Electromagnetic Field Theory. A Student's Manual [1 ed.]

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Electromagnetic Field Theory A Student's Manual G. M. Tattersfield (1998 Edition)

Introduction This handbook has been written to accompany t he third-year module in

Electromagnetic Engineering, given at t he University of Cape Town . The main motivation for providing these notes in handbook form has been that student s come \o the course with a very wide variety of educational. backgrounds; in particular, there are considerable differences in the a.mount of formal physics that they have covered. The notes for t he course therefore aim to provide a description of elect romagnetic field theory which can be used either as a first introduction or as a coherent revision programme for the subject. The emphasis is necessaril y on t he use of electromagnetic field theo ry by electrical engineers, and so the topics chosen for coverage have always been with that slant in mind.

Throughout t he notes , it is assumed t hat students have a working familiari ty with the vector algebra that is studied as a core course in the second year at UCT. It is not assumed that students have taken second year physics, howe~·er. Only the firs t-year physics course is a prerequisite. The notes are presented in an open format , to encourage the writing of further comments in the margins. This is in the belief that it is better for students to be able to concentrate on the material during lect ures, rather than feeling that they need to write copiously. The basic outlines of the lectures are laid down in the notes, and students are invited t.o read up in advance of ea.ch lecture, and to add to t he handbook only where they feel it necessary. Much of this should preferably be done after the lecture, with reference to some of the many textbooks that are a\·a.ilable on the subject. At the back of the handbook there are a few test and examination quest ions , which were given as part of the course in 1995 and 1996. They are included as a sam ple, indicating the standard required at the end of the course. l am very grateful to Richard Lord for helping to compile the text in 0-TE,X, and for producing the diagrams. The materi al for these notes is derived from Sf'Veral texthooh, hut errors in interpretation are, of course, my own . I am always very grateful to be informed of any errors encountered in the use of the handbook so that they can be corrected in lat.er editions.

To the St ude nt This set of notes is designed to help you in your study of electromagnetic fields as part of the third year course at UCT. It is very important, however , that you appreciate the spirit in which the notes were produced: they are not intended to substitute for lectures, and they should stimulate you to look for more information in textbooks. Textbooks will give you much more informa• tion than is possible in a short handbook of t his nature. and will enrich your understanding of the subject, particularly if you make a point of dipping into several works of references from time to time.

It is also important for you to study t his material on a regular and continuous basis. rather than to t ry to learn it in short bursts before tests! These notes art' designed to help you to work steadily on improving your understanding, going forward at the rate of about one chapter per lecture. You may find that you need lo brush up on your second-year mat hs. Put some time into this early in the course, if necessary-it will enhance your enjoyment of the later chapters! Field theory is a highly theoretical part of the background education of any ele("t rical engineer. The fundamentals of field theory help us to understand many phenomena at a basic, physical level. However, there can be few practical experiments in a course of this nature, and students often find it difficult to come to terms with t he large amount of theory. Persevere, and the course will offer you explanat ions of many matters that are relevant to electrical engineers. Among t hese, we will tackle the following questions: • \.Vhat exactly is meant by voltage and current? • What is the theoretical basis of capacitance and inductan ce, and how can they be calculated? • How do electric and magnetic fields store and release energy? • Do ~1axwell"s equations describe all that we know about eled romagnetic phenomena?

• Why do electromagnetic waves travel through a \·acuum at a velocity given by the well-known constant c? • Why are metals shiny, and why are radio waves reflected from layers in the ionosphere? As you can see from this short list, we will address some very fundamental issues in electr(cal engineering. I hope that you will enjoy using this handbook as part of your studies in this area, and that the skills which you gain in taking this course will be of value to you later in your career. G.M.T.

References The textbooks that were used in the production of these notes, and to which you may want to refer, are the following: • D. Halliday and R. Resnick , Fundamentals of Physics, John Wiley & Sons, 3rd edition extended , 1988. • J . A. Edminister, Th eory and Problems of Electromagn etics, Schaum's Outliue Series, McGraw- Hill , 1979. • E. M. Purcell, Electricity and Magnel ism, McGraw- Hill , 2nd edition , 1985. • P. Lorrain. D. P. Corson and F. Lorrain, Electromagnetic Fields and Wa ves. W. H. Freeman, 3rd edition, 1988. • S. Ramo, J. R. Whinnery and T. van Duzer, Fields and Waves in Communica tion Electronics , John Wiley & Sons, 2nd edition , 1984. • E. C. Jordan and K. G. Baima.in, Electromagnetic Wav es and Radiating Syst ems, Prentice-Hall, 2nd edition , 1968.

Contents Part 1: E lectrostatics 1 Electric Fields l.l 1.2

What is a field? . T he force between charges

l.3 1.4 I.5 l.G

Electric field . Superposition of fields Visualising an electric fidd .. Fields of stand ard charge configurations

2 E lectric Flux Density 2.1 2.2 2.3 2.4

Electric Flux and Flux Density Gauss's Law A,

,..,

fl)

~

s

f

f ♦ Af (4.l )

(note that we are so far ignoring the depth of the field into the paper). • From this we have (4.2)

so if the flux per tube, t::.111', the potential difference per division, .6.$ , and the permittivity, (, are all constant, then the side ratio, 6s / D.n will be constant too. • If _vour rectangles are not all simi lar. then you r first guesses about tlie pO!litions of the equipotentials were at fault. Do not be afraid to redraw the diagram several times, until you are satisfied with the result.

4.2

Field strength

• You can read the field strength straight from a field map. For example, if V~ in the drawing on the next page is IOOkV and d = 100m, then E I0~/101 = IOOOV/m at each edge of the drawing. Above the spike. however. this value is much greater because of the tendency of the f>qUipotentia\s to hug the sharp angle.

=

f'HAP'fER 4. TH£ FIELD BETH'EEN ELECTRODES

32

e .g. Draw equipotentia ls a nd electric fie ld Jines b etween these objects

J.

S

100

..,...

rHAPTER 4. THE FIELD BET\.rEEN ELECTRODES

4. 3

33

Cap acitance

• The capacitance between two electrodes of excess charge ±Q with a potential difference between them of ¢, A - ¢>a is defined as C= • •~••

•

~· -~ Jr)® ! ij i~ (4.3)

.,

~~;~

nd ;~: ~~,·:,~:~.~;.~ ~:::::;~;,~,· 11 1111 111111 1 Ignoring fo r a moment the fringing flux that ·_ curves around the edge of the plates, we can fr A ·t obtain, from Gauss's Law, D = ps/2 for each plate. So with both plates considered , D = ps, the surface charge density. Then by definition of potential, and with the plates perpendicular to a cartesian z-axis: 41,4

- 418= -/,A8 E -dJ = - J,A E:..zJ= = -[!!.!.!..]' " = ~( 8 ( l ,_..

