Introduction to Modern Electromagnetics [1 ed.] 0070183880, 9780070183889

Citation preview

numerical constants Velocity of light (o Pem1ittivity of free space µ. 0 Permeability of free space -11-o/lo - Intrinsic impedance of free space e Charge of an electron m Mass of an electron e/m Electronic charge-to-nlass ratio ~,f Mass of a proton c

v

2.998 X 108 m/sec 8.854 X 10-1 2 farad/rn 4rr X 10- 7 henry /1n 376.7 ohn1s - 1.6008 X 10- H 1 coul 9.1066 X 10 - 31 kg - 1.7578 X 101 1 coul/kg 1.679,5 X 10- 21 kg

introduction to modern electromagnetics

introduction to modern electromagnetics Carl H. Durney Associate Professor of Electrical Engineering University of Utah

Curtis C. Johnson Associate Professor of Electrical Engineering University of lVashinglon

l'vlcGraw-Hill Book Company San Francisco St. L-Ouis New York Panama J.fexico Toronto Sydney

London

introduction to modern electromagnetics Copyright @ 1969 by l\lcGraw-Hill, Inc. All rights reserved. Printed in the United States of America. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. L ibrary of Congress Catalog Card Number 69-13605

ISBN 07-018388-0 567890 MAM ivl 765432

To Marie To Katharine and lVilma

preface

Electromagnetics continues to be 0ne of the most fundamental subjects in an engineering curriculum because it provides a basic physical and mathematical understanding of electric, magnetic, and propagation phenomena. The principles of electromagnetics have been applied in an increasing number of research areas in recent years, placing an emphasis on soundness, breadth, and the dynamic aspects of an undergraduate course in electromagnetic field theory. For example, the microwave interest ,vith its base in devices, antennas, and propagation, has moved into microwave phenomena in solid-state materials and gaseous plasmas, where statistical and quantum-mechanical effects are prominent. Interest in the optical spectrum has grown rapidly in step with laser advances into new material interactions, acoustic interactions, propagation, and nonlinear effects. These new research and development interests require a concise and modern undergraduate introduction to electromagnetics. A survey was conducted recently to determine the present-day needs for an electro magnetics textbook. The results indicated a need for mathematical and physical clarity, the use of examples and applicat ions, and a problem-solutions manual to reduce the time required by the instructor. This textbook was prepared ·w ith these objectives in mind. vii

viii

pt'eface

The book is designed for a first course in engineering electromagnetics at t he junior or senior level. There is enough material for a full year's course, and some of the more difficult sections, which have been marked with a star at t heir beginnings and endings, are optional a nd can be omitted with no loss of continuity. This makes the text somewhat adaptable to courses of varying difficulty and length. Problems at the end of each chapter are numbered according to the section to which they are pertinent. Thus Problem 1.5-1 is the first problem of Section 1.5. A number of problems are designed to be solved by computer. We have assumed that students using this book have been introduced to vector analysis. Although an extensive review of vector analysis is given Hl Chapter 1 and the text is mathematically self-contained, the reader who has not had some previous experience with vector analysis will have aifficulty learning the mathematics fast enough. Chapter 1 is meant to be a complete but compact discussion of vector analysis. Since many students have had little experience with cylindrical a nd spherical coordinate systems, these a re discussed in some detail. Physical interpretations of operations such as the divergence and curl are emphasized. We have tried to make tile mathematical level high enough to prepare the student for graduate courses in electromagnetism, while at the same time emphasizing concepts and physical understanding. Our approach differs principally from some of the more traditional a pproaches in t ha.t dy namics is emphasiz-ed over statics. Since fiel d concepts a re more easily introduced in terms of statics, the basic electrostatic a nd magneto&tatic concepts are developed first in Chapter 2. The extension to time-varying fields is made, and :vraxwell's equations are developed in Chapter 3. The remainder of the text is &voted to understanding, solving, and applying 1\1axwell's equations. The rationalized mks system of units is used throttghout t he text. IR Chapter 4 energy and power concepts are discussed. This is followed in Chapter 5 by a description of the interaction of electron1agnetic fields with charges in n1at-erials. The purpose of C hapter 5 is to show in a detailed way how the interaction between t he charges in materials and electromagnetic field s can he described in terms of permittivity and permeabili ty. Since the interaction is easier to understand in the case of dielectrics and conductors t ha n magnetic materials, dielectrics and conducton, are emphasized; th€y a re treated from a common ba-s-e and in tern1s of ti me-varying fields. ChaJnier 6 contai ns a discussion of the wave equations, potentials, and planewaves . This chapter presente t he htndameutal concepts of wave propagation. Refraction a nd reflection at di,electrie bo~ndaries, reflectien at conducting b6undaries, skin effect, radiation fram antennas,

