Table Of Content
Electromagnetism
First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Library of Congress Cataloging-in-Publication Data
Bécherrawy, Tamer.
Electromagnetism : Maxwell equations, wave propagation, and emission / Tamer Bécherrawy.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-355-5
1. Electromagnetism. 2. Maxwell equations. 3. Electromagnetic waves. 4. Field emission. I. Title.
QC670.B37 2012
537--dc23
2012009826
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-355-5
Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1. Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Scalars and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2. Effect of rotations on scalars and vectors . . . . . . . . . . . . . . . . . . 5
1.3. Integrals involving vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4. Gradient and curl, conservative field and scalar potential. . . . . . . . . 8
1.5. Divergence, conservative flux, and vector potential . . . . . . . . . . . . 10
1.6. Other properties of the vector differential operator . . . . . . . . . . . . 10
1.7. Invariance and physical laws . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.8. Electric charges in nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.9. Interactions in nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.10. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter 2. Electrostatics in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1. Electric forces and field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2. Electric energy and potential . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3. The two fundamental laws of electrostatics . . . . . . . . . . . . . . . . . 26
2.4. Poisson’s equation and its solutions . . . . . . . . . . . . . . . . . . . . . 29
2.5. Symmetries of the electric field and potential . . . . . . . . . . . . . . . . 31
2.6. Electric dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7. Electric field and potential of simple charge configurations . . . . . . . 38
2.8. Some general properties of the electric field and potential . . . . . . . . 39
2.9. Electrostatic energy of a system of charges . . . . . . . . . . . . . . . . . 42
2.10. Electrostatic binding energy of ionic crystals and atomic nuclei . . . . 48
vi Electromagnetism
2.11. Interaction-at-a-distance and local interaction* . . . . . . . . . . . . . . 50
2.12. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Chapter 3. Conductors and Currents . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1. Conductors in equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2. Conductors with cavities, electric shielding . . . . . . . . . . . . . . . . . 64
3.3. Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4. Mutual electric influence of conductors . . . . . . . . . . . . . . . . . . . 72
3.5. Electric forces between conductors . . . . . . . . . . . . . . . . . . . . . . 73
3.6. Currents and current densities . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.7. Classical model of conduction, Ohm’s law and the Joule effect . . . . . 79
3.8. Resistance of conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.9. Variation of resistivity with temperature, superconductivity . . . . . . . 82
3.10. Band theory of conduction, semiconductors* . . . . . . . . . . . . . . . 84
3.11. Electric circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.12. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter 4. Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1. Effects of dielectric on capacitors . . . . . . . . . . . . . . . . . . . . . . . 97
4.2. Polarization of dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3. Microscopic interpretation of polarization . . . . . . . . . . . . . . . . . . 100
4.4. Polarization charges in dielectric . . . . . . . . . . . . . . . . . . . . . . . 102
4.5. Potential and field of polarized dielectrics . . . . . . . . . . . . . . . . . . 103
4.6. Gauss’s law in the case of dielectrics, electric displacement . . . . . . . 105
4.7. Electrostatic equations in dielectrics . . . . . . . . . . . . . . . . . . . . . 106
4.8. Field and potential of permanent dielectrics . . . . . . . . . . . . . . . . . 109
4.9. Polarization of a dielectric in an external field . . . . . . . . . . . . . . . 113
4.10. Energy and force in dielectrics . . . . . . . . . . . . . . . . . . . . . . . . 115
4.11. Action of an electric field on a polarized medium . . . . . . . . . . . . 116
4.12. Electric susceptibility and permittivity . . . . . . . . . . . . . . . . . . . 118
4.13. Variation of polarization with temperature . . . . . . . . . . . . . . . . 120
4.14. Nonlinear dielectrics and non-isotropic dielectrics . . . . . . . . . . . . 122
4.15. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Chapter 5. Special Techniques and Approximation Methods . . . . . . . . . 127
5.1. Unicity of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2. Method of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3. Method of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4. Method of separation of variables . . . . . . . . . . . . . . . . . . . . . . 135
5.5. Laplace’s equation in Cartesian coordinates . . . . . . . . . . . . . . . . 136
