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EDITION
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ELECTROMAGNETIC
FIELDS
ROALD K.WANGSNESS
PROFESSOR
OF
PHYSICS
UNIVERSITY
OF
ARIZONA
JOHN WILEY
&
SONS
A
NEW
YORK CHICHESTER BRISBANE TORONTO SINGAPORE
O A /y
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V J-Sh
— V> f~!^ / X J o 7 /
2 N D EDITION
* W B 0 2 0 9 9 1 *
ELECTROMAGNETIC
FIELDS
ROALD K.WANGSNESS
P R O F E S S O R O F PHYSICS
U N I V E R S I T Y O F A R I Z O N A
JOHN WILEY & SONS A
N E W Y O R K C H I C H E S T E R BRISBANE T O R O N T O S I N G A P O R E
O A / y
Copyright © 1979, 1986, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons.
Library of Congress Cataloging in Publication Data:
Wangsness, Roald K. Electromagnetic fields. Includes indexes.
1. Electromagnetic fields. I. Title. QC665.E4W36 1986 537 85- ISBN 0-471-81186- Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
VECTOR OPERATIONS
R E C T A N G U L A R C O O R D I N A T E S
du du du V « = x - — I - y — + z—~ (1-37) dx dy dz dAx dAv dA. V - A = — + (1-42) dx ay dz
IdA. dAv\ I dAx dA.\ ldAv dAA ^ ^ ^ ~ - j r ) +^ A - j t - ^ } +^ " { - i t ( ,^ - 4 3 )
d^2 u d^2 u d^2 u V 2 w^ =^ ^ +^ ^ +^ ^ T (1-46) dx dy^2 dz^1
C Y L I N D R I C A L C O O R D I N A T E S
du 1 du du J I 1 dA.
V K - P ^ - + Cp- — dp p d(p
1 d^ ,. •- V • A = (pAp) + - p dp P d<p
J l d A (^) t dAA V X P (^) ( p dtp dz "
dA: dz
I dA, <p; dz
V z^ u = -
i a / 3h\ i d^2 u d^2 u
p dp^ dp) p^2 dcp^2 dz^2
dA. dp
+^ i^ A 2^1 9 / P d p ^ ^
( 1 - 8 7 )
1 ^ P "
p d<p ( 1 - 8 8 )
( 1 - 8 9 )
S P H E R I C A L C O O R D I N A T E S
dw „ 1 du^1 <9* v (^) " = f (^) a 7 + e (^7) ^ <P r sin 6 dtp I d , 1 V • A = — —(r^2 Ar) + ——j-{smdAd) r~dr r sin 6 do
V X A =
r sin 6
¥
d , (^) x d A (^) e <P'
1 dA,
dArA^ dd
_1 d I du_ 1
r sin 6 dtp
sinf? dtp *
d I du \ 1 02 u
r^2 s'm8 38 \ 30) r^2 s\n^28 3<p^2
PREFACE
This is a textbook for use in the usual year-long course in electromagnetism at an intermediate level given for advanced undergraduates. I have done my best to make it as student oriented as possible by writing it in a systematic and straightforward way without any sleight of hand, and minimum use of " I t can be shown t h a t.... " I have also tried to make clear the motivations for each step in a derivation or each new concept as it is introduced. At appropriate points I have pointed out the source of many of the simple mistakes commonly made by students and have suggested how they may be prevented. There is extensive cross referencing in the text so that there should be n o doubt as to the detailed source of any specific result or its relation to the rest of the subject; this will also make the book much more useful as a reference source well after the course has been completed. In preparing this revision, I have concentrated on adhering to these goals while trying to improve the clarity and organization of the material, since my aim is to produce a useful text with an adequate, rather than encyclopedic, coverage at a reasonable mathematical level. Consequently, most of the changes are scattered throughout the text. For example, I have improved the discussion of electric and magnetic energies in the presence of matter in Sections 10-8, 10-9, and 20-6. In addition, the beginning treatment of radiation has been simplified somewhat in Sections 28-1 and 28-2, and the proof about the electric field in a cavity in a conductor has been corrected in Section 6-1. The principal and most noticeable change has been the addition of a completely new Chapter 27 on Circuits and Transmission Lines. I hope that this will turn out to be useful and