Theory and Computation of Electromagnetic Fields
THEORY AND
COMPUTATION OF
ELECTROMAGNETIC
FIELDS
THEORY AND
COMPUTATION OF
ELECTROMAGNETIC
FIELDS
Second Edition
JIAN-MING JIN
Department of Electrical and Computer Engineering
University of Illinois at U
ana-Champaign
Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved
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Li
ary of Congress Cataloging-in-Publication Data:
Jin, Jian-Ming, 1962- author.
Theory and computation of electromagnetic fields / Jian-Ming Jin (Department of Electrical and Compute
Engineering, University of Illinois at U
ana Champaign). – Second edition.
pages cm
Includes bibliographical references and index.
ISBN XXXXXXXXXXcloth)
1. Electromagnetic fields–Mathematics–Textbooks. I. Title.
QC665.E4J56 2015
530.14’1–dc23
XXXXXXXXXX
Cover image courtesy of ArtyFree/iStockphoto.
Typeset in 10/12pt, TimesLtStd by Spi Global, Chennai, India.
Printed in the United States of America
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1 2015
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CONTENTS
Preface xv
Acknowledgments xxi
PART I Electromagnetic Field Theory 1
1 Basic Electromagnetic Theory 3
1.1 Review of Vector Analysis, 3
1.1.1 Vector Operations and Integral Theorems, 4
1.1.2 Symbolic Vector Method, 6
1.1.3 Helmholtz Decomposition Theorem, 9
1.1.4 Green’s Theorems, 9
1.2 Maxwell’s Equations in Terms of Total Charges and Cu
ents, 11
1.2.1 Maxwell’s Equations in Integral Form, 12
1.2.2 Maxwell’s Equations in Differential Form, 17
1.2.3 Cu
ent Continuity Equation, 17
1.2.4 The Lorentz Force Law, 18
1.3 Constitutive Relations, 18
1.3.1 Electric Polarization, 19
1.3.2 Magnetization, 21
1.3.3 Electric Conduction, 22
1.3.4 Classification of Media, 23
1.4 Maxwell’s Equations in Terms of Free Charges and Cu
ents, 25
vi CONTENTS
1.5 Boundary Conditions, 27
1.6 Energy, Power, and Poynting’s Theorem, 31
1.7 Time-Harmonic Fields, 33
1.7.1 Time-Harmonic Fields, 33
1.7.2 Fourier Transforms, 35
1.7.3 Complex Power, 37
1.7.4 Complex Permittivity and Permeability, 42
References, 46
Problems, 46
2 Electromagnetic Radiation in Free Space 53
2.1 Scalar and Vector Potentials, 53
2.1.1 Static Fields, 54
2.1.2 Time-Harmonic Fields and the Lorenz Gauge Condition, 58
2.2 Solution of Vector Potentials in Free Space, 61
2.2.1 Delta Function and Green’s Function, 61
2.2.2 Green’s Function in Free Space, 62
2.2.3 Field–Source Relations in Free Space, 63
2.2.4 Why Use Auxiliary Potential Functions, 64
2.2.5 Free-Space Dyadic Green’s Functions, 66
2.3 Electromagnetic Radiation in Free Space, 69
2.3.1 Infinitesimal Electric Dipole, 69
2.3.2 Finite Electric Dipole, 72
2.3.3 Far-Field Approximation and the Sommerfeld Radiation
Condition, 73
2.3.4 Circular Cu
ent Loop and Magnetic Dipole, 76
2.4 Radiation by Surface Cu
ents and Phased A
ays, 78
2.4.1 Radiation by a Surface Cu
ent, 78
2.4.2 Radiation by a Phased A
ay, 81
References, 84
Problems, 85
3 Electromagnetic Theorems and Principles 89
3.1 Uniqueness Theorem, 90
3.2 Image Theory, 94
3.2.1 Basic Image Theory, 94
3.2.2 Half-Space Field–Source Relations, 99
3.3 Reciprocity Theorems, 101
3.3.1 General Reciprocity Theorem, 101
3.3.2 Lorentz Reciprocity Theorem, 102
3.3.3 Rayleigh–Carson Reciprocity Theorem, 103
3.4 Equivalence Principles, 107
3.4.1 Surface Equivalence Principle, 107
3.4.2 Application to Scattering by a Conducting Object, 109
3.4.3 Application to Scattering by a Dielectric Object, 114
3.4.4 Volume Equivalence Principle, 116
CONTENTS vii
3.5 Duality Principle, 120
3.6 Aperture Radiation and Scattering, 121
3.6.1 Equivalent Problems, 121
3.6.2 Babinet’s Principle, 124
3.6.3 Complementary Antennas, 127
References, 128
Problems, 129
4 Transmission Lines and Plane Waves 135
4.1 Transmission Line Theory, 135
4.1.1 Governing Differential Equations and General Solutions, 135
4.1.2 Reflection and Transmission, 138
4.1.3 Green’s Function and Eigenfunction Expansion, 140
4.2 Wave Equations and General Solutions, 144
4.2.1 Wave Equations and Solution by Separation of Variables, 144
4.2.2 Characteristics of a Plane Wave, 146