(4.4)

• If each plate is of area A then evidently Q = p5 A , so we get the wellknown equation fo r parallel-plate capacitance:

C= 41,4~ $ 8

= ;sd~(

=7

(4.5)

• Capacitance per unit length can be calculated from graphical field maps with two electrodes. The charge between the electrodes is the same as the flux ( 'Ir = C), and both are equal to the number of "flux tubes" , N1, multiplied by the flux per t ube, 6 111 . Similarly, the potential difference between the electrodes is just the number of potential divisions, N,,, t imes the potential difference per division , .6.41 . Hence

Q

C -

¢, A -

N1.6. '1t N1 $ B - N,,.6. ¢, - N,,

( ,1 ~ lfls

(in U/m)

M

(4. 10)

where the total current, 1, is simply N1Cl. J (because the current flow s along the flux lines). Hence

!!J..

G = _ I _ = N,Cl.J = (o-Cl. $!:J,..,) ::::: o-t!.1_ (4.1 1) 4> A - 4> B N,8 41 N, Cl.nCl.$ N, so long as the rectangles on t he graphical field map are small squares for wh ich !:J,..5/t:J.n::::: I

• We arri ve finally at the useful relation

G=~C

(4. 12)

which vou will meet in vour work on transmission lines. Note t hat t he n.,s1,,/u~in of a conducti.ve medium is just R = 1/G (in 0). This gives us a way to find the resistance in a field map if u is known.

Chapter 5 Divergence and Laplace's Equation 5. 1

Inte gral and differential forms

• We have seen electric fie ld int..ensity and electric potential related by equations in both integral and differential form. Thus

=

• Gauss's La.w in integral form is fs D · dS Q, where we recall that the charge, Q. may be written as Q = fv pdV. Differentiating both sides amounts to dividing through by t::.V and taking the limit as t::. V ...... 0: lim fs D ·dS t::.V

.O.u - 0

=

Jim fvpdV f:!.V

t.v-o

(5.2)

\,\."e recognise the RHS of the above as simply p, and we propose to write the LHS as \7 · D . pronounced div D . Hence: v'- D =p

We call this the differential form of Gauss's Law.

35

(5.3)

CHAPTER 5. DIVERGENCE AND LAPLACE'S EQUATION

5.2

36

T he meaning of divergence

• Di vergen ce refers to t he m:I amount of a \'ector quantity per unit volume that emerges from an infinitesimal volume surrounding some point. It is therefore an expression of whether the quantity has a source at the point of interest. We can sec what this mca.ns by considering a small cubic volume, 6V , that lies within a field as shown.

• If the small cube lies inside an electrostatic field , it should be dear that the flux will be the same through any two of its opposite walls unless the flux density varies between two other walls, perpendicular to tl1e first two walls. In fa.ct, for a small volume, the difference in any vector between two opp~ite ~aces is ap~roximately the gradient o_f the function times the distance between faces. This approximation becomes exact in the limit as the sides of our small cube tend to zero length.

~

-- , -: ~ !.

~ 1

~ \l ..L _ "'

~

Al!

•y

• Let the small cube be cent red at (x , y , .r) and have dimensions 6:z: x !ly x 6=. Considering the x-component of D , written Dz, the above theorem gives us expressions for the flux density in the x-direction at the sides of the cube:

D,(x+T) D, (,-t) • The flux out o/the right-hand face , from the first flux density, is: !ly!J.:: D~ ( x

t,.,8D, (x)) + "'') 2 = !:::. y!:l:: ( D,..(:z:) + 2 ~

and the flux into tl1e left hand face is:

ti

__

(5.5)

CHAPTER 5. DIVERGENCE AND LAPLACE'S EQUATION

37

He111:t' tl1e nd fi«:r out of tl1e cubic volume in the x-direction is the difference of these two: Net flux out (x direction)= 6.y6. z6.x

f)!z:

(5.7)

You can then argue simil arly for they- am.I z- com po11ents of D to see that the total flux out of the small volume is: Total flux out= t.:i:6.yt.z

{}!,r +t.x6.y6. z {}~~+6.x.6.y6.z 8~ •

(5.8 )

• :-'1ow , by G11uss·s L11w. the total lluA out of our s mall volume must equal

the cl1arge that it encloses, which is just pt.rt.yt.z, so in the limit we can now write: (5.9)

But we can define di L• D as the vector operation:

and so we again conclude t hat 'v - D

=p

• Note t he compact11ess of this elegant expression: it says that "the source of flux density is charge", something we began with back in lecture I, but which we now see to be another aspect of Gauss's Law. Note also that you can get back to the int egral form of Gauss's Law by using the divergence theorem, a mat hematical identity t hat holds for any vector. F :

Lv'- F dV= fs F - dS

lntegrati11g p

= v'

D and using this tl1eorem , we get

(5.11)

38

CHAPTER 5. DI VERGENCE AND LAPLACE'S EQUATION

• \ D can also be defin ed in cylindrical and in spherical co-ordinates, using arguments much like the one presented abo\'e for the ca.rtesian case. You should make sure t hat you have researched these arguments , and are capable of 5howing the following from first principles:

V. D

V -O

~~ (r2D,) + rs:n8-!e(D, sin0) + rs:n9 {)~•

;i(rD,)+;8:•+ 8!•

(sph.)

(5. 13)

{cyl. )

e.g. l Find 'v · D for a solid sphere of charge de nsity p and radius a. , embedded in a vacuum Inside the sphere, using a gaussian sphere of radius r < a, Gauss's Law gives D

'f

= e,

(5.14)

Then, using spherical co-ordinates,

exactly as expected in a region of charge density p. Outside the sphere we work with a gaussian sphere of radius r > a to get

ThPn. using spherical co-ordinates.

(5. 17 ) again as expected in a region of zero charge density.

39

CHAPTER 5. DIVERGENCE AND LAPL.4. CE"S EQliATIOf,,'

5.3

The equations of Poisson and Laplace

• Using the th ree relations V · D now write

= p, E = -V iti and

D

= t.:E,

we can (5. 18)

and this is generally seen as a differential equation called Poisson's equation , which is given below in several fo rms:

V. V:: V 2 where V 2 it>

=

::r; + !:; + ~::: = -~

(5.19)

= div(grad it> ) is called the Lap/acian of $.

