physics.

–

Einstein. He said: in the scientific thinking, there is always a factor in poetry, real science and real music requires the same thinking process.

Q&A

–

297

Weil (French social philosopher). The real theme of science is the world’s perfection.

–

Kovalevskaya (famous Russian mathematician). If one cannot be a poet in heart, then he cannot become a mathematician.

–

Chen-Ning Yang (famous American-Chinese scientist). If you can simplify or summarize many phenomena as some equations, then it is indeed a kind of beauty. What is poetry? Poetry is a highly condensed thought, is the essence of thought. A few lines could tell their own inner voice, bare their own ideas. The result of scientific research is also a beautiful poem. The equation we are looking for is the poetry that nature has given us. This is a beautiful poem. When we encounter these concentrated and pithy structures, we will have the feeling of beauty. When we find a secret of nature, a sense of fear arises spontaneously as if we were looking at something with reverence that we should not pay attention to.

–

Copernicus (Polish astronomer). In the Copernicus’s De Revolutionibus Orbium Coelestium, the first sentence is: in the science and art of nurturing a variety of human diverse talents, I think we should use all energy to study what is associated with the most beautiful things.

–

Bohr and Feynman even cleverly combined together mathematics, physics and music. A musician can distinguish between Mozart, Beethoven, and Schubert’s works by listening to the first several syllables. Similarly, a mathematician can also tell articles written by Cauchy, Gauss, Jacobi, Kirchhoff, and Helmholtz by reading the first few pages. French mathematicians are graceful and elegant; and the English, especially Maxwell, is amazed by his extraordinary judgment. For example, is there someone who doesn’t know Maxwell’s papers on kinetic theory of gases? . . . The variables of the velocity are shown at the beginning seriously, but the equation of state is cut from one side, and the motion equation of the central field is cut from the other side. The confusion of formula grows more and more. Suddenly, there are four syllables “n = 5” out of timpani. An unfortunate elf u (relative velocity of two molecules) cross-fades; at the same time, just like the melody, bass which has been very prominent suddenly become silent, and something that originally does not seem to be surmounted now is excluded after the waving of the wand . . . At this time, you don’t have to ask why or why not? If you can not understand the music, then put aside the article. Maxwell does not write the title music with notes . . . The conclusion comes one after another, and finally, the climax suddenly arrives: the heat balance conditions and the expression of transport coefficient appear, and then, the curtain falls.

–

Dirac (British theoretical physicist, one of the founders of quantum mechanics). The laws of physics must have the beauty of mathematics. People who study physics don’t have to worry about the meaning of the physical equations, and all they need to focus is the beauty of physical equations.

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18 Beauty of electromagnetic theory

Kant. Beauty should not be expressed in words.

We can conclude: science is not art, and science is art, its highest level is pursuing its beauty! Q: Is the beauty of science, or the beauty of electromagnetic theory subjective or objective? A: The question is really hard to answer. I can only talk about some views from my own understanding. Beauty has the objectivity. But the beauty of science tends to be deeply “latent,” and many things remain to be explored and extended. The “true” law of the universe must be the most beautiful. In addition, beauty must be subjective. Take the simplest dressing for example. For different nationalities, different ages and different occupations, the understanding of beautiful clothes varies. Therefore, the so-called esthetics or the appreciation of beauty is exactly the same in the field of science. Einstein once said: “In the field of mathematics, my intuition is not enough because it can not identify what are really important researches and what are just unimportant topics. However, in physics, I quickly learned how to find the basic problems to work.” The above discussion about subjectivity and objectivity is not just playing the word game, but it is of important practical significance, that is, using the subjectivity to judge the beauty of the objective things is likely to cause an accident, make a mistake or have a lesson! An old saying from childhood says: poisonous flowers tend to grow particularly bright, and toxic food is often particularly fragrant, and all of those are luring people to “get hooked”! In 1956, young Chen-Ning Yang and Chung-Dao Lee took over the issue about “parity conservation.” All people unanimously believe the conservation of parity. So beautiful, so good! However, there is an objective law in the world, and it will not transfer for the will of any person – finally, they were determined to declare that parity is not conservative, and it is further confirmed by Wu Jianxiong’s experiments. Thus, Y Chen-Ning Yang and Chung-Dao Lee both have won the Nobel Prize in physics. In particular, it should be pointed out that the greatest difficulty in solving this problem lies in the fixed psychological resistance about beauty. In fact, “In the universe created by God,” being not perfect is also a beauty! * The introduction of displacement current ∂D=∂t in Maxwell equations has a * symmetrical pursuit of beauty, but so far the magnetic flow J m has not been found, which proves that there is still “the breaking of symmetry.” Q: When studying the beauty of electromagnetic theory, what misunderstandings should be paid attention to?

