Introduction to Electromagnetic Fields and Waves [1 ed.]

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INTRODUCTION TO ELECTROMAGNETIC FIELDS AND WAVES

A SERIES OF BOOKS IN PHYSICS Editors: HE NRY M. FOLEY A N D MALVIN

A.

R U DERMAN

Concepts of Classica l Optics John Strong Thermo physics Allen L. King Introduction to Elec tromagnetic Fields and Waves D a le R. Corson and Paul Lorrain Modern Quantum Theory Behram Kur~unoglu On the Interaction Between Atomic Nuclei and Electrons (A Golden Ga1e Edi1io11) H.B. G . Casimir X-Ra y Diffrac1ion in Crysta ls, Imperfect Crysta ls, a nd Amorphous Bodies A. Guinier Symmetry Principles at High Energy : Coral G ables Conference Behram Kur~unoglu and Arnold Perlmutter, Editors

Spacctimc Physics Edwin F. T aylor and John Archibald Wheeler

INTRODUCTION TO ELECTROMAGNETIC FIELDS AND WAVES

Dale R. Corson Paul Lorrain

coR N ELL u N1vERs1TY

uN 1VER s 1TY or ~10 N TREAL

W. H. FREEMAN AND COMPANY SAN

FRANCISCO

AND

LONDON

© Copyright 1962 by W. H. Freeman and Company The publisher reserves all rights to reproduce this book, in whole or in part, with rhe exception of the righr to

use short quotations for review of the book.

Library of Congress Catalog Card Number: 62-14193 Printed in the United States of America

(C6)

PREFACE

This introduction to electromagnetic theory is intended for students having a good background in elementary electricity and in differential and integral calculus. It should also be useful for scientists and engineers who wish to review the basic concepts and methods of the subject. It has been taught for many years by the authors both at Cornell University and at the University of Montreal at the advanced undergraduate level. To reduce the mathematical requirements, we have included discussions of various topics in mathematics which are essential to a proper understanding of the text. For example, the first chapter deals with vector analysis in Cartesian and in other coordinate systems. There is also an appendix on the technique which involves replacing cos wt by exp (jwr) for dealing with periodic phenomena, and another on wave propagation. The solution of partial differential equations by the separation of variables and the solution of the Legendre differential equation are discussed in Chapter 4. The mathematical level may be adjusted to a considerable extent either by delaying or omitting various sections, as indicated in footnotes . Our aim has been to give the student a good grasp of the basic concepts and methods. This meant covering fewer subjects more thoroughly. For example, we have dealt with the solution of Laplace's equation in rectangular and spherical coordinates, but not in cylindrica l coordinates. For the same reason, the discussion of electromagnetic waves in the second half of the book is concerned with only three types of media : dielectrics, good conductors, and lowpressure ionized gases. The discussion has been kept as systematic and as thorough as possible, and hazy " physica l" arguments have been avoided. We have also stressed the internal logic of the subject, and we have clea rly stated all assumptions . Thi s approach should accelerate the learning process and help the student gain self-confidence. In most cases we have illustrated the theory by means of examples which are accompanied by figures showing the main characteristics of the fields. V

vi

PREFACE

The illustrations have all been designed and executed with exceptional care, in the hope that they would give the student a much better grasp of the subject than would the usual blackboard-type sketch. Wherever possible, the illustraa lions are quantitative, and three-dimensional objects or phenomena are represented as such. In particular, fields such as those shown in Figures 4-9 and 13-10 are quantitative. Short summaries are provided at the end of each chapter to give a general view of the subject matter, and, throughout the book, new results are tied in with previously acquired results so as to consolidate, as well as to extend, the reader's knowledge. The problems form an essential part of the book. Many are designed to help the student extend the theory expounded in the text. They should be done with care. A few words are in order concerning the units and the notation used in this book. As to the units, we have used exclusively the rationalized m.k.s., or meterkilogram-second system. A table is provided in Appendix C for converting these units into the various c.g.s. systems. The notation used is that suggested by the Commission on Symbols, Units, and Nomenclature of the International Union of Pure and Applied Physics, with the following three exceptions. We have used the term "dielectric coefficient" and the symbol K,, instead of the relative permittivity •/ , 0 • We have also used K., instead ofµ / µ0, but have called it the relative permeability, as suggested by the Commission. Thus , has been replaced by K,, 0 and µ by K .. µ 0 , which has the advantage of stressing the effects of the properties of the medium. In the latter part of the book , however, from Chapter 10 on , this consideration is pedagogically less important and we have used , and µ so as to simplify the notation. Finally, we have used the operators v, v ·, and v X instead of grad, div, and curl, respectively. The v notation is pedagogically much preferable to the other for Cartesian coordinates. For example, a student recogn izes immediately that v-vV = v'V and that v -v X A = 0 (the two top rows of the determinant are identical), whereas div grad V and div curl A seem quite meaningless. The student is made to realize clearly that the operator v can be defined only in Cartesian coordinates. The exponential function for periodic phenomena can be chosen to be either exp (}wt) or exp ( - }wt), since the real part of both of these functions is cos wt. We have chosen the positive exponent. This, we believe, is essential at this level; otherwise, in circuit theory, an impedance Z becomes equal to R - jX instead of the conventional R + jX. We have also used extensively the radian length Ji = ),.j21r instead of the wave length A. This cons iderably simp lifies the calculations. The situation here is similar to that with respect to the frequency f. In both cases, the quantity which

Preface

vii

has an intuitively obvious meaning, namely, frequency and wave length, is not the one which enters into the calculations, but rather 21rf = w and "/,./ 2,r = i\ . The book starts with Coulomb's law and ends with the electromagnetic field of a moving charge. As indicated below, there are many possible ways of utilizing only part of this material. The first chapter deals with vector analysis in Cartesian, orthogonal curvilinear, cylindrical , and spherical coordinates. The following three chapters then cover electrostatic fields , first in a vacuum, and then in dielectrics. These are investigated at length from the "molecular" point of view. Chapters 5 and 6 deal, respectively, with the magnetic fields associated with constant and with variable currents. The first is based on the force between two current-carrying circuits, whereas the second is based on the Lorentz force on a charged particle moving in a magnetic field. Chapter 7 contains a discussion of magnetic materials which parallels to a certain extent that of Chapter 3 on dielectrics. At this point, all four of Maxwell's equations have been found, and Chapter 8 is devoted to a short discussion of these equations, as well as to some new material which follows directly from them . Chapters 9 to 14 are all based on the Maxwell equations. The first subject discussed is the propagation of plane electromagnetic waves, in a vacuum, and then in dielectrics, in good conductors, and in low-pressure ionized gases. Chapter 11 then discusses at length the phenom~na of reflection and refraction at the interface between two dielectrics and between a dielectric and a good conductor. The reflection in a low-pressure ionized gas such as the ionosphere is also discussed at some length. This chapter should make the student thoroughly fami liar with the use of Maxwell's equations. The next chapter is concerned with guided waves. It covers some general considerations and includes a discussion of two relatively simple cases, that of the coaxial line and that of the TE wave in a rectangular wave guide. Finally, the last two chapters deal with the radiation of electromagnetic fields. They are based entirely on the electromagnetic potentials V and A, which, in turn , follow from Maxwell's equations. Chapter 13 covers some general considerations and then discusses electric and magnetic dipole and quadrupole radiation. Chapter 14 contains a simplified discussion of the fields of moving charges using only elementary methods. This chapter should provide a useful background for a course in relativity. Many will be of the opinion that the book contains more material than can be discussed thoroughly in the time avai lable in class. There are, however, many ways in which parts of the book may be omitted without losing continuity. For example, a relatively elementary course could be li mited to the first eight

viii

PREFACE

chapters leading up to and including a discussion of Maxwell's equations. Or, for a more advanced course, the first seven chapters could either be omitted or reviewed rapidly by making use of the summaries provided at the end of each one. Chapters 3 and 7, on the molecular approach to dielectrics and magnetic materials, could be omitted if necessary. This would limit the discussion to the macroscopic theory and the Maxwellian point of view. It would not be advisable, however, to omit Chapter 3 and not Chapter 7, since the latter rests rather heavily on Chapter 3. Chapter 11 , "Reflection and Refraction," could be treated briefly ; it is not essential to Chapter 12, " Guided Waves." Chapter 12 is instructive both because of the insight it provides into electromagnetic wave propagation and because of its engineering implications, but none of its results are required for the following chapters. Chapter 13, " Radiation of Electromagnetic Waves," is of course fundam ental, but it should not be studied without having a thorough grasp of Chap ters 8 and 9. Chapter 13 is necessary for a proper understanding of Chapter 14, " Electromagnetic Field of a Moving Charge." It is a pleasure indeed to acknowledge the help of the many persons who cooperated in producing this book. We are indebted to Mr. Gilles Cliche, who was responsible for the numerical calculations required for the many quantitative figures. The K'A surface (Figure 13-10) was drawn at his suggestion. Mr. Gaelan Marchand also took part in the calculations and helped in the preparation of the manuscript. All the illustrations were designed and partly executed by Mr. Paul Carriere. The cooperation of Miss Evanell Towne in putting them in their final form was much appreciated. The typing was ably done by Miss Huguette Boileau, Mrs. Judith Barnes, Mrs. Marjorie Kinsman, Mrs. Therese Fournier , and Mrs. Yolande LeCavalier. Last but not least, we are indebted to the many students , both at Cornell University and at the University of Montreal , who provided the main incentive and many stimulating discussions. June 1962

DALE

R. CORSO N

PAUL LORRAIN

CONTENTS

Chapter

1.

I. I. 1.2. - 1.3.

-

-

Vector Algebra

I

The Time Derivative

5

The Gradient

6

1.4.

Flux and Divergence. The Divergence Theorem

8

1.5.

Line I ntegral and Curl

1.6.

Stokes's Theorem

15

1.7 .

The Laplacian

17

Curvi linear Coordinates

17

ELECTROSTAT I C FIELDS I

28

- 1.8.

Chapter

VECTORS

2.

10

2.1.

Coulomb's Law

28

2.2.

The Electrostatic Field Intensity

29

2.3.

The Electrostatic Potentia l

30

2.4.

Gauss's Law

33

2.5 .

The Equations of Poisson and of Lap lace

36 37

-

2.6.

Conductors

-

2.7 .

Fields Produced by Some Simple Charge Distributions

38

2.8 .

The Dipole

48

2.9.

The Linear Quadrupole

51

2.10.

Multipoles

51

2.11.

The Electrostatic Potentia l V Due to a11 Arbitrary Charge Distribution

52

2.12.

The Average Field Intensity Inside a Sphere Containi11g an Arbitrary Charge Distribution

57

2.13.

Capacitance

59 ix

X

CONTENTS

2.1 4. 2.15. 2.16.

3.

Potential Energy of a Charge Distribution Forces on Conductors Summary

66 71 75

DIELECTRICS

82

- 3.1.

The Polarization Vector P

- 3.2.

Field at an Exterior Point Field at an Interior Point

83 84 87

The Local Field

93

Chapter

3.3. 3.4. - 3.5. 3.6. - 3.7. 3.8. 3.9.

The Displacement Vector D The Electric Susceptibility

x,

Calculation of Electrostatic Fields Involving Dielectrics The Clausius-Mosotti Equation Polar Molecules

96 99

IOI 105 107

3.10. 3.11.

Frequency Dependence of the Dielectric Coefficient

112

Solid Dielectrics

113

3.12.

Nonlinear, Anisotropic, and Nonhomogeneous Dielectrics

113

3.13.

Potential Energy of a Charge Distribution in the Presence of Dielectrics

114

3.14.

Forces on Dielectrics

3.15.

Summary

116 122

ELECT ROST A TIC FIELDS II

129

Chapter

4.

- 4.1. - 4.2.

Boundary Conditions

130

The Uniqueness Theorem

- 4.3.

Images

132 134

Images in Dielectrics. Charge Near a Semi:infinite Dielectric

143

General Solutions of Laplace's Equation

147

4.4. - 4.5. 4.6.

Solutions of Laplace's Equation in Spherical Coordinates. Legendre's Equation. Legendre Polynomials

4.7.

Solutions of Poisson's Equation

154 167

4. 8.

Summary

170

xi

Contents

Chapter

5.

MAGNETIC FIELDS OF STEADY CURRENTS

176

Magnetic Forces

176

5.2.

T he Magnetic I nduct ion B. The Biot-Savart Law

178

5.3 .

The Lorentz Force on a Point Charge Moving in a Magnetic Field

182

5.4.

The D ivergence of the Magnetic Induction B

184

5.5.

The Vector Potentia l A

186

5.6.

T he Line Integra l of the Vector Potential A Over a Closed Curve

190

5.7.

The Conservation of Charge and the Equation of Continuity

191

The Charge Density

19 1

- 5.1.

--- 5.8.

p

in a Conductor

The Divergence of the Vector Potential A. The Lorentz Cond ition

192

5. IO .

The Curl of the Magnetic Induction B. Poisson's Equation for the Vector Potential A

194

5.1 1.

Ampere's Circuital Law

197

5.12.

The Magnetic Scalar Potential Vm

202

5. 13.

The Magnetic Dipole Moment of a Current Loop

209

5.14.

Ampere's C ircuita l Law a nd the Sca lar Potential

21 I

Summary

2 11

5.9.

- 5. 15.

Chapter

6.

INDUCED ELECTROMOTANCE AND MAGNETI C ENERGY

219

The F araday Induction Law

219

6.2.

Induced Electromotance in a Moving System

226

6.3.

Inductance and Induced Electromotance

230

6.4.

Energy Stored in a Magnetic Field

239

6.5.

Self-inductance for a Volume Distribution of Current

245

6.6.

M agnetic Force

246

6.7.

Magnetic Torque

252

6.8.

Summary

252

-- 6.1.

xii

CONTENTS

Chapter

Chapter

Chapter

MAGNETIC MATERIALS

259

7. 1.

The Magnetic Polarization Vector M

259

7.2.

The Magnetic Induction from Polarized Magnetic Material at an External Point

260

7.3.

The Magnetic Induction from Polarized Magnetic Material at an Internal Point

265

7.4.

The Magnetic Field Intensity H

276

7.5.

Measurement of Magnetic Properties. The Rowland Ring 280

7.

7.6.

Hysteresis

282

7.7.

Magnetic Data for Various Materials

284

7.8.

Boundary Conditions

285

7.9.

Magnetic Field Calculations

287

7.10.

Maxwell's Fourth Equation

297

7. 11.

Summary

299

MAXWELL'S EQUATIONS

304

8. 1.

Maxwell's Equations

304

8.2.

Maxwell's Equations in Integral Form

307

8.3.

E-H Symmetry

310

8.4.

Lorentz's Lemma

310

8.5.

Summary

3 11

8.

9.

- 9.1. - 9.2 . 9.3.

Chapter 10.

IO. I.

PLANE ELECTROMAGNETIC WAVES IN FREE SPACE

315

Electromagnetic Waves in Free Space

315

The Poynting Vector in Free Space

321

Summary

324

PROPAGATION OF PLANE ELECTROMAGNETIC WAVES IN MATTER

326

The Wave Equations for the Field Vectors E , D, B, and H for Homogeneous, Isotropic, Linear, Stationary Media

327

xiii

Conienrs

10.2.

Propagation of Plane Electromagnetic Waves in Nonconductors

329

10.3.

Propagation of Plane Electromagnetic Waves in Conducting Media

331

10.4.

Propagation of Plane Electromagnetic Waves in Good Conductors

336

10.5.

Propagation of Plane Electromagnetic Waves in Ionized Gases

34 1

10.6.

Summary

352

REFLEC TI ON AND REFRACTION

357

Chapter JI.

I I. I.

The Laws of Reflection and Snell's Law of Refraction

358

11.2.

Fresnel's Equations

361

11.3.

Reflection and Refraction at the Interface Between Two Dielectrics

365

11.4.

Total Reflection at an Interface Between Two Dielectrics

372

I 1.5.

Reflection and Refraction at the Surface of a Good Conductor

384

I I .6.

Radiat ion Pressure

396

11.7.

Reflection of an Electromagnetic Wave by an Ionized Gas

400

11.8 .

Summary

403

GUIDED WAVES

409

12.1.

The General Case of a Wave Propagating in the Positive Direction Along the z-Axis

409

12.2.

Coaxial Line

418

12.3.

Hollow Rectangular Wave Guide

420

12.4.

Summary

433

Chap/er 12.

Chap/er 13.

RADIAT I ON OF ELECTRO MAGNETIC WAVES

438

13.1.

The Electromagnet ic Potentials V and A

43 8

13.2.

E lectric Dipo le Radiation

446

13.3.

Radiation from a Ha lf-wave Antenna

46 1

xiv

CONTENTS

13.4.

Antenna Arrays

468

13.5. 13.6.

Electric Quadrupole Radiation Magnetic Dipole Radiation

471 473 479

13.7.

Magnetic Quadrupole Radiation

13.8.

The Electric and Magnetic Dipoles as Receiving Antennas

13 .9.

The Reciprocity Theorem Summary

13.10. Chapler 14.

14.2.

485

ELECTROMAGNETIC FIELD OF A MOVING CHARGE

14.1.

480 481

491

The Electromagnetic Potentials V and A for a Line Charge Moving with a Constant Velocity along Its Length 491 The Lienard-Wiechert Potentials for a Small Moving Charge 494

14.3.

The Field Vectors E and H for a Small Moving Charge

497

14.4.

Summary

506

Notation

509

APPENDIX B.

Vector Definitions, Identities, and Theorems

512

APPENDIX C.

Conversion Table

514

APPENDIX D .

The Complex Potential

515

APPE N DIX E.

Induced Electromotance in Moving Systems

526

APPENDIX F.

The Exponential Notation

533

APPENDIX G.

Waves

537

APPENDIX A.

INDEX

549

CHAPTER

1 Vectors

We shall discuss electric and magnetic phenomena in terms of the fields of electric charges and currents. For example, the force between two electric charges will be considered as being due to an interaction between one of the charges and the field of the other. It is therefore essential that the student acquire at the very outset a thorough understanding of the mathematical methods required to deal with fields. This is the purpose of the present chapter on Vectors. It is important to note that the concept of field and the mathematics of vectors are essential not only to electromagnetic theory but also to most of present-day physics. We shall assume that the student is not familiar with vectors and that a thorough discussion is required. Mathematically, a field is a function which describes a physical quantity at all points in space. In scalar fields this physical quantity is completely specified by a single number for each point. Temperature, density, and electrostatic potential are examples of scalar quantities which can vary from one point to another in space. For vector fields both a number and a direction are required. Wind velocity, gravitational force, and electric field intensity are examples of such vector quantities. Vector quantities will be indicated by boldface type; lightface type will indicate either a scalar quantity or the magnitude of a vector quantity. We shall follow the usual custom of using a right-hand coordinate system as in Figure 1-1: the positive z direction is the direction of advance of a right-hand screw rotated in the sense that turns the positive x-axis into the positive y-axis through a 90 degree angle.

1.1. Vector Algebra A vector can be specified by its components along any three mutually perpendicular axes. In the Cartesian coordinate system of Figure 1-1, for example, the vector A has components A,, A, , and A,.

2

VECTORS

[Chap. l ]

y

Figure 1-1

A vector A and the three vectors Ari. A yi, A, k, which , when placed end-toend, are equivalent to A.

The vector can be uniquely expressed in terms of its components through the use of unit vectors i, j , and k, which are defined as vectors of unit magnitude in the positive x, y , and z directions, respectively:

A = iA,

+ jAu + kA,.

(1-1)

The vector A is the sum of three vectors of magnitude A., Ay, and A,, parallel to the x-axis, y-axis, and z-axis, respectively. It is clear that the magnitude of A is given by (/-2) A = (A ,' + A/ + A,')'"· The sum of two vectors is equal to the sum of their components :

A + B = i(A,

+ B,) + j(Au + Bu) + k(A, + BJ.

(1-3)

Subtraction is simp ly addition with one of the vectors changed in sign:

A - B =A+ (-B) = i(A, - B,)

+ j(A" -

Bu)+ k(A , - B,).

(1-4)

Figure 1-2

Two vectors A and B in the xy-plane. The vector C is th eir vector product A X B.

[I.I]

Vector Algebra

3

There are two types of multiplication: the scalar, or dot product; and the vector, or cross product. The scalar, or dot product, is the scalar quantity obtained on multiplying the magnitude of the first vector by the magnitude of the second and by the cosine of the angle between the two vectors. In Figure 1-2, for example,

A-B = ABcos(P.(cos 8).

n=O

Figure 4-25. A ring of radius a carrying a total charge Q.

(4-193)

We shall proceed as we did in the latter part of Section 4.6.1. On the axis, where 8 = 0 and r = z, we have P,.(cos 8) = 1 and Bu B1 B-, (4-194) V(z, 0) = -; + -;, + + · · ·.

1

We can, however, calculate the potential on the axis directly from Coulomb's law , and if we expand the resultant expression in inverse powers of z we may match coefficients term by term with Eq. 4-194 to determine the B,.'s. The axis thus provides us with the equivalent of a boundary condition. Following this procedure, we have V(z 0) '

=

41rEu(a2

Q

+ z2)1 12

= _Q_ ( I 41r1: 0

z

+ ~)2 z

12 ' ,

(4-195) (4-196)

[4.7]

167

Solutions of Poisson's Equation

On matching coefficients with Eq. 4-/94, we find that Bo= _Q_, 4ir,o

(4-197)

B, = 0,

( 4-198)

B., ·

= _ _Q_~,

(4-199)

B,

= 0, ···,

(4-200)

4ir,o 2

and , from Eq. 4-193, V(r 0) '

I 2I ar (32 cos-,0 - -2I) + · · ·]·

n [= 4rr, _x_ r 0

2

-

- - 3

-

(4-201)

Figure 4-26 shows the equipotential lines in this case. The components of the field intensity may be found , as usual , by ca lculating -V V.

Figure 4-26

Equipotentials for a charged ring. None are shown in the vicinity of the ring, where they are too close t ogether

to be depicted grap hically. At about two diameters from the ring the equipotentials are approximately circular, and the field is quite similar to that of a point charge.

L/.( 7.

Solutions of Poisson's Equation

We have as yet dealt on ly with solutions to Laplace's equation, since we have concerned ourselves on ly with cases in which the charge density p is zero. As we pointed out earlier, however, there are important fields in which a space charge exists and in which p is not zero. For these, we must find a solution of

168

ELECTROSTATIC FIELDS II

[Chap. 4]

Poisson's equation, and again the solution must be consistent with the boundary t:onditions which obtain in the particular problem. We have already shown in Section 4.2 that the solution is unique. 4.7.1 . The Vacuum Diode. As an example of such a field let us find the poten-

tial distribution between the plates of a vacuum diode whose cathode and anode are plane parallel surfaces separated by a distance which is small compared to their linear extent. The anode is maintained at a positive potential V0 relative to the cathode whose potential we shall take to be zero. The cathode is heated in order that electrons will be emitted thermionically and will be accelerated toward the anode under the action of the electric field. We shall assume that the electrons are emitted with zero velocity and that the current is not limited by the cathode temperature but can be increased at will by increasing V0 • Since the electrons move in the space between the plates with finite velocity, they constitute a space Figure 4-27. A plane-parallel vacuum charge whose density p is given by diode. The cathode is grounded, and the anode is at a potential V 1 . An electron of charge e moves to ward the plate with

p

= J-, u

(4-202)

where J is the current density in amperes/ meter' at a point where the electron velocity is u meters/ second. The space charge density p is then measured in coulombs/ meter' ; since the electron carries a negative charge, p is negative. The current density J is also negative, since we take the velocity u as positive for motion from cathode to anode, as in Figure 4-27. Since the potential can depend only on the coordinate x in the direction perpendicular to the plates, Poisson's equation reduces to

a velocity u.

d'V = _E.. (4-203) dx' Eo Thus the second derivative of V with respect to x is everywhere positive, since p is a negative quantity, and , for a given potential difference between the plates, V is everywhere lower than the corresponding free space value. Expressing p in terms of the current density J and of the velocity u, we have d'V J (4-204) dx'

[4.7]

169

Solutions of Poisson's Equation

where we now take J to be the magnitude of the current density , as read on a meter, without regard to sign . From the conservation of energy, the velocity u is given by

!2 mu'=

eV

(4-205)

'

where m is the mass of the electron. Substituting this value of u into Eq. 4-204 gives 11 cl'V = !_ '. dx2 " is given by 2:\a

4rr,0 V = --;- Po(cos 0)

2:\"' P,(cos 0) + s,:, 2:\a' +~ P,(cos 0) + · · ·.

4-18. With the electrode arrangement of Section 4.5.2 calculate the potential at x = 0, using the first five terms of the series. Perform the calculation for y = 0. lb, 0.2b , and so on . Repeat the calculation at x = b, using only the first term of the series. 4-19. With the electrode arrangement of Section 4.5.3 calculate the potential at x = 0, using the first five terms of the series. Calculate V for y = O. lb, 0.2b, and so on. Set V, = V,. 4-20. Show that V = -£0 [1 - (a'/ r')]r cos 0 is the potential in the vicin ity of a grounded, infinite, circular cyl indrical conductor introduced into a previously uniform electrostatic field of intensity E0 , provided the axis of the cylinder is perpendicular to E,. 4-21. A small hemispherical bump is raised on the inner surface of one plate of a parallel-plate capacitor. Find the resulting potential between the plates. 4-22. A grounded conducting sphere of radius a has point charges + Q and - Q situated on an extended diameter at distances D > " to the right and left, respectively. Find the magnitude and position of the image charges within the sphere. Now let D and Q approach infinity in such a way that Q/ D' remains constant. Find the potential and field outside the sphere due to the image charges. Find the field in the vicinity of the sphere due to the charges ±Q. Superposition of the two fields gives the field due to all the charges. Find the surface charge density induced on the outside of the sphere. This is another way of calculating the field around an uncharged, conducting sphere introduced into a previously uniform field. 4-23. A charge Q is uniformly distributed throughout the volume of an ellipsoid of revolution whose semi-major axis is a and whose semi-minor axes are b. Find the electrostatic potential at an y point in space outside the ellipsoid. 4-24. A hollow dielectric sphere with inner and outer radii a and 2a, respectively, and dielectric coefficient K, = 3 is placed in a previously uniform field E0• Show that E = (27 / 34)E,, in the hollow.

CHAPTER

5

Magnetic Fields of Steady Currents Our discussion of electromagnetic field theory has been limited so far to the effects of charges at rest. We shall now discuss in this chapter the fields produced by charges in uniform motion, that is, by steady currents. In general , our results will also be applicable to time-dependent currents, provided that the rates of change are not too great. We shall point out the limits of applicability of our results where necessary.

5.1. Magnetic Forces It is common laboratory experience that circuits carrying electric currents exert forces on each other. For example, the force between two parallel wires carrying currents /" and I, is proportional to I.I,/ p , where p is the distance between the wires (Section 5.2.2). The force is attractive if the currents flow in the same direction and is repulsive if they flow in opposite directions. There is good evidence that all magnetic effects have their origin in charges moving in some way, the motion being either a translation, as in an electric current, or a rotation about an axis, as in a spinning electron. In the general case of a pair of currents, as in Figure 5-1, the force which one current exerts on the other when both are in free space is given by a more complex expression , but it is again proportional to the product I . I,. It bas been found experimentally to be F = ~II J:,. J:,. di. X (dh X r,), (j-J) ab r'!. 471'" a b 'J:i

'Jt

where F., is the force exerted on current I. by current I,, and where the line integrals are evaluated over the two wires. This is the magnetic force law. The vectors dlu and di, point in the directions of positive current flow; r, is a unit 176

[5 .1 ]

177

Magnetic Forces

vector pointing from di• to di,,; and r is the distance between the two element s di, and di.. The force is measu red in newtons, the current in amperes, and the lengths in meters. The meaning of the above double integral is as follows. We choose a fixed element di., on circuit a and add the vectors di, X (di. X r1)/ r' correspondTb ing to each element di, of circuit b. We then repeat the operation for all the other elements di, of circuit a and, finally , calculate the overall sum (In general, this integration cannot be performed analytically. We then divide the circuits into small finit e elements and evaluate the sum numerically). The constant µ 0 is called the permeability of free space and is arbitrarily taken to be exactly

41r X 10- 1 newton/ a mpere' in rationalized m.k.s. units. Since the constant of proportionality Figure 5-1. Two currents l a and 1• . is fix ed with abso lute accuracy, Eq. 5-1 is used to define the magnitud e of the a mpere, as we shall see later on. ILis also used to define the co ulo mb, since the ampere is a current of one coulomb/ seco nd. Coulomb's law , Eq. 2-1 , gave us the force of interaction between stat io nary charges. The magnetic force law now states the force between charges moving with uniform velocities. In both laws there are co nstants of proportionality, , 0 and µ 0 , and it is the latter which is defined arbitrarily. The coulomb is thu s defined not from Coulo mb's law but from the magnetic force law. Thus it turns out experimentally that the va lue of , 0 in Coulomb's law mu st be 8.85 X 10- 12 farads/ meter, as stated previously. The force F,, , is exp ressed above in such a fashion that di. and di• do not play symmetrica l roles. This is quite disturbing, since, from N ewton's third law, we expect F, . to equal F., . The force F,. can be expressed in a sy mmetrical a nd somewhat simpler form by expa nding the triple vector product under the integral:

di, X (di• X r1) = dh(dl,• r1) _ r1(dl. ·dh). r' r' r'

(5-2)

We can now show that the double integral of the first term on the right is zero:

~ ~di la 'ft

ri· dl,, = ~di ~!:!·di b

r'!.

'ft

b

ra

f'!.

a,

(5-3)

178

MAG N ETIC FIELDS OF STEADY CURRENTS

[Chap. 5]

or, from Problem 1-21, 1

(5-4)

¢.~di, r\~ • = -~di,¢. v (~) · di.,

where the gradient involves derivatives with respect to the coordinates of a point on circuit a. Then , from Stokes's theorem,

J., J.,

'fa 'ft

dlo ri. ~I,, = r

J., di, ( v tb }sa

X v

(l)r ·da,

=

0,

(5-5)

where S, is any surface bounded by circuit a. The last integral is zero, since the curl of a gradient is identically equal to zero. We are thus left with the double integral of only the second term for the triple vector product di, X (dlo X r1), and

F = -~ l I 4,r

ab

a b

J., j._ ri(dt-dl,)_

fa J,,

r2

(5-6)

We now have F,,, = -F,,, since the unit vector r 1 is directed toward the circuit on which the force is to be calculated, with the result that it is oriented in one direction for F, , and in the opposite direction for F00 • Newton's third law therefore applies.

5 .2. The Magnetic Induction B. The Biot-Savart Law Despite the fact that the above integral for F,, is simpler and more symmetrical than that of Eq. 5-1 , it is not as interesting. The reason is that, with the above integral , the force cannot be expressed as the interaction of current a with the field of current b. We can perform such an operation on Eq. 5-1, however, since

F,, = I ,

¢.

di, X

=I.¢. di.

(i lo¢ di , r1), r~

X B,,

(5-7) (5-8)

where (5-9)

can be taken to be the field of circuit b at the position of the element di, of circuit a. The vector Bis called the magnetic induction. It is expressed in webers/ meter', the weber being a volt-second: I weber/ meter' = 10·1 gauss.

[5.2]

The Magnetic Induction B. The Biot-Sa vart Law

179

The gauss is not an m .k.s. unit, but it is frequently used because of its convenient order of magnitude. The above equation for B is ca lled the Biot-Savart !a,v. The integration can be performed analytically only for the simplest geometrical forms. It shows that the element of force dF on an element of wire of length di carrying a current / in a region where the magnetic induction is B is given by (5-10) dF = /di X B. If the current/ is distributed in space with a current density J amperes/ meter', then / becomes J da and must be put under the in tegra l sign. Then J da di can be written as J cfr, where cfr is an element of volu me. Thus, in the general case, the magnetic induction B at a point in space is given by B =

/4u :'.Q. 1

7

J X r, I r2

CT ,

(5-ll)

where the integration is carried out over any volume r which includes all the currents. It may well be asked here whether the a bove integral can be used to calculate B at a point inside a current-carrying co nductor. Since r is the distance between the point of observation where B is measured and the point where the current density is J, it appears, at first sigh t, that the contribution of the local current density will be infinite because of the 1/r' factor. The integral does not, in fact , diverge; it does apply within current carrying conductors. This can be seen by analogy with electrostatics, where the sa me problem arises in calculating the electric fi eld intensity E in side a charge distribution. The components of B and of E both vary as I /r'. Since those of E do remain finite within a charge distribution , those of B must a lso remain finite within a current distribution . It is assumed that both the charge den sity and the current density are finite. Just as in electrostatics, where we co nsid ered lines of force to describe an electric field , we can draw lines of magnetic induction which are everywhere tangent to the direction of B. Sin1ilarly, it is convenient to use the concept of flux , the flux of the magnetic inductio n B through a surface S being defined as the normal component of B integrated over S: cJ>

= ( B-da.