• In regions of no charge, another import ant equation, called Laplace's equation, results :

(5.20) Solving Laplace's eq uation to find t he potential function 41 amounts to solving a se,:ond order, partial differential equation in which $ may vary in 1, 2 or 3 dimensions, and may be expressed in ca.rtesian , cylindrical or spherical co-ordinates. As yo u can imagine, the mathematics can vary from very easy through to rather complicated! We restrict ourselves to some very simple examples below. • In order that yo u can he comfor t able wit h Laplace's equation in cylindrical and spherical co-ordinates, you should , in your own time, take the expressions that you derived for V · D in those co-ordinate systems and show that Laplace's equation becomes (sph. ) (5 .21 )

CHAPTER 5. DIVERGENCE AND LAPLACE'S EQUATION

40

e .g.2 Potential between parallel conducting plates

If the region between the para1lel plates is charge-free, then we can write:

(5.22)

9 =

Neglecting fringing, t he potentia1 is a funct ion of;;: only, 50 0, from which we can integrate once to get ~ A and integrate again to obtain

=

4> =Az+B

(5.23)

We can evaluate the constants A and B by using the known values of t he potential given to us as the voltages of the plates. Using this information is known as applying tht lunmdary conditions. So, since at ;;: = 0 we know cJ> = 0, we can see that B = 0 and since at.:= d we know cJ> = 100, we also obtain A= 100/d. T hus (5.24)

This gives us an expression for the potential at any point (z,y,z) between the plates, and it suggests that equipotential levels would be equally-spaced, which we know to be the case. We can now find the expression for the electric field intensity, like this:

at. M. -a;z at. =-dz d ( z). 100. E=-v'4' =-azx-nyy 100d z= -dz

(5.25)

and you should recognise this as having exactly the magnitude and direction that we would expect for an ideal. parallel-plate capacitor

CHAPTER 5. DIVERGENCE AND LAPLACE'S EQUATION

41

e.g. 3 Potential between coaxial cylinders

If the region between the coaxial cylinders is charge.free. then we can wri te Laplace's equation in cylindrical co-ordinates:

(5.26)

Assuming no variation in potential with changing angle t/,, and also assuming that the cylinder is long enough that there is no variation of potential with .;, Laplace's equation reduces to

~!_ rdr

(r~) dr

= 0 so

so

.!:._ dr

(r~) dr

½!- = Afr

= 0 so

so

4'

r~ dr

= A

= A ln lrl + B

(5.27)

Next we apply the boundary conditions: Since at r

= a we know 4' = 0, we can see that

A In lal + B

since at r = b we know 4> = V.,. we also get A ln lbl

+B

= 0 and

= V0 ,

and we can then easily obtain A

= __v,_, -

In lbl - In l•I

and

B =

~ (In lbl - In l•IJ

(5.28)

The expression 4> = A In lrl + B can now be used to find the potential at any radius r. knowing only the dimensions a and b of the coaxial cylinders, and thP vohage 011 thE' outer cylinder with respect to the inner.

Chapter 6 The Uniqueness Theorem and Energy 6.1

T h e unique ness t heorem

• There is an important t heorem which states that if we are given a region wl1ere Laplace's equation l1olds, and if we know the boundary conditions. the n the solut ion to the equati011 is the unique expression of the potential of the field in t hat region . • \Vt' can p rove this uniqueness Lheonm by contradict ion. im agi ne t hat we have a charge-free region , with potentials Spt!cified on its boundary. Let us also imagine t hat the re are two solutions for this potential, written $ 1 and 41 1 . At the boundary, bot h solutions m ust give the same potential (the potential specified), and so we can write /or the bou11dary:

(6.1) • Both potential fu nctions would have to be solut ions of Laplace's equation. and so we can write for ll11: wholf ngi,m:

42

CHAPTER 6. THE l 'NIQlTNESS THEOREM AND EN ERG\"

43

• Now recall the divergence theorem: for any vector F (6.3)

We choose to use the divergence theorem with F This gives

= (4' 1-4'2 )'i7(4' 1-4'2) -

• Another mathematical identity (which you may find it very instructive to prove to yourself) states that for a vector A and a scalar IV , V ·( $A ) = $ V · A + A · V$

(6.5)

Noting that ¢11 - 4' 1 is a scalar and V( 4' 1 - 4> 1 ) is a vector, t he previous equation may now be written

fv (4'1-4'2)V 2(4',- 4'2)dV+ .fv [V(4'1-4'2)12dV

= fs( c>, -4'2)V( c>1- 4'2)·dS (6.6)

• Now the first term 0£ equation 6.6 refers to the whole region and equals zero (by equation 6.2), and the term on the RHS refers to the boundary surface, and equals zero (by equation 6.1 ). We are left with (6.7) The gradient of a real scalar being real , and therefore having a square which is positive or zero, the gra.dient V(4> 1- 4>2) can only be zero. Integrating, we obtain a relationship between the two potential functions which must apply everywhere: 4>1 - 4'2 = constant

(6.8)

CHAPTER 6. THE UNIQ UENESS THEOREM AND ENERGY

44

• But if this applies everywhere, then it applies at. tl1e boundary, where we know the two potentials are equal. We are obliged to conclude t hat the constant in the equa.tion a.hove is zero, and that the two potential fun ctions are equal. Our initial assumption of two different potential solutions has led to a contradiction, and we have proved t hat solutions to Laplace ·s equation wilh boundary conditions given are unique.

6.2

Energy in fields

• Evidently, electric fields store energy (they are capable of accelerating charge q through a potential drop v, and the energy required to do this is w = qv) . We now aim to find ex pressions for t he elect rostatic energy stored in a field, in terms of field quantities such as E , tP and D . • The energy required to assemble a group of n charges , bringing each charge from zero potential at infinite distance to a particular potential , ¢>;, in an arrangement or system of charges is (6.9) where the factor of 1/ 2 compensates for a formula that would otherwise count the contribution of each charge twice: once as it is brought up to the other charges, a nd once as each of them is brought towards it .

• If the cha rge distribution is continuous, with charge density p, we write (6.10)

where the potential , lfl , is a function of t he posit ion within the volume over which the integral is performed. Rtx:alling that p = V · 0 , this becomes

(6. 11 )

CH.4.PTER 6. THE UNI QUENESS THEOREM AND ENERGY

45

and then, since fo r any scalar iJ, and vector A we have

(you have recently checked the proof of this for yourself) , we can write

• The first term of this, by the divergence theorem , now reduces to I•

~/.,V •( t D)dV=~ f, tD - dS =O

(6.14)

si nce we are interested in the total energy due to every part of all of the fields of the charges present , and this involves integrating over a surface at infinity, at which distance the product of D and 4> will have shrunk to nothing compared with the area of the surface of integration. • Wearethereforeleftwith

This equation confirms that total energy resides in the field represented by D or E , and we can use it to express the energy present in each small \·olume element of the region by differentiating to get dU=~ D E dV

(6. 16)

CHAPTER 6. THE UNIQ UENESS THEOREM AND ENERGY

46

• Considering now a parallel-plate capacitor. in which - the plates have area A and separation d; - the voltage drop between the plates is \I;, ; - the elect ric field has constant \'alue E - flu_x density D

= ~ / d;

= fE = €Vc/d is constant, as f is constant:

the energy stored in the capacitor's field {neglecting fringing) is

u,

~DE

Iv dV

(6,J;)

H~Ht)(Ad)

(618)

~(~)v' 2 d '

(6. 19)

But we recognise from equation 4.5 that fA/d = C, the capacitance of the parallel-plate system; and so we have, just from field theory considerations. the well-known result (6.20)