satisfactory to those who have expressed a desire to have this material included. The emphasis continues to be on the properties and sources of the field vectors, and I hope that I have succeeded in making clear the shifts in concepts and points of view that are involved in the change from action at a distance to fields. The overall treatment is generally that of a macroscopic and empirical description of phenomena, although the microscopic point of view is presented in the discussion of conductivity in Sections 12-5 and 24-8. However, Appendix B briefly surveys the microscopic origins of electromagnetic properties, and it is written and organized so that, if desired, it can be taken up section by section at an appropriate intermediate point. Thus Section B-l could be discussed anytime after Section 10-7, and most of Section B-2 can follow after Section 20-5 while the last part on ferromagnetism can follow Section 20-7; finally, Section B-3 could be covered after Section 24-8 has been mastered and the student has worked Exercise 24-28. Similarly, even more flexibility is possible since separate sections of Appendix A that deals with the motion of charged particles can be studied anytime after the corresponding force term involving E a n d / o r B has been obtained. SI units are used throughout; in practice, this means we use MKSA units. It is virtually certain, however, that at some time a student will encounter material in Gaussian units and will need some guidance on what to do about it. This is the purpose of Chapter 23 in which other unit systems are discussed, but only after the general theory as given by Maxwell's equations has been systematically described. I have written this chapter primarily in terms of the purely practical aspects of how to
v i i
CONTENTS
I N T R O D U C T I O N
1 VECTORS (^) Us 3
[ O
O
IT3I
EH]
i m
ll-ll Definition of a Vector 3 Addition 4 Unit Vectors 5 Components 5 The Position Vector 7 Scalar Product 8 Vector Product 9 Differentiation with Respect to a Scalar 11 |1-9| Gradient of a Scalar 12 I1-10I Other Differential Operations 14 II-111 The Line Integral 15 I1-12I Vector Element of Area 17 I1-13I The Surface Integral 20 H-14I The Divergence Theorem 21 II -151 Stokes' Theorem 24 H-161 Cylindrical Coordinates 28 11-171 Spherical Coordinates 31 11-181 Some Vector Relationships 34 11-191 Functions of the Relative Coordinates 35 11-201 The Helmholtz Theorem 37
2 C O U L O M B ' S LAW 401 IE ixercises 12^1 40
F4l
Point Charges Coulomb's Law 41 Systems of Point Charges Continuous Distributions of Charge 44 Point Charge Outside a Uniform Spherical Charge Distribution 46
43
3 THE ELECTRIC FIELD 51 pT> Definition of the Electric Field 51
E 3 Field of^ a Uniform Infinite Line Charge 52 Field of a Uniform Infinite Plane Sheet 53 What Does All of This Mean? 55
4 G A U S S ' LAW 58 [E ixercises 14-11 Derivation of Gauss' Law 58 H ] Some Applications of Gauss' Law 60 |4-3| Direct Calculation of V • E 65
5 THE SCALAR POTENTIAL 68 1 ixercises
5-
M l Definition and Properties of the Scalar Potential 68 Uniform Spherical Charge Distribution 72 Uniform Line Charge Distribution 73 15-41 The Scalar Potential and Energy 79
C O N D U C T O R S IN ELECTROSTATIC FIELDS 8 3 [Hi l£H
E l
6-
Some General Results 83 Systems of Conductors 88 Capacitance 90
7 ELECTROSTATIC ENERGY 98 1 ixercises o
r n i
Energy of a System of Charges 98 Energy of a System of Conductors 100 Energy in Terms of the Electric Field 101 Electrostatic Forces on Conductors 103 IX
X C O N T E N T S
8 ELECTRIC MULTIPOLES 1 1 0 |Exercisesl The Multipole Expansion of the Scalar Potential 110 The Electric Dipole Field 119 The Linear Quadrupole Field 121 Energy of a Charge Distribution in an External Field 123