Q&A

299

A: There are three main misunderstandings which need to be aware of. (1) Blind imitation Just as the popular clothing in streets, as long as the actors or celebrities wear, many enthusiasts (fans) will scramble for or follow it. The condition of science is the same. It is necessary to be clear that the essence of science beauty lies in innovation, so only innovation can create beauty, and blindly imitation can only lead to worse and worse. (2) Sheer imagination It is important to know that the beauty of science must be built on a solid foundation, which is based on practice, and beauty could only be created from reality. Whether it is predecessors such as Newton, Maxwell or Einstein is all so. When Kepler was establishing the law of celestial bodies, at the very beginning he started from the imagination, and then he can only use the regular polyhedron nested. Thus, the results of the many experiments did not agree with his observation data. After taking a roundabout way, he calmed down and started from the practice. Finally, he deduced the world-known three great laws of Kepler and also constituted the most important basis for the creation of Newton mechanics. (3) Subjective likes and dislikes Avoiding subjective likes and dislikes in research on scientific theory and the study of electromagnetic theory is a very important law, which also works for great scientist Einstein. When he was a student, Einstein not only despised but also hated “matrix theory.” He did not attend the class taught by mathematician Minkowshi, only called his disabled girlfriend to write down some notes. When Einstein’s Special Theory of Relativity was published (1905), his teacher Minkowshi attached great importance to learning it. He was amazed that Einstein what he called “lazy dog” even was so capable! It was Einstein’s teacher Minkowshi who applied the matrix theory to the magic place – the introduction of the space–time four-dimensional matrix ½x y z ict from which is derived the matrix Lorentz transformation, and all this is complex. Seeing the results of his teacher, Einstein sighed continually. He “handed over” such a great achievement owing to his dislike of mathematics. He could no longer go on and finally was determined to learn Riemannian geometry. A few years later, he created the most beautiful general theory of relativity independently. This example gives us great enlightenment. In order to cultivate our own esthetic ability of science, we must learn more, see more, start more from experience. We are opposed to blindly imitation, but we need to start from learn and imitation in order to innovate.

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18 Beauty of electromagnetic theory

Recommended scholar In this chapter, Mr. Huang Zhixun (Fig. 18.9), I think, is the most appropriate person to introduce. He has two notable features for us to learn. (1) Long-term study and hard work. From my personal feeling, people as hard-working as Mr. Huang are rare. (2) Long-term concern about science and the forefront of science.

Fig. 18.9: Huang Zhixun.

His own unique insights on the cutoff attenuator, super-speed areas and other fields and his fruitful achievements are admirable. I often come across an almost naive question in my mind: if there are more people like Huang Zhixun around us, maybe scientific progress will be much faster than it is now.

19 Some thoughts on the electromagnetic theory This chapter will discuss several basic problems. In present teaching and research, there is a lack of not only excellent textbooks but also pertinent remarks and in-depth thoughts. This part, it is hoped, can be helpful.

19.1 Introduction The electromagnetic theory discussed here is based on Maxwell’s theories, whose time table is shown below: Maxwell (1864–1865) 8 * * * > ∇  D = ρ; D = ε E ðKulun, 1785Þ > > > * * * > > ∇  B = 0; B = μ H ðMitchell, 1750Þ > > < * * * ∂D * * J =ρ υ ðAmpere and Maxwell, 1825Þ ∇× H = J + > > ∂t* > > > > * > ∂B > : ðFaraday, 1831Þ ∇× E =  ∂t It can be said that the modern electromagnetic theory has developed for more than 200 years. However, in recent two decades, due to the fast development of computers, we are capable of solving complex problems of boundary values. So, for the classic electromagnetic theory, is there anything that we can study, think about and develop? The answer is affirmative and this article will briefly discuss this issue based on several questions.

19.2 Symmetry and asymmetry Symmetry dominates interaction. —Chen-Ning Yang It is asymmetry that creates phenomena. —Pierre Curie

As a mature discipline, there is much literature related to the research and discussion of the electromagnetic theory, including its development and contributions by some prominent scholars. However, if we consider it from the perspective of scientific methodology, we will find that the idea of symmetry acts as a mainline throughout the history of electromagnetic theory and plays an important role in it. Dr. Chen-Ning Yang called the field and symmetry “two physical concepts of great importance in the 20th century.” This chapter briefly reviews the development history of the electromagnetic theory and discusses its symmetry in detail. https://doi.org/10.1515/9783110527407-019

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19 Some thoughts on the electromagnetic theory