)s

(5-12)

5.2.1. The Magnetic Induction Due to a Current Flowing in a Long Straight Wire. In a long straight wire carrying a current ! , as in Figure 5-2, a n element

I di of the current will produce a magnetic induction dB as shown in the fi gure:

dB=µ ,! dlsin . 41r ,2

(5-13)

180

MAGNETIC FIELDS OF STEADY CURRENTS

[Chap. 5]

Figure 5-2

The 111ag11e1ic induction dB produced by an element I di of an infinitely long straight current. The vector dB lies in a plane 1/rat is perpendicular to the wire and which passes through P.

d.f J__

Expressing di, sin, and r' in terms of the angle 0, we find that B

= -µ i, l

4~p

J

+'

-~

2

/ cos 0 c0 = -µof ·

2~p

(5-14)

The magnitude of B thus falls off in versely as the.first power of the distance from an infinitely lon g wire and is in the direction perpendicular to a plane containing the wire. The lines of B a re circles lying in a plane perpendicular to the wire and are ce ntered o n it.

p

Figure 5-3

Two long parallel wires carrying currents in 1he same direction. The elem ent of force dF acting on the e lement din is in the d irection shown.

[5. 2] Tl,e Magnetic Induction B. Tl,e Biot-Savart Law

181

5.2.2. Force Between Two Long Parallel Wires. Definition of the Ampere.

Let us now exa mine the force between two infinitely long parallel wires carrying currents / 0 and l b, separated by a distance p , as in Figure 5-3. The current l b produces a magnetic induction Bb as in the above equation at the position of the current /, . The force dF acting on an element d/0 of this current is then

dF = //di. X Bb),

(5 -15)

dF = I, dl, µofo,

(5-16)

2rrp

and the force per umt len gth is

cj£ - µ, l ,, f.. di, -

(5-/7)

2rrp

The force is attractive if the currents are in the same direction and is repulsive if they are in opposite directions. This equation provides us with a definition of the ampere: two long parallel wires separated by a distance of one meter exert on each other a fo rce of 2 X I0- 1 newton per meter of length when the current in each is one ampere. This assumes that the diameters of the wires a re negligible compared dB to their separatio n. 5.2.3. The Circular Loop. As a second example of the ca lculation of the magnetic inductio n vector, we shall determine the magnitude an d direction of B on the ax is of a circular loop of radius a ca rrying a current / , as in Figure 5-4. Points off the axis will be considered in Section

5.12 .3.

Figura 5-4. The magnetic induction dB produced by an element I di at a poitll on the axis of circular currelll loop of radius a. Th e projection of dB on the axis is dB;:.

An element I di of current produces a magnetic inducti on dB as indicated in the figure. By symmetry, the total magnetic induction will be along the axis, and we need to calculate only dB,: dB. ~

=

µ, I cjj cos 8

41r r?.

'

(5 - /8)

hence B _ !!!>!_2rra

,-4,r r' cose.

(5-/9)

182

MAGNETIC FIELDS OF STEADY CURRENTS

B,

=

2(a'

[Chap. 5]

+ z')'"

(5-20)

The magnetic induction is maximum in the plane of the ring and drops off as z3 for z2 >> a' .

5.3. The Lorentz Force on a Point Charge Moving in a Magnetic Field Let us now return to Eq. 5-8. The force on a current element / di is

dF = /di X B

(5-21)

when the magnetic induction is B. Now, in terms of the current density J and of the cross-sectional area da of the conductor, /di= J da di, (5-22)

= J dr,

(5-23)

where dr is an element of volume with cross-sectional area da and length di. Then dF = J X Bdr, (5-24) or

dF dr

=

J X B.

(5-25)

If we set

J = nQu,

(5-26)

where n is the number of charge carriers per unit volume, Q is the charge on each one, and u is the average velocity of each carrier, then

I dF

~ dr

= Q(u X B).

(5-27)

Since n dr is the number of charge carriers in the volume element dr, dF / (n dr) is the force per charge carrier, which we shall call f. The magnetic force on an individual charge is therefore

f = Q(u X B).

(5-28)

This is known as the Lorentz force. It is perpendicular both to the velocity u and to the magnetic induction B. If we add to th is the force which may arise from the presence of an electric field E, the total force on a charge Q moving with a velocity u in both an electric and a magnetic field is (5-29) f = Q [E (u X B)].

+

5.3.1. The Parallel-plate Magnetron. As an example of the motion of a

charged particle under the action of both electric and magnetic fields, let us

183

[5.3 ] The Lore/1/z Force on a Point Charge consider the parallel-plate magnetron field E = V/ s is established between together with a uniform magnetic field B perpendicular to E. Electrons of charge - Qare released with negligible velocity from the lower plate and are accelerated toward the upper plate by the electric field. Since the magnetic force is always at right angles to the velocity u, the electrons describe a curved path. Let us find the velocity u of an electron as a function of the coord inates, of the potential V of the a node, and of the magnetic induction B. We consider an electron moving with a velocity u as indicated in the figure . The electric force f, and the magnetic force f,, are as indicated , thus

f, =

(5-30)

QBuu,

shown in Figure 5-5. A uniform electric the plates of a parallel-plate capacitor,

X

XB

X

X

X

Figure 5-5. Para lle l-plare mag netro n. The

lower plate is gro unded , a nd th e upper plnte is maintained at a potential V. An electro n of c harge -Q m oves with a ve/oci1y u . Th e e/eclric fi eld exeris a f orce fe 0 11 the e lectro n; and th e magne tic field , a fo rce f 111 •

or 111

du,

dt =

Q

(5-31)

Buu.

Integrating this, we obtain

- cit= - B lo ' dt o loo'du, Q

n1

u cit u

u, = Q By . 111

'

(5-32) (5-33)

The x-component of the velocity is thus proportional to y . It is zero at the cathode, and it increases with y. We can now find the y-component of the velocity u from the conservation of energy. Since the magnetic force is always perpendicular to the velocity, it does no work and does not contribute to the kinetic energy of the electron. Then

lmu'

2

= Q_!'.: y

s '

(5-34)

where m is the electron 's mass, which we assume to be the rest mass. Thus 1 ( .,

.,)

2111 u; + u;

V = Q -; y,

(5-35)

184

MAGNETIC FIELDS OF STEADY CURRENTS

u2 = _Q II

111

(2 _!:'. y- QB' y')· m

S

[Chap. 5] (5-36)

They-component of the velocity is zero at the cathode. It increases at first; and then decreases to zero for a ma ximum y given by

V m Ymax = 2 -; QB2 •

(5-37)

At this distance the electrons move toward the right in a direction parallel to the surface of the cathode. The magnetic force is then directed toward the cathode, and the electrons curve back to it, the y-component of their velocity still being given by Eq. 5-36. It can be shown that the trajectory described by an electron is a cycloid. It is interesting to consider the case in which the maximum value of y given by Eq. 5-37 is just the distance s between the plates. The potential of the anode is then (5-38)

If V is larger than this critical value, all of the electrons emitted by the cathode are collected by the anode. However, if V is smaller than V""' the electrons never reach the anode but return to the cathode.

Figure 5-6. A current element I dl at a source point P' produces an

element of magnetic induction dB at a fi eld point P.

5.4. The Divergence of the Magnetic Induction B. We shall show that the divergence of the magnetic induction B is always zero. According to the Biot-Savart law, Eq . 5-9, B=

/!!!l i+, di 41r

r

X r,,

,2

(5-39)

and (5-40)

185

[5.4] The Divergence of the Magnetic Induction B

where B is the magnetic induction at a field point P(x, y , z) and the element of conductor di carrying the current / is at a source point P'(x' , y', z'), as in Figure 5-6. Now it makes no difference whether we form the vector sum (di X r,)/ r'

at the point P where we compute Band then take the divergence of the resultant vector or whether we compute first the divergence of the elementary vector (di x r,)/ r' and then sum the resultant scalar quan tity over all the circuit elements di. That is, the differentiation and integration operations are interchangeable, thus V ·B

=

µ 0/

4,r

/4_ V 'Y

. (di >:

,-

r1)·

(5-41)

From Problem 1-23 , we have V . (di

X ~) = ~. r r

(v X di) -

di.

(v X r~)-

(5-42)

The first term on the right is zero, since di is not a function of the coordinates x , y , and z of the field point P where we wish to calculate v •B. Also, from Problem 1-21 , (5-43)

= 0,

(5-44)

since the curl of the gradient is always zero . Thus both terms on the right-hand side of Eq. 5-42 are zero, and . from Eq. 5-40, "v"-B

=

0

(5-45)

always. This is the second of the four Ma xwell equations. The first one we found was Eq. 3-58. The present one follows directly from the Biot-Savart law for the magnetic force between currents. The net flu x of magnetic induction through any closed surface is always equal to zero, since lsB-da=

Jcv-B)dr.

= 0.

(5-46)

(5-47)

This result follow s from the definition of the magnetic induction B, which itself is deduced from the empirical law describing the forces between current elements. There are no so urces of magnetic induction at which the divergence of B would

186

MAG NE TIC FIELDS OF STEADY CURRENTS

[Chap. 5]

be different from zero, and, from Gauss's law (Section 2.4), there are no free magnetic charges corresponding to the free electrical charges in an electrostatic field.

5.5. The Vector Potential A We have seen in Chapter 2 that the electrostatic field intensity E can be derived from the potential V through the relation E = -VV. We shall now show that the magnetic induction B is related to a certain quantity A through the equation B = V X A, where the vector A is called , by analogy, the vector potential. This is an important quantity ; we shall have occasion to use it repeatedly both in this chapter and in Chapter 6. Later on , in Chapter 13 we sha ll find that V and A play fundamental roles in electromagnetic theory. According to the Biot-Savart law, stated in Eq. 5-9, the magnetic induction B at the point P of Figure 5-6 is given by B = µ 0/ 4,r

,{_

'f

di X r,. r'

(5-48)

Now, from Problem 1-21, (5-49)

where the gradient operator implies differentiations with respect to the coordinates x , y , and z of the field point P. Then ! B = µ, 4 rr

1- V (') 'f ; X

di.

(5-50)

We have removed the minus sign by inverting the order of the vectors in the vector product. We may transform this equation through the vector identity of Problem 1-22: V X (JC)

= /(v X C) - (C X v/).

(5-51)

Taking the scalar function to be 1/ r and the vector function to be di, we obta in (5-52)

where the v X operators again involve derivatives with respect to the coordinates x , y , and z of the field point P. Since di is not a function of these coordinates, V X di is zero, and (5-53)

To obtain B, we therefore compu te V X

~r at the point P for

every element

[5.5 ]

187

The Vee/or Po1e111ial A

of the circuit and then add all the resulting vector elements together. Again , it makes no difference if we interchange the order of differentiation and integration , thus (5-54)

=

V X

(~':¢~),

(5-55)

= V XA,

(5-56)

A =~ ,{.,~

(5-57)

where

4rr'f r

is the veclor polelllial measured in webers/ meter. If the current is distributed with a current density J, then (5-58)

This integral , li ke that for B, appears to diverge inside a current-carrying conductor because of the I /r factor , but it actually does not. This ca n be seen from the fact that its components vary as I /r, like the electrostatic potential V, which does not diverge within a charge distribution. Equation 5-56 relating B and A is a completely general result and is true under all circumstances, even for time-dependent currents and for points inside a conductor , where the current den sity is finite. The result expressed as Eq. 5-45, namely, that

v-B = 0 under all circumstances, follows immediately from the fact that B can be expressed as the curl of the vector potential A, since the divergence of the curl of a vector is always zero, as we saw in Sectio n 1.6. The vector potential A is not uniquely defined by Eq. 5-57 or 5-58 in that we can add to it a ny term whose curl is zero without changing the value of B in any way . We have no reaso n at the moment to add such a term , however , and we shall use Eq s. 5-57 and 5-58 for defining the vector potential. This point wil1 be discussed furth er in Chapter 13. It must also be noted that B involves only the space derivatives of A, and not the va lue of A itself. The va lue of Bat a given point can thus be calculated only if A is known in the region around the point considered. 5.5. 1. The Long Straight Wire. W e have already found the magnetic induction B for this case in Sect ion 5.2.1, starting from the Biot-Savart law; we shall now calculate it from the vector potential A.

188

MAGNETIC FIELDS OF STEADY CURRENTS

[Chap. 5]

Each element / di of the current contributes to the vector potential an element

dA = µ,l_c!!,

(5-59)

4,r r

where dA is in the same direction as di. The elements / di thus all contribute elements dA in the same direction. From the fundamental definition of the curl in dA terms of line integrals (Eq. 1-69), and from the azimuthal symmetry of the p field , v X A = B is in the azimuthal direction around the conductor. For an infinitely long conductor, dA is proportional to di/ / for values of I 1 where r » p. Thus A -+ oo logarithmically . At first sight this is disturbing, but th e fact that a function is infinite does not necessarily mean that its derivatives are also infinite ; that is, B can be d£ finite even though A is infinite. Let us calculate A and B for a current of finite length, and then we can let the length go to infinity . Referring to Figure 5-7, we have

The element of current J di of a long straight current produces an element of vector potential dA at the

figure 5-7.

(5-60)

point P.

(5-61)

For L' » p', we can expand the square root in terms of p'/L' and keep only the lowest order term. Then 4 A, = µ,I In [ I (5-62) 4,r p2

+

L'J·

To compute B = V X A, we use cylindrical coordinates, keeping in mind that A is parallel to the z-axis and is independent of ed ek:Gtr:-i~· fidid Jn.t.e:m .ltv :o.o~ :;c~n;. '.LilkliJg th'0 Jl.r1t; in~e&n-1l! fl:tt X t.hr; d,ockriLst:; di:recho:)~ R.nd Jrn tb.~ dinx:tkm r:b.n,·-:,_,n h1 the ·,If.E;Ulti~ ~:;pe,; 1

y

X

@')

l

X

~fg~r;ce:., 0> TI A cct:dt-:c:it!t,;8' wlre ah slfoics wifir. c ·'J ,~:fcr:i.1.y JJ :i..•k;-,ng t:!1'1·1,;dfr"'.i,,-:tiJ·tg: ;·,"1{l's fr-1 a. ,"',tgio:';, of urdfovm m:'1.3i-'?!tk f.ndi..:2rlozr,; m. '][ h(,: JL..,Jttn.U, fo re?: 0/% th-!: (-!i'r:Ct?"Oi~/.] ,ii~; th!: Wfr":2 pvDduces rtl cu.rrr,~·~t J i~ 1-ic! cir~ c;.dtc TJ1,,~ ,f::Zec.tnJ,'!f-fr:.: cih:arg,t: ts tc:JikE:\i 10 c

bt! Q.

'fJ1e rn~:.Enit1(k:; tht; 1::lgbt ~Ert.J i:Bdir; fo t:be proC'.ui::t of thze:, sr~re.:i ~rs~e_pt b.)· the \){ii'.r-e ;_Jer u::r1it. tirne and the 111.iltl,'.· i1el.ic iodu.cti.J"i:~1 ft. fa tbt:15 fhf; HJ,!:.~·• nrJ,ir; flnx ~IV/Cpt r~:r ouh tU:nt:: h:;nc•~ r

.jJ .:S•!lE

~

]It fa im;_::;o:rta:.1~.t tn note tt.1i::t the direr!tion in l11bid;: thr; :'.inc i:n.t.Ggrl?J :b r:;Y:ilt:atc·l, an::_: tl.te drl'c:cliGD. '.m which t.hc '.ftr.rn. ® :i~ htk.~n to be ;;,rxdtf.v,e, .~,re :rt:~,,>J:.ted rr1.Ge0:rdfo.~ tbr: rjg)~1t-J:.1a.:nd scn•;•,N :~u~,e.

,.-_,r;o1cnc,?./ '-'iil1.e.re g is tJ1e im]ur;e,j ele.r:tnk: :field intensLty. TJr.e ~11.1-rent :tJ.r:\q:{Ltt tl[tr:::rn.g:h tbe cil'cuh :1s o:pJ.a.1 to tb0 :\J:1du1cr;;J dr;etro1Jl•:,::ita:rr.cc; diwidt.;d by tbs.~ n~s:ista:acf; oJ ~:tr.c: cl~tcr.Jit -cit tha~ ir.1on::1r::n.1\ r.xactly ,i,~s if tbe ir:tdU(!ed :~f. /d: :in 8\Ich cates. 'l:b.::.s t.ypf:: of b:r.ducr.: 11), where Nit> is the total flux linkage.

6.1.3. The Faraday Law in Differential Form . The Faraday law as stated in Eq. 6-6 says nothing about the induced electric field intensity E itself. It only

gives the electromotance in the complete circuit. Using Stokes's theorem to transform the line integral into a surface integral, we have ( (v X E)-da = _ !!._ ( B -da,

}s

cit

J.s

(6-9)

where S is any surface bounded by the closed integration path. For the case in which this path is fixed in space, we may in tercha nge the order of differentiation and integration on the right-hand side. Then ( (V X E) · da = -

}s

( ilB . da,

}s

ill

(6-10)

where we have written the partia l derivative of B because we now require the rate of change ofB with time at a fixed point. In a later section we sha ll exam ine the more general case in which the path moves . The restriction to a fixed path means that the electric fie ld intensity E and the magnetic induction vector B must be measured in the same coordinate system. Since the above equation is va lid for arbitrary surfaces, the integrands must be equal at every point, thus

VXE=

ilB

(6-11)

ill

This is the third of the four Maxwell equations. The other two which we have found so far are Eqs. 3-58 and 5-45. Equation 6-11 is a differential equation wh ich relates the space deriva tives of the electric fie ld intensity Eat a particular point to the time rate of change of the magnetic induction Bat the same point. The equation does not give E itself unless it can be integrated. 6.1.4. The Induced Electric Field Intensity E in Terms of the Vector Poten-

E induced by a changing magnetic field can be expected to be related to the vector potential A of Section 5.5. The relationship can be found as follows. Since

tial A. The electric field intensity

B=V XA

(6-12)

always, then , from Eq. 6-11, V X E

= - .!!_ (V X ill

A)

'

(6-1 3)

224

INDUCED ELECTROMOTANCE AND MAGNETIC ENERGY

(Chap. 6]

or, interchanging the order of differentiation, V

XE= -v X ~ ,

at

(6-14)

+ ~~)

= 0.

(6-15)

V X (E

The term between parentheses must equal a quantity whose curl is zero. Thus

E = -~ - vV

at

'

(6-16)

since any vector whose curl is zero can be represented as the gradient of a scalar function. For steady currents, A is a constant, and Eq. 6-16 reduces to Eq. 2-10 of electrostatics : E = -VV. (6-17) This E is the electrostatic field intensity, which, because it is of no interest to us at the moment, we shall disregard . We sha ll return to Eq. 6-16 for the general case in Chapter 13. Thus the induced electric field intensity at a point is given by (6-18)

We could also have arrived at this same result in the following way. According to the Faraday induction law, the induced electromotance for any given path is given by /4 dif> (6-19) 'f E-dl = -dt ,

_!!__ /4A,dl dt'f '

(6-20)

from Eq. 5-83. Interchanging the order of differentiation and of integration on the right , which we may do , since we are considering fixed paths, we find that

¢E-dl = -¢~~·di.

(6-21)

We have used a partial derivative under the integral sign because the time derivative of A must be evaluated at a given point on the path. Since this equation must be valid for any arbitrary path ,

E = -~

at

(6-22)

at each point, as long as Eis an induced electric field intensity. 6.1.5. The Electromotance Induced in a Loop by a Pair of Long Parallel Wires Carrying a Variable Current I. As an illustration, let us consider Fig-

[6.1]

225

The Faraday Induction Law

/[\ Ia

A

Figure 6-4

Pair of parallel wires carrying a cur~ rent 1 in opposite directions in the

plane of a closed rectangular loop of wire. When the current I increases, 1/ze induced e/ectromotance gives rise to a current I' in the direction shown. The vector potentials A and the induced electromotances -OA/ iJt are shown on the vertical wires. The induced current I' flows in the counterclockwise direc-

E

tion because E is larger on tl,e left tl,an on the rig/,/.

ure 6-4. The wires carry a current I in opposite directions, and I increases at the rate dl/ dt. We shall first calculate the induced electromotance from the Faraday induction law, Eq. 6-6, and then from Eq. 6-21 . The current I in wire a produces a magnetic induction in the azimutha l direction such that

B = B!__ u

(6-23)

21rpu

A similar relation exists for wire b. The flu x through the loop is thus if>=

f; (1'"+"' It dp,, -1"+"' It dp' ) • r

Pu

9

=

µ 0h/

In [ro(r,,

21r

r,,(r ,

+

(6-24)

Pb

fb

iv)],

(6-25)

+ w)

and it points int o the paper, as shown in Figure 6-4. The induced electromota nce is given by

J. E

'f

di = _µ 0/t

clj_ In [ ro(r,,

21r dt

=

µ 01,

r,,(r,

+ 1v)],

(6-26)

w)]·

(6-27)

+

clj_ In [r.(r, + ro(r,, + iv)

21r cit

1v)

It is negative, and , from Lenz's law, produces a current I' in the direction shown in Figure 6-4. Let us now use Eq . 6-2/ to calculate this same induced electromotance from

226

I NDUCED ELECTROMOTANCE AND MAGNETic ENERGY

[Chap. 6]

the time derivative of the vector potential A. From Section 5.5.2, A is parallel lo the wires, and (6-28) A n = _ B. In (r" + iv) 2rr r. + w

(6-29)

along the left- and right-hand sides of the loop , respectively. The positive direction is taken to be upwards. Thus the induced electric field intensities are respectively EL

= - ~!!!In r__,,_, 2rr cit

En = B. cJ!. In 2,r cit

(6-30)

r.

(r, ++ w), r,.

IV

(6-31)

and the electromotance,

/4 E-dl = µoh cJ!. In [ro(r, + w)], 'f 2rr cit r.(r, + w)

(6-32)

produces a current I' in the direction shown in the figure , just as in Eq. 6-26. lf we wish to find the electromotance induced by a changing current in a single conductor, we choose rb >> w, and then

/4 E-dl = 'f

µoh cJ!. ln (_______!E.._), 2rr cit r. w

+

(6-33)

6.2. Induced Electromotance in a Moving System Our differential form of the Faraday law , Eq. 6-11 , was limited to systems at rest. We shall now consider systems moving with velocities which are small compared to the velocity of light. This limitation in velocity is again made to avoid complications from relativistic effects. We return to the integral form of the Faraday law , Eq. 6-6, and consider

Figure 6-5

(d.fxu) dt

A path of imegration moves from C to C in th e time cit. The displacement is general and invol ves a translation., a ro1ario11 , and a distortion. The point P is assumed to move with a veloc ity u in a region wh ere the magnet ic induction is B.

228

I N DUCED ELECTROMOTANCE AND MAG N ETIC ENERGY

[Chap. 6]

gained through the sides of the volume traced out by the moving path, a nd (b) the change of flux by virtue of the change of B with time. Thus, from the Faraday law, (6-43)

or, using Stokes's theorem, (6-44)

Since this equation must be va lid for any surface S bounded by any curve C, the integrands must be equal at every point, and

V XE

aB

= - 81

+ V X (u X B).

(6-45)

In this equation , E is the induced electric field intensity as measured in a coordinate system moving with a velocity u relative to that in which the magnetic induction is measured as B. For example, if E is induced in a moving conductor, u is the velocity of the conductor relative to the laboratory ; B is measured with an appropriate instrument which is fixed with respect to the laboratory. This does not exclude the use of a search coil and ballistic galvanometer in the usual way. As we saw at the beginning of this chapter, the electric field intensity E in the moving conductor is the force on a unit charge at rest with respect to the conductor.* Figure 6-6. Fixed loop in a time-dependent magnetic field B. The vector n is normal to the loop.

6.2.1.

The

Electromotance

In-

duced in a Fixed Loop in a Time-

To illustrate the above equatio n, we consider the square loop of Figure 6-6. Let us first suppose that the loop is at rest, its normal making an angle 0 with the z-axis as indicated in the diagram. There is a uniform magnetic field B that is parallel to the z-axis and which varies with time:

dependent Magnetic Field.

B = E0 sin wt. • See Appendi x E fo r a disc ussion of Eq. 6-45.

(6-46)

[6.2]

Induced Electromotance in a Moving System

229

Since the loop is at rest , u = 0, (6-47)

and

¢

E-dl =

ls

(V X E)-da,

- ( aB . da

J.s ar

'

= -BowS COS 0 COS wt,

(6-48)

(6-49)

(6-50)

where S = wh is the area of the loop. For example, for a square loop 10 X 10 centimeters normal lo a magnetic field va rying at 60 cycles/ second with a maximum va lue of I0- 2 webers/ meter' (100 gauss), the induced electromotance has a frequency of 60 cycles/ second and a maximum value of about 38 millivolts/ turn . 6.2.2. The Electromotance Induced in a Rotating Loop in a Fixed Magnetic Field. Consider now the same loop of Figure 6-6 rotating with an angular velocity w about the x-axis in a uniform, time-independent magnetic field B parallel to the z-axis. Then , from Eq. 6-45 ,

V X E = V X (u X 8),

(6-51)

and

E = u X B,

(6-52)

as in Section 6.1. We have neglected any field due to the electrostatic potential V, as in Section 6.1.4. Thus

¢

E-dl =

=

¢

(u X B) -dl ,

2w

w

.

2 hBsm wt.

(6-53)

(6-54)

The only contributions to the integral are on the vertical sides, (u X B) being perpendicular to di along the top and bottom parts of the integration path. Hence

¢

E-dl = BSw si n wt,

(6-55)

where Sis again the area of the loop. The electromotance must go to zero when the plane of the loop is perpendicular to B, since at this instant the free charges inside the wire are moving parallel to B and there is no force on them.

230

INDUCED ELECTROMOfANCE AND MAGNETIC ENERGY

[Chap. 6]

6 .2.3. The Electromotance Induced in a Rotating Loop in a Time-dependent Magnetic Field. Let us consider again the loop of Figure 6-6, letting B depend

on time as in Eq. 6-46 and letting the loop rotate with angular velocity w about the x -axis, its normal being parallel to the z-axis at t = 0. Then V X E

=

aB -at + V X (u X B).

(6-56)

Integrating over the plane surface S bounded by the loop, we find that

( (V X E)-da = -;· i!___ iJB · da

)s

s

1

+ ( [V X (u X B))-da, }s

(6-57)

and , applying Stokes's theorem to the curl terms, we obtain

/4E-dl = -

'f

=

(@ · da )s at

+ /4(u X B)-dl, 'f

B0Sw(sin 2 wt - cos' wt),

= - B0 Sw cos 2wt .

(6-58) (6-59) (6-60)

In this case the electromotance alternates at double the frequency w/ 2,r. We may also calculate this electrnmotance from the total rate of change of flux through the loop, according to Eq. 6-8. The flux is given by = BS',

(6-61)

S' = Scos0

(6-62)

where is the area of the loop projected on a plane normal to the direction of B. Then

qi' = B dS' dt

dt

+ S' c!J!. dt

(6-63)

On evaluating these derivatives we find the same electromotance as above.

6.3. Inductance and Induced Electromotance In computing the electromotance induced in one circuit when the current changes in another, it is convenient to calculate the flux through the first circuit in term s of: (a) the current in the second and (b) a purely geometrical factor involving both circuits. This factor is known as the mutua l inductance. The same procedure can be used to relate the linking flu x and the current for a single circuit, in which case the geometrica l factor is known as the self-inducta nce. 6.3.1. Mutual Inductance. Let us seek an expression for the magnetic flux

which link s one circuit but arises from the current in another. This flux is

(6 .3]

231

/11d11ctance and Induced E!ectromotance

Figu ra 6-7. Two circuits a and b. The flux 0 , shown linking a and

originating in b is positive . This is because its direction is related by the rig/a-hand screw rule to the direction chosen to be positive around o.

required for the ca lculation of the induced electromotance in the linked circuit. The current Io in a circuit b produces a magnetic flu x , b linking a circuit a, as in Figure 6-7, thus ab

=

r B. -da,,,

(6-64)

)s.

where da, is an element of area on an arbitrary surface S, bo unded by circuit a and where B. is the magnetic induction fr om current / • at a point on S, . We ca n now ca lcu la te B, from the vec tor potent ial A. prod uced by Io:

,, =

h.(,

(6-65)

X A,) · da"

¢. ¢. (~:•¢. d~•) . di., A. -dl"'

= µ,fo 4u

=

J.. J..

'fa 'ft

M ab fo ,

dlo-dl

0 ,

r

(6-66) (6-67) (6-68) (6-69)

where (6-70)

This is the Neumann equation . It is rather remarkable: M ab is a quantity

depending §ok}y fJll tbs ge,ormetry the cmre11t in ~msurnc:d th8,t them

Ln ln:en:i:y~Q

current of

01t-~

-w,:hCT'of·1w:n 1lill

(c,)o~~L2L

the fJther,,

%@~1°U@rllDDCl:@iri10V~'"

fim,x r;mJD,

r:o:rfl_p~ite:d in the

w:hf~:rc. rr lcltt'; of ,the sa,ns

[6.3]

233

Inductance and Induced Electromotance

at P. We then form the scala r product of this vector with di, at P, and, finally, we sum the resultant scalar quantities for all the dl,s around the circuit. As for mutual inductance, the total flux linkage is if>= LI,

where

t =

i

(6-74)

~ (¢, d,1")

· di•

(6-75)

is called the se/f-induc/ance of the circuit. It depends solely on the geometry of the circuit. Self-inductance, like mutual inductance, is measured in henrys. It is always positive. A circuit has a self-inductance of one henry if a current of one ampere produces a flux linkage of one weber-turn. If the circuit is truly filamentary , that is, if the cross-sectional area of the conductor is infinitely small , the flu x if> and the inductance L become infinite, since di,./,-+ oo as di,. approaches the point P. The infinite flux and inductance can also be explained as follows. As the radius , of the conductor tends to zero, B tends to infinity in the immediate neighborhood of the wire. The region where B is infinitely large is itself infinitely small , but the flux tends to infinity logarithmically. This flux clings infinitely close to the wire. In practice, currents are found to be distributed over a finite cross section, thus the flux linkage, and therefore the inductance, does not in fact diverge. We shall calculate self-inductance for two cases in wh ich the curren ts are effectively distributed over a surface. Then both the flux and the inductance are finite . Volume distributions of current wi ll be discussed in Section 6.5. Circuits designed to possess self-inductance are called inductors, and pairs of circuits designed to possess mutual inductance are called mutual inductors. We shall now use these concepts to calculate the self-inductance of a long solenoid, the self-inductance of a toroid , and the mutual inductance between two coaxial solenoids. 6.3.3. Self-inductance of a Long Solenoid . It was shown in Section 5.11.3

that the magnetic induction inside a long solenoid , neglecting end effects, is constant and that (6-76) B = µ 0 N'I, where N' is the number of turns per meter. Thus we can write = µ 0 NI 1rR'

s

,

(6-77)

where N is the total number of turns, s is the length of the solenoid, and R is its radius. Then (6-78)

[6.3]

Inductance and Induced E/ectromotance

L

=

=

In [ 2R

µ 0 NI w µ0 N w

2,,-

+w ],

2R -

2r.

2

235

In [ 2R

(6-83)

w

+w ]·

2R -

(6-84)

w

» w, 2R + w ~ (t + _l!'_)( i + _I!'_),

For the case in which R

2R -

w

2R

2R

(6-85) (6-86)

In [2R + 2R-

w] ~ R~,

(6-87)

w

and (R

»

w).

(6-88)

As with the solenoid , the self-inductance is proportional both to the square of the number of turns and to the cross-sectional area enclosed by the winding and is inversely proportional to the length of the winding. The two results are identical because we have assumed for the solenoid that end effects could be neglected and that B was uniform inside and zero outside.

Figure 6-10.

Coaxial solenoids. The two radii are taken to be approximately equal.