111-51 Separation of Variables in Spherical Coordinates 190 Hl-61 Spherically Symmetric Solution of Poisson's Equation 198
1 2 ELECTRIC CURRENTS 202 IKxercisesl
B O U N D A R Y C O N D I T I O N S AT A SURFACE OF DISCONTINUITY 1 3 2
[Exercises]
19^
9-
Origin of a Surface of Discontinuity 132 The Divergence and the Normal Components 133 The Curl and the Tangential Components 134 Boundary Conditions for the Electric Field 136 Boundary Conditions for the Scalar Potential 138
m i
ITT2l e m i E S S 0 3
1 2 - 6
Current and Current Densities 202 The Equation of Continuity Conduction Currents 207 Energy Relations 211 A Microscopic Point of View 212 The Attainment of Electrostatic Equilibrium 213
205
1 3 AMPERE'S LAW (^217) 1 xercises E ]
O H
1 0 ELECTROSTATICS IN THE PRESENCE OF MATTER 140
f i T 3 l Ifcxercisesl
The Force between Two Complete Circuits 217 Two Infinitely Long Parallel Currents 220 The Force between Current Elements 222
m
Em]
E D
E M ]
EM]
Em]
_m _
E m
Polarization 140 Bound Charge Densities 142 The Electric Field within a Dielectric 145 Uniformly Polarized Sphere The D Field 151 Classification of Dielectrics Linear Isotropic Homogeneous (l.i.h.) Dielectrics 156 Energy 161 Forces 165
1 4 THE M A G N E T I C I N D U C T I O N^225 [&
E 1 3
148
154
14-
14-
14-
Definition of the Magnetic Induction 225 Straight Current of Finite Length 227 Axial Induction of a Circular Current 229 Infinite Plane Uniform Current Sheet 231 Moving Point Charges 233
11 SPECIAL M E T H O D S IN ELECTROSTATICS 171 Exercises 111-11 Uniqueness of the Solution of Laplace's Equation 171 I11 -2I Method of Images 173 I11-3I " Remembrance of Things Past" 183 HL-4| Separation of Variables in Rectangular Coordinates 185
1 5 THE INTEGRAL FORM O F AMPERE'S LAW 2 3 7 15-
15-
15-
Exercises
Derivation of the Integral Form 237 Some Applications of the Integral Form 242 Direct Calculation of V X B 248
x i i C O N T E N T S
25-2 E Perpendicular to the Plane of Incidence 411 25-31 E Parallel to the Plane of Incidence 415 I25-4I Total Reflection (n 1 >n 2 ,6i>9e) 418 25-5 Energy Relations 420 I25-6I Reflection at the Surface of a Conductor 421 25-71 Continuously Varying Index of Refraction 424 |25-8| Radiation Pressure 425
2 6 FIELDS IN B O U N D E D R E G I O N S 4 3 0
I Exercises]
2 6 - 1
2 6 - 2
26- m m 26-5] [26^
Boundary Conditions at the Surface of a Perfect Conductor 430 Propagation Characteristics of Wave Guides 431 Fields in a Wave Guide Rectangular Guide 435 TEM Waves 441 Resonant Cavities 444
433
2 7 CIRCUITS A N D TRANSMISSION LINES 4 4 9 ll '.xercisesl 27-3 Kirchhoff's Laws 449 I27-2I The Series RLC Circuit I27-3I More Complicated Situations 457 |27-4| Transmission Lines 461
453
2 8 R A D I A T I O N 4 6 9 (^) Exercises
H y ] [2^
28-
Retarded Potentials 469 Multipole Expansion for Harmonically Oscillating Sources 472 Electric Dipole Radiation
28- HEU
128^
Magnetic Dipole Radiation Linear Electric Quadrupole Radiation 484 Antennas 487
482
2 9 SPECIAL RELATIVITY 4 9 4 | Exercises I 29-
129^
HOI
[293]
29-
Historical Origins of Special Relativity 494 The Postulates and the Lorentz Transformation 499 General Lorentz Transformations, 4-Vectors, and Tensors 507 Particle Mechanics 514 Electromagnetism in Vacuum 518 Fields of a Uniformly Moving Point Charge 523
M O T I O N O F C H A R G E D PARTICLES 5 3 0
lExercisesl
IA-1I Static Electric Field 530 |A-2| Static Magnetic Field 531 |A-3l Static Electric and Magnetic Fields 538 A Time-Dependent Magnetic Field 543