preface

ix

and propagation between parallel conducting plates are some of the topics considered. Some examples involving laser beams are given. The connection between field theory and circuit theory is discussed in Chapter 7. The quasi-static approxin1ation is used to show how Kirchhoff's laws are related to l\,faxwell's equations. This helps to show the connection between planewave propagation, described in Chapter 6, and the wave propagation on two-conductor systems, described in Chapter 8. Transmission-line concepts such as impedance, reflection coefficient, and standing-wave ratio are discussed in Chapter 8. Also included are the Smith chart and optional sections on tapered transmission lines and transients on transmission lines. The last chapter is on boundary-value problerns. In this chapter mathematical techniques for solving the wave equations are discussed. Propagation in wave guides is treated as a boundary-value problem. An optional section describes how a partial separation of variables in 1\1axwell's equations leads to a description of wave-guide propagation in terms of transmission-line equations. Then wave-guide propagation is discussed in terms of transmission-line concepts, including the use of the Smith chart. The development of this text was influenced by many sources. We are grateful to the students who worked through the text when it was in the form of course notes and offered corrections and valuable suggestions. Our colleague, Professor Om P. Gandhi, and several reviewers have also offered many valuable suggestions. We appreciate the excellent work of Mrs. l\1arian Swenson and her staff in typing and reproducing the manuscript through several revisions. Carl H. Durney Curtis C. John son

contents

-preface

vii

CHAPTER 1 MATHEMATICAL INTRODUCTION

1.1 1.2

1.3 1.4

1.5

Introduction 1 Review of Basic Properties of Vectors 2 Vector Addition and S ubtraction 3 Vector Products 4 Unit Vectors 5 Scalar and Vector Fields 5 Coordinate Systems 7 R ectangular Coordinate S ystem 8 Cylindrical Coordinate System 11 Spherical Coordinate S ystem 13 Addition and Multiplication of Vectors Represented in the Three Standard Coordinate Systems 15 Generalized Orthogonal Curvilinear Coordinates 19 xi

xii

contents

19 1.6 Transformations between Coordinate Systems 19 R ectangular-w-C ylindrical Trans!orm,ation 23 Rectangula-r-to-S pherical Tran s!ormation 24 Tran sforrnations fro m Another Point of View 1.7 Differentiation of Vectors 25 1.8 Integrat ion of Vectors 29 Line I ntcgrals 29 Surface I ntegrals 35 Volurne Integrals 39 1.9 Gradient 39 1.10 Divergence 41 1.11 Curl 45 *l.1 2 Derivation of an Expression for t he Curl of a Vector 1.13 The D ivergence Theoren1 (Gauss's Theorem) 53 1. 14 Stokes's Theorem 55 1.15 Other Operations I nvolving V 56 1.16 Complex Vector Notation 59 1.17 Source-point and Field-point Notation 61 1. 18 The D elta F unction 64

48

*

CHAPTER 2 ELECTROMAGNETIC FIELDS

2.1 Introduc tion 71 2.2 The Electric Field i3 2.3 Coulomb's Law 76 2.4 E lectric P otential 84 2.5 The :tviagnetic Field 87 Current R elations 89 2.6 Ampere's Law of Force 95 2.7 Magnetic Field-generating Systems 99 * 2.8 The l\1agnetic Dipole 101 2.9 R eview of the F orce and Field Relations