4.2.3 Wave Velocities and Attenuation, 147
4.2.4 Linear, Circular, and Elliptical Polarizations, 151
4.2.5 Wave Propagation in Metamaterials, 154
4.3 Plane Waves Generated by a Cu
ent Sheet, 156
4.4 Reflection and Transmission, 159
4.4.1 Reflection and Transmission at Normal Incidence, 159
4.4.2 Reflection and Transmission at Oblique Incidence, 161
4.4.3 Total Transmission and Total Reflection, 164
4.4.4 Transmission into a Left-Handed Medium, 168
4.4.5 Plane Waves Versus Transmission Lines, 170
4.5 Plane Waves in Anisotropic and Bi-Isotropic Media, 174
4.5.1 Plane Waves in Uniaxial Media, 174
4.5.2 Plane Waves in Gyrotropic Media, 179
4.5.3 Plane Waves in Chiral Media, 183
References, 190
Problems, 191
5 Fields and Waves in Rectangular Coordinates 199
5.1 Uniform Waveguides, 199
5.1.1 General Analysis, 200
5.1.2 General Characteristics, 204
5.1.3 Uniform Rectangular Waveguide, 208
5.1.4 Losses in Waveguides and Attenuation Constant, 215
5.2 Uniform Cavities, 220
5.2.1 General Theory, 221
5.2.2 Rectangular Cavity, 223
5.2.3 Material and Geometry Pertu
ations, 226
5.3 Partially Filled Waveguides and Dielectric Slab Waveguides, 229
5.3.1 General Theory, 229
5.3.2 Partially Filled Rectangular Waveguide, 231
5.3.3 Dielectric Slab Waveguide on a Ground Plane, 236
viii CONTENTS
5.4 Field Excitation in Waveguides, 241
5.4.1 Excitation by Planar Surface Cu
ents, 242
5.4.2 Excitation by General Volumetric Cu
ents, 243
5.5 Fields in Planar Layered Media, 245
5.5.1 Spectral Green’s Function and Sommerfeld Identity, 245
5.5.2 Vertical Electric Dipole above a Layered Medium, 247
5.5.3 Horizontal Electric Dipole above a Layered Medium, 249
5.5.4 Dipoles on a Grounded Dielectric Slab, 251
References, 257
Problems, 257
6 Fields and Waves in Cylindrical Coordinates 261
6.1 Solution of Wave Equation, 261
6.1.1 Solution by Separation of Variables, 262
6.1.2 Cylindrical Wave Functions, 263
6.2 Circular and Coaxial Waveguides and Cavities, 266
6.2.1 Circular Waveguide, 267
6.2.2 Coaxial Waveguide, 273
6.2.3 Cylindrical Cavity, 276
6.3 Circular Dielectric Waveguide, 279
6.3.1 Analysis of Hy
id Modes, 279
6.3.2 Characteristics of Hy
id Modes, 283
6.4 Wave Transformation and Scattering Analysis, 287
6.4.1 Wave Transformation, 288
6.4.2 Scattering by a Circular Conducting Cylinder, 289
6.4.3 Scattering by a Circular Dielectric Cylinder, 293
6.4.4 Scattering by a Circular Multilayer Dielectric Cylinder, 296
6.5 Radiation by Infinitely Long Cu
ents, 300
6.5.1 Line Cu
ent Radiation in Free Space, 300
6.5.2 Radiation by a Cylindrical Surface Cu
ent, 304
6.5.3 Radiation in the Presence of a Circular Conducting Cylinder, 306
6.5.4 Radiation in the Presence of a Conducting Wedge, 309
6.5.5 Radiation by a Finite Cu
ent, 312
References, 319
Problems, 320
7 Fields and Waves in Spherical Coordinates 325
7.1 Solution of Wave Equation, 325
7.1.1 Solution by Separation of Variables, 325
7.1.2 Spherical Wave Functions, 328
7.1.3 TEr and TMr Modes, 329
7.2 Spherical Cavity, 331
7.3 Biconical Antenna, 335
7.3.1 Infinitely Long Model, 335
7.3.2 Finite Biconical Antenna, 339
7.4 Wave Transformation and Scattering Analysis, 341
7.4.1 Wave Transformation, 342
CONTENTS ix
7.4.2 Expansion of a Plane Wave, 344
7.4.3 Scattering by a Conducting Sphere, 347
7.4.4 Scattering by a Dielectric Sphere, 352
7.4.5 Scattering by a Multilayer Dielectric Sphere, 357
7.5 Addition Theorem and Radiation Analysis, 360
7.5.1 Addition Theorem for Spherical Wave Functions, 360
7.5.2 Radiation of a Spherical Surface Cu
ent, 362
7.5.3 Radiation in the Presence of a Sphere, 368
7.5.4 Radiation in the Presence of a Conducting Cone, 370
References, 377
Problems, 377
PART II Electromagnetic Field Computation 383
8 The Finite Difference Method 385
8.1 Finite Differencing Formulas, 385
8.2 One-Dimensional Analysis, 387
8.2.1 Solution of the Diffusion Equation, 387
8.2.2 Solution of the Wave Equation, 389
8.2.3 Stability Analysis, 390
8.2.4 Numerical Dispersion Analysis, 392
8.3 Two-Dimensional Analysis, 393
8.3.1 Analysis in the Time Domain, 393
8.3.2 Analysis in the