B ELECTROMAGNETIC PROPERTIES OF MATTER 5 4 6 lExercises] B-l Static Electric Properties B-2 Static Magnetic Properties lB-31 Response to Time-Varying Fields 562
546 554
A N S W E R S T O O D D - N U M B E R E D EXERCISES 5 6 9
(^477) INDEX 5 7 7
INTRODUCTION
... Faraday, in his m i n d ' s eye, s a w lines of f o r c e traversing all s p a c e w h e r e t h e m a t h e m a t i c i a n s s a w c e n t r e s of f o r c e a t t r a c t i n g at a d i s t a n c e : Faraday s a w a m e d i u m w h e r e t h e y s a w n o t h i n g b u t d i s t a n c e : Faraday s o u g h t t h e seat of t h e p h e n o m e n a in real a c t i o n s g o i n g o n in t h e m e d i u m , t h e y w e r e s a t i s f i e d that t h e y had f o u n d it in a p o w e r of a c t i o n at a d i s t a n c e i m p r e s s e d o n t h e electric fluids. — J. C. Maxwell, A Treatise on Electricity and Magnetism
It has been more than 100 years since Maxwell wrote the above in the preface to his now-famous book. His aim was to put the field concepts, which Faraday had been so instrumental in developing, into mathematical forms that would be convenient to use and would emphasize the fields as basic to a coherent description of electromagnetic effects. At that time, it had been only slightly more than 50 years since Oersted and Ampere had shown the relation between electricity and magnetism—subjects that had been studied and developed completely separately over a long period. The emphasis had been primarily on the forces exerted between electric charges and between electric currents and the idea of shifting to electric and magnetic fields as the primary features had little acceptance and was often, in fact, viewed with outright hostility. As the title of this book indicates, times have changed, and our main interest here is the study of the nature, properties, and origins of electromagnetic fields, that is, of electric and magnetic vector quantities that are defined as functions of time and of position in space. Forces, and associated concepts such as energy, have not disappeared from the subject, of course, and it is desirable to begin with forces and to define the field vectors in terms of them. Nevertheless, our principal aim is to express our descriptions of phenomena in terms of fields in as complete a manner as we possibly can. This emphasis on fields has proved to be extremely rewarding and it is difficult to imagine how electromagnetic theory could have been developed to its present state without it. This book contains more material than is normally covered in the usual one-year course; all of it, however, is of interest and value to a serious student of physics.
The points of view of all authors are generally not the same, and no book discusses every detail of a given subject. Here is a short list of relatively recent books on electromagnetism that are written at roughly the same level as this one.
D. M. Cook, The Theory of the Electromagnetic Field, Prentice-Hall, Englewood Cliffs, N.J., 1975. P. Lorrain and D. R. Corson, Electromagnetic Fields and Waves, Second Edition, Freeman, San Francisco, 1970. M. H. Nayfeh and M. K. Brussel, Electricity and Magnetism, Wiley, New York,
1
1
VECTORS
In the study of electricity and magnetism, we are constantly dealing with quantities that need to be described in terms of their directions as well as their magnitudes. Such quantities are called vectors and it is well to consider their properties in general before we meet specific examples. Using the notation and terminology that has been developed for this purpose enables us to state our results more compactly and to understand their basic physical significance more easily.