6.3.5 . Mutual Inductance Between Two Coaxial Solenoids. Let us now add a second winding over the solenoid as in Figure 6- 10. We assume for simplicity that both windings are long with respect to the commo n diameter, in order that end effects can be neglected. We also assume that the pitches of the two windings are equal, with the resu lt that (6-89)

To calcu late the mutual inductance between the two coils, let us assume a current I,. in coil a (we do not use the Neumann equation . Eq. 6-70. because it

236

INDUCED ELECTROMOTANCE AND MAGNETIC ENERGY

[Chap. 6]

would require a much more complicated calculation). Then the flux cf>,, linking coil b and produced by I, is cI> ba

=

2

N,Ja, s,

µ01rR

(6-90)

and the mutual inductance is

= N,cf>" ,

M, a

I,

(6-91) (6-92)

Let us now calculate the mutual inductance M a, by assuming a current I. in coil b. In this case .._ _ µ,1rR' N ,I, . (6-93) '¥a b s, This flu x now links only N, turns of coil a, since B falls rapidly to zero beyond the end of the solenoid, as we saw in Problem 5-7. Then , M (lb = µ01rR2N'f.,

s,

(6-94)

and (6-95)

from Eq. 6-89. Thus (6-96)

This is true even if we take the end effects of the two solenoids into account: a current of one ampere in coil a produces the same flux in coil b as a current of one ampere in coil b produces in coil a. 6.3.6. Coefficient of Coupling . Let us consider a coil a through which a cur-

rent I, produces a flu x 'Paa and a coil b arranged such that only a fraction k, of 'Paa passes through it: (6-97) We consider single-turn coils for si mplicity . The self-inductance of coil a is (6-98)

and the mutual inductance between the two coils is M oa = N,k a'Paa. I.,

(6-99)

Thus (6-100)

Likewise, M.,, = k , NN, L,. • b

(6-101)

[6.3]

23 7

/11ducta11ce and Induced E/ectro111otance

Then, since M 00

= M ,, = M , (6-102)

M ' = k"k' L"L,,

M = ± k( L,L,)

112

,

(6-/03)

where k = ±(k"k,) 11'

(6-104)

is called the coefficient of coup/i11g between the two coils. It can have va lues ranging from - I to I. The maximum mutual inductance between two coi ls is thus the square root of the product of their self-inductances. We have seen in Section 6.3 .2 that the self-inductance of a circuit tends to infinity as the wire radius r tends to zero. The mutual inductance, however , does not tend to infinity if one or both of the circuits is fil a mentary, since the magnetic induction at some d istance from a wire is hardly sensitive to r. This can be seen from the integral for B in the general case, Eq. 5-9. We have also seen that B outside a long straight wire depends only on the current flowin g through it. The intense flu x very close to a thin wire does not link a nother circuit some distance away. That is, as r - + 0, L - + co, k - + 0, and the mutual inductance remains a bout the same. 6.3.7. Inductance and Induced Electromotance. We ret urn to the Faraday

law and express the induced electromotance in terms of inducta nce. Consider a current I, in circuit b producting a flu x ,, through circuit a. If/, changes, there is a corresponding change in ,,, a nd the induced electromotance in circuit a is (6-/05 )

(6-/06)

where the minus sign comes from Lenz's law, the sign of the mutual inductance being determined as in Section 6.3 .1 . This equatio n is co nvenient for computing the induced electromotance, since it involves on ly the mutua l ind uctance Mand d!,/ dt , both of which can be measured. If the current cha nges in a circuit , the induced electromotance within the same circuit is

¢

E-dl =

-dt,

"'~

(6-107)

-L r}!_, dt

(6 -108)

238

INDUCED ELECTROMOTANCE AND MAGNETIC ENERGY

[Chap. 6]

where L is the self-inductance of the circuit. This induced electromotance adds to whatever other voltages are present. These equations can be used to define both mutual and self-inductance. In the single circuit, for example, the self-inductance L is I henry if a current changing at the rate of I ampere/ second induces an electromotance of I volt. We have calculated (Section 6.3.5) the mutual inductance between coaxial solenoids. It is paradoxical that a varying current in the inner solenoid should induce an electromotance in the outer one, since we have shown in Section 5.11.3 that the magnetic induction outside a long solenoid is zero! The explanation is that the induced electric field intensity at any given point is equal to the negative time derivative of the vector potential at that point and that the vector potential A does not vanish outside an infinite solenoid, despite the fact that B = V X A does. We can actually calculate the vector potential at the surface of the inner solenoid from the mutual inductance found earlier. If we consider the direction of the current 1. in the primary as positive, then the electromotance induced in the secondary is 'lJ

= -Md!,,

(6-109)

dt

where the negative sign comes from Lenz's law. Then 'U•

µo7fR2N,N, di,

=

Sa

dt'

(6-1 JO)

and the induced electric field intensity is ~ _ µ 0 RN. di, 2-rrRN• - - ~ di'

aA

at at

p

(6-1 II) (6-112)

= R. Integrating this expression, we obtain (6-113)

in the azimuthal direction at the surface of the solenoid. This can be shown lo be correct. In fact , at any radius p outside the solenoid, A is azimuthal and is given by A

=

µ,R' N, 1•. 2p Sa

(6-114)

This can be found by calculating A for a single turn and integrating over the length of the so lenoid. But since this calculation involves elliptic integrals , it will not be given here. However, it is easy to verify that B = V X A = 0 in this case.

[6 .4]

Energy Stored in a Magnetic Field

239

6.4. Energy Stored in a Magnetic Field To calculate the work which must be done to establish a magnetic field, we shall calculate the energy supplied by a source to an isolated circuit when the current density increases from zero to some value J. At a given point in the conducting medium, which we assume to be nonmagnetic, there is a current density Janda corresponding electric field intensity (6-115)

where ,, is the electrical conductivity in mhos/ meter at the point considered. This equation is always true, whatever the origin of E. It states that, under the action of an electric field intensity E, the electrons drift through the conductor at such a velocity that the resulting current density is J/ -, This is Ohm's law (Eq. 5-87). The electric field intensity results from (a) the field -v V produced by accumulations of charge on the term inals of the source and on the surfaces of the conductor and from (b) the fi eld - aA/ at induced by the vector potential A in the conductor, if it is time dependent. It will be remembered from Section 5.8 that the net charge density p can be set equal to zero inside a conductor. In a wire, V V adjusts itself such that the total field intensity E is along the axis. Inside a source, we also have a third electric field which comes from -VV the local generation of energy . This field can be written as E, ; for example, inside a battery,

J/ -, = E

+ E,.

(6-116)

The work done per unit time and per unit volume on the moving charges at any point outside the source Figure 6-11 . Rectangular parallelepiped parallel to the currenr density vector J can be calculated as follows. Consider in a conductor. The electric field intensity an element of volume havin g the form Eis rhe sum of -VV and -aA/ at. of a rectangular parallelepiped oriented such that one set of sides is parallel to the total current density J, as in Figure 6-11. In one second , a charge J da goes in through the left-hand face a nd a similar charge comes out at the other end. The source which maintains a difference in potential of v V •di across these faces supplies to the element of volume dT an amount of power given by dW

d,- =

( - VV)-dl Jda ,

(6-117)

[6.4]

24 1

Energy Stored in a Magn etic Field

the volume T being any volume which includes all poi nts where the current density J is not zero, and S being the correspond ing surface. As in the corresponding electrostatic case , Eq . 6-125 becomes simple if we choose r to include all space, in which case the surface S is at infinity. The magnetic induction B falls off as 1/ r' at large distances. This was shown for the case of a current loop in Eqs. 5-191 and 5-192. The induced electric field intensity E, also falls off as 1/ r', as was shown in Problem 5-25, where we found that the vector potential A, and therefore E ;, falls off as 1/ r' at large distances from a current loop. Since the surface area S increases only as r', the surface integral decreases as l / r 3 and vanishes as the surface of integration becomes infinite. Thus dWm = _I_ dt µ0

f (B .~) J at

w

= _I_~ 2µ 0 dt

dr,

(6-126)

B 2 dr.

(6-127)

B' clr .

(6-128)

w

Setting Wm = 0 when B = 0, we have W ,,, =

f-f -µo

00

The quantity W,,, is the total work which must be done to establish a magnetic fi eld in terms of the magnetic induction B either in free space or in nonmagnetic matter. It should be noted that the magnetic energy varies as the square of the magnetic induction B. If severa l fields are superposed , the total energy is therefore not just the sum of the energies calculated for each separate field. Just as in electrostatics, Section 2. 14.1 , we may define an energy density dW,,, = B2 dr

(6-129)

associated with each point in space. This matter wi ll be discussed further in the next section. 6.4.2. The Magnetic Energy in Terms of the Current Density J and of the Vector Potential A. It will be recalled from Sections 2.14 and 2.14. 1 that we expressed the energy density in an electrostatic field either in the form ,,£2 / 2 m as pV/ 2. We have already expressed the magnetic energy density as B'/2µ,; we shall now express it in terms of the current density J and of the vector potential A. We rewrite Eq. 6-128 as follows:

W,,.

=

J.. 2µ,

f

w

(B·V X A)dr.

(6-130)

242

INDUCED ELECTROMOTANCE AND MAGNETIC ENERGY

[Chap. 6]

Using the vector identity of Problem 1-23 and the divergence theorem, we obtain 1

W=(A-~X~~- ~(~X~-~2µo }oo 2µo }oo

(6-131)

The surface integral vanishes again as in Eq. 6-125, and when v X B = /l-OJ (Section 5.10), W,,, =

!1

(J·A) dr,

(6-132)

where r is any volume which includes all regions where J is not zero. It is convenient to assign an energy density dW., =

dr

! (J-A) 2

(6-133)

to conductors carrying a current density J. Again as in electrostatics, the assignment of an energy density to a point in space is quite arbitrary and meaningless, except as a means of computing the overall magnetic energy W.,. Equations 6-129 and 6-133 are clearly contradictory in that the former assigns a finite energy density to all points where B "" 0, whereas the latter makes the energy density zero wherever there is zero current density. Does the energy reside in the field, or does it reside in the current? These are meaningless questions, even though the "densities" we have found do provide convenient methods for computing the total energy stored in a magnetic field. 6.4.3. The Magnetic Energy in Terms of the Current I and of the Flux .

For a filamentary circuit, we may also express the field energy in terms of the current and of the flux linking the circuit. If we replace J dr by I di in Eq. 6-132, di being an element of the circuit which carries the current /, then (6-134) With real conductors the current is not truly filamentary but is distributed over a small but finite area. We can then use the mean value of A over the cross section. Since, from Eq. 5-83, the integral gives the flux linking the circuit,

w.. =

I

2/.

(6-135)

The directions in which I and are taken to be positive are related as in the right-hand screw rule. 6.4.4. The Magnetic Energy in Terms of the Currents and of the Inductances.

It is possible to express the energy stored in a magnetic field in still anothe,.

[6.4]

243

Energy Stored in a Magnetic Field

way, in terms of the currents and of the inducta nces. In the above equation , the magnetic flu x can be replaced by the product of the self-inductance L and the current J, from Section 6.3 .2. Then

W,. = ~LI'.

(6-136)

For two circuits carrying currents I . and h, W .,,

I

I

= 2 f a•• + 2 hbb + h,,,

(6-142)

or , from the definition of self- and mutual inductance,

LJ, , = L ,h , =

+ Mh,

(6-143)

+ Mia,

(6-144)

and so (6-145)

The fir st two term s on the right are self-energies a risin g fr om the interact ion of each current with its own fi eld , whereas the third term is an interactio n energy arising from the mutual inductance. To sum up , we have found four ways to calcu late the magnetic energy associated wit h a current. We have expressed magnetic energy ( I) in terms of the magnetic induction Bin Eq. 6-128, (2) in term s of the current density J and the vector potential A in Eq. 6-132 , (3) in terms of the current 1 and the flux linking the circuit in Eq. 6-135 , and (4) in terms of the currents/ and the selfand mutual inducta nces of a pair of circuits in Eq. 6-145. 6.4.5. The Magnetic Energy for a Solenoid Carrying a Cur rent /. We shall

consider aga in the relat ively simple case of the lo ng soleno id of length s a nd rad iu s R with negligible encl effects and ca lcu late W,,, by each of the four methods give n above.

{]) 'C..:Nf'- J1~nrc frJBt1d I:r1 S;;c:tio.n 5. I I .3 ~liat~ Degl1s~tb1g cn:d~ c'/fc0t2., the r:11a;~1:r~h\c £i0d-11ction is urdf(;.nn throughoi:..1t tht. intc:tfr;w -o:Y the ~;c,{cuoidl Knd thmt

wbtxi~ 1."Vf fo tJie. nurrnh~r of tnrr1z per :n1et~r. T'I1t· :magnetic irnluctf.011 1§ z:enr Urc~s supply nn -::ncrgy l() Lhc, systc.111. e\c0pt fL)r the

Joule losses, which wc arc. negk.cting. Then F,,. -dr

=

-(ciW, , ).1,.

(6-187)

In this case the mechanical 1.Yc11·k is accompanied by a corresponding decrease of magnetic energy. It \\ill be sho\\n in Pr,,blem 6-19 that the above result is really the same as that found pre,iousl) in Eq. 6-177.

Figure 6-14. T1·,;o cooxial solenoids u/ df{f'cre nr si-::,t"!;. 1hr: smolla one pt'netrming a div1un ci.: I inside th :.~ ot!1cr. Titer!! is on a.ri£1! a11ruui·1•e /urn~ F w hen the two currenrs f ,1 and I r, are in the .rn111e direc tion. The m1111/ie.r of wrn\· 11er rneler on euch

solenoid is S~ i:- nd :\"_~.• 6.6.3. Force Betwee;i Two Coaxial Solenoids. The cas-'C of two long sole-

11nicL, nnc

[6.6]

251

Mag11e1ic Force

calcu lation would be vastly more complex. We therefore assume that the solenoids are long and thin. Let us first calculate the force from the rate of change of the mutual inductance, using Eq . 6-179. By symmetry, Fis along the common axis of the solenoids. Setting S to be the cross-sectional area of the smaller solenoid,

M = (N:,ryp., = (N ;l),., I,

I,

where

N; and N; are the number M

=

(6-188)

nf turns per meter on the two solenoids. Thus

µ,N;N:,hSI

µ,N:,N(I. SI, 1.

=

h

(6-189) (6-190) (6-191)

and F

=

µ0N;N£SJJ,.

(6-1 92)

The force is attractive, since M increases with I. It is interesting to repeat the calc ulation by using the rate of change of the magnetic energy when the currents are kept co nstant, as in Eq . 6-181. We have, from Eq. 6-128, 1 (6-193) W ,,. = B 2 dr, 2µ , hence 1 (6-/94) w,. = -2 [BJ(ra - SI)+ Blh - SI)+ (Ba + B,)' SI] ,

J ~

µ,

= 2J_ (B~Ta + B;r , µ,

+ 2B, B,SI),

(6-195)

where B. = µ,N;I. is the magnet ic induct ion originat in g in solenoid a and , similarly, where B, is the magnet ic induction origina ting in solen oid b. These are constan ts, since the curren ts /,. a nd hare agai n assumed to be constant. Then dW,,.

=

F di

=

B. B,S di, µ,

(6-196)

and (6-197)

as in Eq. 6-192. Note that the magnetic energy W,,. is a fun ction of I because the magnetic energy density depends on the square of the magnetic induction B. If it depend ed on the fir st power of B, W,,. would not be a function of I, a nd the force F would be zero.

252

I NDUCE D ELECTROMOTANCE AN D MAGNE TIC ENERGY

[Chap. 6]

6.7. Magnetic Torque In many cases a circuit is submitted to a torque, and not to a single force. fhe movin g co il of a galvanometer is a n obvious example. The same procedure can then be used to calculate the torque, and Eqs. 6-179 to 6-181 become

aM M, = IJ,ao,

(6-198)

= I ( iJ ) ,

(6-199)

ae

r

(6-200)

where is the flu x linking the circuit. For example, the torque on a loop, such as th at shown in Figure 6- I 5, carrying a current I and set at an a ngle 8 in a uniform magnetic field Bis M8 = I

aoa ( -

BS cos 8) ,

(6-201)

where is the flu x produced by the current / in the loop a nd S is the area of the loop . The flu x BS cos 8 is negati ve beca use it links the loop in the negat ive direction with respect to the current I . Thus

M = JBS sin 8,

Figure 6-15. Loop carrying a cu rrent I in a uniform magnetic field B. With th e current in the direction shown, the loop is Sllbjec1ed to a torque tending to increase

1he angle 0.

(6-202)

and the torque is positi ve, or is in the directio n of increasing 0, for 0 < O < ,r. This result can be easily verifi ed fr om the directi on of the elementary forces /(di X B) on a rectangular coil.

6.8. Summary This chapter is co ncerned with (a) the nonconserva tive electric fi eld s associated with time-dependent ma gnet ic field s, (b) the energy stored in magnet ic fi eld s, and (c) the forces and torques exerted on current-carrying ci rcuits situated iD magnet ic field s.

[6.8]

253

Summary

If a conductor is moved w ith a velocity u w ith respect to a mag net ic field B. it s electrons of cha rge Q ex perience a Lorentz.force f given by

f = Q(u X B),

(6-2)

where u X B is called the induced electric field intensity. F o r a closed circuit,

J.E.dl =

'f

_ cf:!'..

(6-5)

dt

The directions in which the line integral and are taken to be positive a re related according to the right-hand screw ru le . The above line integra l is the induced electromotance. Induced electromotance can a rise as above fr o m the relative motion of a circuit and a magnetic fi eld . It is also o bserved when a fi xed ci rcuit is linked by a variable flu x, as in a transformer. The Faraday induction law is

J. E · di 'f

= - i_ ( B da, dt ).,

(6-6)

which is rea lly the same as Eq. 6-5. The nega ti ve sign mea ns that the induced electromota nce tends to oppose changes in the flu x linking the circuit. This is Lenz's laiv. In differen tia l for m ,

VXE=

aB

(6 -1 I )

a1

This is one of Maxirel/'s .fundame111al equario11s of electromagnetism . It leads to the genera l express ion for the electric field intensity E:

E = _aA - vV

at

'

(6-1 6)

where the first term is the electric fi eld intensity induced by a changing magnetic field a nd where v Vis the electric field intensity produced by acc umula tions of cha rge, as in electrostatics . F or a conductor moving wit h a velocity u in a magnetic fi eld B, Eq. 6-11 becomes

v X E = -

aB

DI + v

X (u X B)

(6-45)

when Eis measured wi th respect to a coord inate system moving with a velocity u relative to that in wh ic h the magnetic induction is measured as B. The mutual inductance M between two circuits is equal to the magnetic flu x linking one circuit per unit current flowing in the other : (6-69)

w here

., is the flux

linking circuit a and originating in circ uit b. It is found tha t

254

INDU CED ELECTROMOTANCE AND MAGNETIC ENERGY

[Chap. 6] (6-70)

This is the Neumann equation. It shows that M depends solely on the geometry of the two circuits, if there are no magnetic materials present. By symmetry, (6-72)

Self-inductance Lis a similar quantity, but which applies to a single circuit:

= LI,

and L =

(6-74)

~ J. (J. di.) - di,.

471"

r,,

'ft

r

(6-75)

The coefficient of coupling k between two circuits is defined by (6-103) M = ±k(L, L,) 1' '· + 1. It is zero if none of the flux of one

It can have values ranging from - I to

circuit links the other. In terms of inductance, the induced electromotance is given by

J. E-dl = 'f

-L c_l!_ . cit '

(6-108)

a similar relation applies to mutual inductance. A magnetic field involves energy, as does a n electric field . This energy is equa l to that required to esta blish the field , and we find that the energy supplied to a n element of volume dT of conductor per unit time is c/Wm = dt

1~ . • i/1

J c/T.

(6-121)

If there are no magnetic materials present, and if the displacement current is negligible compared to the conduction current, then (6-128)

or W,,,

=

~

1

(J-A) cir,

(6-132)

where T is any volume which includes all regions in which J is not zero. We also find that (6-135)

where is the flux linking the current /. As usual , the positive directions for I a nd are related accord ing to the right-hand screw rule. In terms of the selfinductance L ,

255

Problems

w.. =

1u'.

(6-136)

The self-inductance of a volume d istribution of current follows from the above. It is defined as (6-/59)

The mag11e1ic forc e on a current-carrying circu it can be ca lculated from the principle of conservation of energy applied to a virtual disp lacement of the circuit. The x-component of the force F,b exerted on a by b is found to be

aM

(6-179)

Fabx. = 11.Jb ax'

=

I,

(a~')' a,x I

=(awax.. ),

(6-180) (6-181)

I

the index I indicating that the currents are kept constant. This leads us back to Eq. 5-6 , which is an a lternative form of the magnetic force law stated in Eq. 5-1 and which was the starting point for our entire discussion of magnetic fields. Mag11e1ic Torque is given similarly by M, =

(dW. ,) _ c/0 I

(6-200)

Problems 6-l. A conducting bar slides with a constant velocity u along conducting rails in a region of uniform magnet ic induction B, as in Figure 6-l. The total resistance in the circuit is R. What current flows in the circuit? How much power is required to move the bar? How does this power compare with the rate at which Joule heat is developed in the circuit ?

6-2. Compute the magnitude and direction of the electromotance induced in a square loop of side a moving in the magnetic field of a long straight wire carrying a steady current /. The loop and the wire lie in the sa me plane, and the loop moves toward the wire with a ve locity u.

6-3. An eiectron is at rest in vacuum at a distance r from the axis of a wire l » r meters lon g ca rryi ng a current /. If the current increases at the rate dl/ dt, what is the magnitude and direction of the force on the electron ? Disregard the field due to the rest of the circuit. 6-4. In the betatron , electrons are held in a circular orbit in a vacuum chamber by a magnetic field B. The electrons are accelerated by increasing the magnetic flu x link ing the orbit. Show that the average magnetic induction over the plane of the orbit mu st be twice the induction a t the orbit if the orbit radius is to remain fixed as

256

INDU CED ELECTROMOTANCE AND MAGNETIC ENERGY

[Chap. 6]

the electron's energy is increased. Hint: Relate the centripetal acceleration to the magnetic force acting on the electron, and then find the condition which lets the linear momentum of the electron increase with fixed orbit radius. Use Newton's second law with the tangential force on the electron given by the Faraday induction law. 6-5. An electron revolves with angular momentum h/ 2rr in an orbit about a proton, and an externa l magnetic field B is applied in a direction perpendicular to the plane of the orbit. What happens to the motion of the electron by virtue of the electromotance induced as the magnetic field is established? Calculate the change in the electron's angular frequency, assuming that its radius remains fi xed. What is the change in the electron's orbital magnetic moment? In which direction is this change? This is the phenomenon of diamagnetism. Hint: The total centripetal force acting on the electron after the magnetic field is established consists of both electrostatic and magnetic forces . Equate the change in centripetal force to the magnetic force. 6-6. A conducting rod of length L rotates with angular frequ ency w about an axis perpendicular to the rod and through one end in a uniform magnetic field B parallel to the ax is. What is the electromotance developed between the ends of the rod? What happens to the conduction electrons in the rod? What forces hold them in equilibrium? 2 6-7. A magnetic field is described by B, = Bo sin ;Y sin wt. In this field a square loop of side A/4 lies in the yz-plane with its sides parallel to the y- and z-axes. The loop moves in the positive y direction with a constant velocity u. Compute the electromotance induced in the loop as a function of time if the trailing edge of the loop is at y = 0 at t = 0. Assume w = 2rr11/ "/,.. 6-8. A thin flat conducting disk of thickness h, diameter D, and resistivity p is placed in a uniform alternating magnetic field B = Bo sin wt parallel to the axis of the disk. Find the induced current density as a function of distance from the axis of the disk. What is the direction of this current? 6-9. Show that both the normal and tangential components of the vector potential A must be continuous across the interface between two media if the currents are constant. 6-10. Two long parallel wires of radius a are separated by a distance D and carry a current I in the same direction. Calculate the magnetic indsction B al a point that is located between the wires and which lies in the plane containing the two wires. What is the magnetic flu x per unit length linking the wires? What is the flux per unit length when the currents flow in opposite directions in the two wires? Calculate the inductance per unit length for a parallel wire transmission line. Whal effect does the flux within the conductors have on the inductance? 6-11. Show that the self-inductance of a close-wound toroidal coil of radii Rand r is given by L = µ.011(R R' - r'),

v

where" is the number of turns.

Problems

257

6-12. It is known that high freq uency currents do not penetrate into a conductor as do low frequency currents. This is called the skin effect. Would you expect the selfinductance of a coaxial line to increase or to decrease with increasing frequency? What would you expect the a pproxima te percentage change to be between very low and very high frequencies? 6-13. The coefficient of coupling k between two coils was defined in Eq. 6-104, where k. and k , are defined as in Eq. 6-97 . Show that k.

L,

k,, = T.' so that k.

,;,!

k,, in general.

6-14. Two long parallel rectangular loops lying in the same plane have lengths /1 and /, a nd widths w, and w,, respectively. The loops do not overlap, and the distance between the near sides is s. Show that the mutual inducta nce between the loops is given by

M = µ 01, In 2 ,r

if/,

< /1, and

+ iv, + S~+ IV1 )

s s( I

if the loops have but a single turn. Neglect end effects.

6-15. A wire bent in the form of a circle of rad ius R is placed so that its center is at a distance D = 2R from a long straight wire, the two being in the same plane. Show that the mutual inductance is

M = 0.268µ , R. Wha t is the mutual inductance in microhenrys when R

= 10.0 centimeters?

6-16. Compute the mutual inductance between two single-turn circular loops of radii R and r, where R » r, when the sma ll loop is o n the ax is of the large one a t a distance x from its center, with the planes of the loops parallel. Find how the mutua l inductance va ries as a function of the angle between the norma ls to the two pla nes. 6-17. Find the vector potential A just outside a close-wound toroidal co il. Hint : Proceed as in Section 6.3.7. 6-18. Compare the energies per unit volume in (a) a magnetic field of 1. 0 weber/ meter' and (b) a n electrostatic field of 10' volts/ meter. 6-1 9. It was shown in Eqs. 6- / 77 and 6-187 that the mechanical work F-dr done when one ci rcuit is displaced a distance dr with respect to another is either luh dM or -dWm, depending on whet her the currents or the flu xes are assumed to rema in consta nt during the virtu a l displacement. Show that these two expressions are equa l. 6-20. Consider a general system of 11 rigid fixed circuits in whic h the ith circ uit carries a current / ;. Show tha t the magnetic energy associated with the system can be written as a sum of self-energy terms of the form (l / 2)L;//, plus a sum of mutual energy terms of th e fo rm M iiJi, each pa ir or curren ts appearing once. 6-2 1. Starting with the definition of self-inducta nce in terms o f energy, find the self-inducta nce per unit length for a long straight conductor of rad ius a carryi ng a uniform current density.

258

I NDUCED ELECTROMOT AN CE AND MAGNETIC ENERGY

[Chap . 6]

Is this likely to be a major contribution to the inductance per unit length of a transmission line consisting of two parallel wires? See Problem 6-10. 6-22 . From the computation of the mutual inductance between two single-turn ~urrent loops in Problem 6-16, find the magnetic force acting on the small coil when / 1 flo ws in the large coil and I , in the small one. Compare this result with a direct ,;alculation from the force law (Eq. 5-8) , assuming that th e induction at the position -of the small coil due to the current in the large coil is unifo rm and of the same magnitude as on the axis.

CHAPTER

7 Magnetic Materials

Thus, far, our discussion of magnetic phenomena bas been limited to free space and to nonmagnetic materials, that is, to magnetic fields arising solely from conduction currents. Now, on the atomic sca le, all bodies contain electrons which move in orbits, thereby constituting currents in the usual sense, and which spin about an ax is. These moving electrons produce magnetic fields which add to those produced by the co nduction currents. For example, the electron spin is believed to be responsible for ferromagnetic effects. Our purpose in this chapter is to exam ine the magnetic fields of such atomic currents and to express these fields in macroscopic terms. The situation in magnetic materials is quite si milar to that in dielectrics. Individual charges or systems of charges possess magnetic moments, and these moments can be oriented to produce a resultant magnetic moment in a finite volume. When there is a net orientation of this sort the material is said to be magnetica lly polarized. The polarization process by which the dipoles are oriented is beyond the scope of this book and wil l not be discussed here.

7 .1. The Magnetic Polarization Vector M In discussing the fields of the magnetic dipoles associated with atomic systems we shall fo ll ow much the same pattern as for dielectrics. We saw in Section 5. 13 that, at distances which are la rge compa red to the size of the loop , the magnetic induction B of a circular current loop bas the form of a dipole field. The dipole moment is the current multiplied by the area of the loop. This concept of magnetic dipole moment can be extended to loops of arbitrary shape, and we can a lways associate a magnetic dipole moment m, measured in ampere-meters', with a current loop . Although a magneric dipole produces a magnetic field which , at large distances, is identical in form to rhe eleclrostaric dipole.field, rhe existence of111ag11eric c!,arges is 1101 implied. There is co mpelling evidence that a ll magnetic 259

260

MAGNETIC MATERIALS

[Chap. 7]

effects arise from moving electric charges. The a nalogy wi th the electrostatic dipole extends only to the mathematical similarity of the two fi eld s. The magnelic po!ariza!ion M is the magnetic dipole moment per unit volume, and , if m is the average magnetic dipole 1110111e111 per atom and N is the number of atoms per unit volume, then M=Nm . (7-1) The vector Mis measured in amperes/ meter. Later we shall express the magnetic induction B due to the atomic dipoles entirely in terms of M, but, just as for dielectrics, this procedure must be justified by first examining the phenomenon from a molecular point of view. The above definition of M is subject to the same limitations as is the corresponding quantity P (Section 3.1) for dielectrics. The volume T in which M is defined must be large enough such that there is no significant statistical fluctuation in M from one T to a neighboring one, or from one instant to the next. On the other hand , the volume , must not be so large that we make significant errors in using the smoothed-over, macroscopic distribution M rather than the discrete distribution of elementary dipoles m. As for dielectrics, the smallness of atomic dimensions ensures the validity of the macroscopic approach.

7.2. The Magnetic Induction from Polarized Magnetic Material at an External Point Let us calculate the magnetic induction B at a point P external to a magnetic core polarized by a conduction current / , as in the solenoid wound around the core shown in Figure 7-1.

XcJl.g

;-,

Figure 7-1 A coil carrying a current 1 polarizes a core of magnelic

material. The vector M is the magnetic dipole moment per unit volume; Pis a point owside the core.

262

MAGNETIC MATERIALS

[Chap. 7]

due to surface and volume distributions of electrostatic charge. The quantity (7-8)

which is the normal component of the polarization vector M at a point on the surface, is called the surface magnetic pole density, and (7-9)

is the volume magnetic pole density. It is important to remember that magnetic charges do not exist. The term magnetic pole has no more significance than that associated with the definitions of,,.;,, and µ;,,. Thus V

M

Since B

=

=le!!. {~d 41r } s r a

+lc!!.j0-;l 41r r r c. r.

-VVM , from Eq. 5-160, and since

(7-10)

VG)= r,/r', (7-11)

where r1 is the unit vector from the source point to the field point, as in Figure 7-1. We can thus find the magnetic pole densities ,,.;,, and p;,, from the polarization vector M, and from these we can calculate B"'. In terms of the poles, BM varies as 1/ ,, , as does the electrostatic field intensity E. 7.2.2. The Equivalent Current Formalism. It is also possible to calculate the magnetic induction B"' at the external point P starting with the vector potential

A. From Problem 5-25, the vector potential for a current loop is (7-12)

where m is the magnetic dipole moment of the loop and where r, the distance from the loop to P, is large compared to the size of the loop. In this case the element of dipole moment M ch gives

dA JI = 4rr 1!Q. (M X r2 at P, and

AM = .!!!!. 41r

1 r

r,) dr

(M ,2X

r,) d

(7-13)

r,

(7-14) (7-15)

This equation can also be put in a form more amenable to physical interpretation through the vector identity of Problem 1-22: V X JC

= Vf X C + f{V X C),

(7-16)

A.11 are am equivalent , but lEq. inter~u:tatllon: both ir2t1e:3r.Knd.s have t:1e forrr. tc, U3 {frorrt §e:ctio:i 505) 1 ~u:n :nt de:nsity in ·~ he se;cond. le:;-r.n.

density "A;' ww,ch that

= lv'tl

II>

Tin JJXn:pi;res/zneter anC I' in 2.mpt:re:;/;11eter", :[::1 1h:es1e ,d:1e to tt e polc.dzed rnate:daJ

the v1::ctor ;;,otentfa!. a t the point

current den~it:es conO:uction cvxrents.

'fhey are fictittous r:urrents wb!ch ;~:aic·ul:z:te the ve::;to1 potential, and tteTefr1r,e the rm:e;r{etk [ndur:J.ion" jast as we.: vvould for coa = 0.uct l.on c·J.1 rr; nt;, lfhe;y provkie a

c0r!Yt.~1ien/ .rrn°:~hod oC viscaii:t:ing the i:iiJkc'.:kl;:1 c.uc to the pofar ~

264

MAGNETIC MATERIALS

[Chap. 7]

The physical nature of the equivalent or Amperian currents can be understood from a model due to Ampere. Consider Figure 7-3 , and imagine currents X' flowing around square cells in a piece of material whose length in the direction perpendicular to the plane of the paper is one meter. The current in one cell is nullified by the currents in the adjoining cells, except at the periphery of the material. The net current density on the surface of the rod is thus ·,: . Equating the magnetic moment of the current ·,: around the periphery to the sum of the magnetic moments of the current in the cells, we have

·,!S = MS,

(7-23)

where S is the cross-sectional area of the rod and M is the magnetic moment per unit volume of the material, or ">-.'= M . (7-24) The origin of the equivalent surface current density X' can now be understood by extending this model to currents on the atomic scale. The volume current density arises when currents in adjacent cells are unequal. Note that, since the equivalent volume current density J' is equal to V X M , V ·J' is identically equal to zero, thus charge cannot accumulate at a point by virtue of the equivalent volume current density J'. Furthermore, the equivalent currents do not dissipate energy and therefore do not produce heating, since equivalent currents do not inFigure 7-3 . Ampere's model for the volve electron drift and scattering equivalent currents. processes of the type associated with conduction currents. Let us use the equivalent current concept to calculate the magnetic induction B,11 • Since B,,1 = V X A.u from Eq. 5-56, where A,,1 is given by Eq. 7-22, then µo

{ (

BM = ~ ) s V

1

X') da + ~µ, , V X--;:J' d r, X--;:

(7-25)

where we have interchanged the order of differentiation and integration. The curl is calculated at the field point P by uring the vector identity of Problem 1-22, (7-26)

The second term on the right is zero, since the curl involves derivatives with respect to x, y, and z, whereas X' is a function of x', y', and z'. Finally, setting

[7.3J

265

Magnetic lnduc1io11 at an Internal Point

(7-27) we obtain

B = 41r~ h. . (X' r2.X r,) cal + 41r~ f (.I' ,.2X r,) I .11

.,

"

C.T.