Chapter 7 Electrostatic Boundary Conditions 7.1 Electric field due to de current • When a de potential is applied lo a conductor, a steady de current will flow. The electrons that participate in the current give rise to an electric field which, although not strictly static, is of interest to us now because it does not vary in time. Such fields are called quasistatic fields, and analysis of them proceeds similarly to our study of electrostatic fields. • In an ohmic conductor (a conductor obeying Ohm's Law ), the current dmsity, J , is related to the applied field E via the conductivity, a. Furthermore, the relationship between electric field and potential allows us to write (7.1) • A stationary (i.e. non-time-varying) current will remove exactly as

much charge from any closed region as it adds to it, per unit of time. There can be no build-up of charge either spontaneously, or due to a stationary current. This principle of continuity can be written

fs J · dS =O

(7.2)

or. di\·idiug through by ~V and taking the limit as ti.V-+ 0,

'v -J 47

=0

(7.3)

CHAPTER i . ELECTROSTATIC BOUNDARY CONDITIONS

• We t herefore have 'v (-cr'v'IJJ )

48

=0, so with er a constant we can write (7.4)

In other words, the potential t hat is due to a stationary current satisfies Laplace's equation , ju.st like the potential due to .static charge. • We have already seen that solving Laplace's equation a.mounts to solving a differential equation in several variables . The re:st of this chapter investigates the boundary conditions tha.t we use to evaluate the arbitrary constants in such calculations.

7.2

The boundary between dielectrics

• Since the field that we wish to study is often confined to a. region occupied by one materiaJ and surrounded by other, different materials, we often gain useful information from knowing how E , D , cJl and J will behave at the boundary. • Consider an ubitrary boundary between b

(9.4 )

This explains why coaxial cable is used to transmit any signal from which com ponents or systems that lie near t he outer sheath of the transmission line need to be electromagnetically shielded.

CHAPTER 9. AMPERE'S CIRCUI TAL LAW

61

e.g.3 The magnetic field of a sole noid • If current / flows in an infinitely-long coll (known as a solenoid) , which is wound with n turns per metre, then it is acceptable to regard the current as flowing in a sheet, ent irely circumferentially around the solenoid. We ignore the tiny component of current flow in the z-direction that is due to the progress of the solenoid winding. The sheet's current density is rd Am- 1 , and so the current that flows in a differential strip of length dz metres is nl dz amperes.

• By the methods of the last lecture, we can write the field due to this differential element (which is just a ring of current) as n /a 1 dz dH~ = 2(a1 + =1)f

(9.5)

and so, if the solenoid has length 2b and is centred at the origin,

H,

16 nJa2dz -b2(a2 nfal

2

16

I

- ~(a1+ z1) ! dz

nfo ' [

2a'

(9.6)

+ .::l)½

2 [(o/b)' + !]l

l

(9.7) 1

(9.8)

CHAPTER 9. AMPERE'S CIRCUITAL LAW

62

• T he integral in the line above is relati vely easy to obtain, using the substitution z 'cc a tan Q. Having found the expression for H, for a finite solenoid. we now let lbl _, OCJ to arrive at a simple expression (purely in the z-direction) fo r the for the magnetic field of an infinite solenoid at any point on its axis: H,

= r1 /

on the axis

• The diagram shows a sectional view of a part of an infinite solenoid , with the cu rrent directions indicated by dots and crosses. Now if we consider the I-met re-long path shown on the left-hand-side of this diagram, we can use Ampere's circuital law to give us

f H · dl =nl

(9.9)

G)(:)0000000 (9. 10)

But we already know from the above that Hz = nl on the solenoid's axis, and that since H is purely Z-directed there a.re no contributiom

to the integral from the radial sections of t he path . Thus, all of the c011tribution to f H ·di com!"..s from the axial section of the chosen path, and hence lh t fidd outside the solenoid is ::ero. • Furt hermore, if we look the other I-metre-long path shown in the diagram , then by Ampere's circuital law we know that f H • di = 0 (since t he path encloses no current). But we know that JH · di = nl for t he axial section , so the contribution from the other horizontal section must be -n/ . In addition, we note that the choice of where the upper horizontal section is drawn is arbitrary. In other words, th e magnetic field is constant cvcrywhcTI! inside the solenoid, and of \'alue n) Am- 1 .

Chapte r 10 Inductance 10 .1

Definition of inductance

• In circuit analysis. we would describe any structure which stores energy in a magnetic field as an inductor. Recall from b~ic physics how any current has a magnetic flux (4>) associated with it. Wheie two (or more)

conductors are separated by free space, or by a material medium, the magnet ic fluxes due to each conductor are said to be linked. Thus, for a coil of N turns for example, the total flux linkagf. is written A= N 4>

(ID.I)

• Since the flux , 4>, is usually proportional to the current,/, flowing in the i11ductor, so too is the total flux linkage,,\_ Thus we may write

(10.2) where the constant of proportionality, L in llenries, is known a.s the inductance of whatever arrangement of conductors is being studied .

63

CHAPTER JO. TNDUCTANCE

64

• Now the total flux linkage, >., can be found by simply adding up the magnetic flux density over the area in which the flux linkage exists:

>-=fs B-dS

(10.3)

So. where the surface S has been specified, we have the impo rtant expressi_on for inductance:

L=ifs B·dS

(10.4)

• Take, fo r example, the coil of N turns shown here. Almost all of the flux due to the current passes through the shaded surface S. A little, however, passes through the wires of the coil itself. We often distinguish these as giving rise, respectively, to the external inductance and to the internal inductance of the coil. As suggested in the case of the coil of N turns, the internal inductance may be very much sm aller t han the external induct ance, and is sometimes negligible.

s

CHAPTER 10. INDUCTANCE

65

e .g. 1 The inductance of a parallel-plat e trans mission line

• Consider a pair of parallel plates, such as. for instance, in a microstrip transmission line. Let the plates have width w and have separation d. The separating medium has permeabilityµ. Each strip is of thickness h, considered negligible, and a current J flows in the one strip and returns in the other.

• \Ve make the important assumption that the strips are wide enough, compared to their distance apart , that the magnetic field lines between them are parallel. We write the magnetic field strength between the strips as // and note that we have made t he assumption that d ., obeys Laplace's equation. There are methods available (as we saw in electrostatics) for solving Lapke's equation, so yet another way of calculating magnetic field may be to obtain 4'm in a given situation , and then to find H = -V¢>,,. .