1-1 I DEFINITION OF A VECTOR
The properties of the displacement of a point provide us the essentials required for our definition. If we start at some point Pl and move in some arbitrary way to another point P 2 , we see from Figure 1-1 that the net effect of the motion is the same as if the point were moved directly along the straight line D from Pl to P 2 as indicated by the direction of the arrow. This line D is called the displacement and is characterized by both a magnitude (its length) and a direction (from Pl to P 2 ). If we now further displace our point along E from P 2 to still another point P 3 , we see from Figure 1- that the new net effect is the same as if the point had been given the single displacement along F from P 1 to P 3. Accordingly, we can speak of F as the resultant, or sum, of the successive displacements D and E, so that Figure 1-2 shows the fundamental way in which displacements are combined or added to obtain their resultant. A vector is a generalization of these considerations in that it is defined as any quantity that has the same mathematical properties as the displacement of a point. Thus we see that a vector has a magnitude; it has a direction; and the addition of two vectors of the same intrinsic nature follows the basic rule illustrated in Figure 1-2. Because of the first two properties, we can represent a vector by a directed line such as those already used for displacements. A vector is generally printed in boldface type, thus, A; its magnitude will be represented by |A| or by^4. A scalar is a quantity that has magnitude only. For example, the mass of a body is a scalar, whereas its weight, which is the gravitational force acting on the body, is a vector. Because of the nature of a vector as a directed quantity, it follows that a parallel displacement of a vector does not alter it, or, in other words, two vectors are equal if they have the same magnitude and direction. This is illustrated in Figure 1-3 where we see that A = A'. Now we can investigate what mathematical operations we can perform with and on vectors.
3
4 VECTORS
A D D I T I O N
According to our basic rule we find that, if we take A and add B, we obtain the sum C shown as a solid line in Figure 1-4. We also see that, if we take B and then add A, we get the same vector C. Therefore addition of vectors has the commutative property that
C = A + B = B + A (1-1)
By proceeding in the same manner, one can establish the associative property of vector addition:
D = (A + B) + C = A + ( B + C) = (A + C) + B (1-2)
and so on. If we reverse a displacement such as D in Figure 1-1 by retracing it in the opposite direction, the net effect is then no displacement; hence it is appropriate to define the negative of a vector as a vector of the same magnitude but reversed in direction, for then we should obtain A + ( — A) = 0, as we would want. Then we can easily subtract a vector by adding its negative:
A — B = A + ( — B ) (1 -3)
Figure 1-4. The sum of t w o vectors does not depend on the order in which they are added.
6 VECTORS
y (^) Figure 1-7. A is the sum of the rectangular vector components.
we write Ax = Axx, and so on, and the above expression becomes A = Axx + Ay y + Azi (1-5)
The three scalars Ax, Ay, Az are called the components of A; hence we see that a vector can be specified by three numbers. The components can be positive or negative; for example, if Ax were negative, then the vector A (^) x of Figure 1-7 would have a direction in the sense of decreasing values of x. From Figure 1-7, it is seen that the magnitude of a vector can be expressed in terms of its components as
A = |A| = (Al + Al + A])
1 / 2 (1-6)
In Figure 1-8, we illustrate the fact that A makes specific angles with respect to each of the axes; these angles a, fi, y are called the direction angles of A and are measured from the positive directions of their respective axes. Figure 1-9 shows the plane containing both A and x and we see that Ax is given by Ax = A cos a. Combining this with (1-6), we get
I = cos a = (A^2 X + A^2 y + A^2 z)
where Ix is called a direction cosine. Similar expressions hold for the other two direction angles /? and y and their associated direction cosines ly and lz , so we see from (1-6) and (1-7) that, if we know the rectangular components of a vector, we can calculate its magnitude and direction.
y
Figure 1-8. Definition of direction angles.
1-5 THE POSITION VECTOR 7
xL Figure 1-9. Ax is the x component of A.