105

CHAPTER 3 ELECTROMAGNETIC FIELD LAWS

3.1 3.2 3.3 3.4

3.5 3.6 3.7 * 3.8

In t roduction 108 Gauss's Law 109 Applications of Gauss's Law 111 The lvlagnetic Source Law 114 Faraday's Law 115 A Stationary Loop in a Time-varying Magnetic Field 117 A ~1oving Loop in a Constant Magnetic Field 119 A Moving Loop in a Time-varying Magnetic F ield 122

contents

xiii

3. 9 3.10 *3.11

The Continuity Equation 125 Ampere's Circuital Law 126 Derivation of Ampere's Circuital Law from the Biot-Sa.vart Law 129 3.12 Displacement Current and the General Time-varying Form of Ampere's Circuital Law 131 Displacement Current 133 3.13 Maxwell's Equations 135

CHAPTER 4 PHYSICAL PROPERTIES OF FIELDS

Introduction 140 4.2 Field Force on Charges 141 4.3 Energy As.5ociated with an Assen1bly of Electric Charges 4.4 Power-Energy Conservation Theories 149 Complex Power and Energy 153 *4.5 The Electric Force Field 156 *4.6 Derivation of the Electric Surface-force-Density R elation *4.7 The Magnetic Force Field 163 *4.8 Derivation of the Magnetic Surface-force-Density Relation 4.1

CHAPTER 5 ELECTROMAGNETIC FIELDS AND MATERIALS

5.1 Introduction 171 5.2 Interaction of Charges and Fields 172 5.3 Effects of Fields on Nonmagnetic Materials 175 Polarization of Bound Charges · 175 Drift of Conduction Charges 182 5.4 Effects of Nonmagnetic Materials on Fields 183 Polarization Charge Density and Current 183 Conduction Current 190 5.5 The Displacement Vector D and Permittivity E 190 5.6 Dielectrics 194 5.7 Conductors 196 5.8 Semiconductors 201 5.9 Complex Permittivity for Sinusoidal Time Variation 201 5.10 Magnetic Materials 203 M agnetization-cu,:rent Density 204 Magnetic Fiela1nten8ity Hand Permeability 208 5.11 Magnetic Scalar Potential 210 5.12 Materials and Maxwell's Equations 213 5.13 Boundary Conditions 214 N ONnal Components 214 Tangential Components 216

143

162 166

xiv

contents

Interdependence of Boundary Conditions of N orrnal and Tangential Co·mponents 217 218 Boundary Conditions for Good Dielectrics 218 Boundary Conditions for Perfect Conductors 219 Boundary C ondilions f-0r Magnetic Materials Summary of Boundary Conditions 219

5.14 5.15

Materials and Power and Energy 222 Power Dissipation in a Dielectric with Losses

224

CHAPTER 6 THE WAVE EQUATION AND PLANEWAVE PROPAGATION

6.1 6.2 6.3 6.4 * 6.5 6.6

6.7 * 6.8 6.9 6.10

6.11 6.12 6.13 6.14 6.15 6.16

6.17

• 232 Introduction One-dimensional Electromagnetic Planewaves 233 Inhomogeneous Wave Equations for the Fields 238 Potential Functions 240 Potential-function Wave Equations 242 The Lorentz Condition 244 Retarded Potentials 244 * R etarded Potentials as Solutions to the Potential Wave E_quations 245 The Electric Dipole Antenna 247 The Two-dipole Array 250 Power Radiation from the Electric Dipole Antenna 253 General Planewave Solution to the Field Homogeneous Wave Equations 256 Field Relations in a Planewave 258 Power and Energy in a Planewave 259 Planewave Reflection at a Perfect Conductor: Normal Incidence 261 Planewave Reflection at a Perfect Conductor: Oblique Incidence 264 Planewave Incidence upon a Dielectric Interface 266 Wave Propagation in Homogeneous Dielectrics 272 Effects of Conduction Current in Diilectrics 274 Wave Propagation in Conductors 275 ... Tenuous Gaseous Conductors 276 Solid-state Conductors 276 The Skin Effect 281