Chapter 13 Energy in Magnetic Fields 13.1

Energy storage in st atic m agnetic fields

• You may recall that we derived this express ion for the energy contained in a small volume of a region in which there is an electrostatic field : dUE = ½D · E dV . One can reason in a di rectly analogous way wi th static magnetic fields, provided that B and H are proportionately related- that is , provided t hat /J is constant for the material medium, in wh ich case the medium is described as being linear. We may then write B = µH where µ is a constant and, with this proviso, we have (after integration and if B II H ):

UH=ifv B -HdV = ~fv H~dV

BL

• Many magnetic materials are not linear-i n other words, the \"alue of 11 depends upo n t he strength of the magnetic field applied. The result is that, if you plot IB I against ., 11 1 IH I as t he field strength varies in this material, you will I obtain a curve like the one shown. As we shall see, this complicates the picture of the energy stored in t he field . In the non-linear case. the extra energy stored in a rolume 1· for a small increase in B is dUH = fv H dB dV

78

(13. 1)

:

1

: ; H ( 13.2)

H

79

CHAPTER 13. ENERGY JN MAGNETIC FIELDS

• If H

II B , which is a feature of dUH

isotropic materials, we can then write

= HdB [dV

(13.3)

which means that tbe shaded area in the graph above represents the extra energy stored per unit volume for a small increment in the B -field.

13.2

Energy loss in hysteretic materials

• We have therefore seen that a non-linear, isotropic material will respond to an increase in field strength from zero by travelling from a to b in the 8 -H graph shown below.

8

H

• By the time the material is brought to point b, small magnetic domains within the material may have been aligned by the applied field . There is a tendency for this alignment to be semi-permanent, with the result that the material is somewhat magnetised, i.e. it will act as a permanent magnet.

• If the appl ied field is now decreased, the materiaJ does not retrace the path from b to a, but will travel the path cde as the applied field is brought first to zero and is then made negative. Note that at point c. where the applied ff -field is zero, there nevertheless remains a B field due to the magnetisation of the material. This is known as the rtmanence or retentivity of the material. It then requires a furthe r applied fie ld in the negative direction to bring the B-field to zero at point d. This value of H is known as the coercive force.

CHAPTER 13. ENER.G\' IN MAGN ETIC FIELDS

80

• Similar arguments apply by symmetry as the applied H -fie]d is again increased, and it is found that the material will trace a path from e through/ and g back to b. The tendency of the material to "remember" its most recent state of magnetisation and not to retrace its B-H path until it has traversed the full loop bcdefg is known as hysteresis, and the loop is called a hysfrresis loop. • Applying our law for energy storage as the B-field varie5, we see, for example, that as the material moves from 9 to b there is a point (shown on the figure) where the increase in B-field leads to the storage of energy H 1 dB per unit volume. Then , as the material moves from b to c, at the same B -field strength there is a release of stored energy equal to H~ dB . The difference between the energy added to the stored energy and the energy taken from storage is (H 1 - H 1 ) dB , which is shown in cross-hatched shading on the diagram. This energy is effectively wasted, being dissipated as heat during the hysteresis process. • It should therefore be clear that ii, one transit of the hysteresis loop bcdef 9b. the energy dissipat ed per unit 11-0/ume of the material is equal to the area enclosed by the hysteresis loop. Materials with narrow hysteresis loops are termed "soft", and are of value in electromagnets, transformers and motors because their energy wastage is relatively small. "Hard" magnetic materials have broad hysteresis loops and are used for making permanent magnets .

13.3

Magnetic field e nergy and inductance

• In t he analysis of linear circuits, we are familiar with the expression ½Ll 2 for the energy stored in the magnetic field of a circuit element. Since we ha\'e already seen how to calcula~e the induc\ance of certain symmet rical structures directly from field considerations, we now have the opportunity to check that our two definitions for stored energy are the same.

CHAPTER 13. ENERG}' IN MA.GNETIC FIELDS

81

• The examples which follow will demonstrate and explore the usefu l relationship: (1 3.4 ) • NoLe carefully that the inductance L refers to the exterrwl inductance of the structure if we consider only the fie ld H that lies outside the conductors, a.nd refers to the internal inductance if we consider only the ff .field within the conductors. Because the external field generally occupies a much larger volume than the internal field, the external in • ductance usually far outweighs the internal inductance. The equation above applies equally to both situations. however, and , because of linearity, will also apply for the total inductar1ce, if the whole volume both within and outside the conductors is considered.

e.g.1 Show that the relation fvI H 2 dV = ½LI2 holds for the region between the two

conductors in a coaxial cable • We already know from Ampere's ci rcuital law t hat the external inductance is L = f; In H/m and that the field between the conductors is H. = ~- For a I-metre-long cylinder V = -irr 2 xl, and sodV = 2:irrdr.

U)

• Hence, if we consider a !-metre length of coaxial cable which has an inner conductor of radius a and outer conductor of inner radius b:

!:...!!.... 10 2 2:ir

(!) = !u a

2

2

(1 3.5)

CHAPTER 13. ENERGY IN MAGNETIC FIELDS

82

e .g.2 Find the inductance per unit length of an infinite solenoid of radius a. which has n turns per metre and is filled with a material of p ermea bility I'· • We recall that the field is flz = n/ everywhere inside an infin ite solenoid. The volume per unit length is V 1ra2 x l. Hence for any one-metre length we can write the stored energy a.s

=

• But the energy stored is also !LI1,

so

(13.7)

from which

where A is the cross-sectional area of the solenoid. • Note that the inductance of a coil is related to the square of the number of turns per metre.

Part 3

Maxwell's Equations

83

Chapter 14 Time-Varying Electric & Magnetic Fields 14.1

Summary of imp ortant re lationships

• Up to this point, our analysis of fields has included no dynamic effects. In all of our examples, the sources of electric or magnetic field have fallen into one of the following categories: I . They have been stationary in space and invariant in time. 2. T hey have varied in time very slowly (or at very low frequency),

so that over a given period of interest they have been effectively static.

3. They have moved in space, but, as with a steady current in a ..n , and so we can effectively neglect displacement current altogether and write

v 2 E =iw11a-E

(17 .7)

I07

' CHAPTER Ji. THE Sh.IN-DEPTH PHENOMENON 11 : Similarly. for the magnetic field intensity,

H, we might begin

'v x H =(er+ jw() E

::,;j

crE

with (17.8)

Taking the curl of both sides now gives v' x \7 x

H=

v'(v' • ii) - v' 2 H = crv' x

E

(17.9)

But 'v x E = -jwµ H. by the t hird Maxwell phasor equation, and 'v - ff = 0 by the second, and so

(17 .IO) Ill : Finally, for the conduction current density, j , a possible starting point is to take the curl of both sides of Ohm 's Law and then to apply the third Maxwell equation, 'v x E = -jwpH , to obtain: v' x j = cr'v x

E=

-jwµcr 'H

(17.ll )

:\ext. we can take the curl of both sides again, and this time replace \' x H lo get v' x v' x j

= -jwµcr v' x H = -jwµ cr(cr + jw,)E w,

(17.12)

Now, remembering that u > for a conductor, we can neglect the displacement current term jwt E and replace the other term, crE , with j (by Ohm·s Law ) to get 'v xv' x j = 'v(v' · i ) - v'1 J = -jwµcr i

(17.13)

Then, finally, we must ap peal to the concept of continuity, in which we saw that 'v · j = - ~ ::,;j O for a good conductor in the st eady state. This then leads us to (11.14) • The differential equations governing E, ff and j are therefore of identical form in a good conductor. We now turn our attention to solving for E: the other solutions follow the same pattern.