If we now combine (1-4), (1-5), and (1-7), we find that the unit vector a can also be written as
a = lxx + ly y + lzz (1-8)
so that the components of a unit vector in a given direction are simply the direction cosines associated with the direction. If we now apply the general result (1-6) to the specific vector a, we get the important relation involving direction cosines
ll + I) + l] = 1 (1-9)
which can also be obtained from (1-7) and its analogues. The addition of vectors illustrated in Figure 1-4 is easily expressed in terms of the rectangular components. From Figure 1-10, we see that a component of the sum C = A + B is given by the sum of the corresponding components, that is, CX = AX + Bx Cy = Ay + By CZ = AZ + Bz (1-10)
THE POSITION VECTOR We now consider a simple specific example of a vector. As shown in Figure 1-11, the location of a particular point P in space can be specified by the vector r drawn from the origin of a suitably and conveniently chosen coordinate system; this vector r is called the position vector of the point. In terms of the rectangular coordinate system of Figure 1-6, the components of r are just the rectangular coordinates ( jc , y, z) of the point; thus we have
r = x x + yy + zz ( l - l l )
Similarly, another point P' with coordinates (x', y', z') will be located by its position vector r' = x'x + y'y + z'z as shown in Figure 1-12. Now that we have located the two points individually, we can describe the position of one with respect to the other by drawing a vector from P' to _P_ this vector R is called the relative position vector of P
1-7 VECTOR PRODUCT 9
— — F i g u r e 1-13. The angle involved in the scalar product.
We see from Figure 1-13 that we can get a simple interpretation of the scalar product: (2? cos 4^) ,4 = component of B along the direction of A times the magnitude of A = (A cos <k)B = component of A along B times the magnitude of B. It is clear from (1-15) that the order of terms does not change the scalar product, that is, A B = B A (1-16) and that if two vectors are perpendicular then A • B = 0 and conversely. Furthermore, the square of a vector can be interpreted as the vector dotted with itself; the result is the square of its magnitude and we can write A^2 = A • A = A^2 (1-17) If we knew only the rectangular components of A and B, it would be inconvenient to calculate A • B from (1-15) since this would necessitate finding the angle between A and B. Fortunately, it is possible to express A • B directly in terms of the rectangular components. Since the angle between each pair of unit vectors defined in Figure 1-6 is 90°, we easily find from (1-15) that x - y = y - z = z - x = 0 (1-18)
and from (1-17) that x - x = y - y = z - z = l (1-19) Writing A and B each in the form (1-5), we can multiply them together term by term to get A B = ( A (^) x x + Ay y + Az z) • (Bxx + By y + Bz z)
_= AxBx • X + AxBy • y + AxBA • Z +..._**
and, after we use (1-18) and (1-19) to simplify the resulting nine terms, we find that A B = AXBX + AyBy 4- AZBZ (1-20)
Suppose now that e is a unit vector in some specific direction. If we let Ae be the component of A in this direction, we see from (1-15) that it can be found from Ae = A • e (1-21)
VECTOR PRODUCT
This is also called the cross product and is written A X B. It is a vector perpendicular to both A and B and its magnitude is defined as | A X B| = AB s i n ^ (1-22) Its direction is given by the following right-hand rule: if the fingers of the right hand are curled in the sense necessary to rotate A through the smaller angle into coincidence with B, the thumb points in the direction of A X B. This rule is illustrated in Figure 1-14.
1 0 VECTORS
A x B
A s i n \ Figure 1-15. Interpretation of the magnitude of a .A cross product as an area.
If we look at the plane containing A and B shown in Figure 1-15, we can get a simple interpretation of the cross product. We see f r o m the figure and (1-22) that the magnitude of the cross product is equal to the area of the parallelogram with A and B as sides. F r o m the definition of the direction of a cross product shown in Figure 1-14, it is evident that the order of terms is important, since it is seen that B x A = - ( A X B) (1-23) If A and B are parallel, then it follows f r o m (1-22) that A X B = 0, and conversely. In particular, A X A = 0 (1-24) For the unit vectors along the axes shown in Figure 1-6, if we use (1-22), the right-hand rule, the facts that they are mutually perpendicular, and that the cross product is perpendicular to both vectors, we find that x X y = z y X z = x z X x = y (1-25) and, f r o m (1-24), that x X x = y X y = z X z = 0 (1-26)
The vector product can also be conveniently written in terms of the rectangular components. Proceeding similarly to what we did in obtaining (1-20), we write A and B in the f o r m (1-5), multiply them together term by term, and use (1-23), (1-25), and (1-26) to simplify the results. We find that
A X B = (AyBz - AzBy)x + (AZBX - AxBz)y + (AxBy - AyBx)z (1-27)
This can be written as an easily remembered determinant
X^ A yA z a
A X B = Ay Bx By Bz