CHAPTER 7 CIRCUIT THEORY FROM FIELD THEORY

7.1 Introduction 286 7 .2 Quasi-static Fields

287

conte_nts

7 .3 Fundamental Circuit Relations 288 The RL Circuit 288 The RLC Circuit 290 A Generator and an External RL Circuit Lumped-circuit Theory 295 Kirchhoff's Current Law 296 7.4 Inductance 297 Mutual Inductance 298 . Inductance and Energy 300 Finite Conductors 303 7.5 Internal Impedance 305 7.6 Capacitance 308 Capacitance and Energy 313

xv

292

CHAPTER 8 TRANS MISSION OF ELECTROMAGNETIC ENERGY

8.1 8.2 8.3 8.4 8.5 8.6

8.7 8.8

8.9

8.10

* 8.11

.8.12

Introduction 316 Diffraction of Planewaves 318 The Distributed-circuit-element Transmission Line 322 The Lumped-circuit-element Transmission Line 327 Normal-mode Analysis of the Transmission-line Equations 328 Transmission-line Progagation Characteristics 332 Characteristic Impedance 333 Propagation Constant 335 Lossless-transmission-line Discontinuities and Impedances 339 Lossy-transmission-line Discontinuities and Impedances 349 The Smith Chart 351 Lossless Lines 352 Lossy Lines 360 Impedance Transformation and Matching 360 R eactive Elements 361 Matching with Shunt Elements 362 Jfat.ching with Series Elements 363 Quarter-wave Transformers 364 Half-wave Transformers 365 Nonuniform and Tapered Transmission Lines 365 368 Low-reflection Tapered Transmission Lines Exponential Impedance Taper for Matching 368 Transmission-line Transients 371 Tlie Laplace Transf or1n of tlie Transmission-line Equations 371 Quiescent Lines 378

xvi

contents

CHAPTER 9 BOUNDARY-VALUE PROBLEMS

9.1 9.2 9.3 9.4 9.5 9.6 9. 7 9.8

9.9 9.10 *9.11

9.12

*9.13

Introduction 386 A One-dimensional Boundary-Value Problem 387 Fields between Infinite Parallel Plates 389 Separation of Variables in Rectangular Coordinates 394 Orthogonality and Eigenfunction Expansions 397 Uniqueness of Solutions for Field Equations 400 Method of Images 402 The Parallel-plate Wave Guide 405 Solution of },faxwell' s Equations 405 Satisfying the Boundary Condition3 409 Parallel-plate-wave-guide Modes 410 Propagation Characteristics of the Modes 411 Field Configuration of the Modes 412 Separation of Maxwell's Equations 414 The Rectangular \Vave Guide 415 Derivation of Transmission-line Equations for \Vave Guides 420 Transmission-line Equations for T EM Modes on Guiding Structures 422 Transmission-line Equations for TE}.J ...Modes in Free Space 427 Transmission-line Equations for TE and TM Modes 428 Transverse Boundary Conditions on Hz and Ez 433 Transmission-line Characteristics of ,vave Guides 437 Characteristic Impedance and Propagation Characteristics 438 lVave-guide Transmission-line Discontinuities and Impedances 440 Transmission-line Power and Energy 445 Energy-flow Velocity (Group Velocity) 450 The k/3 Diagram 451

APPENDIX

VECTOR RELATIONS

bibliography

459

index

461

455

introduction to modern electromagnetics

CHAPTER

1 mathematical introduction

1.1 INTRODUCTION

Vector calculus is not strictly necessary to the study of electromagnetic field theory, but without it the presentation of the principles of electromagnetic theory would be awkward and cumbersome. Vector calculus not only provides compact and meaningful expression, it also provides a powerful means of comprehension. Hence the use of vector calculus in field theory has become standard. It is assumed that the reader has previously obtained a basic understanding of vector analysis. Consequently, this chapter is not meant to be a rigorous mathematical development of the principles of vector calculus. Rather, it is meant to provide a review of important principles which are used repeatedly throughout the text, and to provide a common basis and point of departure for the study of the remainder of the text. Hence application and interpretation rather than rigorous mathematical proofs are stressed. Some of the very elementary principles are included for convenience of the reader and for completeness. Since it is difficult to provide a good physical interpretation of the mathematical relations before the corresponding physical principles are 1

2

lntrod~tion to modern electromagneti-cs

chapter one

introduced, the reader may not feel at first that he has gained -a good appreciation for all the points developed in this chapter. It is hoped, however, that this chapter will provide a good foundation and that a deeper understanding and appreciation will be obtained as the relevant physical principles are encountered. 1.2