CHAPTER 1i. 1'HE Sldl\'- DEPTH PH£NOMEN0,\1

17.3

108

Field penetration into a good conductor

• The following analysis will show that the E-field (and, si milarly ff and j ) only penet rates with significant amplitude to a limited depth into a good conductor. At high frequencies. it will be found that this depth is small in comparison to the wavelength. This implies that the performance of high frequency fields near a conductor is not affected much by ·the shape, edges or cur\'ature of the conductor, because typical conductor dimensions are a lot higher than the penetration depth of the field. We therefore study an £-field of magnitude £ at the surface of a conductor, and can make the assumption that the conductor is infinite in extent below the surface. 0

• Y_

-

• We also assume that the E-field is a plane wave, which is directed in the .;-di rection and which propagates in the x-direction. There is therefore no variation of electric field with ~ or z. As this_mca.ns th~t I; f. = O. our differential equation for E, namely V 2E jwµ11 E, becomes

=

=

(17.15)

for which

where the factor~ is recognised as the posit ive square root of j, the frequency in hertz , and 6 in metres.

="#

f is

CHAPTER 17. THE SKIN-DEPTH PHENOMENON

• The differential equation for

£,

109

can now be written (li.17)

=

and its characteristic equation , s 2 - r ~ 0, implies that (11 + r )(s - r ) = 0 and sos= ±r. The solution is therefore (17.18)

where A and B are constants to be determined from the boundary conditions. We quickly see that, since £, certainly cannot increase to infinity as :r increases, B must etiual zero and since, when :r 0, £, = EP, A must equal E0 • Hence our solution for the differential equation is the decaying exponential

=

• Likewi se, you could arrive at very similar developments.

iJ,

M'

= H0 e- t1:- 1 i or j , = J0 e- te- 1 t by

• Th. e interpretation of the E -field solution is shown in this figure . You will see that the magnitude of the i -componcnt of Eis £ 0 /e at depth x = D, which is known as t he skin depth, and t hat the envelope of the magnitude of the £.field decays exponentially with depth. Furthermore, the decayi ng sinusoidal wave associated with the E-field t ravels into the conductor in t he i-direction.

~ ~ _ _ .. -:

• Note also that, in all cases the E (or D ) and H (or B} fields have direction which is uniquely determined by the right-hand set [E, H ,i ], where i is iu t.he direction of propagat ion of the wave, and E = [£_., E~, E,]7.

_j

,. ::11

CHAPTER 17. THE SJ-;JN-DEPTH PHENOMENON

17 .4

110

R esistance and internal impedance

• Let us now consider a section of the semi- infinite plane of unit width and length (hut effect ively infinite depth). We can calculate the total curren t Rowing i11 this conductor by integrating the exponentially-decaying current density down to infinite depth. Thus

• The magn itude of the su rface electric field is then simply found from the surface current density to be: ( 17 .21)

The internal impedance of the conductor per unit length and width is t he ratio of the field and of the current at the surface, which is

Z,

= R, + jwl; = ~ = ~ + fo

(17.22)

- Note t hat this expression gives us the surface resistivity, R. , of the conduct.or {multiply it by length and divid e by width to get the resistance of the conductor) , and also the internal reactance , jwL;. The latter does not include any cont ri bution to total inductance from fields that are external to the conductor. - Note also that for a plane conductor of unit lengt h and widt h, the resistance and internal react.ance are of equal magnitude or, equivalently, that the pl1ase angle of the internal impedance, Z., is 45°. - F inally, note that the resistance of the conductor is inversely proport ional to skin depth , 5, which is itself inversely proportional to the square root of t he frequen cy, /, Thus, at high frequencies we understand that fie lds penetrate less deeply into a conductor, and that the resistance of the conductor increases. This is in line with our expectation for a conductor of decreased cross-sectional area..

Chapter 18 The Basis of Kirchhoff's Laws 18.1

Lump ed elements

• In e~·eryday electrical engineering, the great majority of electromagnetic interactions are analysed and evaluated using standard circuit theory. Th purpose of this chapter is to s how that all circuit theory ha.s its foundations in field theory, a11d that tl1erefore the use of ci rcuit analysis techniques is justified upon solid theoretical grounds.

• In lumped•element circuit analysis, we make the assumption that all of the circuit elements used are small compared with the wavelength of the electromagnetic fields within them. This assumption allows us to trf"at the conditions around each element as quasistatic. simplifying our treatment considerably. At high frequencies (e.g. in microwave ci rcuits) or where very large ci rcuit elements are being considered (e.g. an inter-city power transmission line), the quasistatic assumption is invalid. In such ca.ses, it is necessary to resort to t ransmission-line theory in preference to standard circuit analysis. We concern ourselves here on ly with the validity of the latter at sufficiently low frequencies. • The three ideal lum ped elements most frequently dealt with by circuit analysis (see Module A of EEE22lW) are t he resistor, the capacitor and lhe inductor. We now consider each of these in turn from the sta11dpoint of field t heory.

111

CHAPTER 18. THE BA.SIS OF l\IRCHHOFF'S LAWS

18.2

112

R esjstors

• The resistor is a dissipativ e element: that is, the interactions which occur within it at a subatomic level cause it always to absorb net electrical po\\·er from its surroundings. It converts much of the energy that it absorbs to the generally unusable form of heat . • Resistors conduct current , and are charact erised by a conductiuity, u in Um- 1 . Hence, as is well-known , they obey Ohm 's Law, so that if any field Eis ap plied across a resistor, t hen a current of density J will flow , where J =uE ( IS.I) • The voltage across the resistor can be expressed using the simple relation for electrostatic potential difference:

Vii=

4>2 -4>1

=-

fi

2

E • dl

fi

fi

= - 2 ~ • dl = - 2 ~ (18.2) = / / A, and that A represents the

where it is clear that J II di , that J cross-sectional area of the resistor.

• If we now assume t hat the current , /, the conductivity, u and the cross-sectional area, A, are all constant between points l and 2, we can write

Vii =

-h [

dl =-~=-JR

(18.3 )

where L is the length of the resistor, and R = L /( a A) is known as the rt si,,;lancc. We have thus recovered the fa miliar form of Ohm 's Law. • In the case where / , u and/or A arc not constant over the length of the resistor, we simply define (18.4) to arrive at

t-'2 1(t)

= -Ri(t)

(18.5)

Cl-lA.PTER 18. THE BASIS OF KIRCHHOFF'S LAWS

1s. 3

113

Inductors ,; J I ~

a

'

'

• Inductors, as we know, store energy in the magnetic field that is associated with the current that passes through them. Ideal inductors have zero resistance in their coils, and so all of the stored energy may be recovered at a later time. • Taken around the path shown in the diagram, clockwise from point 2, the integral of the electric field around the path is evidently

where the fi rst term on the right-hand side integrates the field along the perfectly-conducting coil, and is therefore equal to zero, while the second term integrates the field along the return path between the terminals. We t hus have, by Faraday's Law,

• We defined inductance much earlier in the course

a.,

(18.8)

so we may now write. for the quasistatic case, vi:i(t ) = -

l

E ·di =

1ls

B · dS = ~(Li(t)) = l d~~t )

( 18.9)

if the current through the inductor is taken to be the time-varying i(t). Again , this is a very familiar result from circuit theory.