(7-28)

It will be observed that we could have written this eq ua tio n directly from t he Biot-Savart law on the assumption th at B.1 1 is produced by the surface current density X' and by the volume current density J'. Thus, if we know the polarization vector M , we can find the equivalent current densities and then treat them as conduction current densities for calculating the magnetic induction B.1 1 . In practice we never know the magnetic polarization Ma priori, and we must eventually find a practical way of dealing with magnetic problems. This will require the concept of magnetic circuits discussed at the end of this chapter.

7.3. The Magnetic Induction from Polarized Magnetic Material at an Internal Point We must now calculate the macroscop ic magnetic induction, that is, the space and time average of the magnetic induction on the molecular scale at an internal point in the polarized mag netic material. We shall see that the magnetic poles alone do not give B.11 at an internal point, whereas the equivalent currents do. 7 .3.1.

Calculation

of B., 1 at an

Internal Point Using the Pole Formalism. We sha ll proceed exact ly as with dielectrics and divide the material into two regions, o ne nea r the point P and one farther away . The near region is the vo lume r" within a small sphere of radius R, as in Figure 7-4, and the far region includes all the volume r' of the material outside this sphere. Thus, at P,

Figure 7-4. A small imaginary sphere of radius R, surface S", surface pole density a~:i, and volume , 0 cen tered at P within a magnetic material. The remaining volum e ,' has a volume pole densiry p:11 and a surface pole density u{,1 on its outer sur-

face S'.

(7-29)

where B,11 is the total magnetic induction caused by the magnetic ma terial, B\1 is the part contributed by the polarized material in the volume r', and B'.,; is

266

MAG N ETIC MATERIALS

[Chap. 7]

the part contributed by the material in r" . The radius R is taken large enough such that , for all the dipoles outside the spherical surface S", the point Pis an external point. Using the pole formalism , we have

B

_ ~ (

.11 -

41r ) -:'

p;,, dr + r2

r1

/!!!. ( er:,, da , 2 r1

41r }.,;,

+~ ( 41r

}s"

er:,: da

,2

r1

+ B" Jt,

(7-30)

where p~ is the volume pole density in r', er:,, is the surface pole density on the external surface S', and er;:, is the surface pole density on the imaginary spherical surface S".

Figure 7-5

The small imaginary sphere

of Figure 7-4 is shown here in greater dewil for the calculation of tl:e ir.ducrion B ; 13 due to the poles on the surfa ce S". The surface charge density is equal to M · n.

u;,;

,

Let us first evaluate the third term on the right: ,

µo

BM,= -

4 7r

ls

.,,

CTm

,

r-

dar,.

(7-31)

From Figure 7-5 , the surface pole density is

a;:=

M-n ,

and an element of area da contributes at the center of the sphere an element of induction ,

dB,1/ 3 =

µ0

M-n

;r,;:R' da n.

(7-32)

By symmetry, all the poles on the strip of width R dO shown in Figure 7-5 will contribute a resultant induction in the axial direction, and , 3 = -µ , B.11

4rr

lo' M cos e2 R'

µ,M

=3 in the sa me direction as M.

-R 2

1r

. e cos e de , -sin

(7-33)

0

(7-34)

[7.3]

Mag11e1ic lnd11c1ion at an /n/erna/ Point

267

268

MAGNETIC MATERIALS

[Chap. 7j

We must also calculate B'.,; , which is the magnetic induction at P due to the near dipoles. Furthermore, we must average the magnetic induction on the molecul ar sca le over a volume containing many atomic systems in order to find the macroscopic induction. For dielectrics we made this calculation by finding the average field inside a sphere containing polarized molecules, each having a resultant dipole moment (Section 2.12). This average field intensity turned out to be proportional to the dipole moment and in the opposite direction. In the magnetic case, the same calculation gives an entirely different result because the form of the magnetic induction field in the vicinity of a current loop is quite different from that of the electrostatic field in the vicinity of an electric dipole. In fact, the average magnetic field produced over a spherical volume by a current loop is in the same direction as the polarization vector M , as we shall see. Figure 7-6 shows the difference between the magnet ic field associated with a magnetically polarized molecule and the electrostatic field for an electrically polarized molecule. At large distances from the molecules both fields have exactly the same form. 1n Section 7.4 we shall define a magnetic fi eld intensity H whose behavior in magnetically polarized material is strictly ana logo us to that of the electrostatic field intensity E in electrically polarized matter.

Figur~ 7-7

Current loop of magnetic moment m at the center of a sphere

of radius R large enough for the dip ole approximation to be valid on the surface.

Let us compute the average magnetic induction produced over the volume of a sphere by a small current loop within it. For simplicity let us place the loop at the center of the sphere, and let us assume that the radius of the sphere is large compared to the size of the loop. At the surface of the sphere the magnetic induction is then accurately given by the dipole approximation. In Figure 7-7 the loop is represented by a dipole of moment m at the center. By definition, the average field in the sphere is

[7.3]

Magnetic Induction at an Interned Point

- 11

B=-T = ;

T

269

Bl C.T,

(7-35)

1

(V X A) dr,

(7-36)

where r is the volume of the sphere. We do not know A in the immediate vicinity of the loop , but, on the surface of the sphere, where the dipole approximation is valid, we have, from Problem 5-25, A= ~ m X r 1_ 4n r2 But , from Problem 1-25,

1

(V X A) d, =

1(n

(7-37)

(7-38)

X A) da,

where Sis the surface bounding the volume T and n is the outward drawn normal to S. The volume integral can thus be eval uated without A being known within the volume, provided it is known at all points of the surface. In this case, therefore, µ 0 / ,..( n X m X r 1) da. (7-39) B d, = ~ ~

1

.,

Expanding the triple vector product according to Problem 1-9,

1

B d, =

i

1 (n · m

~)da -

1

(7-40)

~(n-m)da.

We have taken the component of the elementa ry vector along the axis, that is, in the direction of m, by virtue of the sym metry. Or

JB

_~ { 4 ,r }s m

I CT

(n•n), _ }o{'mcos0) R' c,a R'

_,r

R' .

,

s1n0cos0c,0,

(7-41) (7-42) (7-43)

rnd the average magnetic induction in side the sphere is

B=

~

1

B cir,

(2/ J)µ om

= (4/ 3)1rR'

(7-44) (7-45) (7-46 j

If the current loop is not centered in the volume, B has the same magnitude

270

MAGNETIC MATERIALS

[Chap. 7]

and direction as above. This will be shown in Problem 7-2. Thus the average induction over the volume of the sphere depends only on the magnitude and direction of the loop producing the field, and not on its position inside the sphere. If there are many small current loops within the volume, each one produces an average field as above, hence (7-47)

since

M = Nm ,

(7-48)

where N is the number of molecules per unit volume, m is the average magnetic dipole moment of each molecule, and M is the magnetic dipole moment per unit volume. The resultant total dipole moment of all the molecules within the sphere is (7-49) P"' = M(4/ 3)1rR3 • The above result is open to question because some of the loops will be too close to the spherical surface for the dipole approximation of Eq. 7-37 to be valid . However, the error introduced from this source can be made arbitrarily small by increasing the size of the sphere. Molecules are so small that it is easy to choose a sphere which is large enough to make this error negligible but which will sti ll be macroscopically small. This type of problem was encountered previously in Section 3.3. We still have to average the field on the molecular scale over a volume containing a large number of molecules. This average is calculated exactly as it was for dielectrics in Section 3.3. As the point at the center of the sphere moves about, taking the sphere with it, the contribution from the far dipoles does not ~hange significantly for small displacements. The average induction produced by the nea r dipoles has already been calculated . Consequently we need make no further calculation to obtain the space average of the magnetic induction on the molecular scale. Finally , then, with the third term B'.\1 3 of Eq. 7-30 evaluated, and with B'.,; ca lculated a nd averaged over the volume T", we have

B M -

lli_

471"

f

r'

p;,, ch r2

r1

+ 41r lli_ ( ,,.;,, da + µ,M + J~ µoM. }s, r 2 r1 J

(7-50)

The volume integral is to be taken only over the volume T' of Figure 7-4. Again, as for dielectrics (Section 3.3.2), it makes no difference if we include the vo lu me T" in the integrat ion. Thus, substituting T for T' , and writing S for the real surface of the material, we obtain (7-51)

[7.3]

Magnetic Induction at an In ternal Point

(a)

271

(b)

Figure 7-8 . (a) The field inside a permanently polarized dielectric. (b) Th e field

inside a permanently polarized magnetic material.

Only the first two terms represe nt the contribution of the magnetic poles to BM. The poles therefore do not give the correct value of BM inside the polarized material , but only BM - µoM. To illustrate this fact, and to indicate the contrast with dielectrics, let us consider uniformly polarized blocks of dielectric and of magnetic material , as in Figure 7-8. Let us assume that these blocks are permanently polarized such that the polarization is not caused by an external field . Since the polarization is uniform , both the volume distribution of bo und electric charge p' and the volume distribution of magnetic poles p;,, are zero. The macroscopic electric field E of the dielec tric arises only from the bound charges and is thus in the direction Figure 7-9. Permanently polarized rod of opposite to the polarization vector magnetic material. The surface poles on the P. In the magnetic material, the shaded element of area produce a magnetic poles contribute an induction in induction dB on the axis. the direction opposite to M, but µoM must be added to this in order to find the total induct ion B.11 . The resu ltant induction is in the same direction as the polarization vector.

272

[Chap. 7]

MAGNETIC MA TE R I A L S

As a further example, let us calculate B ,11 along the length of the uniforml y polarized magnetic block. Let us consider a cylindrical rod of material whose diameter and length are both 2a, as in Figure 7-9. The pole density is M-n = +Mon the right-hand face and -M on the left-ha nd face. On the axis, at a distance I fr om the right-hand face , the poles of magnitude M da on the small shaded area produce an induction dB' as indicated. By symmetry, the resultant induction must be along the axis, as indica ted by dB, and the magnetic induction due to the poles on the right-hand face is l4l

B.,. = ~ µ0

(

}o

0

M/21rsds + i')"''

M( t -

=2

( 7-52)

(s'

I

)

(7-53)

Va'+ I'.

The poles on the left-hand face contribute in like manner a nd in the same direction . The induction B.,. due to both sets of poles is plotted in the negative direction in Figure 7-10, which a lso sli ows the total induction, that is, the magnetic induction due to the poles plus the quantity /4JM.

B total

QI----'---- ' - - - ~ - - - ' - - - ~ - - ~- ~ - ~

05

LO X a

15

2.0

Ber•

- µOM Figure 7-10. The magnetic induction due to the poles, and th e total mag-

netic induction in a permanent magnet.

Let us return to the comparison with dielectric materials. When the polarization resu lts from the action of an external fi eld , the electric field intensity E inside the dielectric is weakened by the bou nd charge field which is in the opposite direction to that of the external field . In magnetic materia ls, on thf

773 ~l p,11',:Kluced :rne poladzed. ffie)d adds to iL

is

1til:d!\l~rtf5~rrn :ErJ?~ '.[n. dfar.uz,sine 1b~ x:rrn.glaw fn;r

1

whi,d1 armliogous lo the dxtrost~i:ic lt'.0gr,"i.tf:YJig tbe firnx over the Vlk.tole surfac~(; .S~ ha,.v,e

r

d'f8pcq~:m

~

,VoP:·: dr,,

JS

1( )Bp,,1lJ&l J

= .,,,

Ir p;,, J-r

a1rr, ghr,stration'> consf.dex an i11fin'.lte; pfame \;;;1itJL a. :rntago.e1ic pole d:err1sil'ly

az in Jriguxt~ 7., J:2._. Syrrnae1ry (>DD§idews;tilf.ms th~ ntagut;ttc :induit!t:\on t'.O:crtribmted by r.':w po;es i.o per;icndicular to tl:m plan~:, ff we take: a cyli:m:lr:ieat s11rfa,~~ as ,hown, ,bc:I ,be o d y flux is thrn·:1.e;h tJ1,c c:1ds, and.

274

h1AGNETIC MATERIALS

[Chap. 7]

Figure 7-12 .

Gaussian cylinder at the surface of magnetized material.

sma ller and cannot be calculated in this way, since they are relatively far away. To get the total F.\/, we must add µoM to B,,. at every point. It is important to understand clearly that this Gauss's law applies only to that part of the magnetic induction which is associated with the fictitious poles. The divergence of the total magnetic induction is a lways zero and the total flux of B through any closed surface is always zero. 7 .3 .3. Calculation of BM al an Internal Point Using the Equivalent Current Formalism. Let us also investigate the macroscopic induction at a point in the interior of a magn e: ic ma :erial in terms of the equ iva lent currents, starting with the vector po:ential A. As in Section 7.3.1 , we consider a block of magnetic material and divide it in to two par:s separa ted by an imaginary sp here of radius R. This rad ius is aga in chosen large enough such that , at the center of the sphere. the magnetic induc.i J n due to atomic current loops in the outer volume ,' is adequately given by the dipole approx imation. If we use the same notation as in Figure 7-4, and if we use Eq. 7-28 fo r th~ magnetic induction produced by the dipoles in the outer volume .', we find that , at the center of the sphere,

Let us again leave the first two terms as they are for the moment and evaluate the third. The equivalent current on the imaginary surface S" is >.." = M X n and is in th: direction indica ted in Figure 7-13. This current produces a magnetic induction at the center of the sp here wh ich is in the direction opposite to M, and (7-62) >, " = IM X nl = M sin 0

[7.3]

275

Magnetic Induction at an Internal Point

Figure 7-1 3

The magnetic induction B.111 at the center of the imaginary

sphere of Figure 7-4 can be calculated /ram th e equivalent currents flowing on its surface.

is the current per unit length on the surface. Then the current di" on a strip of width R c/0 is di" = M sin OR dO. (7-63; From the Biot-Savart law, the induction direction opposite to M , ~

- 4..

('M.

)o

sm

B.11 3

produced by this current is, in the

OR 102,rRsinO. 0

c

R'

(7-64)

sm ,

-3? µr,1\1.

(7-65)

The contribution B'.{i from the current loops within the small sphere of radius R can be calculated in exactly the same way as in Section 7.3.1. Finally,

B,11 = -µ, 4w

1 ~

J' X -r1 cir --., r

+ -4µo7

ls

5,

')I.' X., -r1 --

r-

da - -2 µr,1\1 3

"" + 23- µ01u.

(7-66)

If we take the volume integral over the whole volume and write S to indicate the real surfaces of the material , then

(7-67) where J' = V X M and ')I.' = M X n. The equivalent currents therefore give the correct value for the magnetic induction at points inside the polarized material as well as outside. The magnetic induction on the axis of the uniformly polarized cylinder of Figure 7-9 is thus the same as inside a short solenoid, and the result is identical with that shown in Figure 7-IO.

276

MAGNETIC M A TERIALS

[Chap. 7]

7.4. The Magnetic Field Intensity H We are now in a position to calculate the macroscopic magnetic induction B, either inside magnetic material or outside, for any distribution of conduction currents and of magnetic materials, provided that the polarization vector M is known at all points in the magnetic material. So far we have oo way of finding M. We could, in principle, determine M by measuring the total flux through each unit cross section of the system, but the experiment would be impossibly complicated and wholly impractical. Our discussion is therefore not yet satisfactory. In developing magnetic concepts for free space and nonmagnetic materials, we found in Eq. 5-127 that (7-68)

for quasi-stati onary currents. In this chapter we have seen that magnetically polarized materials can always be replaced by equivalent currents and that these currents produce magnetic inductio ns in exactly the same manner as do conduction cu rrents, both in side and o utside magnetic ma terials. Consequently the above equation sho uld be rewr itten as V X B

=

µ 0(J

+ J')

(7-69)

to account for the presence of magnetica lly polarized material, J' including both surface and volume equivalent current densities. Now, since (7-70) J' = V X M, then (7-71) V X B = µr,J + µ,(v X M), or (7-72)

The vector in the parentheses is such that its curl depends o nly o n the conduction curren t density J at the poi nt , and not on the equivalent currents. This is the magnetic field i11te11sity vector: H = ~-M. µ,

(7-73)

The vector H has the same units as M: Di pole moment Ampere turns X Area Volume = Volume ' _ Ampere turns. M eter

(7-74)

(7-75)

Thus V X

H= J

(7-76)

277

[7.4] The Magnetic Field Intensity H

for quasi-stationa ry currents, regardless of whether magnetic materials are presen t or not. The magnet ic field intensity H it self can be fo und only when this differential equation can be solved, but even then the magnet ic inductio n B remains unknown. A more genera l expression for V X H will be found in Sect ion 7.10. 7 .4.1 . Gauss's Law for the Magnetic Field Intensity H. Before proceeding with a general method of calculating H , let us derive a Gauss's law for H , as we have already done for Bin Section 7.3.2. We have

B = µoH

+ µoM

a nd V·B = µoV·H

(7-77)

+ µoV·M,

(7-78)

but, from Eq . 5-45, V B = 0, and V·H = -V·M .

(7-79)

From the definition of pole density in Eq. 7-9,

(7-80) or, integrating over a volume

T

bounded by a surface S, we have

J V·H dr= JP~, cfr,

(7-81)

and, using the divergence theorem, we find

J,

H-da =

Jp;,,

dT .

(7-82)

Thus Gauss's law for H is ana logous to the corresponding law for E in electrostatics. 7.4.2. Calculation of the Magnetic Field Intens ity H. Let us concentrate for

the moment on the ca lculat ion of H. On in tegratin g v X H of Eq. 7-76 over a surface S , for quasi-stationary currents,

J,

(v X H)- da =

J,

J -da,

(7-83)

or , if we use Stokes's theorem ,

¢H -dl = /,

(7-84)

where the line integral is to be evaluated around the curve which bounds the su rface S. The line integra l of H- dl is called the magnetomotance, and Eq. 7-84 is Ampere's circuital law (Section 5.11) in its more genera l form for quasistationary currents. This equation is applicable in the presence of magnetic

278

MAGNETIC MATERIALS

[Chap. 7]

materials. It can be used to calculate H, at least for simple current distributions. Using the pole formalism for expressing the magnetic induction as in Eq. 7-51, and taking into account both volume and surface distributions J and X of cond11ction currents, we have from Eq. 7-73 that

H = _I_

1 f' p',,,

41r • r·

r,

+ _I_

1

( u',,, far,+ _I_ J X; r, dr r· 41r • r·

41r ) s

+ _l_

( X

41r ) s

>;r- r, da.

(7-85)

Thus H depends only on the poles and on the conduction currents. The contribution from the poles is calculated just as one would calculate an electrostatic field from electrostatic charges, except that the constant multiplying the integrals is l/ 41r rather than l/ 41r,o. The contribution from the conduction currents is calculated just as one would calculate the magnetic induction from conduction currents using the Biot-Savart law, except that again the constant is 1/ 4,r rather than µo / 4,r. In terms of equivalent currents, Eqs. 7-67 and 7-73 give

H=..!...J~+~x~~+_i_(~+~x~~-~ 41r

T

41r

r-

)s

r-

(7-86)

where we have included the conduction as well as the equivalent currents. Thus, from Eqs. 7-85 and 7-86, the vector H depends on (a) the conduction currents and the poles if we use the pole formalism , or (b) on the conduction currents, the equivalent currents, and the magnetic polarization vector M if we use the equiva lent current formalism. For some purposes it is simpler to think in terms of poles, whereas for other purposes it is preferable to use the equivalent currents. 7.4.3. Magnetic Susceptibility Xm and Relative Permeability Km. Again, as for dielectrics, it is convenient to define a magnetic s11sceptibility Xm such that

M =

x.. H,

(7-87)

where x.. is a dimensionless number, M and H having the same dimensions. This equation holds only if the polarization M and the magnetic field intensity Hare in the same direction , that is, if the material is isotropic. Many magnetic materials which are interesting from an engineering point of view are not isotropic and must be dealt with by considering the individual components of B and of H , as for dielectrics (Section 3.12). In permanent magnets B, M , and H may all be in different directions. There is one important distinction to be made between magnetic and dielectric materials : many interesting dielectrics are linear, hence the electric susceptibility Xe is a constant, whereas n1ost interesting magnetic materials are nonlinear, thus x,,, is not a constant but depends on H. The magnetic susceptibility is nonetheless a useful quantity .

[7.4]

The Magnetic Field Intensity H

From Eq. 7-73,

27,

B = µoH + µoM, = µ,( ! + x,,.)H, = K,,,µ,H,

(7-90)

+ Xm

(7-91)

where

K,,.

=]

(7-88) (7-89)

is another dimensionless quantity called the relative permeability. In general. Km is not a constant but is a function of H. If a given material can have a magnetic polarization M different from zero, it must consist of atomic systems which possess magnetic moments capable of orientation. In terms of the magnetic susceptibility or the relative permeability we may place magnetic materials into the following three classes. PARAMAGNETI C MATERIALS. In most atoms the magnetic moments arising from the orbital and spinning motions of the electrons cancel. In some, however, the cancellation is not complete, and there exists a residual permanent magnetic dipole moment. The so-called transition elements, such as manganese, are examples of this. When such atoms are placed in a magnetic field, they are subject to a torque which tends to align them , but thermal agitation tends to destroy this alignment. This phenomenon is entirely analogous to the alignment of polar molecules in dielectrics (Section 3.9). Consider a gas or liquid with N atoms or molecules per unit volume, each with a magnetic dipole momen t m. The fraction dN / N of the atoms or molec ules which have their moments lying in the angular interval between O and O do is

+

dN ti =

sin o do

- 2

(7-92)

in the absence of an external field , as in Eq. 3-122. If, however, the magnetic dipoles are placed in an external field with a magnetic induction B, the solid angle elements are no longer equally probable, and

dN

= Cexp[m-B / kT] si n OdO,

(7-93)

as in Eq. 3-123. The normalizing constant C is chosen so that the total number of atoms or molecules per unit volume is N. Then , just as in Section 3.9.1 , the resultant magnetic dipole moment per unit volume is M = Nm ( coth 1118 - - -kT) · kT 1118

(7-94)

As for dielectrics, the magnetic energy m8 « kT. The elementary dipole moment is approximately 10- es ampere-meter', thus in a field of one weber/ meter' , (1118/ kT) ~ 2.5 X J0- 3 • Expanding the expo nentials as in Sectio:i 3.9. 1, we find that

280

MAGNETIC MATERIALS

[Chap. 7]

2

M ~N111 8 3k T

(7-95)

~

and that M Xm;::::; µo

B=

µ 0 Nm'

JkT •

(7-96)

The magnetic susceptibility x.. is small compared to unity and varies inversely with the absoluie temperature. Table 7-1 shows typical values of x,,,. DI AM AG NET IC MATER I A LS . In these materials the elementary moments are not permanent but are induced according to the Faraday law. The resultant polarization is in the direction opposite to the external field ; the relative permeability is less than unity and is independent of temperature. All materials are diamagnetic, but orientational polarization may predominate, in which case the resultant permeability is greater tha n unity. FERROMAGNET I C

MATERIALS.

Jn these ma terials there is a strong magnetic polarization, and the relative permeability can be large compared to unity, reaching magnitudes of many thousands in some materials. Such large polarizations are the result of group phenomena in the material, in which all the elementary moments in a small region , known as a domain , are aligned. The resultant polarization in one domain may be oriented at random with respect to Figura 7- 14. Raw /and ring for the determination of the magnetic properties of a the polarization in a neighboring domain. The large polarizations charJerronwgneric material. acteristic of ferromagnetic materials are the result of the orientation of whole domains . The phenomenon is complex and will not be discussed further here.*

7.5. Measurement of Magnetic Properties. The Rowland Ring To determine the properties of a particular ferromagnetic material it is convenient to use a torus of the material on which a coi l is wound, as in Figure 7-14. The lines of B and Hare nearly azimuthal if the turns are closely spaced. If the * See, fo r examp!e, Kittel , fm roducrion 10 Solid S rate Physics (Wiley, New York , 1956), 2nd ed., p. 402 ff.

[7.51

281

Measurement of Magnetic Properties

material is isotropic, the polarization M is also azimuthal , and there are no surface poles. Let us examine the polarization M on a cross section of the torus. This will show whether or not there exists a volume distribution of poles. From the circuital law, (7-97)

along a path of radius r, as in Figure 7-14, and, by symmetry, His everywhere the same on this path . Thus

H = NI, 2r.r

(7-98)

M= xmH,

(7-99)

and

M

=

NI Xm ,.; 2

(7-100)

Then (7-/01)

Since there is cylindrical symmetry with no dependence on th e is thus given by ii>= NI_ (7-158) R

F or example, the flu x in a toroidal core, with a mean diameter of 20 centimeters, a cross-sectional area of IO centimeters', and a relative permea bility of 1000, wound with I 00 turns carrying a current of I 00 milliamperes is

d~-l

ii>= IOO X rrX .

X 1000 X 4.- X 10- 7 X 10-3 ,

(7-159) (7-/60)

= 2.0 X 10- 5 weber, and the self-inductance L

= N

=

100 X 10- 1 X 4.- X 10- 1 X 10-3 (0 .2.-/ 1000) + 10-3 ,

= 7.7 X I0- weber, 0

(7-162) (7-Jq3)

[7.10]

Maxwell's Fourth Equation

297

and

L = IOO X ~ ~,x Io-,= 7.7 X 10- 3 henry.

0

(7-164)

The 1.0-mm gap has therefore decreased the flux and the inductance by a factor greater than two .

7.10. Maxwell's Fourth Equation Although we have introduced the magnetic fi eld intensity H as an aid in computing the induction in magnetic materials, its usefulness is of much broader scope. Let us examine Eq. 5-126 for v X B. We have

For this equation to be completely general , such that it applies to fields in dielectrics and magnetic materials as well as in free space and in nonmagnetic materials, J must include all forms of current. It must include not only the usua l conduction current but also convection currents such as are found in a vacuum diode or in an electrolytic cell. We shall call this current density (conduction plus convection) J,. The current density J must also include dielectric polarization currents J, and equivalent magnetic ct:rrents J."• The dielectric polarization cur rent density is given by aP/ at (see Problem 7-16), and we have already see n in Eq. 7-21 that the equivalent magnetic current is given by V X M. Thus V X B = µ o ( J,

aP + ai + V X M ) + µo aD ai'

v X (;, - M) = J,

+

fi (,,E +

P).

(7-166) (7-167)

In terms of the magnetic fi eld intensity H and the electric displacement D we have (7-168)

where we have dropped the subscript from J, which from now on will always represent conduction plus convection current densi ty. This is the fourth and last of the Max well equations. The term aD / at is the displacement current density and is expressed in amperes/ meter' (Section 5.10). The displacement current obviously does not correspond to current in the usual sense of the word. For exa mple, aD/ at can have a finite value in a perfect vacuum , where there are no cha rges of any kind . It serves to make the total

298

MAGNETIC MATERIALS

[Chap. 7]

current continuous across discontinuities in conduction current. For example, consider a current charging a capacitor, as in Figure 7-27. The current produces a magnetic field , and for the path indicated we have, from Ampere's circuital law (Eq. 7-84), (7-169)

Figure 7-27. Current I flowing through a conductor and charging the capacitance at the gap . The displacement current through th e gap is equal to the conduction current through the conductor, and the line integral of H ·di around the path shown is equal to I even if the surface limited by the path is chosen to pass through the gap.

where / is the current crossing any surface whose boundary is the path of integration. If we choose the surface such that it passes through the gap, then the conduction current through the surface is zero. However, if we compute the displacement current in this case (see Problem 7-17) we find that it is just equal to / . Thus, if the circuital law is written (7-170)

the result is independent of which surface S we choose. It is instructive to estimate the ratio of displacement current density aD/at to conduction current density Jin a conductor. For an alternating field £ 0 cos wt within a conductor of conductivity rr, the conduction current density is

J = rrEo cos wt,

(7-17 I)

whereas the displacement current density is

aD at=

"'0 sm . wt, wK,,o,,

(7-172)

where K, is the dielectric coefficient of the conducting material. Thus

l?I=w:,•o.

(7-173)

The dielectric coefficient K , for a conductor is not readily measured, since any polarization effect is completely overshadowed by conduction. However, the atoms of the material are polarizable, and , for purposes of estimation, we may ·ake K, :=:co I. Setting rr :::co 107 mhos/ meter for a good conductor, we have (7-174)

(7.1 I]

299

Summary

where f = w/ 2..-. Thus the displacement current in a good conductor is completely negligible compared to the conduction current at any frequ ency lower than optical frequencies, wheref R::: 10"/second. It is interesting to note that, although the conduction current is in phase with the electric field intensity, the displacement current leads the electric fi eld by ..-/ 2 radians. In free space ,; = 0, and there is no conduction current-only displacement current.

7 .11. Summary Electron motions within atoms produce atomic magnetic moments. A material possessing a resultant moment is said to be magnetically polarized, and the polarization vector M is the magnetic dipole moment per unit volume. The tota l magnetic induction B in or near polarized magnetic material is the sum of the induction produced by the currents and that produced by the oriented dipoles of the material. The magnetic inductio n contribu ted by the polarized material may be calculated by using either the magnetic pole formalism or the equivalen t current formalism. The former is based on the sca lar magnetic potential V,,, and the latter on the vector potential A. At a point o u tside the polarized material the induction due to the material is B

_ B! ( ,;,;, da 41r s r 2 r1

M -

J

+

B!

J

4r. .,

p,;, ciT ,2

r i,

(7-11)

where (7-8)

and (7-9)

are the s111face a nd volume pole densilies, respecti vely. Using the poles, we calculate the induction B.11 at an externa l point exact ly as we would calculate an electrostat ic field inten sity from electric cha rge d ensities. At the same externa l point the induction can a lso be written (7-28)

where

X' = M X n

(7-20)

and

J' =

V X

M

(7-21)

a re the equivalent s111:face c11rre11/ de11si1y and the equivalent volume currenl

300

MAGNETIC MATERIALS

[Chap. 7]

density, respectively. Using these equiva lent currents, we calculate the induction exactly as we would for conduction currents-from the Biot-Savart law. At an internal point the poles by themselves do not give the induction B"' ; in stead , they give B"' - µoM. The equivalent currents give the correct value of the induction B"' at an internal point. A Gauss's law for magnetic poles can be written for the magnetic induction B:

ls

B,,-da = µ,

1

p:,,ciT.

(7-58)

However, the induction B,, in this equation is only that part of the induction associated with the poles. The total flux of B through any surface bounding a volume is always zero. Since equiva lent currents produce magnetic fields in the same manner as do conduction currents, V X B = µo(J + J') (7-69) for steady currents . This leads to the definition of the magnetic field intensity : (7-73)

whose curl depends only on the conduction current: V X H = J.

(7-76)

The corresponding integral statement,

¢H-dl = /,

(7-84)

is Ampere's circuital law. We can also write a Gauss's law for magnetic poles in terms of H:

lsH -da =

f

(7-82)

p:,, d, ,

which is always true. The magnetic field intensity H can be written in terms of poles and conduction currents:

H= ~ (p :,, d,r 41r .IT r'2

+~ ( 47r

t

+~1JXr1,fr+~

,,:,, dar

J.s r 2

I

41r

41r T ,-?.

(7-85)

(-,.Xr1da

)s r2

'

or in terms of conduction currents, equivalent currents, and M:

H= ~JW+~x~~+~;· ~+~x~~-M 41r

T

r-

41r s

r-

(7-86)

In ma ny magnetic materials B and H are in the same direction, and M is proportional to H , thus (7-87) M = x,,,H,

-'!

[7.11 ] and

Summary

301

B = µo{ l + x,,,)H, = K,,,µol-1,

(7-89) (7-90)

where Xm is the magnetic suscep tibility and K,,, is the relative permeability . Bot h x,,, and K,,, a re dimensionless numbers. Materials fall into three general classes: (a) diamagnetic materials, whose susceptibility Xm is slightly less than unity, (b) paramagnetic materials, whose susceptibility is slightly greater than unity, a nd (c) ferromagnetic materials, whose susceptibility is large comp ared to uni ty . In ferromagnetic materials B depends not only o n H but also on previous va lues of H , a nd the B vs H cu rves a re loops . This phe nomeno n is called hysteresis. When a sa mple of such ma ter ial is subjected to a cycl ic variatio n in H a nd B, the energy di ssipated is W=T¢HdB.