REVIEW OF BASIC PROP ERTi ES OF VECTORS

In contrast to a scalar, which is characterized by only a magnitude, a vector is characterized by both a magnitude and a direction . A familiar example of a vector quantity is velocity. It is convenient to represent a vector quantity graphically as a directed line segment, as shown in'Fig. 1.1. The length of the line segment represents the magnitude of the vector, and the arro,v represents t he direction of the vector. It is sometimes also convenient to think of t he displacement of a point as representing a vector quantity. The magnitude and direction of the displacement t hen represent the magnitude and direction of t he vector. In this book vectors are denoted by boldface type, as indicated in Fig. 1.1. The basic vector operations, which are only briefly reviewed here, are addition, subt:raction, and two kinds of multiplication- the dot and cross products. It is important to note that these operations are independent of any coordinate s~rstem which might be used to locate the vectors in space. A detailed discussion of the representation of vectors in several coordinate systems is given in, Sec. 1.4, but the basic operations are defined independently of any coordinate system. Before discussing the basic operations, we need the definition of "equal" vectors. Any two vectors which have t he same directions and the same magnitudes are equal-vectors. Because of this definition, we are free to "slide" a vector to any position in space if we do not change its direction or magnitude. For example, the vector A in Fig. 1.1 may be used to denote the velocity of a projectile. Although it is customary to locate the base of t he vector at the projectile, it is not necessary to do so. The length and direction of the vector specify the velocity, not the location of the vector in space. Thus any of t he vectors A shown in Fig. 1.2 may be used to denote the projectile velocity. In some instances, the idea of being able to slide a vector is confusing. For example, supp~se the wind velocity in some region of space is to be represented by vectors. This could be accomplished by assigning a vector to each point in space

Fig. Ll

segment.

A graphical representation of a vector: a directed line

sec. 1..Z

mathematical Introduction

3

Projecti le

Any of the vectors A shown may be used to denote the projectile velocity. Fig. 1.2

(see Sec. 1.3) . The vector at a given point would t hen represent the wind velocity at that point. In this case it would be confusing to think of sliding the vector from one point to another because it is important to associate one certain vector with a given point. In cases like these we do not usually think of sUding vectors, although we could if we were careful somehow to keep track of which vector belongs to ,vhich point. VECTOR ADDITION AND SUBTRACTION

The addition of two vectors may be conveniently defined in terms of the displacement of a point. Thus, if A represen ts the displacement from point 1 to point 2 and B the displacement from point 2 to point 3, as shown in Fig. 1.3, then C represents the displacement from point 1 to point 3. I t is obvious that C is equivalent to A + B; that is, A + B = C. The sum of two vectors may thus be found by a graphic construction . I t is easy to show by such graphical construction t hat (1.1)

A+B = B+A and

A + (B

+

(1.2)

C) = (A + B) + C

These two characteristics of vector addition are called the commutative law and the associative law of vector addition, respectively. The negative of a vector is defined as a vector with the same magnitude but opposite direction. With this definition, vector subtraction is easily defined in terms of vector addition, (1.3)

A - B = A+ ( -B)

3

Fig. 1.3

The sum of two vectors.

1

4

chapter one

introduction to modern electromagnetics

VECTOR PRODUCTS

As constrasted with a single kind of scalar mutilplication, two kinds of vector multiplication are defined. They are the dot and cross products, also called the scalar and vector products of two vectors. The dot product of t\vo vectors is a scalar quantity and is defined as A·B

=

AB cos o

(1.4)

where 8 is the smaller angle between the two vectors, as indicated in :Fig. I .4. The notation A means the magnitude of the vector A. The dot product is commutative and distributive; that is, (1.5)

A·B = B·A

A · (B

+ C)

= A·B

+ A •C

(1.6)

The cross product of two vectors, written A X B, is a vector whose magnitude is AB sin O and whose direction is perpendicular to the plane defined by A and B and is in the direction a right-hand Bcrew would travel when turned in the same direction as A turned into B through the angle 0. The graphical representation of AxB = C -

is shown in Fig. 1.5. I t is easy to sho\v from the definition that the cross product is not commutative, but that it is distributive; that is, AxB ~ BxA but

A X (B

+ C)

=A

X

B

+AX

C

Some vector identiti(:?8 involving dot and cross products of n1ore than two vectors a re listed in the Appendix.

A

The angle behvcen two vectors used in the -,, r',

z ,,..________