CHAPTER 18. THE BASIS OF KIRCHHOFF'S LAWS

114

• Note that if t he resistance of the coil were not zero, but instead had value RL, we would then have

f

E · di = +RLi(I) - vz-3(l)

=

-:i ls

B · dS

(18. 10 )

so that. as we might expecl, a time-varying current would now gi~·e us

v,,(t)=R,i(t)+Ld~;t)

(18.11)

• Note also that a voltage may appear across an inductive coil in a circuit as a resuh of magnetic flux changes that are due to some other source of magnetic flux. Thus, if coil Y lies close to coil X and has a timevarying current iy(t) flowing in it, then the magnetic flux that links coil.\' due to this current is written IPxY • By di rect analogy with the concept of self-inductance, L, the mutual inductance, M , of the coils is defined as the ratio of the flux linkage and t he current in coil )' that produces it. Thus

Mxy

= T/J~y >y

(18.12)

The voltage induced across coil X by this flu x linkage is then written vxy(t) = d~:y = MXY di~~t)

(18. 13)

• The phenomenon of mutual inductance is sym metric.a.I in the sense that the same value for M is used in cakulating the voltage in coil Y due to changes in the current in coil X . Note that dots drawn at the ends of the coils in the ci rcuit diagram show the directions of the windings, and so give the polarity sign for the voltage induced.

CHAPTER 18. THE BASIS OF ldRCHHOFF'S LAWS

18.4

115

Capacitors

• C11pa.citots store energy in the electric fields between their plates, and we established much earlier on that, with a charge Q on either plate, the capacitance is found using the potential difference between the plates, (18.14) • Any c.urrent flowing in the capacitor will, by continuity, merely raise or lower the value of Q. Thus we can write i(I) =

~

=

i(Cv,,,(I)) = Cdv:,(I)

(18. 15)

or. equivalently,

V:,o{t)= hli(t)dt

(18.16)

if the capacitance is constant over time, and if the capacitor is initially uncharged. We have therefore recovered the familiar expressions for the behaviour of an ideal capacitor in a low-frequency circuit.

18 .5

Kirchhoff's Voltage Law

• We can now put together the three elements considered in this chapter, and drive them with a time-varying voltage source, as shown in the cliagram below.

'D l r,_'.I

11

..... m

l-

o

C

l

CHAPTER 18. J'HE BASIS OF KIRCHHOFF'S LAWS

116

Taking the line integral of electric field around the whole path , clockwise from point 0, now gives us

• The surface S referred to in the equation above is, of course, any surface bounded by the loop of the circuit. If we assume that the magnetic flux in t he inductor is largely confined to the inside of the coil {a fair assumpt ion , a.s we know from our study of the solenoid ). we can see t hat tbe component of stray magnetic flu x perpendicular to the surface S is negligible. Insofar as it needs to be considered at all, it could in any case be added to the effect of the inductor, L. Hence, t he term on the right-hand side of the above equation can be regarded as equalling zero. The equation thus becomes Vio+t•21 (t ) + v31(t)+ Vro( t)

v0 (t ) - Ri(t) - L d~~t) 0

hl

(18.18} i(t )dt (18.19) (18.20)

• This is what we know as h"irchhoff's Voltog ~ Law, which we use habitually in electrical circuit analysis. You can now do so with confidence, knowing that it is a result which is formally derivable fro m electromag• netic field theory. The general statement of the Jaw for N elements around a closed path in a circuit is (18.2 1)

CHAPTER 18. THE BASIS OF 11:JRCHHOFF'S LAWS

18.6

117

K irchhoff 's Curre nt Law

• Consider a node in an ideal circuit at which a number of conductors (assumed perfect) all meet. To be completely general, let us suppose that there is a charge Q at the node: evidently, this charge rnust be maintained by a capacitive effect, or else it would instantly be dissipated in the form of currents along the perfect conductors. • We know that the principle of continuity states that

~-J=-'!£

or

t J -dS=-~!vpdV

{18.22)

and we can give meaning to these expressions in this situation by imagining setting a surface S around the node, enclosing volume V , which would then have charge density p Q/V as shown above. In the integral form of the continuity expression,

=

i J -dS +~fvpdV= O

(18.23)

the first term clearly represents the sum of the currents, / 1, h , /3, /◄ .• which fl ow in the perfect conductors. The second term is dQ/dt, the current leaving the node via the capacitor. This current is not included. in the first term, since J denotes conduction or convection cu rrent only, and the capacitor current is a displacement current. • Nevertheless, the equation shows that the sum of all the currents of all types out of the node is zero. We know this as A"irchhoff 's Current Law, and we use it implicitly on a daily ba.sis when we work with ci rcuits. In general form , it may be written

(18.24)

CHAPTER 18. THE BASIS OF K IRCHHOFF'S LAWS

118

• With Kirchhoff's Laws established from fie ld theory, you may recall (see EEE221 W) that the whole structure of circuit theory now follows from them. We t herefore have the theoretical basis upon which to analyse circuits and t heir equivalents, thanks to the theory of electromagnetic field s. To be specific, Kirchhoff's two laws are necessary to t he proofs of the following well-known circuit analysis techniques: - Nodal analysis - Mesh analysis - The theorems of ThCvenin and Norton - AC phasor analysis

Chapter 19 Reflection of Travelling Plane Waves 19.1

Intrinsic impedance of a medium

• In introducing t he concept of plane electromagnetic waves in free space,

we saw how Maxwell's equalions reduced (with J =-

p

=0) to

'v x E =-µ~ and 'v x H =t~

(19.l)

We can extract component expressions from these equations, and , fo r a pl.inc EM wave propagating in the i-direction , these are: ( 19.2) ( 19.3) (19.4)

Fiirther study of these equations quickly led us to t he conclusion t hat the E and H fields of a travelling wave a.re transverse to t he i -direction in which t he wave propagates, and also led us to the wave equat ion, which we are now able to solve for any of Ez , Er , Hz or Hr119

CHAPTER 19. REFLECTION OF TRAVELLING PLANE WAVES

120

• We then found, for exam ple, that the general solution to the wave equat ion for E:r gave us a wave travelling in the direction of +i (which we will now denote Er+) , and also a wave of the same velocity propagating along -Z (which we ca.n similarly ca.II Er-) , Thus the solution was

where v = 1/ ffi was the velocity of the wave, which equals the vacuum speed of light, c, in free space. • In the commonly-encountered situation where the time-variation of the fields is sinusoidal {with frequen cy w), we used phasors and solved the Helmholtz equation to arrive at the similar solution for E:r:

where (19.7)