(7-112)

For a unit volume a nd for one cycle this energy is equal to the area enclosed by the hysteresis loop. At a n interface between two magnet ic med ia the tange111ia/ component of H a nd the normal componen t of B are continuous. T o find the magnetic induction B in magnetic ma terials one can, in principle, ca lcula te the magnet ic fi eld intensity H from either Eq. 7-85 or 7-86 and then , knowing the relative permeability for the ma teria l, find B. In practice, however, to find the inductio n B at so me point, or to find the flu x th rough some cross section , we resort to the magnetic circuit concept. We assume (a) that H and B a re either tangential or perpendicular to all the ferromagnetic surfaces, (b) that H a nd B are uniform over a ny cross sect ion of the material, a nd (c) that the rela tive permeability is constan t thro ughout each type of ma terial in the circuit. The effec ts of magnetic poles o n the surfaces of the ma terial help ma ke these assumptio ns genera lly valid. In such a magnetic circuit = NI, R

(7-158)

where N I is ca lled the magnetomotance a nd where R

=I: -._!;_

(7-157) ; K,,, ;µoS ; is the reluctance of the circuit. The util ity of the magnetic field in tensity vector H is not limited to the calc ul a tion of the in duc ti on in magnet ic mate ria ls. When we write the genera l equ ation fo r V X B (Eq. 5-126) and includ e dielectric polarization curren ts and equi valent magnetic cu rrents \ Ve fine! that (7-168)

302

MAGNETIC MATERIALS

[Chap. 7]

where i!D/ at is the displacement current density. This is the fourth and last Maxive/1 equation.

Problems 7-1. Two magnetic dipoles of moments m, and m, are separated by a distance D. ff the dipoles are aligned along the line joining them , what is the force of interaction? If m, lies along the line and m, is perpendicular to the line, what is the force exerted by each on the other? What about Newton's third law? 7-2. A small current loop of moment m is arbitrarily oriented at an off-center point within a spherical surface of radius R. What is the mean value of the induction B within the sphere? 7-3. An iron rod is magnetized in the azimuthal direction (that is, M is always perpendicular to the radius and to the axis of the rod). What dependence on distance from the axis can M have in order that there be no net magnetic charge in the system? 7-4. The magnetic polarization of iron can contribute as much as 2 webers/ meter' to the magnetic induction in iron. If each electron can contribute a magnetic moment of 0.927 X 10- 23 ampere-meter' (one Bohr magneton), how many electrons per atom, on the average, co,,tribute to the polarization? 7-5. A steel sphere of radius R has a uniform magnetization M parallel to the z-axis. What is the total magnetic moment of the sphere? Calculate the magnetic induction B and the magnetic field intensity H at the center of the sphere. How would you find B at points outside the sphere at a distance large compared to R? How does B depend on the distance to the sphere? 7-6. A thin disk of iron of radius rand thickness I is permanently magnetized with a magnetic dipole moment per unit volume M parallel to the axis of the disk. Calculate H in the iron at a point on the axis. Calculate Bat the same point and at a point on the axis outside the disk at a distance D from the center of the disk. If the disk was magnetized by carrying it around a hysteresis loop, at what point on the loop was it left after the magnetizing current was removed? 7-7. A permanent horseshoe magnet has its poles connected with a " keeper" of soft iron when it is not in use. What is the effect of the keeper? 7-8. A permanent magnet consists of a ring of iron of mean radius a, with a small gap of height d cut out. If the iron is magnetized uniformly with magnetic moment M per unit volume, calculate the magnetic field intensity H in the gap and in the iron. 7-9. A permanent magnet (consider a torus with a gap cut out) operates at a point in the second quadrant of its hysteresis curve-the curve in the second quadrant portion being known as the demagnetization curve-since the poles produce a demagnetizing H. The Bo and H 0 in the iron at the operating point can be adjusted by varying the relative lengths of the iron and of the gap in the magnetic circuit. Show that a minimum

303

Problems

volume of iron is required to produce a given B, in the gap if the operating point is selected to make the so-called energy product B,Ho a maximum. Assume that the flux in the iron is the same as that in the gap. 7-10. A long wire of radius a carries a current / and is surrounded coaxially by a long hollow iron cylinder of relative permeability Km. The inner radius of the cylinder is band the outer radius c. Compute the total flu x of B inside a section of the cylinder I meters long. Find the equivalent current density on the inner and outer iron surfaces, and find the direction of the equivalent currents relative to the current in the wire. Find the equivalent current density inside the iron. Find B at distances r > c from the wire. How would this value be affected if the iron cylinder were removed? 7-11. A toroidal solenoid has a mean radius R, has a cross section of radius r, is wound with N turns, and carries a current /. Calculate the magnetic flu x through the solenoid by integrating the induction B over a cross section. Calculate the magnetic flu x by finding a mean value of B and multiplying by the cross-sectional area. At what radius does B have its mean value? 7-12. An iron rod of square cross section and of relative permeability Km is bent to form a ring, and its ends are welded together. Wire is wound toroidally around the ring to form a coil of N turns. Compute the total flu x of B in the ring when the current in the wire is / amperes, taking into account the variation of B over the cross section of the iron. Calculate the inductance of the coil. 7-13. A coil of 300 turns is wound on an iron ring (Km = 500) of 40 centimeters mean diameter and 1O centimeters2 in cross section. Calculate the magnetic flux in

the ring when the current in the coil is one ampere. Calculate the flux when there is a gap of 1.0 millimeter in the ring. 7-14. Show that the_energy stored in a magnetic circuit ca n be written as W= {'R ,

where is the flux and R is the reluctance of the circuit. 7-15. A region of space may be "sh ielded" from magnetic fields by surrounding it with high permeability iron. Show this for a magnetic field B perpendicular to the axis of an iron cylinder of inner and outer radii a and b, respectively, and of relative permea bility K,,,. Solve Laplace's equation for the magnetic potential outside the iron, in the iron, and in the enclosed space. Use the cylindrical harmonics of Problem 4-16 in a suitable combination to match the boundary conditions. By how large a factor can the field in the inner region be reduced with this technique ' 7-16. Show that a time-dependent dielectric polarizat ion vector P is accompanied by an effective electric current whose densit y is aP / at. This is the polarizatio11 curren t density. 7-17. A constant current charges a capacitor as in Figure 7-27. Show that the displacement current is equa l to the charging current.

CHAPTER

8

Maxwell's Equations We have established at this stage the four basic principles of electromagnetic theory which are stated in mathematical form as the fo ur equations of Maxwell. Our object in this chap ter is to re-examine these four equations as a group and to draw from them several conclusio ns of fundamental importance.

8.1. Maxwell's Equations Let us group the four equations of Maxwell which we found successively as Eqs. 3-58, 5-45, 6-11 , and 7-168:

v·D =

P,

(8-1)

,,B = 0,

(8-2)

v XE+ aB = 0 a1 '

(8-3)

iJD

v X H - at = J.

(8-4)

As usual , D is the electric displacement in coulombs/ meter' , Bis the magnetic induction in webers/ meter' , Eis the electric field intensity in volts/ meter , H is the magnetic field intensity in amperes/ meter, pis the free charge density in coulombs/ meter' . and J is the current density in amperes/ meter' . The current density J is meant to include both conduction and convection current s (Section 7. 10). For points outside the sources, and for conductors obeying Ohm's law, J = aE, (8-5) where" is the conductivity of the medium in mhos; meter. Inside a source such 304

[8. 1]

305

Maxll'elts Equa1io11s

as a battery , there is also another electric field intensity E, due to the local generation of energy , a nd (8-6) J = o-(E -I- E.,). Maxwell's eq uatio ns are partial differential eq uations involving space derivatives of the four fie ld vec/ors E , D , B, and H, the time derivatives of Band of D, the free charge density p, a nd the curren t density J. They do not, of course, yield the va lues of the field vectors directly, but only after integrat in g with the boundary conditions appropriate to the fi eld under conside rat io n. Since these equations are linear, each charge or current distribution produces its own field independently; for example, the resulting E vector at a given poi nt is the vector sum of the E vectors prod uced at that point by the va rious sources. This is the princ1j;le of s11perposi1io11 (Sect io n 2.2). These are the four f,mdamental eq11a1ions of electromagnetism. They are completely general and apply 10 all electromagnetic phenomena in media which are at res/ 1Vilh respecl 10 1he coordina1e sys1em used. They are validfor nonlwmogeneous, nonlinear and even for 11011iso1ropic media. All of !he remaining clwplers IVill be based on them. It is worthwhil e to rewrite Maxwell 's equations in sca lar for m fo r Cartesia n coordinates:

aD, -I- aD,, + aD, =

ax

ay

az

+ aB,, + ~B, =

P,

(8-7)

0,

(8-8)

+ i!B, = O

'

(8-9)

ac _ aE, + as,, = 0

'

(8-10)

aB,

ax

ay

aE, _ aE,,

ay az

az

ax

aE,, _ iJ E,

ax

ay

az a1

a1

+ as, = 0 a, '

aH, _ aH,, _ aD, =

(8-JI)

1

(8-12)

aH , - aH, - i! Dy = J

(8-13)

ay

i)z

az ax

a1 at

"

y,

i! H,, _ i!H, _ aD, = 1-.

ax

iiy

at

-

(8-14)

M axwell 's equations, Eqs. 8-1 to 8-4, are not a ll independent of each other. If we take the divergence of Eq. 8-3, reca lli ng that the divergence of the curl of any vector is zero, then (8-15)

306

MAXWELL ' S EQUATIO NS

[Chap. 8]

or, inverting the order of the operations,

(8-16) The quan tity v •B is therefore independent of the time at any point in space. We ca n set the divergence of B equa l to zero everywhere if we assume that, for each point of space, it becomes equal to zero at any time, either in the past or in the future. Under this assumption, Eq. 8-2 can thus be deduced from Eq. 8-3. Equations 8-2 and 8-3 are sometimes called the first pair of Maxwell's equations. Similarly, taking the divergence of Eq. 8-4, we have

(8-17) If we assume that there is conservation of charge, as we did in Eq . 5-86, then V .

iiD = iJ.l'., iii

where

p

is the charge den sity, and V·D = p

iii

+

C,

(8-18) (8-19)

where C is some quantity which can be a function of the coordinates, but which is independent of the time. If we further assume that, at every point of space, at some time either in the past or in the futur e, both v · D and p become equal to zero, then the constant of integration C must be zero, and we are left with Eq. 8-/. Under these two assumptio ns, Eqs. 8-1 and 8-4 are therefore not ind epend ent. They form the second pair of Maxwell's equations. It is possible to eliminate the vectors D a nd H from Maxwell's equations by substituting for these quantities the values given in Eqs. 3-57 and 7-73:

D = ,.,E

+ P,

H=_!!-M µ,

'

(8-20) (8-21)

where P is the polariza tion vector in a dielectric in coulombs/ meter', and M is the magnetization vector in a magnetic medium in ampere-turns/ meter. These two quantities acco unt for the presence of matter at the point considered. Then Maxwell 's equations ta ke the following form : V·E = _l_(p - V·P),

,,

(8-22)

= 0,

(8-23)

X E + iiB = 0

(8-24)

v-B V

ii t

'

(8-25)

[8. 2]

Maxwell's Equations in Integral Form

307

The a bove are again M axwell's equati ons in completely genera l fo rm , but expressed in such a way as to stress the contributi ons of the medi um. It will be o bserved that the presence of matter has the effect of adding the bound cha rge density - V. p (Sectio n 3.2.1 ), the po la riza tion c urrent density aP/at (Pro blem 7-16), a nd the equi va lent c urrent densi ty V X M (Sectio n 7.2.2.). Usually, in linear isotropic media,

D

=

K,oE,

(8-26)

H

=

__!!_,

(8-27)

and

K,,,µ,

where K. is the d ielectric coefficie nt a nd K,,, is the relative permea bi lity. The n, in terms o f E a nd H , M axwell 's equat ions become (8-28)

V X E

+

V·H = 0, aH K,,,µ, at = 0,

X H - K,,o

V

aE

at = J.

(8-29) (8-30) (8-3 I)

In the remain ing ch apters we sha ll be concerned mostly with p henomena in which the fie ld vectors a re sinuso ida l fun ct io ns of time. Then, instead o f d ifferentia ting with respect to time, we can m ul tiply by jw, acco rd in g to Append ix F , a nd M axwell 's eq ua tio ns ca n then be rew ritte n as

V ·D = V

P,

(8-32)

, -B = 0, X E + jwB = 0,

(8-33)

V X H - j wD

=

J.

(8-34) (8-35)

8.2. Maxwell's Equations in Integra l Form We have stated Ma xwell's eq ua tio ns in d iffe re nti al for m ; let us now state them in their integra l fo rm in order th at we may a rrive a t a better understa nd ing o f their physica l mea ning. Integrating Eq. 8-1 over a vo lume r , we obtain

J,

v-D dr = J p dr,

(8-36)

or, fr om the d ivergence th eorem,

ls

D-da =

J p

cir = Q,

(8-37)

308

MAXWELL 'S EQUATIONS

[Chap. 8]

Figure 8-1

Lines of D e:nerging from a volume T , containing a net cha rge Q. The out~ ward flux o f D is equal to Q.

where S is the surface bounding the volume , and where Q is the net charge contained wit hin T. The outward flu x of the electric displacement D through any closed surface S is therefore equal to the net charge inside, as illustrated in Figure 8-1. This is Gauss's law (Section 2.4). Integrating Eq. 8-2 in a similar manner , we find that the outward flux of the magnetic induction B through any closed surface S is equal to zero. This is shown in Figure 8-2. Equation 8-3 can be integrated over a surface S bounded by a curve C: ( V

)s

X E-da = - ( aB. da,

)s a1

(8-38)

Figure 8-2

Lines of B 1hrough a closed surface S. The nel ou /ward flux of B is equal 10 zero.

[8.2]

Maxwell's Equations in Integral Form

309

Figure 8-3 .

The direction of the electromotance induced around C is indicated by an arro w for th e case where the magnet ic iw:lucti on 8 is in the direc tion sh o wn and increases. T he e/ectromotan ce is in the same directio n if B is upward and decreases .

or, if we use Stokes·s theorem o n the left and inve rt the operations on the right , we have

):, E di = _ j_ ( B-da.

'Yc

at )s

(8-39)

Then the electromotance induced aro und the curve C is eq ual to minus the rate of change of the magnetic flu x linking the curve C, as in Figure 8-3. The positi ve direction s for B a nd around Care rela ted according to the right-hand screw rule. Finally, if we also in tegra te Eq. 8-4 over a n area S bounded by a c urve C, we fine! that

):, H-dl 'fc.

.da, .Is + aD) at

= ( (1

(8-40)

and the magnetomota nce a ro und the curve C is eq ua l to the total current linkin g the curve C, aD/ at bein g the disp lacement c urren t density. This is illustrated in Figure 8-4. The positive direct io ns a re again rela ted by the right-hand screw rule.

Figure 8-4

The direction of the magnetomotance around C is indicated by an arrow for the case wh ere th e torn/ currenl J +

(aD/ut) is in th e direction sho;vn. Th e displacement current is d o wnward if (iJ D/Ut) is downward and increases, or if it is upwa rd and decreases .

310

MAXWELL'S EQUATIONS

[Chap. 8]

8.3. E-H Symmetry In a linear isotropic medium where the charge density ,; are both zero, Maxwell's equations reduce to

V X E

p

and the conductivity

V·E = 0,

(8-41)

V·H = 0,

(8-42)

+ K.,µ, aH at = 0,

(8-43)

aE

V X H - K,-.

Ao

1.05 X I0- 5 ( _l_

crK.,

)1''·

(10-114)

A striking featur e of this expression is the strong dispersion which occurs in good conductors: the wave velocity varies as the square root of the frequency.

~

.~0 TABLE 10 - 1.

hi

Skin Depths O for Cond uctors •

"',.,c:,°

::;

Skin Depth 0

~

Relative

Cond uctor

Conduc tivity u

Permeabi lity

ofl l~

60 cps

1000 cps

I Mes

300 0 Mes

~ ::,

(mhos / meter)

K.., t

{me ters/ second 112)

(centimeters)

(mill imeters)

(millimetersl

(micr'lns)

~-

7

Aluminum

3.54

X 10

1.00

0.085

1. 1

2.7

0.085

Bross (65 .8 Cu, 34 .2 Zn}

1.59 3.8

X 10 7 X 107

1.00 1.00

0. 126

1.63

3.98

X 10 7

1.00

1.0 0.85

2.6

Gold

5.80 4.50

0.081 0.066

0.126 0.08 1

X 10'

1.00

0.075

0 .97

Graphite

1.0

1.00

1.59

Magnetic iron

1.0

X 10'' X 107

2 X 10'

0.0 1 1

20.5 0.14

Chromium

Cop p er

2. 1 2.38 50.3 0.35

0.066 0.075 1.59 0.01 1

1.6 2.30 1.5 1.2 1.4 29.0 0.20

X 107

2 X 10•

0.0029

0.037

X 10'

1 X 10' 1.00

0.014

0.18

Nickel

0. 16 1.3

Seo water

5 .0

Silve r

6.15

X 10 7

Tin

0.870 X 10 7 1.86 X 10 7

Zinc

~

;,g

"s· C)

c:, c:,

"-

g;i ::,

9-

Mumetal (75 Ni, 2 Cr,

5 Cu, 18 Fe)

.,,

1.00

2 X 10'' 0.064

3 X 10' 0.83

1.00 1.00

0. 17 1 0.117

2.21 1.51

0.092 4.4

0.0029 0.014

7 X 10' 2.03 5.41

2 X 10'

3.70

0.1 17

0.064 0.171

0.053 0.26

;;

c ~

1.2 3.12 2. 14

* Adopted from the American Institute of Physics Handbook (McGraw-Hill, New York, 1957), Section 5, p. 90.

t At S = 0 .002 weber/ meter~. ** At this fre quency l:J is about 2, and

sea water is not o ''good" conductor (K,

z

70).

(;.7 (;.7

"'

340

PLANE ELECTROMAGNETIC WAVES IN MATTER

[Chap. 10]

The velocity is also low, at least as long as Q is small, as required by our calculation. For example, at a frequency of I megacycle/ second in copper, the velocity turns out to be about 500 meters/ second and the index of refraction c/ 11 about 7 X 105• The wavelength in the copper is only about 0.5 millimeter, whereas it is 300 meters in air. Within the limits of our approximation, that is, for Q ;;;;

-fci, the wavelength in the conductor is always much shorter than in

free space. For copper, (10-115)

The ratio of the electric to the magnetic energy density is

(10-116)

wp- We may use the wave number k for a medium of conductivity u as given by Eq. 10-55. We set K, = I, K"' = 1, and substitute the value of u from Eq. 10-147. Then

(/0-170)

(/0-171)

and

E=

E oei (w!- kz ) i.

(/0-172)

It is to be understood that we must use the positive value of k to describe a wave traveling in the positive direction of the z-axis. From the general relationship for £/ H , Eq. 10-24, we have H = !5:._ E wµ,

(/0-173) (/0-174)

350

PLA NE ELECTROMAGNETIC WAVES IN MATTER

[Chap. 10]

Now let us consider the case of high frequencies where w > wp . Then the wave number k is real and the E and H vectors are in phase, the ratio E/ H being larger than in free space by a factor of 1/[ I - (wp/w)']. As w--+ wp, H--+ 0 as expected, since the total current density J + (a D/a t) --+ 0, and the Poynting vector S --+ 0. The index of refraction is given by n

= =

[

I-

[I -

:cE ) ' ] "' ' ( w w

80.5

N,]"' f' ,

(10-175) (10-176)

and the phase velocity by II =

C

[1 - (-w")']"'

(10-177)

w

For high frequ encies where w > wp, the phase velocity u is therefore greater than the velocity of light, the wave number k is real, and there is no attenuation. Figure 10-6 shows both n and 1/n as functi ons of w/wp, Since the phase velocity increases with increasing ion density, waves tend to bend away from regions of high ion density. For high frequencies where w 2 >> w2p, the transmissio n is unaffected by the presence of ionized gas. The index of refraction and the wave velocity which we have discussed above refer to the phase velocity, that is, to the velocity at which a given phase is propagated. This is not the velocity at which a signal can be transmitted. The reason for this is the following. A signal can be trans4 5 mitted only if the wave is modulated w/wp either in amplitude or in frequency . Figure 10-6. The index of refraction n for an ionized gas and its inverse I / n as For example, the source may operate in short bursts of variable length as functions of the ratio c,,/,,,v . in telegraphy, or the amplitude may cha nge at an acoustic frequency as in amplitude-modulation radio broadcasting, or the frequency may change as in frequency-modulation broadcasting. Since changes in amplitude necessarily involve frequencies other than the carrier frequency , a signal necessarily involves more than one frequency. (n other

[10.5] Electromagnetic Waves in Ionized Gases

35 1

words, a single frequency corresponds to a pure sine wave extending from t= -co to 1 = +co which can transmit no information . Since the phase velocity in an ionized gas is frequency dependent , the various frequency components of a signal are transmitted at different velocities. The result is that a signal travels at a velocity which is different from those o f its component waves. It has been show n tha t a signal can never be transmitted at a velocity exceeding the velocity of light in free space, c. * Moreover, since the co mponent waves travel at different velocities, the shape of the signal. that is, the envelope of the wave, changes with time as the waves progress through the dispersive medium.

< wp. For w < w.p, the wave number k of Eq. 10-171 is a pure imaginary number. The vectors E and H are then out of phase by 7r/ 2 radians, and £ / His again la rger than in free space. The Poynting vector S = ( 1/ 2) Re (E X H*) is zero, and there is no energy transmission. The index of refraction and the wave velocity are also imaginary, and 10.5.4. Wave Propagation at Low Frequencies Where w

E = E,eiwt- k',, H = H oeiwt- k'z,

(10-178) (10-179)

where k ' = jk is a real number. Thus, there is no wave, the phases of E and of H being i11depe ndent of z, and the amplitude decreases expo nentially with z. An electromagnetic wave is therefore either transmitted without attenuation or not tra nsmitted at all through an io nized gas, depending on the ratio w/ w,,. High frequencies are transmitted , whereas low frequ encies a re not. Our theory is however much simplified. In pa rticular, we have neglected all energy losses, which is justifiable only a t low gas press ures. I0.5.5. The Ionosphere. Thi s regio n or the upper atmosphere ranges in altitude from approximately 50-1000 kilometers, where the ionization is sufficie nt to interfere with the propagatio n of electromagnetic waves. The ioni zation is attributed mostly to the ultravio let radiation of the sun. On the whole, the ion density fir st increases with altitude a nd then decreases, but it shows " ledges" where the ion density varies more slowly with altitude. These ledges are commonly called the D, E, F 1, and F, layers. These maxima arise from the fact that both the nature of the solar radiation a nd the composition of the atmosphere change with altitude. Both the heights and the inte nsities of ionization of these layers change wi th the hour of the day, the season, the sunspot cycle, and so on. It a ppea rs tha t the conductivity " is due almost exclusively to the electron density , except possibly near the lower limit of the ionosp here.

* Leon

Brillouin, Wm'e Propagation and Group Velociry (Academic , New Yo rk, 1960).

352

PLANE ELECTROMAGNETIC WAVES IN MATTER

[Chap. 10]

The free electron density is typically 10 11 / meter3 , varying by about 1010 to 1012/ meter 3 from the lowest to the highest layer. Over this range of altitudes, the number of molecules per cubic meter varies from about 1022 to 10 15 • The percent ionization therefore increases very rapidly with altitude, but it always remains low. For N, = 1011 / meter', the plasma frequency f, ~ 3 megacycles/ second. Frequencies lower than f, are not transmitted . At frequencies somewhat higher than/,, waves are bent back toward the earth, since ion densities generally increase with increasing height and since the phase velocity increases with increasing ion density . The assumption that there are no collisions between the electrons and the gas atoms or molecules is satisfactory except in the lowest regions of the ionosphere where the pressure is highest, and at frequencies of the order of I megacycle/ second or higher. In the presence of the earth's magnetic field the ionized gas becomes doubly refracting, with the result that there are two distinct phase velocities, depending on whether the wave is polarized parallel to the magnetic induction vector or perpendicular to it.

10.6. Summary In homogeneous, isotropic, linear, and stationary media, a plane electromagnetic wave has the following characteristics :

(I) p = 0 (Eq. 10-18); (2) both E and Hare transverse (Eqs . 10-17 and 10-20); (3) E and H are mutually perpendicular ; (4) £/ H = wµ. / k (Eq. 10-24) ; (5) E and H are oriented such that their vector product E X H points in the direction of propagation. In nonconductors, the phase velocity is U

=

C

(K,Km)l/2'

(10-27)

and , in nonmagnetic media (Km = I), the index of refraction n is related to the dielectric coefficient by the relation (10-29)

The vectors E and H are in phase, and the electric and magnetic energy densities are equal (Eq. 10-38). In conductin g media we define the Q of the medium as

Q=

w, (J

(10-49)

[I0.6]

Summary

353

This is the modulus of the ratio of the displacement current density to the conduction current density. The wave number k is complex (Eq. 10-58), and the E vector leads the H vector (Eq. 10-67). The average value of the Poynting vector is then given by

i

S.,. =

Re (EX H*),

(10-88)

which is the product of the average energy density by the phase velocity. Good conductors are defined as media for which 0

I -.,; the other has a slightly different circular frequency w, and a wavelength >-.,. At a given time I there exist values of z for which the two waves are in phase and values of z for which they are opposite in phase. What is the distance between the maxima? What is their velocity? This velocity is called the group 1·elocity 11,. Show that, in the limit,

Show that where 11 , is the phase velocity given in Eq. 10-177. Calculate the phase velocities and the group velocity for /i = 5.3 megacycles/ second, J, = 5.4 megacycles/ second, and N, = 5 X !010 electrons/ meter'. Calculate the distance and the number of waves between two mininia. What happens to the group a, Ne increases?

CHAPTER

11 Reflection and Refraction

In Chapter IO we co nsidered the propagation of electromagnetic waves in infinit e, continuous med ia. We shall now examine the effect of a discontinuity in the medium of propagation , as show n in Figure 11-1. We sha U invest igate again the same three types of media as in Chapter 10: lossless dielectrics, good cond uctors, a nd low-pressure ionized gases. We assume an idea lly thin , infinite, plane interface between two ho mogeneo us, isotrop ic media. Then a n incident wave alo ng n,, as show n in Figure 11-1 , gives rise to both a refl ected wave along n, a nd a trans mitted wave a long n,. We a lso assume that media I a nd 2 a re infinite so that there are no multiple reflect io ns. We shall show tha t, in general, all three waves a re required to satisfy the condi tion of continuity for the tangential compo nents of E a nd of H. F or the time being, we shall excl ude the subject of total refl ection ; this will be discussed later in Section 11 .4. This genera l type of phenomenon is frequently enco untered in other fi eld s. For example, a so und wave incident upon a wall results in both a reflected wave which comes back into the room and a tran smitted wave which proceeds in to the wall. This phenomenon is also well known in electrica l tra nsmissio n lines a t points where one type of lin e is co nnected to a nother, for exa mple, at a junction between two different types of coaxial cable. Waves on stri ngs show the same type of behavior (Append ix G). It is not suggested , however, that the student attempt to treat acoust ical a nd electrica l reflections by the methods used in this chapter. 357

358

REl'LECTION AND REFRAC,ION

[Chap. 11]

1 I. I. The Laws of Reflection and Snell's Law of Refraction If we assume that the incident electromagnetic wave is planar, then its electric field intensity E, is of the form E, = E,, exp [}(wt - k1n , ·r)]

(J 1-1)

(see Appendix G). The time I = 0 and the origin r = 0 can be chosen arbitrarily. This equation defines a plane wave for all values of I and for all values of r, that is, a wave which extends throughout all time and all space. However, it will be taken to be applicable only in medium I. If the incident wave is planar, then both the reflected and the refracted waves from a plane interface must also be planar, since the laws of refl ection and of refraction for any given incident ray must be the same figure 11-1. An electromagnetic wave in at all points on the interface. Thea the m edium I is incident on the interface be- reflected and the transmitted waves tween media I and 2 and giv es rise to are given by both a reflected and a transmiued wave. The vectors n 1, n,., and n 1 are unit vectors normal to the respective wave fro nts, and point in rhe direcrion of propagarion. The angles 01, 0 r, and 0, are, respectively, rh e angles of incidence, o f refle crion, and of refraction. (Radio engineers use a different convention and call the comple111entary angles (-;; / 2) - 0 1, (,r / 2) - 0,, and (,r/ 2) - 0 1 the angles of incidence, of reflection, and of refraction, respectively.) Total reflection will be treated separately in Section 11.4.

E, = E,, exp [j(w, t - k1n,•r

+ A)], (J 1-2)

E, = E,. exp [j(w,t - k,n, · r + B)], (J 1-3)

where the constants A and B allow for possible phase differences with the incident wave at the interface. Note that we have made no assumption whatever as to the amplitudes, phases, frequencies , or directions of the reflected a nd transmitted waves. We shall be able to determine the characteristics of both the reflected and the transmitted waves from the fact that the tangential component of E and the ta ngential component of H must both be continuous across the interface (Sections 4.1 .3 and 7.8). Then the sum of the tangential components of E, and E, just a bove the interface must be equal to the ta ngential component of E, just below the interface; a simil ar situation holds for the magnetic field intensity H . Tht>

[11.1]

The Laws of Reflection and Snell's Law of Refraction

359

reflected and the transmitted waves could also be obta ined fr om the continuity of the normal components of D and of B across the in terface. To obtain continuity o f the tangential components of E and of H at the interface, some va lid relation must exist between E,, E., and E, for all time t a nd for all vectors r, which terminate o n the interface, as in Figure 11-2. S uch a relation will be possible (a) if a ll three vectors E ,, E,, and E, a re identical func tions of the time t and of position r, on the interface, and (b) if there exist certain relations between E, ,, E,,, a nd E,,. Let us start with the first conditio n. Figure 11-2. The projection of r 1 on n 1 nr is a constant at any point o n th e interWe must have fa ce, that is, for any value of r 1. Th e vecw ;t - k,n ,•r, = w,t - k,n, •r, + A

=w,t - k 2n ,• r,+B (11-4)

tor n is normal to th e int erface.

for a ll t and for all r,. It follows , then, that

(I 1-5)

All three waves must therefore be of the same frequ ency. This is intuitively quite obvious, since they are a ll superpositio ns of the wave emitted by the source a nd of those waves em itted by the electrons exec utin g forced vibrations in media I and 2. It will be reca lled from mechanics that forced vibrations have the same frequ ency as the app lied force. From Eq. 11-4 we must a lso have, a t any point r, o n the interface,

k1n i r1 = k1nr r1 - A,

= k,n, r1

B.

-

(I 1-6) (I 1-7)

Then, from t he first of these equat io ns,

(n , - n,)•rt =

A

- k-, ,

(I 1-8)

where the term A / k, o n the right-hand side is a consta nt for all r, . That is, the projection on (n, - n,) of a ny vector r, terminating on the interface must be a constant. The vector (n,. - n,) is also a consta nt for any given incident plane wave. Then n,: - n, must be normal to the interface, as in Figure 11-2. The interface itself is defined by (11-9) n •r, = Constant, where n is a unit vector normal to the inter face.

360

REFLECTIO N A N D REFRACTIO N

[Chap. 11]

Since n, - n, is normal to the interface, the tangential components of these two vectors must be equal and opposite in sign as in Figure 11-2. Then (11-10)

or the angle of reflection is equal to the angle of incidence. We also conclude that the vectors n,, n,, and n are coplanar, since (n, - n,) and n are parallel. ThesJ are the laws of reflection. The plane of the above three vectors is called the plane of incidence; it is normal to the interface. Considering now Eq . 11-7, we have (11-11)

hence the vector k,n , - k,n, must also be normal to the interface, so that n,, n,, a nd n are coplanar; thus all four vectors n,, n,, n,, and n are in the plane of incidence. Moreover, the tangential components of k,n, and of k,n , must be equal , hence k, sin 0; = k, sin 0,. . (11-12) The quantity k sin 0 is therefore conserved in crossing the interface. We can also write that sin 0, k, (11-13) sin 0; = k,' (11-14)

since the wave number k = n/ X0, where n is the index of refraction . This is Snell's la w. It is important to note that this law, as well as the laws of reflection, are perfectly general. They apply to any two media-they even hold true for total reflection, as will be shown later on. The constants A and B in the above equations are related to the choice of the origin, as can be seen from Eqs. 11-2 and 11-3 : at a given point on the interface and at a given time, the reflected and the transmitted Jy waves have definite phases which are Figure 11-3. System of axes used for the represented by the coefficients of j study of reflection and refraction. in the exponents; if the origin is displaced, A and B must be adjusted accordingly. It turns out that if the origin is chosen in the interface, A and B both become eq ual to zero. Thi s will be shown in Problem 11-1. If we choose the origin in the in terface and place axes as shown in Figure

•

[11.2]

361

Fresnel's Equations

11-3 , then A = B = 0, and the expressions for E,, E,, and E, can be written as follows:

E, = E,, exp {j[wt - k1(sin 0; x - cos 0,- z)]},

(I 1-15)

+ cos0 , z)]},

(I 1-/6)

E, = E,, exp {j[wl - k,(sin 0, x - cos 0, z)]}.