19.2.1 The “first handshake” between electricity and magnetism In history, the studies of electricity and magnetism had existed as two independent branches for a long time. It seems that people knew magnetism earlier than electricity. Although western scholars tend to illustrate De Magnete by William Gilbert as regard to magnetism, we should also be aware that the compass, as one of the four great inventions of ancient China, has always been the pride of Chinese nation. Nevertheless, the magnetic field of the earth was the major cause that started the development of magnetism. The early studies of electricity include electrification by friction, the uniformity test of lighting and static electricity conducted by Benjamin Franklin and Coulomb law. The man who first connected electricity with magnetism is the Danish scholar – Hans Christian Oersted. In 1820, he found that an electric current produces a circular magnetic field as it flows through a wire. The experiment he conducted was plain and straightforward. The most extraordinary thing about this experiment is the realization that magnetism can be generated from electricity (or current I). In this way, these two previously independent branches are connected, thus opening up a new chapter for the development of electromagnetics (the word was created by Oersted himself). There are some details that should not be ignored. The magnetism generated in Oersted’s experiment is actually magnetic field. Although Oersted did not have the concept of field, he used a piece of paper to isolate current from magnetic needles, but the needles still deflect. And, the concept of field passed him. But we should also acknowledge that the success of Oersted’s experiment originates from his acute observation as well as the conditions at that time (i.e., the current I generated by voltaic batteries produces a magnetic field stronger than geomagnetic field and deflects magnetic needles). In the history of physics, facts of advanced ideas that cannot be proved for the lack of experiment conditions are very common.

19.2.2 Faraday’s symmetry concept The electromagnetic induction discovered by Michael Faraday has always been one of the most successful examples that people enjoy talking. Faraday gained his talent by self-study. Having a direct, clear and straight scientific character, he was the first to propose the idea of force line and field and he stick to that all the time. Many researchers have noticed that the electromagnetic induction was discovered consciously. Faraday accepted the instruction of symmetry idea. According to the symmetry of natural law, he thought, now that Oersted proved magnetism can be generated from electricity, it was likely that electricity can be generated from magnetism. It was the firm belief in symmetry that helped Faraday insist on this research area despite all those failed experiments in the depressed 10 years.

19.2 Symmetry and asymmetry

303

The morning of August 29, 1831 was indeed a memorable time in the history of physics. The discovery of electromagnetic induction was finally accomplished as the condition of electromagnetic field motion and conductor wire cutting magnetic lines was finally satisfied. Those countless failures in the past 10 years came to fruition. The interesting part is that the experiment designed by Faraday, who insisted field all the time, tended to generate current I rather than field. So even in modern perspective, this is a great discovery. However, there is indeed something that should not be ignored. The electromagnetic induction discovery by Faraday is different from what he thought. In the original design, Faraday did not take the motion of magnet or coil into consideration. He thought that magnetism can be generated from current, so the symmetrical idea should be current generated from magnetic stream (Fig. 19.1). But regretfully, a magnet is not the same as magnetic stream and has never been found. Oersted’s idea

Electricity I

Generated magnetic field H

Faraday’s symmetry concept

Magnetic stream M

Generated electrical field E

Fig. 19.1: The symmetrical idea of electromagnetic relationship.

Faraday possessed the most important character to be a great physicist – seeking truth form facts. He was good at adjusting his ideas from experiments and performing further research. Finally, he found that the key to electricity generation was magnet* ism varying with time (∂ B =∂t).

19.2.3 The “second handshake” between electricity and magnetism For most people, the connection between electricity and magnetism was accomplished after Faraday’s discovery. But it is not true. James Clerk Maxwell, 40 years younger than Faraday, was the first to notice the problem and did intensive research. It was a historical coincidence that Maxwell was born in 1831 when electromagnetic induction was discovered. Maxwell himself also said that he had never done a single experiment but studied all Faraday’s experimental data. History endowed him an important job to accomplish a great theoretical synthesis. The reason that Maxwell could fulfill his job comes from two aspects. On the one hand, the experiment foundation mentioned earlier provided a strong support for him. On the other hand, Maxwell had profound philosophical thinking as well as solid mathematical foundation.

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19 Some thoughts on the electromagnetic theory

There are two great contributions that Maxwell had made regarding the electromagnetic theory. 1. Making full use of Faraday’s idea of force line and field, and performing beautiful mathematical expressions. 2. Going deep into the discovery and analysis of the asymmetry between Oersted and Faraday’s research. In addition, he applied the symmetry idea as guidance and came to a conclusion. That is, since Faraday found that time-varying mag* * netic field (∂ B =∂t) generates electric field ( E ), it is likely that time-varying * * electric field (∂ D =∂t) generates magnetic field ( H ). That is the famous notion of displacement current (as shown in Fig. 19.2). Faraday’s electromagnetic induction discovery

Time-varying magnetic field (дB/дt)

Generates electric field (E )

Maxwell’s symmetrical speculation

Time-varying electric field (дD/дt)

Generates magnetic field (H )

Fig. 19.2: Maxwell’s symmetrical speculation.

It is worth pointing out that this is actually the “second handshake” between electricity and magnetism. Different from what Oersted found, the first symmetrical discovery of electromagnetism was accomplished in this time. It is meaningful to learn Maxwell’s equation set. 8 * * *