• Now, one of our component equations above states that 8Er

-

a,

tJH,

=-µ-

a,

(19.8)

which , in phasor form , becomes (19.9)

so that

IF, [t,. -t,-]

(19.12)

CHAPTER 19. REFLECTION OF TRAVELLING PLANE WAVES

121

• Hence we can writeily as

where f/ = .J{µTJ is therefore the ratio of Ez {the x-oomponent of E field ) ana H" (they-component of H -field) everywhere along the wave. You can see that '1 is a constant for any given medium, and , because it has dimensions of (V /m)/(A/ m)= fl, it is known as the intrinsic impedance of the medium in which the wu·e propagates. • In particular, the impedance of free space is

"' = y;; re;_ =376.73"" 120, n

(19.14)

This is a quantity with whi ch you will frequently meet in any analysis of waves travelling in space or in similar media. • As we remarked when studying the wave equation, it is a simple procedure to extract from Maxwell 's equations the wave equation for any of the other quantities E11 , ifz or i/r. You should practi~e doi ng this. You will then find that , if you ha\'e the solut ion for Er such that

then a resubstitution into a component equat ion, similar to the one described above for Ez , will give you

ii,, = Hr++ fl,,_= - E,+ + £"_

" "

(19. 16)

CHAPTER 19. REFLECTION OF TRAVELLING PLANE WAVES

122

• Combining the results of both procedures, we can now see t hat

One imp_ortant interpretation of these equations is that they show that the total E and H fields of a t ravelling plane wave are everywhere perpendicular to one another. This is illustrated in the following diagram, which depicts an electromagnetic wave moving out of the page (in the positive Z-direction ). Once the information contained in equation 19.17 has been included in the diagram, you can sec by a similar-triangles approach that E and H must be perpendicular to each other, and aJso that the wave travels in the direction of E x H , as expected.

CHAPTER 19. REFLECTION OF TRAVELLING PLA NE WAVES

19.2

123

Incidence on a perfect conductor

• \Ve now imagine a plane wave, that has been propagating in a medium such as free space, impinging upon a perfectly-conducting surface. In the simplest case, we can visualise the wave striking the conductor so that its direction of propaga tion is normal to the conductor's surface (i.e. it is in the i-direction as drawn). Now. at the surface of the perfect conductor there can be no x- or y- components of E-field. because such a field would instantly collapse as charge carriers moved in response to it. Hence. the negatively-travelling part of the plane wave must, at the surface, possess x- and y- components of E-field which exactly balance (i.e. they are equal and opposite to) the components of the impinging, positively-travelling part of the wave. Under these circumstances, we would refer to the negativelytravelling part of the wave as a re/frction; and, since its components are f'qual in magnitude but opposite in direction to those of the incident wave. it is clear that the reflected wave is very similar to the incident wave, but that it moves away from the conducting surface, as expected. • Next imagine a plane wave impinging upon the conductor at some angle 9 to the normal line as shown here. Again , the x- and y• components of E-field at the conductor's surface must each total zero, and so we expect there to be a reflected t ravelling plane wave, which is drawn here at an angle of 9' to the normal.

,.

• In this example, we have drawn the ff -field coming out of the paper. and the E-field lies in the plain of incidence. which is ~he plane containing both tht> direction of propagation and t he normal line (it is the plane of the paper a.s drawn here). This geometry is referred to as transverse magnetic, or parallel, po/ari.satio11.

CH,\PTER 19. REFLECTION OF TRAVELLING PLANE H-'AVES

124

I\ote how the diagram preserves the perpendicularity of E and H , and note also the fa.ct that E x H gives the direction of propagation for both the inci dent and the reflected. wa\'es. • Now, the directions of the incident and reflected waves have been denoted ( and (-)(' respectively, so we can write the whole E-field of the (19.18)

The diagram also shows how we can write ( and (' in terms of x and z as follows:

(=::r:sin 8+zcos8 and ('=-::r:sin6'+ zcos 8'

(19.19)

Hence, the whole E-field of the wave may be given as

Just considering the x•component of E-field , and still with reference to the diagram. you can easily s= that taking x-•componenls gives £.,(x,..:: )

= -E+cosoe-i ktroi111+ ..:ool) + £_ cos9'dkl - zsinl'+•cod'J (19.21)

• At 1l1e bou11dary witli the perfect conductor z = 0, and we know that Er must equal zero as well. So, setting E., (x , O) = 0 gives us

E+ cosoe-i hoi.a f

= £_ cosO'e- i boinf'

{19.22)

CHAPTER 19. REfLECTTON OF TRAVEUING PLA N£ 1.\r'AVES

125

The only way t hat t he complex numbers on both sides of this equality can be equal for nll vnlues of x (i.e. anywhere along the surface of the conduc tor) is if they are identical in both magnitude and phase. We conclude, tl1erefore, that hi11 IJ = bin e'

so

B = 8'

and

E+ = L

(19.23 )

and so we see that the reflection at the perfect conducting surface is lossless, and that t he angle of reflection to the normal is equal to the angle of incidence. • You should not fin d it difficult to corn~ 1o the sarne conclusion in studying t his second diagram , in which the incident and reflected waves both have their E -fields d irected straight into the pa.per (perpendicular to the plane of incidence). This is referred to a.s transverse e/edric, or perpendicular, polarisation. Since any other orient ation of t he E-field may be regarded simply as a superposition of parallel and perpendicular pol arisations, we know tha.t t he important conclusions that reflection is symmetric and lossless wi ll be true, irrespect ive of what linear polarisation the incident wave has .

CHAPTER 19- REFLECTION OF TRAVELLING PLANE \-\:AVES

19.3

126

Incidence on a die lectric boundary

• Unlike at the boundary of a perfect conductor, across which a p lane wavf' cannot transmit energy, a travelling wave is not so constrained at the boundary bet.ween two dielectric media. At such a boundary, we know from our early consideration of boundary conditions that the t an· gential components of the E and H fields are unchanged by the change of medium. Furthermore, there is some trc\flsfer of energy from the first medium to the second as the wave crosses the boundary; and yet not all of the incident energy is transmitted hecaui;e t he normal components of E and H field are not the Mame in both dielectrics. Hence, we expect to see both transmission (known as refraction) and also reflection in such a situation.

• Followi11g our approach at the perfect conductor boundary, and with reference to the diagram, we can write an equation for t he x-components of E-field at a point (x, O) anywhere on the dielectric boundary:

'·

,, c,

..

Since the boundary condition diet.ates t hat ( 19.26) since the previous expression must be t rue irrespective of the value of x (i.e. everywhere on the boundary) , we can conclude, si milarly to t he case of the perfect conductor, that t he following terms, known as phast factors. must be C