(11-17)

E, = E ,, exp {}[wt - k 1(sin0 , x

11.2. Fresnel's Equations We shall turn now to the second condition mentioned in the preceding section. We must find the relations between the quantities E,,, E,,, E., which will ensure continuity of the tangential components of E and of H at the interface. We recall from Section 10.1.1 that the E and H vectors in a plane ,y electromagnetic wave are always perpendicular to the direction of Figure 11-4. The incident, reflected, and propagation and to each other. The transmitted waves for the case in which th e incident wave is polarized with its E E vector of the incident wave can vector normal to the plane of incidence. thus be oriented in any direction The arro ws for E and for H indicate the perpendicular to th e vector n ,. directions in which th e E and H vectors It will be convenient to divide the are taken to be positive at the interface. discussion into two parts. We shall first consider the case in which the incident wave is polarized such that its E vector is normal to the plane of incidence and then consider the case in which its E vector is parallel to the plane of incidence. Any incident wave can be separated into two such components.

11 .2.1. Incident Wave Polarized with Its E Vector Norma/ to the Plane of Incidence. The E and the H vectors of the incident wave are oriented as in Figure 11-4. If the media are isotropic, as we assumed at the beginning of this chapter, the E vectors of both the reflected and the transmitted waves wiH also be perpendicular to the plane of incidence, as in Figure 11-4. Considering the electric and magnetic field intensities of the incident wave to be known , we now have four unknowns: £ 0 , , £ 0 1 , H,,, and H,,. We also have a total of four equations which are provided (a) by the continuity of the tangential components of both the E and H vectors at the interface and (b) by the relations between the E and H vectors for plane waves in medium I and in

362

REFLECTIO N AND REFRACTIO N

(Chap. II]

medium 2 as given in Eq . 10-24. It will suffice, for the moment, to calculate on ly the E vectors. Instead of using the continuity of the tangential components of E and H, we could also use the continuity of the normal components of D and of B. This is not desirable, however, because if we did so, and if we wished our results to be applicable to reflection from the surface of a conductor, we would have to take the surface charge density into account. This would introduce another unknown. The continuity of the tangential component of the electric field intensity at the interface requires that (11-18)

at any given time and at any given point on the interface. Similarly, the continuity of the magnetic field intensity requires that (11-19)

or , from Eq. 10-24, (J 1-20)

Thus

(£) Eoi

n2 -111 coso, - Kcoso, 1112

K ml

QI

N

=

O

!!..!.._ K ml COS

and

(£Eoi")

N

i

+ Km _!!J_ O' COS t

(11-21)

2

,,,

2

=

!!!_ K ml COS

x:.. cos O i

0,

+ /(_m 112

0' 2

COS

(11-22)

t

where the index N indicates that £ ,, is normal to the plane of incidence. We have replaced the wave numbers k by the indices of refraction, one being proportional to the other: k = n/'A0, where '-c is the free space wavelength divided by 21r. These are two of Fresnel's eq1101io11s; the other pair will be deduced in the next section . Fresnel's equations give the ratios of the amplitudes of the incident, reflected , and transmitted waves. They are completely general and apply to any two media. We shall show later on that they are valid even for total reflection and for reflection from the surface of a good conductor. 11 .2.2. Fresnel's Equations for the Case Where the Incident Wave ls Po• larized with Its E Vector Parallel to the Plane of Incidence. In this case the E vectors of all three waves must be in the plane of incidence as in Figure 11-5. We have chosen the orientations of E,, and of E,, such that Figures 11-4 and

[11 .2]

36:

Fresnel's E I, then 8, > 81, and cos 8, > cos o,; whereas if ni/n, < I , then 8, < 8,, and cos 81 < cos 8,. The reflected wave is thus either in phase with the incident wave at the interface if 11 1 > 11, or is -rr radians out of phase if n, < n,. Figure 11 -6 illustrates the E vectors for both types of reflection; Figure 11-7 shows the ratios of Eqs. 11-31 and 11-32 for 1.0

1.0

0.8

0.8

~~] E o1 N

0.6

0.6

0.4

0.4

0.2

0.2

0.0

10

20

30

40

-04 -0.6

50 8;

-0.2 Ee, ~~

60

70

80 °

0.0 -0.2 -0.4

N

-0.6

-0.8

-OS

-1.0

-IO

rarios (£,,./£ 0 ,Js and (£ 01 / 01 ) .v as fun crions of rh e angle of incidence 8 1 for n 1 /n 0 I I I .5 . This corresponds to light incident in air on a glass with n = 1.5. Th e wave is polarized with its E ,;ector normal to 1/ie plane of incidence. Figure 11-7. Th e

=

[I 1.3]

Reflection and Refraction Between Two Dielectrics

367

n, / n, = 1/ 1.5. This corresponds, for example, to a light wave incident in a ir on a glass with an index of refraction of 1.5. At normal incidence, O, = 0, and E,,

(;,) -

i'

(Jl-33)

2 (::) .

(I /-34)

E,, = (;,) E., -

+

(;,)+I

E,, -

For an incident wave polarized with its E vector parallel to the plane of incidence, -cos cos

o, + ('!!) cos o, n,

(I 1-35)

o, + (;,) cos e, '

whereas

(tt =

2 (;,) cos 0, COS

8i

+ (n-

1)

(l l-36)

COS

0I

112

The second ratio is always positive. This indicates that the relative phases of E,, and £ ,, are as shown in Figure 11-5, which we used in arriving at this result; that is , the tangential components of the incident and transmitted electric field intensities are in phase at the interface. On the other hand , the ratio for E,, can be either positive or negative, which indicates that E,, can point either as in Figure 11-5 or in the opposite direction. The tangential components of E,, and of E,, can thus be either in phase or rr radians out of phase. The E,, component is in phase with E, , at the interface if

('!!) cos e, -

cose, > O

llz

or if sin

o, cos o, -

cos e, sin

o, > 0,

(I 1-38)

>

0,

(l /-39)

+ 0,) > 0.

(l l-40)

sin 28, - sin 20, sin (O, - 0;) cos (8,

(l J-37)

This inequality will be sat isfied either if

o, > o,

und

o, + 0, < i

(ll-41)

< 0,

and

o, + o, > i

(I 1-42)

or if 0,

368

REFLECTION AND REFRACTION

[Chap. Ill

The phase of the reflected wave in this case does not therefore depend only on the ratio n,/11,; it depends on both 0; and 0,. The ratio £ 0 , / £ 0 , can be either positive or negative, both for 11 2 > 111 and for n, < n,. For normal incidence, Eqs. 11-33 and 11-34 apply as expected. Figure 11-8 shows the ratios of Eqs. 11-35 and Jl-36 again for nifn, = 1/ 1.5. I.O

1.0

0.8

0 .8

[,

06

Ea,

0.6 P

04

04

0.2

0.2

00

10

0 .0

20

-0.2

-0.2

-04

-0.4

Fig ure 11 -8. The

ratios (E 0 JE 0 ;)p and (Eot l 0 ;)p as functions of the angle of incidence 0 1 for n 1 /11 2 JI 1.5. This corresponds to light incident in air on a glass with n = 1.5. The wave is polarized with its E vector parallel to the plane of incidence.

=

11 .3.1 . The Brewster Angle. We have seen in the foregoing that the electric field intensity £ of the reflected wave is either in phase or is ,r radians out of phase with the incident wave, depending on whether sin (0 1 - 0;) cos (0, + 0;) is greater or less than zero. It will be gathered from this that there is 110 reflected wave when this expression is equal to zero, that is, when 0; = 0, = 0 or when 0; + 0, = ,r/ 2. The first condition is incorrect, however; it arises from the fact that we have multiplied the inequa lity 0,

(;,) cos 0, - cos 0; > 0 by sin 0;, which is equal to zero at 0; = 0. Thus, for

e, + 0,

=

1l'

2,

(11-43)

there is a reflected wave only when the incident wave is polarized with its E vector normal to the plane of incidence. This is rather remarkable because it involves th e passage of a wave through a discontinuity in the medium of propagation without the production of a reflected wave. The conditions of con-

[11.3]

Reflection and Refraction Between Two Dielectrics

369

tinuity at the interface are then satisfied by two waves only-the incident and the transmitted waves-instead of the usual three. This is illustrated in Figure 11-9. The angle of incidence is then called the Brewster angle. It is also called

,y

ly

When the incident wave is polarized with its E vector parallel to the plane of incidence , there is no reflected wave at 0i. + 0, = ;:-/2. The angle of incidence 01 is then called the Brewster angle. The position of the missing reflected Figure 11-9.

ray is at 90° to the transmi1ted ray. For any pair of media, the sum of the two

angles 0w is 90°.

the polarizing angle, since an unpolarized wave incident on an interface at this angle is reflected as a polarized wave with its E vector normal to the plane of incidence.* At the Brewster angle,

!1.! = sin O, _ n2 sin 0;n -

sin (~ - 0,n ),

(I 1-44)

sin 01B (11-45)

For light incident on glass with an index of refraction of 1.6, the Brewster angle is about 58°; for light emerging from this glass the Brewster angle is about 32°. At radio frequencies the index of refraction of water is 9, and the angles are about 84° and 6° respectively; water will therefore not reflect a vertically polarized radio wave when the angle of incidence is 84°.

* The Brewster angle is often explained incorrecrly as follows. For this particular angle of incidence, the position of the missing reflected ray is at 90° to the transmitted ray. It is argued that the electrons excited in medium 2 do not radiate in their direclion of oscillation (Section 13.2) and hence cannot give rise to a reAected ray in medium I in this case. This explanation is incorrect , since the Brewster angle is observed for an incident wave polarized with its E vector normal to the plane of incidence in nonconducting magnetic materials (Problem 11-10).

170

REFLE CTION AND REFRACTION

[Chap. II]

11.3.2. The Coefficients of Reflection and of Transmission at an Interface Between Two Dielectrics. It is useful to define coefficients of reflection and of

transmission which are related to the flow of energy across the interface. The average energy flux per unit area in the incident wave is given by Poynting's vector, Eq. 10-40. Setting K,. = I, we find that (I 1-46)

=

; Eo,' n,, 2I('•)'"

(I 1-47)

(I 1-48)

where , = K,, 0, as usual. The coefficients of reflection Rand of transmission Tare defined as the ratios of the average energy fluxes per unit time and per unit area across the interface: R =

IS ,.--. ·D \ = £or Ef, S ia,·.

(I 1-49)

·ll

where n is the unit vector normal to the interface ; T

=

S, av.· n

=

Siav.·D 112

(K, ,)'"E;,cos8i £;, 8,, Kc1 COS

E';, cos 01 Ei1 cos 8;

(11-51)

0

ll1

(I 1-50)

Then, from Fresnel's equations for dielectrics,

'!.!) cos 8, - cos 8,] ' [(-'-'n'-' ' ------

R

(I 1-52)

"' = (;)cos8, +cos8, ' T.v =

("')

4 - cos8, cos8, n1. _ .,' [(;) cos + cos

0,

0,J

RP= [-cos 8, + (~) cos 8,] '•

cos 8, +

T,, =

(;,ll1) cos 8,

(!!.!) cos o, cos 8, n, . [ cos 0,+ (;)cos 0,J'

(J 1-53)

(JI 54)

4

(I 1-55)

In both cases, R + T = I, as expected, since there must be conservation of

[ l l .3]

37!

Rejleclion and Refrac1io11 Be11veen Two Die!eclrics

10

I.Ot------0.8

0.6 0.4

0.2

Figure 11-10. The coefficient of reflection Rx and

1he coefficient of tran smission T x as Junctions of

=

the angle of incidence 0, for 11 1 111, I I 1.5. The E vector of the incident wave is normal to the plane of incidence.

energy. At the Brewster angle, defined by Eq. JJ-45, RP= 0 and TP = I, aga in as expected. For normal incidence, e, = 0, 0, = 0, and

R=

[('!..!) - ']' 112

(I 1-56)

(~)+I ' 4

('!..!) (I 1-57)

T=

Figures 11-10 and 11-11 show the coefficients of reflection Rand of trans1. 0i:::r_ __ _ _ _ _ _ _ _ __

1.0

0.8 0.6

0.4

Rp 00 0

10

20

30

9(}N

40

50 60 70 BO 8; Figure 11-11. The coefficient of reflection R p and the coefficient of transmission Tp as functions of the angle incidence 0, for 11,/11 2 1/1.5. The E vector of the incident wave is parallel to the plane of incidence.

=

372

REFLECTION AND REFRACTION

LO

[Chap. i i]

LO

T

08

08

0.6

06

04

04 R

0.2 0.0

0.2 I

0. 1

02

04

4.0 n 1;n 2

Fig ure 11 - 12. The coefficient of reflection Rand the coefficient of transmission Tat normal incidence as functions o f the ratio n 1 /n:!.

m ission T as funct ions of the angle of incidence 0; for ni/n, = 1/ 1.5, whereas Figure 11-1 2 shows Rand T as functions of 121/ 12, at normal incidence.

11 .4. Total Reflection at an Interface Between T wo Dielectrics We shall now consider the phenome non of total reflection , which we have excluded until now. If 12 1 > 12 2, and if 0; is sufficiently large, Snell's law sin 0, = '!..! sin O1 112

(I 1-58)

gives values of sin 0, which appear to be absurd, since they are greater than unity. The critical angle of incidence, where sin 0, = I and 0, = 90°, is given by (I 1-59)

It is observed experimentally that, when 0, c". 0," the wave originating in medium I and incident on the interface is totally reflected back into medium I as shown in Figure 11 -1 3. This phenomenon does not depend on the orientation of the E vector in the incident wave. For light propagating in glass with an index of refraction of 1.6, the critica l angle of incidence is 38.7°. T otal reflection is an important limiting factor in the collection of light produced in a dense medium, as wi ll be shown in Problem I 1- 17. The critical angle, defined by Eq. J1-59 , is somewhat larger than the Brewster

[l l .4]

373

Total Rej/ectio11 at an Interface Between Two Dielectrics

angle (Eq. 11-45). For exampie. again in the case of light propagating in a glass with an ir.dex of refract ion of 1.6, the wave is tota lly transmitted when the angle of incidence is the Brewster angle, 32.0°, and is totally reflected at the critical angle of 38.7°. Figure 11-14 shows these two angles as functions of the ratio n1/ n,. For large values of 111/ n,, that is, for light incident in a relatively " dense" medium, si n 0;, and tan 0w are sma ll, and 0;, is nearly eq ual to 0,B- For Figure 11-13. For angles of incidence 0; media with more similar indexes of eq ual to or greater than the critical angle refraction , the Brewster angle ap- 0 ic, the wave is totally reffected back into medium I. proaches 45°, whereas the cr itical angle approaches 90°. It is interest ing to note that , for a wave polarized with its E vector parallel to the pla ne of incidence, the amp litude of the reflected wave changes very

60°

50

TOTAL_ _ REF LECTI0N

50

40

40

30

30

20

20

10

10

0 ~ -- -~ ~- - ~ - - ~ - ~ - ~ ~ l - ~-=-----'.c--=--::c-1-c~ I 2 3 4 5 6 7 8 9 10

n 1; n2 Figure 11 ~14. The critical angle and the Brewster angle as functions of th e ratio 11 1/ n:1 of the indexes of refraction on eith er side of the inter/ace. The wave is incident in medium 1. It is polarized with its E vector parall el tv the plan~ of incidence

for the Brewster angle curve.

374

REFLECTION AND REFRACTION

[Chap. 11]

rapidly when the angle of incidence lies between the Brewster angle and the critical angle. It turns out that it is impossible to satisfy the requirement of continuity of the tangential components of E and of H, and of the normal components of D and ofB at the interface, with only the incident and reflected waves in medium I. We therefore conclude that there must exist some sort of transmitted wave in medium 2. The transmitted wave must, however, be of a rather special nature, since it is not observable under ordinary conditions. It must of course satisfy the general wave equation for nonconductors (Eq. /0-25), with the result that

a2E,

a2E,

,,µ,aa,E, 2

ax + az' = 2

(/1-60)

2 '

where we have set the derivative with respect to y equal to zero since, by hypothesis, the field does not vary with the y-coordinate (Figure 11-3). We have also set the permeability Km, of medium 2 equal to unity, since we are considering the interface between two dielectrics. We can use Eq. / /-15 for the incident wave :

E,. = E,, exp {j[w t - k,(sin e, x - cos e,. z)]} .

(11-61)

For the reflected wave, we can write

E, = E,, exp [j(wt - k,.:x - k,,z)],

(11-62)

where E,,, ki., and k, , are unknown constants. We have used the same value of w as for the incident wave, since it can be shown , exactly as in Section 11.1 , that all three waves have the same frequency. We have also used only x and z terms in the exponent, since the derivative with respect to y must, again, be zero. For E,, we choose a more general expression:

E,

=

(E,,.i

+ E,,vj + E,,,k) exp [j(wt -

k,.:x - k 2,z)],

(ll-63)

where E,,,, E,,v, E,,,, k,,, and k,, are also unknown constants. Again we have the sa me w, and we have no y term in the exponent. We let the unknowns be complex in order that the various components of E , and of E , can be out of phase with each other and in order that the dependence on x and on z can be more general than with a simpler expression such as that for the incident wave. Our only assumption is that the waves are independent of the y-coordinate. The wave equati on 10-60 follows directly from Maxwell 's equations. All of the unknowns are independent both of the coordinates and of time, the only dependence on x , z, and t being epresented by the exponential functions. We represent the H vectors by identical expressions, in which the letter E is replaced by the letter H , and for simplicity we choose the origin of coordinates 0

375

[ il

sllrnJlar

rand L

!1 1

r

Kh

Tht: aciffcc!.ed' waiw,

;.a

iij COS

1b:s of the form

IE, = IE,. This ex;tn-ess]c,n~ '-which Jdtmtlcal to thBt. of lEq" repnesents reffiecterl frorr1 ~.he i;:1.tnfacc: sif. an an1glc c:quai~ the a.ngie oil 1rn:m;;nu,, angk·: of J:f;~~cc:ion 01:rn eqrnal t{) the angle of ilrnddcr,tct:o

ffit

!he inte1fiwt {z

~

iEqn·it:h:;:g ;;l111.fJar1y f:filc r:x;v:irn,rr:>.s far ]E;i Rrrrl E1, r,1llJ:1nfu e:r of the trn1rnm.l1t;a;d 0

To the otlher ''-'Navt :em:riber k2"" 'We agai n us,e the wave e;quation J1-60 Using 1:t.r: cxpn:ssRon rn~b

376

REFLECTION AND REFRACTION

[Chap. I!) (I 1-73) (I 1-74) (I 1-75)

=

±j~[(' !..!)' sin 0; Ao 2

n2

112

1]

The exponential function for the transmitted wave is thus exp

V[

(I 1-76)

-

r

(I 1-77) ~ (sin O,)x] + ~ [(~)'sin' O; - 1 z} the ± sign before the z term by a + sign since the electric wt -

We have rep laced field intensity must not become infinite as z -+ - oo . Then k,, can be written with a sign:

+

11

k,, = +j -112[ (11•)' sin 2 0; - 1] ' · Ao

(I 1-78)

112

W e still have to determine £ ,,, £ ,,,, E,,y, E, 1,, a nd the corresponding H components of the trans mitted wave, but it is interesting to exa mine immediately the above exponential function. The transm itted wave is quite remarkable. First of all , it travels una ttenuated , parallel to the interface, with a wavelength

(I 1-79)

= ~, sin Oi

(I 1-80)

A1 being the wavelength in medium I above the interface. The wavelength A. is exactly the distance alo ng the x -a xis between two neighboring equiphase points in the incide nt wave. This was to be expected , since the co ntinuity conditions must be sa tisfied a t all points o n the interface. On the other hand , this result, namely, that the wave travels una ttenuated parallel to the interface, is most surprising if we think of a n incident wave of finite cross sect ion. Does the transmitted wave on the ot her side of the interface extend beyond the illuminated region? Our discussion can not provide us with an answer, since it is based on the assumption that the incid ent wave is infinite in extent. Physically, what happens is this: a given inciden t ray, instead of being reflected abruptly at the interface, penetrates in to medium 2, where it is bent back into medium I. It is this phenomeno n which gives rise to the Htransm itted" wave.* The tra nsmitted wave is also damped exponentially in the direction perpendicular to the interface in such a way that its amplitude decreases by a factor of e over a dist ance

* See A. von Hippe! , Dielectrics and Wares (Wiley, New York ), p. 54 for a brief acco unt of wo rk by F. Goo s and H. Hanchen o n thi s subject.

[ 11.4]

Total Reflection at an Inte1face Between Two Dielectrics

377

DENSE MEDIUM

Y i

AIR

Figure 11-15. "Crests" of E for the incident, reflected, and transmilled waves are

represented here schematically for the case of total reflection. They are sp aced one wave length apart . The 1ra11s111iued wave travels una11e11uated below the interface , and its amplitude decreases exponentially with depth in medium 2. The data used for the figure are the following: n 1 3.0, 11, 1.0, 0.; 75 °. (See Problem

=

=

=

11-13.)

1]'"

(I 1-81)

This is illustrated qualitatively in Figure 11-1 5. The ratio o,/'A 2 is shown Figure 11-1 6 as a function of the angle of incidence 8, for 11,/11, = 1.5.

111

60

60

50

50

40 30

20

- -- - - - ---=I 50

60

70

80

10

90°

8; Figure 11-16. The ratio of

S0 (r/,e depth of pene-

tration) to A:! (the wavelength in medium 2 divided by 2-;;-) for th e transmitted wave when there

is total re flectio n. Tl,e index of refraction of r!, e first medium is 1.50 times that of the second

medium.

378

REFLECTION AND REFRACTION

[Chap. l l)

The transmitted wave, for total reflection , has been . observed both with visible light and with microwaves. * It is interesting to note that we could have arrived at the exponential function simply by substituting in Eq. 11-17 the expression (n, / n,) sin 0,- for sine, (Snell's law), disregarding the fact that it corresponds to a sine function which is larger than unity (see Problem I 1-11). If 0, were real , we would expect to have (I 1-82)

Comparing with Equation 11-78, we sett cos e, = -(I - sin' 0,)'" ,

= -

[

I -

~)' ( 11

sin2 0 1]

(I 1-83) "' ,

(I 1-84)

where we have elected to place the negative sign before the radical to preserve the formalism. 11.4.3. The Electric and Magnetic Fields in the Reflected and Transmitted Waves for the Case of Total Reflection. We have now determined the two

wave numbers k,, and k,, that determine the manner in which the amplitude and phase of the transmitted wave vary with the x- and z-coordinates. It will be recalled from Figure 11-3 that the z-axis is normal to the interface, whereas the x-a xis is parallel to the interface and lies in the plane of incidence. We must still determine the amplitudes of the E and H vectors for both the reflected and transmitted waves , or, from Eqs. 11-62 and 11-63, E,,,, £ 0 ,y, £ 0 , , , H ,,., H ,,u, H .,,, £ 0 , , and H 0 , . This is an imposing number of unknowns. We can dismiss H ,, immediately because the relation between H., and E,, is known from our discussion of plane waves in dielectrics (Eq. 10-34). To simplify the discussion , we shall proceed as before and discuss separa'.ely the cases in which the incident wave is polarized with its E vector normal and parallel to the plane of incidence.

* See, for example, J. Strong, Concepts of Physical Oprics (Freeman , San Francisco), Appendix J by G. F. Hull , p. 516. t It is probably useful to recall here that, if A is some positi ve real number, then (-A) 112

-

jA 112 ,

and A'i' - -j(-A)IJ'. but A 112 a'j(-A)IJ'.

The explanation of this will be found by representing A. -A , and their square roots on the comple x plane. Ir such care is not exercised, it is easy to show that + I = -1:

(11 .4]

379

Total Reflection at an Interface Betiveen Tivo Dielectrics

ITS E VECTOR normal TO THE We can assume here, as in Section 11 .2. 1, that the vectors E., and E., for the reflected and transmitted waves are also normal to the plane of incidence. We can therefore represent the incident and reflected waves as in Figure 11-17. Then I NCIDENT WAVE

POLARIZED WITH

PLANE OF I NCIDENCE.

E, = j£ exp [j(wt - k,n , •r)], 0 ,

(I /-85)

E, = jE0 , exp [j(wt - k 1n, •r)], (/ /-86)

E, = jE,,u exp [j(wt - k,,x - k,,z)].

,y

(//-87)

Figure 11-17. The E and H vectors for

We have wri11en the expo nents in the incident , reflected , and transmitted the first two cases in a more com- waves in the case of total reflection when pact form than previously (Eqs. 11-6/ the incident wave is polarized with its E and l l-62). We have also used k,, vector normal to th e plane of incidence. and k 2, in the last exponent (not their The continuity of the tangential component of H across the interface makes values of Eqs. 11-72 and 11-78) simH oly = 0. ply for brevity. For the incident and reflected waves, His in the plane of incidence, just as in Figure I 1-4, and it has both x- a nd z-components:

H , = H ,, (cos B,i

+ sin O,k) exp [j(wt -

k,n , -r)J,

(I 1-88)

- k,n ,• r)J.

(I /-89)

and

H, = H ., (-cos 8,i

+ sin B,k) exp [j(wt

For the transmitted wave , however, we must use a more genera l expression , since we know nothing as yet concerning its magnetic field intensity. We therefore set (//-90) H, = (H,ixi H.,uj H0 ,,k) exp [j(wt - k,,x - k,,z)].

+

+

We now utilize the fact that the tangential component of the electric field intensity must be continuous across the interface. At the origin , which we have chosen at some arbitrary point in the interface, the expo nential functions for E ,, E,, and E, all reduce toe;"', thus (I 1-91)

Similarly, the continuity of the tangential componen · of H requires that H otu,

(I 1-92)

= H otJ-,

(11-93)

0 = (Hoi - H or) cos Bi

380

REFLECTION AND REFRACTION

[Chap. 11]

or, from Eq. 10-34 which applies to plane waves in dielectrics,

)l/2 111 (E,, , (;'O

E,,) cos 0,

= H ,,,.

(11-94)

From the continuity of the normal component of B (or of H, since Km1 = K,,,, = I) at the interface, we have

('')'" ;;;

(£0 ,

111

+ E ,,) sm. 0; =

H

0" .

(I 1-95)

We are now left with four unknowns (£0 ,, E,,u, H o,,, and Ho1,) and only three equations. We therefore turn to Maxwell's equations, which apply to all electromagnetic fields. We choose one of the simpler ones, namely Eq. 8-2, and apply it to the transmitted wave. This equation states that the divergence of the magnetic induction must always be zero. Since the permeability of medium 2 is everywhere equal to unity, we can set (I 1-96) V·H, = 0, or

(I 1-97)

Solving a nd substituting the values of k 2, and of k,,, we find that

(£") E,,

2

= cos0,+J [ sin 0,-(~)'J'", cos 0, - j [ sin' 0, -

N

(£E,,.,, )

N

(~)'J''

(I 1-98)

(/1'/11 )']1/''

(I 1-99)

2 cos 0,

= cos 0 i

-

. "0 i J. [ sm-

- ( ~ ) 1" 2)111

µ0

-

-

cos 0, [ sin' 0, - (~)']' ",

cos 0, - j

[

sin' 0, -

(/1•)']1 /2 ;;;

(I 1-100)

and H 0 1, ) (-

E,,

N -

-0)

( ,

,u o

1 "

11, sin ---~ -20, ~--cos 0, - j [ sin' 0, - (~)']'''

(I 1-101)

To obtain E,, the above value for £ must be substituted into Eq. 11-70. Similarly, to obtain E,u, H ,,, and H ,,, the above values of £ 0 1, , H , 1,, and H 0 ,, must be multiplied by the exponential fun ction / /-77 for the transmitted wave. We have thus found all of the quantities which are req uired to describe both the reflected and the tra nsmitted waves. Before discussing the physical mea ning of these quantities, let us recapitulate. We have solved this rather complex problem of total reflection by using just 0,

[l 1.4]

Total Reflection at an Interface Between T,vo Dielectrics

381

a few basic ideas. We first used the wave equation, and then the requirements of continuity at the interface: continuity of the tangential components of E and H and continuity of the normal component of B. Finally, we used one of Maxwell's equations; namely, V ·B = 0. It must be realized, however, that both the wave equation and the continuity conditions have themselves been deduced from Maxwell's equations, which represent, therefore, our only basic assumptions. Let us see how the two first expressions above for £,, and £,, compare with Fresnel's equations for dielectrics, Eqs. 11-31 and 11-32. We have found in Eq. 11-84 that the square root which appears in the Eqs. 11-98 to 11-101 is related lo cos 0, as follows: [ sin' 0, - (~)']"'

=

j~ [ I -

(;,)' sin'

0J',

(//-102) (//-103)

Substituting in Eqs. 11-98 and / /-99 , we find that these become Fresnel's equations. Fresnel's equations are therefore valid in the case of total reflection, at least in a formal way, if we again disregard the fact that sin 0, is greater than unity, and if cos 0, is defined as in Eqs. 11-83 and 11-84. We have already found , near the end of Section 11.4.2, that the wave numbers k2, and k,, can be derived directly from Snell's law. Let us now examine the physical meaning of the expressions which we have found above for£,,, E,,u, H ,,,, and H , ,, in Eqs. I 1-98 and/ /-JOI. We first notice that the amplitude of the reflected wave is equal to that of the incident wave, since

£,, = 1£"'1

(I 1-104)

I.

The coefficient of reflection R is therefore equal to unity. The energy is totally reflected , and , on the average, there is no flux of energy through the interface. The wave undergoes a phase jump on reflection , since

£"') ( Eoi N

=

{

exp 2j arc tan

[

0

sin' ; COS

("'YJ'lJ = ;;'.

Bi

exp (}a).

(I 1-105)

The phase jump therefore varies from 0° at the critical angle of incidence (sin 0; = n,/11 1) to 180° at glancing incidence, through positive angles. This is shown in Figure 11-18 . As regards the transmitted wave, it is obvious that its electric field intensity is not zero, despite the fact that the average flt,x of energy across the interface

382

REFLECTION AND REFRACTIO N

180 °

180°

~

135

[C hap. 11)

135

a 90

90

45

45

0 40

50

60

70

8;

80

0 90°

Figure 11- 18. The phase a of the reflected wave

w ith respect to that of the incident wave in the case of total reflection. The incident wave is polarized with its E vector normal to the plane of incidence; the ratio n 1/ n 2 is set equal to 1.50.

is zero, as we have just found. This is to be expected , since the incident wave suffers a gradual reflection within medium 2, as explained near the end of Section 11 .4.2. Medium 2 can be considered to act like a pure inductance fed by a source of alternating current. The average power input to the inductance is zero, the power flow being alternately one way and then the other, but there is nevertheless a current through the inductance. Figure 11-19 shows how the jy

65°

1.0

55°

60°

50°

70° 0.8

75° 45°

0.6 0.4 0.2 0

0 .5

1.0

1.5

2 .0

X

Fig ure 11-19. The ratio (£0 ,,/ £ 0 ;) N of Eq . 11-99 is plolled here in the complex plane for various angles of incidence 0, larger than the critical angle and for n 1/n 2 1.50. The amplitude of the transmiued wave is represented by the distance from a point on the curve to th e origin, and is greatest at the critical angle. The transmilled wave leads the incident wave by an angle equal to th e argu m en t of th e complex ratio-for example, the angle Hotz,

since k,, is much smaller k,, (Eq. 11-132). The wave in the conductor therefore has its E vector parallel to the y-axis and its H vector parallel to the x-axis. It propagates in the negative direction of the z-axis. From the above,

£") = n,K,,,, cos 0; - 8"-o(I - ? (£ ,; n, K"'' cos 0; + Ii""'(I - ") N

(11-139)

J

We would again have obtained the same result by the direct application of Fresnel 's equation(/ 1-21) with cos 01 as in Eq. 11-127. We have shown above in Eq. 11-131 that /:.0 » /5. Then

~ .~

E,, E

"

-I

,

(11-140)

at least if K,,,, is much smaJler than /:.0/ /5. The coefficient of reflection R is thus approximately equal to unity, and E is reflected .,,. radians out of phase as in Figure 11-22

The incident, reffected, and transmitted waves at the interface between a dielec-

t; .

il'.t

X

""

•.

~

"'"

I


(/2-1 04)

c.

It will be observed that £, a nd H, a re in phase bu t th at H, has the sa me phase

at ( z +~ ) as have the two other vectors at z . Figure 12-8 shows the electric a nd Figure 12-8. Elecrric and magnetic lines of fo rce fo r a TE wave with n = 1 propagming in a hollow , ectangular wave guide.

~ "'. ......... -..,,,

........_ '4

~,

~

1/ The ovals are the magnetic lines of

fo rce. The electric lines of force are ven ica/ straight lines and arc repre-

se111ed by dots and crosses.

y

126

GUIDED WAVES

magnetic lines of force for a TE wave with rectangular wave guide.

11

[Chap. 12]

= I propagating in a hollow

12.3 .2. Hollow Rectangular Wave Guide. Internal Reflections. Let us return to Figure 12-6, which shows a typical wave front for a wave zigzagging down the guide. It will be instructive to investigate the field by considering the interference resulting from the multiple reflections.

\

a I

I

\

I

I

I

I

F

I

I

G

I I

,'D

B I

I

b1

/

'

I

,/ 11

v' Figure 12-9. A plane electromagneric wave propaga tin g in a rec-

tangular hollow wave guide along a zigzag pcllh. The lines AB and CD are parallel to wave fronts for the wave propagating to the right and upward. Similarly , BC and DE are parallel to wave fronts traveling to the right and downward. The angle a is the angle of incidence; the broken line FCC represents a ray reflecte d al C.

Figure 12-9 shows the multiple reflections in more detail. Let us assume that along the fi xed line AB the electric field intensity of the wave which is directed upward a nd to the right is £ 0 exp (jwr). The line AB is thus parallel to the wave fronts for this wave. The lines BC and DE are similarly parallel to wave fronts for the wave propagating to the right and downward. These two waves must interfere at B to give zero field intensity at the perfectly conducting wall, since their electric field intensities E are perpendicular to the paper and are parallel to the wall. Then the electric field intensity along BC must be £ 0 exp [j(wt 1r)] = - £ 0 exp (jwr). At the point C, interference must again give zero net field , such that, along CD, E = £ 0 exp [J(wt 21r )] = Eoexp (Jwr). Thus, AB and CD are one free-space wave length :>-., apart, and, from the geometrical construction shown , An (12-105) cos a = -

+

+

211

[12 .3]

427

Hollo w Rectangular Wave Guide

We could a lso have chosen the lines AB and CD to be " wave lengths apa rt. Then we would have had llAo (1 2-106) cos a= b · 2 The angle a can thus have only certain discrete values which permit destructive interference to occur at the walls of the guid e. We have therefore found a geometrical interpretation for the ratio Ao/(2b/ n). From Eq . 12-92,

~=--• Ao

(12-107)

sin a

where the guide wave length Au is equal to AC, CE, or BO. At the critical wave length Ao = 2b/ n, cos" = I, a = 0, and the wave front s are parallel to the axis of the guide. F or Ao« 2b/ n, cos a -+ 0, and a - + 1r/ 2. The TE wave then approaches a TEM mode as a limit. The phase velocity is given by U P

=~C=~>c. f\ s111 a

(12-108}

0

This is the velocity at wh ich the phase propagates along the guide. It is larger than c beca use the in d ividual plane wave fronts are inclined at an angle with respect to the axis of the guide. Thi s ca n be seen fr om Figure 12-9 as follows. Consid er AB to be a wave front propagating parallel to itself and to the right at a veloc ity c. Then the point A moves along the z-axis at a velocity which is larger than c. We can also consider the velocity at which a given signal pr ogresses along the length of the guide. The z compo nent of the velocity of a n individual wave front is only c sin " and is smaller than c. Thus, if we call this velocity u,, then lls

Sin a,

=

C

=

c[ I -

(12- 109) ; ) "']''" - ( llA 2

< c,

(1 2-110)

and (12-11 I ) 12.3.3 . Hollow Rectangular Wave Guide. Energy Transmission . Let us consider the ene rgy tra nsm it ted by a TE wave in a low-loss rectangular wave guide. We shall ass ume the usua l case, in which n = I. The field is then co mpletely described by Eqs. 12-89 to 12-91, with n = I, or by Eqs. I 2-96 to 12- 102. Th e fi rst set is slighlly more convenient for our purpose . From Eq. 10-88, the average va lue of the Poy nting vecto r is given by

S,v.

=

I

2 Re (E X

H *),

(12-112)

428

GUIDED WAVES

[Chap. 12]

where H• is the complex conjugate of H. In the present case,

Sav.

= l Re 2

i

j

E,

0

0

H:

(12-113)

(12-114)

Substituting the values of E,, Hu, H ,, we find that the first term in the parentheses is imaginary, whereas the second term is real. The energy therefore flows only in the direction of the z-axis, and E];.,

S""·

..,

= 2wµo'Au sin-

(,rby)k.

(12-115)

This is the energy flux in watts per square meter in the d irection of the z-axis withi n the guide. The value of S,"·· is independent of x , as expected, since the amplitude a nd phase of the wave are independent of the x coordinate. It is zero at the walls y = 0 and y = b, where E is zero, and it is maximum at y = b/ 2, aga in as expected. The total transmitted power is thus given by Wr

=

1"-• =0

E];,, sin'

7 _wµuA0

("b y)a dy,

E'f:Qfab

= 4wµoA0

(12-117)

1

= E; ,ab [ I _ ( ~)'] 112 0

4cµ 0

(12-116)

2b

(watts),

(12-118)

where £ 00 , is the maximum va lue of E in the guide in volts/ meter, a and b are the dimensions of the cross sect ion of the guide in meters, as in Figure 12-5, a nd ">--u is the guide wave length in meters, as in Eq. 12-103. Let us compare this tra nsmitted power with the average electromagnetic energy per unit length within the guide. The instantaneous electric energy density is (l / 2),0 £ 2, and its average value is (I / 4),oEi. The average electric energy per un it length is thus

(,r )

( " I Eo£'OOI sm. ' by a dy = 'g" ab L""Qox• )o 4

(12-119)

To find the average magnetic energy content per unit length, we proceed similarly for both they- and z-components of H and add the results, since (12-120)

_\ga in the result is (,o/8)abE;0 , , and the average electric and magnetic energies per unit lengt h a re equal. This is reasonable, si nce the plane electromagnetic

[12.3]

429

Hollow Rectangular Wave Guide

waves which produce the field configuration by reflection at the side walls involve equal electric and magnetic energy densities. It is not at all obvious, however, because the interference effects tend to confuse the picture. The total electromagnetic energy content per unit length in the guide is therefore , 0abE;;,x/ 4. Upon dividing the total transmitted power by this quantity, we find (12-121)

=

c[, -Gt)T',

(/2-122)

(I 2-123) The transmitted power is thus equal lo the product of the energy per unit length times the signal velocity, as could have been expected intuitively.

12.3.4. Attenuation in Hollow Rectangular Wave Guides. We have as sumed until now that the walls were perfectly conducting; let us now consider real wave guides of finite conductivity. In the process of guiding electromagnetic waves, conductors waste part of the wave energy in the form of Joule losses. This is because the guided waves always induce electric currents ii, the co nductors. However, a rigorou s calculation of the fi eld for a guide of finite conductivity is difficult and , fortunately, unnecessa ry. The procedure used for calculat ing the Joul e losses is the following. We have performed a calculation on the assumpti on that the gu ide is perfectly conducting. This has led to a fi eld in which there is a ta ngential magnetic fi eld intensity H at the surface of the guide. Since the tangential H must be continuous across any interface, we have a known H inside the conductor. Then, using Maxwell 's equations, we can find the corresponding tangential electric field intensity E, which is not zero unless the guide material is a perfect conductor. Thi s small tangential Ei s then considered to be a perturbation of the ideal field obtained with perfect conductors. The method is entirely satisfactory because this E is so small that it ha rdl y disturbs the wave. We thus have a ta ngential E, a tangential H , and a Poynting vector which is normal to the conducting surface and directed into the metal. That both E and H vectors must exist inside the conducting walls ca n also be shown as follows. T o begin with , we must have a ta ngential H just insid e the wa ll. On the other hand , at so me distance within the wall, there must be zero field , since the attenuation distance Ii (Section I0.4) is quite short a t frequencies which are high enough to propagate in wave guides. For example,

430

GUIDED WAVES

[Chap. 12]

if b = 7.5 centimeters, the frequency can be, say, 3000 megacycles/second, and~ is then only 1.2 microns in copper (Table 10-1). The tangential component ofH thus decreases rapidly with depth inside the conducting wall. Then V X H is not zero, and there is a current density J parallel to the surface and normal to H , since v X H = J. We must therefore have both a tangential E to produce the tangential current density J, and a tangential H. From the Poynting vector directed into the guide wall we can calculate the power W1, which is removed from the wave per meter of length; we then wish to calculate the attenuation constant k ,; of Eq. 12-7. This constant must be such that, when both the E and Hof the transmitted wave are multiplied by e-k,;,, both the Poynting vector for the transmitted wave and the transmitted power Wr must decrease by a factor of

exp(-2k,,t.z),::; I - 2k,,t.z in a distance guides. Then

(/2-124)

t.z. The approximation is excellent for ordinary types of wave (12-125)

or (12-126)

The real part k ,, of k, (Eq. 12-7) can be taken to be the k, obtained on the assumption of perfectly conducting walls. It might be expected at first sight that the attenuation could be calculated from the reflection losses . It will be recalled from Section 11.5.2 that an electromagnetic wave reflected from a good conductor is slightly weaker than the incident wave. This method of calculation is incorrect because, as we shall see, there are also energy losses in the guide faces parallel to the yz-plane. It will be shown in Problem 12-9 that the two calculations agree when the guide height a approaches infinity , for then the losses in the faces parallel to the yz-plane become negligible compared to the reflection losses. Let us examine the process whereby energy is removed from the wave. The tangentia l H produces an electromagnetic wave which penetrates perpendicularly into the wall. Inside the conducting wall,

H -;; )''' ej,r/ 4, E = (µ°"'

(12-127)

as in Eq. 10-105. We can also arrive at this equation by first assuming that H propagates into the wall as a damped wave and then using the two Maxwell equations v X H = aDJat and v XE = -DB/at. We assume agai n that the dielectr ic inside the guide is dry air, and is therefore lossless . We a lso assume that n = I in order that the field be described , as a first approximation , by Eqs. 12-96 to 12-102.

[12.3]

431

Hollow Rectangular Wave Guide

Along the face which lies in the xz-plane, H,

rrEnox =- exp [ ; ·( wt wµo/J

- -Z - " - )] 'A, 2

=

(y

(12-128)

0).

Then£, is not equal to zero for y = 0, as in Eq. 12-96, bul is instead given by the above two equations : E,

=

112 (µ(f,)) rrE.., exp [1 (wt u wµ b 0

!'... -

'A,

'!:)]

(y

4

=

0).

(12-129)

When u approaches infinity, £ , approaches zero. The Poynting vector S.v . (1 / 2) Re (E X H*) directed into the guide wall is equal to ( ,r ~ "") 2

(y = 0).

ul i2(2wµo}1 ' '

This is the energy flowing into the wall (y = 0) per square meter and per second . It is interesting to note that this energy flux is the same at all points on the face y = 0. The power lost to the wall per meter of length is a times the above expression and, for the lwo faces parallel to the xz-plane, (12-130)

This is the power lost by reflection. For the face at x = 0, H has y - and z-components as in Eqs. 12-100 and 12-101 . For they-component, nu ,

("Y) b exp [; ·(

E•o'i-u ., SII1 . = wµ

wt -

z)]

X,: •

(12-131)

and then £ , is not zero , but rather E, =

- ,J ) ( µrf. u

1 ''

("Y ) exp [.; (wt b

£ - s111 00

,

wµo'A 11

•

-z : X,

+

u C

(X',, +KX )"' sin X, I

(13-JJ3)

and these lines pulsate in and out without ever escaping. It is important to note that the KX surface and Figure I 3-9 do not give the modulus of E, but only its direction. For example. in drawing a figure such as 13-9, one naturally selects equal intervals of KX; this leads to a constant density of lines of force for r » X, which appears to indicate that the amplitude of E does not decrease with r. In fact , E decreases as 1/ r for r » X, from Eq. 13-81. There is, of course, no such thing as a distinct line of force; the KX surface provides only the direction of E as a function of the position r, 0, and of the time t.

13.3. Radiation from a Half-wave Antenna The half-wave antenna illustrated in Figure 13-12 is commonly used for radiating electromagnetic waves into space. It is simply a straight conductor whose length is half a free-space wave length. When a current / 0 cos wt is established at the center by means of a suitable electronic circuit, a standing wave

462

RADIATION OF ELECTROMAGNETIC WAVES

[Chap. 13]

P(r, 8)

-

I0

COS

wt

I /

/

l

I

Figure 13-12. Half-wave antenna. The current distribucion shown as a broken line is / 0 cos (// 11-) cos wt. This is the standing wave pattern at some particular time

when cos wl

= 1.

is formed along the conductor such that the current/ at any position I is given by

I = lo COS

I

~ COS wt.

(/3-114)

Each element / di of the antenna then radiates an electromagnetic wave similar to that of an electric dipole, and the field at any given point in space is obtained by integrating over the length of the antenna. In many cases the half-wave antenna is a one-quarter wave length mast set vertically on the ground, which then acts as a mirror. The mast and its image in the ground together form a halfwave anten na . Radio broadcast antennas are often of this type. To achieve good conductivity, the ground in the neighborhood of the antenna can be covered with a conducting screen . This description of the half-wave a ntenna is really contradictory , because the standing wave along the wire can be truly sinusoidal only if there is no energy loss, and hence no radiated wave. It turns out, however, that a rigorous calculation , although too involved to discuss here, leads to nearly the same result as the approx imate one. The current distribution is not quite sinusoidal , but the distortion has little effect on the field. It will therefore be sufficient for our purposes to assume a pure sinusoidal current distribution. The standi ng wave can be expressed in exponential form as follows: I = Re? (exp { J(w1

-0} + exp { i(w1 + 0})•

(13-115)

where "Re" mean s, as usual, "Real part of. " The right-hand side shows that

.l

l

•

[13.3]

463

Radiation from a Half-wave Antenna

the standing wave is the sum of two traveling waves, one in the positive direction , and one in the negative direction , with amp litudes I0/ 2. Then , from Eq. 13-49, using the usual comp lex notation , we can express the electric dipole moment p of the element di as (13-ll6) (13-117) 13.3.1. The Electric Field Intensity E. In calculating the electric field intensity at the point (r , 0, ,p), we sha ll assume that the distance r to the point of observation is much greater than X. Then the electric field intensity c/E from the element di is given by Eq. 13-81 , in which we must substitute the value of Po obtained from the above equations. Thus

dE = - ~ (exp ;81rc, 0Xr

{1(wt - LX - C.X)}+ exp {1(wt + LX - !:.)} ) sin X

0 di j,

(13-118)

where r' is the distance between the element di and the point (r, 0, ,p), as in Figure 13-12, (13-JJ9) r' = r - 1cos 0, and

Io exp

E=

{iw (

I -

D}

j81rc, 0Xr

f

t)dlj.

st

+ • ( exp { I;;(cose 1 I X sine _, - l)J\ + exp { j~(cos0 + l)J 14

(13-120)

We have removed the I /r' from under the integral sign and set it equal to 1/r, since it is assumed that r » X: with this condition , the dE's can a ll be taken to be parallel for a given point of observation and can be assumed to have the same amplitude but different phases, and the integration can be limited to the phases. Integrating, we obtain

Io exp

E=

{Jw (1 -

D}

j81rc,0 Xr 1

Xsine( Xexp {; fccose-1)} + Xexp {i~ (cose+ l)} ):• •_ }(cos 0 - 1)

}(cos 0 + 1)

(1 3 _121 )

_, 14

It is not permissible to expand the exponential functions betwee n the main parentheses in series form , since I is not sma ll with respect to ;\ in this case. We have

464

RADIATION OF ELECTROMAGNETIC WAVES

jloexp { Jw(r - ~)} . E

=

4.-c,.,-

Sm

8

(sin { 1(cos6cos 0 - I

1)1 +

[Chap. 13]

sin ; 1(cos0+ cos 0 + I

I)})· J. (/3-122)

The expression between the braces can be simplified by setting sin

t

(cos 0 -

I)} =-cos (1 cos 0)

(/3-123)

I)} =+cos(}cos0)

(13-124)

and sin { 1ccos0+ and adding the two terms. Then

E

j . ( Jµ. oc = 2.-r lo exp l Jw I

r ) } cos (~- cos 0) • sin 0 J,

- ~

(volts/ meter)

(13-125)

(r » :z, W changes to W + L'>W, and the ratio L'>W/ L'>z is equal to dW / dz for L'>z ➔ 0.

We shall consider functions W(z) such that the derivative dW/ dz exists in the region considered. This condition leads to an important pair of equations. First, let us examine the meaning of the derivative dW/ dz by considering Figure D-2. The point W in the W-plane corresponds to the point z in the z-plane, according to some specified function W(z). If z changes to z + L'>z, W changes sim ilarly to W + L'> W, where the increments L'>z and L'>W are complex numbers. The derivative dW/ dz is the ratio of these increments at the limit L'>z -+ 0. The value of this derivative can take on different values for different values of z, but we wish to have a single value of dW/ dz for a given value of z, that is, for a given point in the z-plane, no matter how dz is chosen. We consider two particular values of dz: dx and j dy. In the first case, dz is parallel to the x-axis; in the second case, dz is parallel to thejy-axis. For both of these particular values of dz, we must have the same value of dW / dz. For the first case, the value of dW/ dz becomes aw;ax, and aw= au +jav_ (D-5) ax ax ax For the second case, dW/ dz becomes aw/j ay, and aw=! au+ av. jay j ay ay

(D-6)

These two expressions must be equal: au +jav =!au+ av_ {)x ax j ay ay

(D-7\

517

The Complex Potential Then

au= av ax ilv

and

au= ay

av ax

(D-8)

These are the Cauchy-Riemann equ{l{ions. The functions U and V are related to each other through these equations and are called conjugate _functions. A function W(z) is said to be analytic if its four partial derivatives exist and are continuous throughout the region considered and, moreover, if they satisfy the CauchyRiemann equations.

jV

jy

w"1w

z~z'

w'

z"

u

0

0

X

(bl z plane

(a) W plane

Figure D-3. The three points W , W' , and W" correspond, respectively, to z, z', and

z". The angles a and {3 are equal when z' ➔ z and z" and is not zero.

➔

z as long as dW /dz exists 1

D.2. Conformal Transformations Consider now a point z and two neighboring points z' and z" in the z-plane. These three points correspond to three other points in the W-plane: the point W and the neighboring points W ' and W " , as in Figure D-3. Since dW/ dz is unique at the point z, then . W' - W . W" - W -dW = hm - - - = hm - - - -· (D-9) dz

z '---, z

z' -

Z

z"---, z

z" -

Z

If dW/ dz a" 0, it can be written in the form Ae;• _ (The argument has no meaning if dW/ dz = 0.) Then, at the limits z' - + z and z" - + z, considering only the arguments of the various complex quantities, we have

+ +

arg (W ' - W) = arg (z' - z) , arg ( W " - W) = arg (z" - z) , arg (W' - W) - arg (W " - W) = arg (z' - z) - arg (z" - z),

(D-10) (D-11) (D-12)

or Ci=

(3,

(D-/3)

where the angles a and fJ are as in Figure D-3. This result does not apply to points where dW/ dz = 0. The angle between two infinitesimal line segments is therefore conserved in passing from the z- to the W-plane, as long as the derivative dW/ dz exists and is not zero. For example, the two families of straight lines represented by U = constant and V = constant in the W-plane are clearly orthogonal. Then, the corresponding curves U = constant and V = constant in the z-plane are also orthogonal. Since there is a one-to-one correspondence between the points in the z-plane and those in the W-plane, we can imagine the W-plane to be distorted into the z-plane

518

APPENDIX D

according to the function W(z). Then a geometrical figure in the W-plane is "mapped" into a corresponding figure in the z-plane, and inversely. This process is called a conformal transformation.

D.3. The Function W(z) as a Complex Potential We have (D-14)

a'W d'W fix'= dz'' aw

ay

dWaz

= dz

(D-15)

.dw

ay = 1 dz'

(D-16)

a'W d'W f!y' = - dz'' and

aw aw 2

(D-17)

2

ax'+ ay'

= 0·

(D-18)

The function W(z) is thus a solution of Laplace's equation in two dimensions. Separating real and imaginary quantities, we find that a'U ax'

+

a'U fiy'

=

0 and

a'V ax'

+

fi'V ay'

=

0.

(D-19)

Thus both U and V independently satisfy Laplace's equation, and either one can be set to he the electrostatic potential which also satisfies Laplace's equation, since we have assumed zero space charge density. If the imaginary part of W is taken to be the electrostatic potential, then the equipotentials are given by V = constant. Since the U = constant curves are orthogonal to the equipotentials, as we have seen above, they therefore define the lines of force, from Section 2.3. The function Vis then called the potential ji111ctio11, U is called the Srream function, and Wis called the complex pote111ial function. It is shown in Problem D-2 that the electrostatic field intensity Eis given by (D-20)

at any point in the field. A 11vo-dime11sio11al electrostatic field wilh zero space charge density is therefore completely determined once the complex potential func tion W(z) is known. It is essential to recall here the uniqueness theorem derived earlier in Section 4.2,

according to which there is only one field configuration which satisfies given boundary conditions. Thus, if in one way or another we ca n find a satisfactory function W(z), then that function is the proper one and the only one. The determination of W(z) is usually intuitive and empirical. However, much work h as been done in this connection, and it is usually possible to determine the proper function for simple geometries.* One can usually arrive at W(z) by using some function of the complex potential for some other known field.

* See, for example, E. Durand, Elecrrosrarique et magnerosratique. (Masson et Cie, Paris, 1953) and E. Kober, Dicriouary of Conformal Represemarians (Dover, New York, 1952).

519

The Complex Potential y Figure

v,

0 -4

parallel-plate capacitor with its lower plate gro1111ded and irs upper plate at potential V,. E11d effects are

T

neglected. The horizontal lines are equipotentials V (V 1/ s)y con-

1

A

=

s

=

stant; the vertical lines are lines of force U (V 1 /s)x constant.

=

=

X

~

As an illustra tion, let us consider t he field inside an infin ite pa rallel- plate capacitor as in Figure D-4. The equ ipotentia ls and the lines of force are given respectively by y = constant a nd x = consta nt. Setting V to be the electrostatic potential,

(~0

V=~~ s

This suggests that the function W(z) must be W(z)

=

U

+ jV = ~s (x + jy)

= ~ s

(D-22)

z.

T hus,

U = ~x. s

(D-23)

We could a lso have determined the stream function U thro ugh the Cauchy-Riemann equations.

D.4. The Stream Function We ha ve seen above tha t the s tream function is a constant a long a line of force . We shall now see that it is quantitatively j ust as importa nt as t he potentia l function. We s ha ll assume, as in Section 0.3, that Vis the potential function . Figure D -5 s hows three eq uipotentia ls and three lines of force in a portion of an

E V-6V

Figure D-5

dn

A ser of lines of force a11d a ser of equipotentia/s in a portion of an electrosta1ic field. The elem entary vecrors do and ds are d irected resp ectiv ely,

V

a/011g tl,e li11e of force and along rl, e equipotential at the point considered. The vector dn p o ints in the direction of E, and the vector ds poi11ts to tl,e left wl,en one looks alo11g dn . For

convenience, th e axes are chosen to be parallel to these vectors a'i shown.

U+6U

u

U-6U

520

APPENDIX D

electrostatic field , The vector dn is an element of length along the line of force normal to the equipotential at the point considered. The vector ds is an element of length along an equipotential and is oriented with respect to dn such that it points to the left when viewed along dn. For convenience, we choose our x- and jy-axes as shown, such that the x-axis is parallel to dn and the jy axis is parallel to ds at the point considered. According to the Cauchy-Riemann equations,

au= au= _av= av as ay ax - an'

(D-24)

Since -av;an is the electric field intensity Eat the point, the positive direction for E being along dn, then , along an equipotential, dU = Eds.

(D-25)

This relation is valid for any point in the field as long as Vis chosen to be the potential function , and E and ds are oriented as in Figure D-5. The stream function is thus related to Eds. This quantity Eds is the flux of E crossing the equipotential in the direction of dn through an element of area on the equipotential surface which is ds wide and whose height, measured in the direction perpendicular to the paper, is the unit of length , namely, one meter. Integrating Eq. D-25 along an equipotential between the lines of force U, and U,, we obtain U, - U, =

f

c_;, dU ds ds =

1u,

U1

Eds.

(D-26)

U1

The line of force for which the stream function is zero is chosen arbitrarily, just as for the equipotential on which the potential is zero. Thus the charge density rJ at the surface of a conductor is C1

= EoE = Eo dU,

(D-27)

ds

where the vector ds points to ward the leji when one looks toward the outside of the conductor into the field. The total charge Q per unit length on a cylindrical conductor whose axis is perpendicular to the paper is obtained by integrating (J ds around the periphery of the conductor in the direction of increasing ds: (D-28)

The stream function is thus useful for determining surface charge densities and total charges on conductors.

D.5. The Parallel-plate Capacitor Let us return to the parallel-plate capacitor of Figure D-4, for which we found the complex potentia l function W(z) in Eq. D-22. The charge density on the lower plate is obtained from Eq. D-22 with the vector ds pointing toward the left when one looks toward the outside of the conductor into the field: O"\o wcr

=

E;:i

dU ds =

- Eo

dU dx =

V, -Eo --;·

(D-29)

The Complex .Potential

521

Similarly, the charge density on the upper plate is found to be +,,(V1/ s). Both of these charge densities can be verified to be correct by using Gauss's law. The capacitance C' per unit area is C' = ..!!..... = EJI,

v,

s

(D-30)

and E =

ldWI I = ~. dz s

(D-31)

as expected. We have thus verified our method of calculation by applying it to a well-known field.

D.6. 1'he Cylindrical Capacitor In the case of the cylindrical capacitor, as in Figure D-6, the equipotentials are concentric circles, whereas the lines of force are radial straight lines. We find empirically the following complex potential function , which is justified below: W(z) = jVi In '"•

(D-32)

r,

~

Jn or, writing

(D-33)

we obtain

(1n r1"._ + j0)•

(D-34)

- ~ 0 + j ~ ln!_, In 0 In 0 r1

(D-35)

W(z) = .iVi In ~ r1

r1

r1

Thus

U= - ~ 0, In 'J

(D-36)

Figure D-6. A section through a cylindri-

cal capacitor of internal radii r 1 and ' '2·

and

The inner cylinder is grounded, and the

V=~ln!_· ln ~

ri

(D-37)

outer one is at a potential V 1 , the charges

being, respectively, -Q and +Q.

r1

The above expression for W(z) is justified as follows: (a) the logarithm serves to give equipotentials and lines of force of the required form r = constant and 0 = constant when the real and imaginary parts of W(z) are set equal to constants; (b) the factor j serves to make V the potential; (c) the ratio z/ r1 makes V equal to zero at r = r1 ; (d) finally, the factor V1 / ln (r, / r 1) serves to make the potential equal to V1 at r

=

r1.

Let us find the charge on the inner cylinder. To do this we must integrate the stream

522

APPENDIX D

function U in the direction of ds or of increasing O between -,rand +,,-, as in Figure D-6. Then (D-38)

The charge on the outer cylinder is found by integrating U in the direction of decreasing 8, and Qouter = - Qinner, as expected. The capacitance C' per unit length normal to the paper is given by C'

=

IQ_ I =

2,,.,,,,

v,

(D-3fl)

In,:; r,

which is correct, and E is also given correctly by

!t

l'½I

E = d. = I-"' In.!... "..1 z r,

(D-40)

(D-41)

D.7. Field Due to Two Parallel Line Charges of Opposite Polarity As an example of the use of the complex potential for determining electrostatic fields, we consider the field due to two parallel line charges of Q coulombs/ meter and of opposite polarity, as in Figure D-7 jy The line charges shown at -a and at +a are presumed to be infinitely long in the direction perpendicular to the paper. The potential V at any point due to this charge distribution can be found by integration, as in Problem 2-2 I: V = __Q_ In ~. 27rto r2

-0 / a Figure D-7. Line charges of -Q and +Q coulombs/meter are situated, respectively, at x = -a and x = +a.

( D-42)

where Q is the charge per unit length perpendicular to the paper, and r, and r, are as in Figure D-7. This equation shows that V - + +«> at r, = 0, V - + - «> at r, = 0, and V = 0 at r1

=

rz, which is correct. Also, for a

given va lue of r., V varies as the logarithm of r, and, sin1ilarly, for a given value of r'l, V varies as the logarithm of r1, which is also correct.

We ca n also rewrite Vas follows: V =

__Q_ 2r.to

lnlzz + a[ - a/

(D-43)

The Complex Potential

523

This suggests W(z)

= j .JL In 27r t:o

2

+ aa,

(D-44)

z -

(D-45)

= j-2Q [1n lzz +a l+ j(O, 7l"Eo - a

0,)],

-

(D-46)

and

u = .JL (0, 27rEo V=

01),

(D-47)

.JL1n 1z+a1= .JL 1n ".r!., z - a 21rt:o

2-irEo

( D-48)

2

The lines of force are given by the equation 0, - 0, = constant. They are thus arcs of circles passing through the line charges and centered on the jy-axis, both above and below the x-ax is. The equipotential V is determined by the equation

~ = exp (2n0 V/ Q). r,

(D-49)

Figure D-8. Lines of force (indica1ed by arrows) and equipotentials for two infinite line charges perpendicular to the paper. All the curves are circles. T he equipotential surfaces are generated, as usual, by rotating th e figure about th e axis identified by the curved arrv w.

524

APPENDIX D

It is shown in Problem D-3 that this equipotential is a cylinder whose radius is lacsch (2,,-,, V/ Q)I and whose axis is situated at x = lacoth 2,,-,0 V/ QI. For V-+

+.,,

the equipotential surface reduces to a line perpendicular to the paper and situated at x = a, as expected. Similarly, for V ~ -oo, we have a line at x = -a. The equipotentials and lines of force are shown in Figure D-8.

D.8. Field Due to Two Parallel Conducting Circular Cylinders of Opposite Polarity The field investigated above was shown to have equipotentials in the form of circular cylinders whose axes are all parallel. We can place an uncharged conducting foil on any of these equipotentials without disjy turbing the field in any way. When we do this, charges migrate inside the conducting foil so as to cancel the electrostatic field within it, and charges of opposite polarity appear on the two +Q ·O surfaces. In this way, the field remains everywhere exactly as it was, except for the conducting region inside the foil, ~--,o-~----~--,---, where the field is zero. If the line charge surrounded by the foil is Q per unit length, a charge - Q per unit length is induced on the inside surface of the foil, Figure D-9. A section through two paral- and a charge of + Q per unit length is lel cylinders of radius R whose axes are induced on the outside surface. These .eparated by a distance D and which carry induced charges are of course due to c-harges - Q and + Q coulombs/ meter, re- the electrostatic field and are due to both spectively. The potentials are -V and of the line charges. We can cancel the + V. The origin of coordinates is chosen + Q and - Q charges surrounded by midway between the axes. the foil by shorting the line charge +Q and the foil , leaving us with a net charge + Q on the outside of the conducting foil and zero charge inside. The field in the internal region limited by the foil is then zero, but the field outside is unchanged. Instead of canceling the charges inside the foil as above, we could similarly have canceled the charges outside, making the field zero outside the region bounded by the conducting foil but leaving it intact inside. We can now find the field due to a pair of parallel conducting circular cylinders by replacing two of the equipotentials with conducting cylinders carrying charge~ equal in magnitude but opposite in sign. We consider two cylinders of radius R as in Figure D-9, with their axes separated by a distance D and carrying known charges -Q and +Q, respectively. Their potentials - V and + V are unknown. These are the pokntials with respect to that along thejy-axis, which is taken to be zero. The potential difference 2V is especially important, since it is required for the calculation of the capacitance per unit length of the system. The function W(z) for the pair of cylinders is the same as in Eq. D-44, except for the fact that now the quantity a is an unknown, such that

+

525

The Complex Potential

Q , (2"''v) D (2n V)· 2 = acoth Q R = acsc h

and

(D-50)

0

(D-51)

Since coth' x - csch' x = 1, we have

z

a= [( D)' -R'

(D-51)

r

(D-53 )

·

Substituting this value o f a in Eq. D-50 for R, recalling that the capacitance per meter C' = Q/ 2V, we find that C' =

7rEo

. 1 ( - D' smh-

4R'

(D-54)

-

(D-55)

cosh- 1 (D / 2R)

It will be observed that the capacitance per meter C' depends only o n the ratio D/ R, and not on the act ual dimensio ns, just as in the case of the cylindrical capacitor. When D = 4R, C' = 21.2 micromicrofarads/ meter.

Problems D-1. Two cylinders of length I and radius a have their axes parallel and separated by a distance Din a liquid with a dielectric coefficient K,. A potential difference V is a pplied between the cylinders. Calculate the force of a tt raction for the case in which I = 1.00 meter, a = 0.500 centimeter, D = 2.50 centimeters, K , = 2.6, and V = 1.00 X 10' volts. Neglect end effects. D-2. Show that the electric field intensity E for a two-dimensional field in the xy-plane is given by

where W is the complex potential and z

=

x

+ jy.

D-3. Show that the equipotentials defined by Eq. D-49 are circular cylinders given by 2 [ x - a coth ( + y' = a' csch' ( 2

71'~oV) ]'

1r3V)-

D-4. Consider an un known two-dimensional field due to two charged conductors A' and B' in the complex z'-plane, and a known two-dimensional field due to the charged conductors A a nd Bin the z-plane. Points in the z'-plane a re related to those in the z-pla ne according to the transformation z' = z'(z), and the corresponding potentials are equal : VA = V,., , and VB = VB' , Show that there is conservation of charge under this transformation and that there is, in consequence, cons~rvation of capacitance.

APPENDIX E INDUCED ELECTROMOT ANCE IN MOVING SYSTEMS*

So as to explore further the meaning of Eq. 6-45, '17 X E

= -

aB at + '17

X (u X B),

(E-1)

in which the induced electric field intensity E is measured in one coordinate system and the magnetic induction B in another, let us consider the following experiments.

Experiment 1 Figure E-1 shows a circular disk DI rotating with an angular velocity w about an axis perpendicular to its plane and parallel to a uniform magnetic field B. The disk is assumed to be both nonconducting and nonmagnetic. B We now station two observers, one in the laboratory and one on the rotating disk, both equipped with a "curl-meter," a "B-meter," and a stop watch. The HcurlFigure E- 1. A circular, nonconduc ting, nonmagnetic disk rotating with angular meter" consists of a small loop of wire velocity w about an axis perpendicular capable of orientation in any direction to its plane and parallel to a u11iform and connected in series with a sensitive, infinite-impedance voltmeter. By definimagnetic field B. tion , the component of '17 X E normal to the plane containing the path of integration is

~ti

('17 X E).

=

f

E-dl

~.':':o--s-•

(E-2)

The voltmeter reading divided by the area of the loop is thus the component of the curl in the direction of the normal to the loop, if the loop is small enough. The "B-meter" can be a cathode ray tube with the deflection of the electron beam on the tube fa ce calibrated in webers/ meter'. The stop watch is used to measure the time rate

* This discussion closel y parallels a published paper by D. R. Corson , Am. J. Phys., 24, 126 (1956).

526

527

Induced Electromotance in Mo ving Syste1m

of cha nge o f B. The o bservers know nothing about the magnetic field except what they meas ure with their own instruments . The laborato ry observer measures a uniform magnetic field: BL = B. He determines its direction a nd magnitude by observing the deflection of his electron beam for a t least two mutually perpendicula r orientations. He also observes tha t, for each orientation , the beam deflection is time-independent, such that = 0. Furthermore, his " curl-meter" reads zero for aU orientatio ns, since there is no changing flu x through the loop, and thus

aBd a1

v

x EL =

a:,'· = o.

-

(E-3)

When the disk observer points his electron beam in the plane of the dis k, he always records t he same deflection , no matter where he is on the disk. When he points it parallel to the ax is of the disk, he a lways records zero deflection. He therefore concludes that the mag netic inductio n is uniform and perpendicular to his disk. His va lue Bm is the same as that of the la boratory observer : Bm = BL = B. The "curl-meter" on the disk sees only a consta nt flux , a nd V X Em = 0

(E-4)

r.-

w

e verywhere on the disk. Thus, the disk observe r finds V X EDI= _ a!;'= 0.

(E-5)

Now let us consider a second similar nonconducting a nd nonmagnetic disk D2 rotating with a n a ngular velocity w abo ut a n ax is in the plane o f the disk and perpendicular to the direc tio n of B, as indicated in Figure E-2. An observer on this disk , if eq uipped wi th the same instrumen ts as the o ther observers, wi ll asc ribe entirely different properties to the field. Let the observer on disk D2 establish Figure E-2. A ,wnconduciing, nonmagpendicular to the plane of the disk a nd netic disk rotating with an angular velocits x -axis para llel to the axis of the rota- i:y l:.l ab o ut an axis in rhe plane of the tion. When he points his electron bea m disk and perpendicular to the direction of in the z direction, the .\·-component of a unifo rm magnetic field B . the beam deflection measures (Boz)u, a nd they-compo nent meas ures (B 0 ,),. He finds that

a coordinate system with its z-ax is per-

( IJ,,,), (BvJ ,

= 0, = 8 sin wt. 0

(£-6) (£- 7)

When he poin ts his beam in the y direction, the .r-deflection measures (B o,:),, and the z-deflection measures (B,,,), : (Bo,),

(Bmt

= 0, = Bo cos wt.

(£-SJ (E-9)

528

APPENDIX B

He finds these sa me fields no matter where he measures on his disk . He can describe the field he meas ures as a uniform field Bo rotating with angular velocity w about his x-axis. If he were to compare notes with the laboratory observer, he would find that Bo = BL= B. What about his "curl-meter"? When the disk observer points it such that the axis of the loop is in the x-direction, there is no flux through the loop, and the reading is zero. Then (E-10)

When he points the loop axis in the z direction, there is a changing flux through the loop, and

= (r!i>.) d1 no

-BSw sin wl,

(E-11)

where S is the area of the loop. Then, for this orientation,

¢

E"2·dl = BSw sin wt,

(E-12)

and, from the definition of the curl, (V X Eno), = Bw sin wl,

(E-13)

a(Br,,). B . (V X E m )z = - ~ = w sm wt.

(E-14)

or

If the "curl-meter" reading is calculated for they-component, we find that ) a(Bv-,)., B ( V X E "'• = -~ = wcoswr. From this experiment we can see that,

110

(E-15)

matter how an observer may be moving

ill a magnetic field,

(E-16)

when he measures both E and Bin the coordinate system in which he is at rest.

Expcri,nent 2 Let us now suppose that each observer is given some conducting wire and is told to try to arrange it so as to induce an electromotance in a closed circuit at rest in his own coordinate system. For the laboratory observer,

v

x EL =

-

aB,, = o a1

(E-17)

everywhere and , from Stokes's theorem, the induced electromotance

¢

EL·dl = 0

(E-18)

fo r any closed path. The observer on disk DI has the same experience, and if DI is made of conducting rr.aterial , there are no eddy currents induced in it.

Induced E/ectromotance in Moving Systems

529

Again the situation is different in D2. Let the conductor be a single loop around the rim of the disk, wit h a voltmeter connected in series. The voltmeter can either be on the disk or in a fixed position in the laboratory and connected to the loop by means of slip rings. The voltmeter is read both by the observer on the disk and by the laboratory observer. The former can calculate what the meter will read by invoking the Faraday law, since he knows aBv,/ ar:

j,_ E 'f'

D2'

di - S i!(Bv,),

_d_t_'

-

= BoSw sin wt.

(E-19) (E-20)

He can also calculate the electromotance from Stokes's theorem, since he has measured V X Em for every point on the disk:

¢

E,,2 -dl = /, {V X Eno)· da,

= 8 0Sw sin wt.

(E-2I) (E-22)

The two results are the same, of course, since

_ aBn,.

V X Ev,=

·

at

(E-23)

Now let us calculate what the laboratory observer thinks the meter on the disk will read according to the Faraday law. He says that dif>/ dt differs from zero because the circuit is rotating in a time-independent field , whereas the observer on the disk says that d/f>/ dt differs from zero because his circuit is in a time-varying field . The laboratory man calculates d/f>/ dt through the rotating loop and gets

¢

En,- dl = BiSw sin wt,

(E-24)

where his Bi is the same as the Bo measured by the rotating observer. The laboratory observer can also calculate the electromotance using Equation E-1: V X E/J2 = -

a:,, +

V X (u X B,,).

(E-25)

He must use this complete expression, since the path around which he calculates the electromotance is moving in his coordinate system. Since

aBi =

at

the equation V X

0

(E-26)

'

Ev, = V X (u X Bi)

(E-27)

holds everywhere on the disk or, excluding terms of zero curl,

Ev,= u X Bi.

(E-28)

The laboratory observer then says that

¢

E", -dl =

¢

(u X B1.)- dl.

(E-29)

530

APPENDIX E

w

Figure E-3

The vector product u X B on the rim of a disk rotating about an axis lying in its plane and perpendicular to a uniform magnetic field B. The vector u is perpendicular to the plane of the disk.

Ux8

From Figure E-3, at an arbitrary point on the rim of the disk, u=

wr

sin 0,

(E-30)

and iu X

Bd -

B Lrw sin 8 sin wt.

(E-31)

Thus (E-32)

=

SBws in wt,

(E-33)

as with the Faraday law. The laboratory observer and the D2 observer therefore agree as to the voltmeter reading, but they disagree as to the reason for the induced electromotance. The laboratory observer says that the magnetic field is static but that the Lorentz force(u X B) on the free charges in the moving conductor produces the electromutance. The disk D2 observer, on the other hand, says that the conductor is at re, t b~t that it is in a time-dependent magnetic field.

Experiment 3 As a hr.al example, let us consider the Faraday disk or, as 1t is also called, the lwmopolur ge11erwvr. Let us return to the disk DI. We place a conducting ring around

its circumference and use a conducting axle. The axle and the ring are connected by a

531

Induced Electromota11ce in Moving Systems

radial conducting wire attached rigidly to the disk, a nd the circu it is completed by brushes and a stationary wire with a vo ltmeter in series, as indicated in Figure E-4. According to the laboratory observer, aBt.1a1 is everywhere zero, and (E-34)

The part of the circuit that is stationary in the laboratory has u = 0. On the rim and on the axle, (u X B,.) is everywhere perpendicular to the path of integration, thus there is no contribution to the integral. Along the radial conductor, the laboratory observer finds that (E-35)

R'w B

= - 2- ·

(E-36)

With the Faraday law, it is importa nt to specify carefully the surface through which the flu x is to be calcula ted. It ca n be a ny surface bounded by the path of integration in the electromotance calculation. For convenience we may choose a surface lying in two planes, as in Figure E-4. The on ly pa rt of this surface in whic h the flu x differs from zero is the part that lies on the disk. The rate of change of flu x thro ugh this part of the surface is readil y calcu lated. If the observe r on the disk calculates the electromotance, he finds that aBm/ a1 is also everywhere zero a nd that the onl y place whe re (u X Bm) differs fro m zero is in the portion of the circuit ex -

ternal to his disk. He sees this pa rt of the circuit rotating with respect to the disk wi th ang ula r velocity w. Aga in, since the only contribution to the electromota nce is in the radia l parts of the circuit, his calculation a lso gives

Figure E-4. Homopolar ge nerator.

j_ R'w B '.f'Edl=~-

(E-37)

The laboratory observer says there is an electromotance in'.!uced in the circuit because o f the Lorentz force on the moving cha rges of the disk, and the disk observer says there is an electrornotance because of the Lorentz force on the moving charges

of the portion of the circuit externa l to the disk, but they a lways agree on the voltmeter reading. If the whole disk is made of conducting ma teria l, the electromotance is calcu lated in exactly the same way. It makes no difference wha t integratio n path we choose from

532

APPENDIX E

the axle to the rim, as long as the path is at rest relative to the disk. It is essential that this part of the path be at rest relati ve to the conductor, since the charges which experience the force resulting in the electromotance are, on the average, at rest with respect to the conductor. The electromotance is independent of the path in the conductor, since only the radial components of the path elements contribute to the integral of(u X B) -dl. The electromotance may also be calculated for the conducting disk from the Faraday law. Again we may choose any path in the moving conductor, as long as it is at rest relative to it. The result is the same as with the wire discussed above. Our discussion is strictly valid only for large magnetic fields and small angular velocities, since centrifugal and Coriolis effects have been neglected. For practical laboratory purposes, however, this is not a limitation. The ratio of the Lorentz force to the centripetal force on an electron in a disk rotating with angular velocity w in a magnetic field B is F,,

e B

Fe =-;;, --;·

(E-38)

For an angular velocity of l 800 revolutions/ minute and a magnetic field of one weber/ meter', this ratio is about 109• This complete separation of electrical and mechanical effects is a consequence of the large value of the ratio e/ m for an electron.

APPE N DIX F

THE EXPONENTIAL NOTATION

The subject of this appendix is a mathematical technique for solving what is proba bly, from the point of view of the physicist, the most important class of differential equations. The sine and cosine functions play a particularly important role in all of physics, mostly because of the relati ve ease wit h which they can be generated and measured by the ordinary types of instruments. T hey are also relati vely easy to manipulate mathematically. All other periodic functions, such as square waves, for example, are vastl y more complicated to use, both experimentally and mathematica lly. The mathematical technique which we shall develo p here is therefore widely used, despite the fact that it can apply only to fun ctions of the form x

= x, cos (w t

+ 0),

(F-1)

where x0 is the amplitude of x, w is the angular f requency, and t is the time. The quantity (wt 0) is the phase, or phase angle, 0 being the phase at I = 0. We shall limit our discussio n to the cosine function, since the sine can be transformed to a cosine by an appropriate cho ice of 0. We shall assume that the origin of time is chosen such that 0 = 0. The procedure for differentiating cos wt , although elementary, is rather inconvenient : the cosine functio n is changed to a sine, the result is multiplied by w, and the sign is changed. To find the second derivative, the sine is changed bac k to a cosine, and the result is again multiplied by w, but this time without changing sign, and so on. Differentiation can be simplified if it is kept in mind that

+

efwt

where j =

v'=l. Then X

=

= cos wt Xo

+ j sin wt,

cos wt = Re Xoefw t,

( F- 2) (F-3)

where Re is an operato r which means " Real part of" whatever follo ws. Let us calculate the first two derivati ves :

),

(G-42)

where the angles 0 and are constant. Comparing this with Eq . G-5, we find that the above equation determines a pair of waves tra veling in opposite directions with a common velocity u. The sepa ration constant k is the wave number, and ku

=

w =

2,r.f.

(G-43)

Since a is a function only of z and of 1, the waves are planar, with wave fronts parallel to the xy-plane. It will be observed that the formal solutio n which we arri ved at by separating the variables z and I has led us to a very special class of waves; namely, sine waves. We did not find a general fun ction of {1 - (z/ u)). This is q ui te disturbing at first sight, since our formal solution is presumably general. There is no contradiction, however, for the following reason. Since o ur differential equation is linea r, that is, since its terms are all of the first degree in a or its derivatives, the sum of any number of solutions is also a solution. Any type of periodic wave form encountered in practice can be expressed as a Fourier series of sines and cosines of the fundamental frequency and of its harmo nics (Section 4.5). Even individual nonperiodic pulses ca n be analyzed in a somewhat similar manner by means of Fourier integrals.* Any wave form can thus be synthesized by combining terms of the form sho wn in Eq. G-42 with appropriate amplitudes and wave numbers.

I(

F

F

Figure G-4. Two strings of densities p1 and Pe fixed together at O and stretched

with a force F.

G.4. Reflection of a Wave on a Stretched St ring at a Point where the Density Changes f rom

p1

to

p2

If two st rings of d ifferent densit ies p, and p, are tied together and stretc hed with a force F, as in Figu re G-4, a wave traveling along the first section wi ll be partly reflected and part ly transmilted at the knot.

* See, for example, J. Stra lton, Electrumag11eric Theo ry (McGraw-Hill , New Yo rk , 194 1), p. 287.

Waves

54.1

The wave velocities are respectively lit =

(pf_,)'"

and

(G-44) ll z

(F)"' '

= -

p,

and the corresponding wave numbers are

and

(G-45)

k2=~=w (~) 112-

Let us assume that a wave travels to the right a lon g string I. We shall call this the incident wave and set (G-46) J' being the late ra l displacement of the string a nd z For the wa ve tran smitt ed to string 2, we have

= 0 being chosen at the kn 9t (G-47)

Finally, for the wave reflected back at the knot , (G-48)

We assume, for simplic ity, that there are no reflected waves formed at the supports, In the above three expressions for Y i, y ,, and y , , the amplitude Yoi of the incident wave can be assumed to be kno wn. T hus we ha ve two quantities to determine: y 01 and Y or• lt is possible to calculate the va lues of Y ol and Y or in terms of Yoi, pi, and P:i by considering the conditions of continui ty which must be satisfied at the knot First, there must, of course, be co ntinuity o f the disp lacement y: the value of y just to the left of the knot must be equa l to its va lue just to the right Then Y ; +Y,

=

= 0),

(z

Y,

or )',; + ) ',,

= y,, ,

(G-49)

Second, there must be continui ty of the slope of the st ring dy/ dz , The reason for this is as fo llows, We ha ve assumed im plicitly that the knot was weightless, Thus the sum of the forces act ing on it must be zero, a nd the two op posing tensio n forces F at that point must be along th e sa me line. Th en

-k ,y, ; + k,y,,

= -k,y,,,

(G-50)

Solv ing these equations, we o bta in Yor k1 - k 2 p\ 12 ~ = k1 k2 = P:12

+

Yot 2k, ~ = k1 + k"!. =

-

2pt pl / 2

pf 2

+ py2'

(G-5/)

12

+ p f2

(G-52)

S ince the ratio y 0 tfy 0 i is always rea l and positi ve, the transmitted wave is always in phase with the incident wave, On the other hand , the ratio y ,,/ y,; can be either positive

544

APPENDIX G

or negative. The reflected wave is in phase with the incident wave if p1 > p,, and .,,. radians out of phase if p 1 < p,. If p 1 = p, there is no discontinuity, no reflected wave, and Y o t = Y oi•

G.5. Waves on a Stretched String with Damping. The Differential Equation for an Attenuated Wave Let us return to the case of the stretched string of uniform density p. We assume now that the string is in a viscous medium that provides a damping force which is proporlional to the velocity. The damping force on the element of string dz is then dFo = -bdziJJ', dt

(G-53)

and, from Eq. G-24, we now have p

dz d'y = F cos 8 d8 - b dz iJJ', ar' dt

(G-54)

or

=

d'y

cJz'

e. 'S!' + '?._ iJJ'. Fa r'

(G-55)

Far

We can show that this is the differential equation for an attenuated wave by trying a solution of the form y = y e i (wt -k'z)_

(G-56)

0

Substituting, we find that k' 2 =

w; ( i ! ),

(G-57)

I -

and the wave number k' is now obviously complex. The above differential equation is therefore that of an attenuated wave. This is to be expected, since it is identical to Eq. G-26, except for the addition of the second term on the right-hand side. This term corresponds to a damping force which dissipates energy. Equation G-55 can also be rewritten as

cJ'y az' -- - \ ; ( I

b) y = -k''y,

-j-:;;i,

(G-58)

or

a, + k'') y = ( az'!

0

(G-59)

This equation is similar to Eq. G-30, except that the wave number k' is now complex. It represents a pair of attenuated waves traveling in opposite directions along the z-axis.

It will be observed from Eq. G-57 that the differential equation is equally well satisfied by +k' or by -k', that is, by waves traveling in either direction along the z-axis.

We can find both k, and k, by reca lling Eq. G-18 and substituting (G-60)

In Eq. G-57, which gives

(.p_)'''{1± (1 + £)

k; = ±w 2F

w'p'

11 ' } '"

(G-61)

Waves

545

r{-1 r{

and

k! = ±w (~

I + ):,tT'·

± (

(G-62)

Since, by definition, k , and k ; are both positive and real (Section G .1), the ± signs must all be replaced by + signs, and

k; = k ';

=

I

w (~

P )' " { w ( 2F

+ (I +

w~:JT'·

b' ) '" }'" · -I+ ( I + w'p'

(G-63) (G-64)

It will also be recalled from Sectio n G. I that the velocity of the wave is w divided by the real part of the propagation constant. This reduces to (F/ p)''' when b = 0, as expected.

G.6. Solution of the Differential Equation for an Attenuated Wave by the Separation of Jlariables The differential equation G-55 which we found above is general, and any attenuated wave traveling along the z-axis is described by (G-65)

We can solve this equation formally by the method of separation of variables, as in Section G.3. We set oi = T'(t)Z'(z), (G-66) substitute in Eq. G-65 , and divide by T 'Z ' . Then

.!_ cf!Z' Z' dz'

=

cf'T' T' dr'

Jf._

+ !!_ dT', T' dt

(G-67)

where the left-ha nd side is a function onl y of z, whereas the right-hand side is a function only of 1. Then .!_d'Z ' = -k'' Z ' dz' '

(G-68)

and fl

T'

d' T ' dr'

I, dT'

+ T' dt

-k''.

(G-69)

D'e- ik'z,

(G-70)

=

From the first of these equations, Z'

= C 'eik'z

+

and k' is the wave number. The second equation takes o n a mo re familiar form when rewritten as g d'T'

d1'

+ t, dT' + k'' T' dt

= 0.

(G-7/)

We can assume a sinusoidal wave witho ut losing generality, as was noted in Section G.3, and set (G-72)

546

APPENDIX G

The quantity w must be real in order that the amplitude of the wave can decrease only with z. Substituting in Eq. G-71, we find that (G-73)

This result is identical to that which we found in Eq. G-57, the coefficients g and h of the general differential equation for an attenuated wave (Eq. G-65) being equal, respectively, to p/ F and to b/ F in the differential equation for an attenuated wave on a stretched string (Eq. G-55). If we again set k' = kt -jkf, (G-74) we find that

, (g)I/' h' )'" }'", 2 fl 1 + ( 1 + w'g'

k, = w

k,'. = w (~)' "

as in Eqs. G-61 and G-62, since g

=

{-1 +(I+;::,)"'}"',

(G-75) (G-76)

p/ F, and h = b/ F.

Finally, from Eqs. G-66, G-70, and G-72, we have a'

= A'C'ei(wt+k'z)

+ A' D'ei(wt-k'z>,

(G-77)

where k' = k; - jk/. The first term represents a plane wave traveling in the negative direction along the z-axis, whereas the second term represents a similar wave traveling in the positive direction. In this general case, the wave velocity II is w/ k., which reduces to (1 / g)'" when h = 0. The wave is attenuated by a factor of e in a distance ii = 1/ k;, which approaches infinily as h approaches zero. These results are the same as those of the preceding section.

C.7. Wave Propagation in Three Dimensions In the case of a wave propagating in space, the wave equation is similar to Eq. G-65, except that the second derivative with respect to z is replaced by the Laplacian V'a = g J'a2 a1

+ h ~a1 -

(G-78)

The coefficients g and h again determine the wave number as in Eqs. G-75 and G-76. For sinusoidal waves, V 2a = (-gw 2 + jwh)a, (G-79) or (G-80)

where k is the wave number. We have omitted the primes which were used previously to identify the attenuated wave. If there is no attenuation, h = 0, g = 1/ 112, and 1 a2a V 2a = - -2 u2

for any waveform,

11

being the phase velocity.

ar

(G-81)

547

Waves

G.8. Wave Propagation of a Vector Quantity As yet, we have only considered waves in which the quantity which is propagated is a scalar. Vector quantities, such as an electric field intensity E, for example, can also propagate as a wave, and then , for a plane wave propagating in the positive direction of the z-ax is,

E=

E oeiCwt-kz J.

(G-82)

Since E, is a vector, we may write that

E - (£,,i

+ £ ,,j + E,,k)eitwt - h >,

(G-83)

where E0 ',C. , E0 y, and Eoa are the components of Eo. These components may conceivably depend on x and on y, but they do not depend on z , because a plane sinusoidal wave is characterized by the dependence on I and on z, which is shown in the exponential

function.

INDEX

Ampere, 181 Ampere's circuital law, 197, 21 1, 277, 295, 298

Continuity, equation of, 19 Convection current density, 297 , 345

Ampere-turns, 197,293 Amperian currents: see Equivalent currents

Coordinates, I , 17, 19, 20

Amplitude, 533, 537 Angular frequency, 533 Antenna: arrays, 468 half-wave, 461 ff. receiving, 480

Coulomb, 29, 177 Coulomb's law, 28 , 37, 38 ff. , 87, 143

Corson , 526

Critical angle of incidence, 372

Curl , 10 ff., 23 , 512, 513 Cut-off wavelength, 424

Attenuation constant, 539

distance, 337, 339, 539 Biot-Savart la w, 178, 184, 196,278 Bo undary cond ition s: for electrostatic fields, 130 for magnetosta tic field , 285 in wave guides, 413 Brackets , square, 445

Brewster angle, 368, 371 , 373 Capacitance: 59, 64 coefficient of, 61 , 64 Capacitor, 60 Carter, 222 Cauchy-Riemann equations, 517 Charge, field of a movi ng, 49 1 ff. , 497 ff. Charge density: bound, polarization , or in-

duced, 85, 87, 98, 101 , l03 free , 87, 98, IOI , 103, 328 Child-Langmuir law, 170 Circuital law : see Ampere's circl!ital la w Clausius-Mossotti equation, 105, 106, I I9, 120 Coaxial line, 245, 4 I 8 Coefficient of coup ling, 236 Coercive force, 282, 284 Complete set of functions, 150 Complex potential function, 518

Complex variable, 515 Conductivity: 339 of an ionized gas, 344

Conductors: 37, 33 I good, 336, 338, 339

Debye equation , 111 Del, 6, to, 13 , 25 Depth of penetration, 338, 339 Diamagnetic materials , 280, 284 Dielectric: coefficient , 99 , 100, 112,298 constant, 100

polarization current, 297, 303, 307 Dielectrics: 82 class A, 99, 101 forces on , 1 I6 ff. liquid, 117 no nlinear, anisotropic, nonhomogeneous,

113 solid , I 13 Diode, 168

Dipole: see Electric dipole, Magnetic dipole Dispersion, 330, 338 Displacement: see Electric displacement Displacement current, 196,297 , 298 Divergence, 8 ff. , 22, 512 Divergence theorem, 8 ff., 513 Durand, 518 Electric dipole: 48 ff. moment of a molecule , 83 , 93 , 98, 99, 112 moment of an arbitrary charge distribution ,

54 oscillating, 446 ff. , 480 Electric displacement: 96 ff. , 98, 101 , 103 boundary conditions, 130

Electric energy density, 69, 32 1, 330, 334,340 Electric fie ld intensit y: 29, 38 ff., 223, 239

Conformal transformation , 517

boundary condi tions, 131

Conservation of charge, I91 , 306, 442

induced, 220, 224, 238

549

550

INDEX

Electric field intensity (co111inued) inside a dielectric, 87 ff., 98 local, 93 ff., 99, 105, 111 macroscopic, 87 ff., 93 , 98, 99 near a dielectric, 84

Induction, coefficient of, 61 , 64 Inductors, 233 Ionized gases, 341 ff., 400 ff. Ionosphere, 351

Electric force, 28 , 71 , 116

Joule losses, 240, 340

Electric lines

or force: 32

bending at an interface, 132 Electric: monopole, 54

multipole, 51 ff. polarization , 83 , 99, 114

Electric potential, 30, 32, 38 ff., 52, 224, 239 (see also Scalar potential) boundary conditions, 130

Electric quadrupole : linear, 51 moment, 55, 56 oscillating, 471 ff. Electric susceptibility, 99, 114 Electromotance: induced, 220, 230, 237 motional, 220 transformer , 220

in a moving system, 226, 526 ff. Electromotive force, 220 Energy density: electric (see also Electric)

magnetic (see also Magnetic) Equation or continuity, 191

Equipotential surface, 32 Equivalent current: 261 , 263, 276, 297, 307 formalism, 262, 274 Exponential notation for solving differential equations, 535 Farad, 59 Faraday induction law, 219, 223, 526 ff. Ferromagnetic materials, 280 Field, I , II , 16 Flux, 8 Flux linkage, 222, 233 Fourier series, 150 Fresnel 's equations, 361 , 381,388, 400

Gauge invariance, 44 1 Gauge transformation , 441 Gauss's law: for the electric displacement, 97 , 308 for the electric field intensity, 33, 37, 38 ff. for the magnetic field intensity, 277 for the magnetic induction , 273 Gradient, 6, 21 , 512 Homo polar generator, 531 Hysteres is, 282 Images, 134 ff., !43 ff. Index of re frac tion: 329,333, 360 of a conductor, 340, 386 of an ionized g:1s, 350 Inductance: 230 mutu al, 230, 237, 243 self, 232, 238, 243 , 245, 296

Kober, 518 Langevin equ:1.tion, 108, 110 Langmuir, 170 Laplace's equation: 36, 37, 129, 130, 518 rectangular coordinates, 147 ff. spherical coordinates, 154 ff. Laplacia n: 17, 24, 512, 513 of a vector, 17, 25, 512 Lawton, 29 Lechatelier's principle, 222 Legendre equation, 155 Legendre polynomials, 155, 157, 158, 172 Lenz's law, 221 Lienard, 494 Line integral, 10 ff. . Local field: see Electric field intensity, loc1! Loop, current: 205, 207, 209 ; see also Magnetic dipole Lorentz: condition, 192, 439 ff., 443 force , 182, 219,397, 530,531,532 lemma, 310 local field (see Electric field intensity, local) "l\lagnet, permanent bar, 288 Magnetic circuits, 296 Magnetic currents, equivalent: see Equivalent currents Magnetic dipole: 210 moment of a molecule, 260 moment of an arbitrary current distribution , 210 oscillating, 473 ff., 480 Magnetic energy: 239 ff. density, 241,321 , 330,334, 340 Magnetic field calculations, 287 Magnetic field intensity : 268, 276, 277, 282, 296, 297 boundary conditions, 285 Magnetic flux , 179, 185, 220 ff., 236,242,250, 296 Magnetic force , 176, 246 ff. Magnetic induction: 178, 240, 282, 296 boundary conditions, 285 curl, 194 due to m3.gnetized m1terial, 260, 265 lines, 179. 203, 290 s:Huration , 282. 28..1 Magnet ic lines of force: 179, :ms, 287, 290, 291 bending at an interface, 287 Magnetic polarization, 259 ff.

551

I ND EX

Magnetic pole: 261 formalism. 261 , 265 den sity, 262 Magnetic quadrupole, oscillating. 479 Magnetic scalar potential, 202, 204, 206, 211, 261 Magnetic susceptibility: 278, 284 torque, 252 Magnetic vector potential: 186. 190, 192, 194, 223, 238, 239, 241 , 262; see also Vector potential Magnetization curve, 282 Magnetomotance. 277, 293 , 296 Magnetron, I 82 Maxwell's equations, 97,185,223,297,304 If. Moulin, 291 l\"eumann equations, 231 Notation, 509 Ohm's law , 191 , 239, 304, 326, 396 Paramagn etic materia ls, 279, 284 Penetration of a wave through a co nducting sheet, 392 Permeability: 326 or free space, 177, 317 relative, 278, 280, 284, 339 Permittivity: 326 or free space, 29, 177, 317 rel ati ve, 100 Phase, 533, 537 Plasma angu lar frequenc y, 346 Plimpton. 29 Poisson's equations : 36, 37, 129, 130, 132, 167 for Class A dielectrics, IO I for the veclOr potenti a l, 194 Polarizability, molecular , 99, 106 Polarization: elect ric , 83 , 99, 114 magnetic, 259 ff. molar, 106 of a wave, 320 Polar molecules, I 07 Potential, complex, 515 ff. Potential energy of a cha rge dist ribution , 66, 114 Potentials, electromagnetic: see Sca lar potential, Vector potential retarded , 443 Poynting vector: and radi ation pressure. 399 for the half-wave antenn a , 465 for the oscillating dipole, 454, 478 for total reflec tion, 383 in conductors, 334, 340 in free space, 321 in gu ided waves, 4 17, 419, 427 in nonconductors, 33 1 Pressure of radi ation: see Radi ation pressure Pressure rise inside a liquid dielectric, 117

Propnga tion co nst:mt , 539 Q or a medium, 33 1 Quadrupole : see Electric q uadrupole , Mag netic quadrupole Quasi-stat ionary currents, 196

Had ian length, 412, 538 Radiation pressure, 396 ff. Reciprocity theorem, 48 1 Reflection: 357 laws, 358, 400 be tween two dielectrics, 365 ff. coefficient, 370, 372. 38 I. 389 total. 372 ff., 388 from a good conductor, 384 ff. from an ionized gas. 400 Refrac tion: 357 Snell's law, 358, 378, 386, 400, 401 between two dielec trics, 365 ff. into a good conducto r, 384 ff. Relaxation time, 191 Reluctance, 296 Remanence, 282, 284 Re operator, 533 Retenti vity, 282 Riem ann , 517 Rowla nd ring, 280, 282 Sava rt, 178 Scalar potential :438 ff. , see also Electric potential for a moving charge, 49 l ff. fo r the oscillating dipole, 447, 474 Lienard-Wiechert, 494 magnetic (see Magnetic scalar potential) wave equation, 441 Skin depth , 338, 339, 387 Snell's law : see Refraction Spectrum of elec tromagnetic wa ves, 319 Spherical harmonic functions, 154 Slokes's theorem, 15, 513 Stream function, 519 String. waves on , 540 ff. Superposit io n. principle of, 30. 305 Symmetry be tween£ a nd H , 3IO T ra nsmission coefficien t, 370, 372 Uniq ueness theorem , 132 Units, conversion ta ble, 514 Va ri ab le separation , 147, 541 Vector potentia l: 438 ff., see also Magnetic vector potential for a movi ng charge, 49 1 ff. for the oscillating dipole, 450, 474 Lienard-Wiechert , 494 wave equa tion 1 441

552 Vectors: components of, 1 cross, or vector product of, 3

derivatives of. 5 dot, or scalar product of, 3

formulas for, 512 unit, 2, 19 Velocity, phase, 538 in conductors, 332, 338 in free space, 316 in ionized gases, 350 in nonconductors, 329 in rectangular wave guides, 427 Velocity, signal, 350, 427, 429 Wave equations, 316, 327, 332, 336, 412, 540 ff. Wave front, 537 Wave guides: 409 If. coaxial line, 418

INDEX

rectangular, 420 If.. 433 Wave impedance, 413 Wave length, 538 Wave number, 330, 332, 336, 349, 4IO, 538 Waves: 537 If. Waves, guided, 409 If. attenuation, 429 cut-off wavelength, 424 modes of propagation, 423 radian length, 412 TE, 412, 420 TEM, 414, 418 TM , 412 Waves, plane: in free space, 315 If. in nonconductors, 329 in conductors, 331 in ionized gases, 341 in good conductors, 336 Wiechert, 494