5.6. Laplace’s equation in spherical coordinates . . . . . . . . . . . . . . . . . 138
TableofContents vii
5.7.Laplace’sequationincylindricalcoordinates. . . . . . . . . . . . . . . . 143
5.8.Multipoleexpansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.9.Othermethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.10.Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Chapter6.MagneticFieldinVacuum . . . . . . . . . . . . . . . . . . . . . . . 153
6.1.Forceexertedbyamagneticfieldonamovingcharge . . . . . . . . . . 153
6.2.Forceexertedbyamagneticfieldonacurrent,Laplace’sforce. . . . . 155
6.3.Magneticfluxandvectorpotential . . . . . . . . . . . . . . . . . . . . . . 157
6.4.Magneticfieldofparticlesandcurrents,Biot-Savart’slaw. . . . . . . . 159
6.5.Magneticmoment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.6.Symmetriesofthemagneticfield. . . . . . . . . . . . . . . . . . . . . . . 165
6.7.Ampère’slawintheintegralform . . . . . . . . . . . . . . . . . . . . . . 167
6.8.Fieldandpotentialofsomesimplecircuits . . . . . . . . . . . . . . . . . 169
6.9.Equationsoftime-independentmagnetisminvacuum,
singularitiesofB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.10.MagneticenergyofacircuitinafieldB. . . . . . . . . . . . . . . . . . 178
6.11.Magneticforces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.12.Questionofmagneticmonopoles* . . . . . . . . . . . . . . . . . . . . . 186
6.13.Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Chapter7.MagnetisminMatter. . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.1.Typesofmagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.2.Diamagnetismandparamagnetism. . . . . . . . . . . . . . . . . . . . . . 197
7.3.Magnetizationcurrent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7.4.Magneticfieldandvectorpotentialinthepresenceofmagneticmatter 203
7.5.Ampère’slawintheintegralforminthepresenceofmagneticmatter. 204
7.6.Equationsoftime-independentmagnetisminthepresenceofmatter. . 206
7.7.Discontinuitiesofthemagneticfield. . . . . . . . . . . . . . . . . . . . . 209
7.8.Examplesofcalculationofthefieldofpermanentmagnets. . . . . . . 211
7.9.Magnetizationofabodyinanexternalfield . . . . . . . . . . . . . . . . 214
7.10.Magneticsusceptibility,nonlinearmediumsand
non-isotropicmediums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.11.Actionofamagneticfieldonamagneticbody. . . . . . . . . . . . . . 218
7.12.Magneticenergyinmatter . . . . . . . . . . . . . . . . . . . . . . . . . . 220
7.13.Variationofmagnetizationwithtemperature . . . . . . . . . . . . . . . 221
7.14.Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.15.Magneticcircuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.16.Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
viii Electromagnetism
Chapter 8. Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.1. Induction due to the variation of the flux, Faraday’s and Lenz’s laws . 233
8.2. Neumann’s induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.3. Lorentz induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.4. Lorentz induction and the Galilean transformation of fields . . . . . . . 239
8.5. Mutual inductance and self-inductance . . . . . . . . . . . . . . . . . . . 240
8.6. LR circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.7. Magnetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8.8. Magnetic forces acting on circuits . . . . . . . . . . . . . . . . . . . . . . 249
8.9. Some applications of induction . . . . . . . . . . . . . . . . . . . . . . . . 252
8.10. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Chapter 9. Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.1. Fundamental laws of electromagnetism . . . . . . . . . . . . . . . . . . . 263
9.2. Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.3. Electromagnetic potentials and gauge transformation . . . . . . . . . . . 270
9.4. Quasi-permanent approximation . . . . . . . . . . . . . . . . . . . . . . . 272
9.5. Discontinuities on the interface of two mediums . . . . . . . . . . . . . . 276
9.6. Electromagnetic energy and Poynting vector . . . . . . . . . . . . . . . . 277
9.7. Electromagnetic pressure, Maxwell’s tensor . . . . . . . . . . . . . . . . 278
9.8. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Chapter 10. Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . 283
10.1. A short review on waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
10.2. Electromagnetic waves in infinite vacuum and dielectrics . . . . . . . 291
10.3. Polarization of electromagnetic waves . . . . . . . . . . . . . . . . . . . 295
10.4. Energy and intensity of plane electromagnetic waves . . . . . . . . . . 299
10.5. Momentum and angular momentum densities, radiation pressure . . . 301
10.6. A simple model of dispersion . . . . . . . . . . . . . . . . . . . . . . . . 304
10.7. Electromagnetic waves in conductors . . . . . . . . . . . . . . . . . . . 308
10.8. Electromagnetic waves in plasmas . . . . . . . . . . . . . . . . . . . . . 314
10.9. Quantization of electromagnetic waves . . . . . . . . . . . . . . . . . . 320
10.10. Electromagnetic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 321
10.11. Emission of electromagnetic radiations . . . . . . . . . . . . . . . . . . 323
10.12. Spontaneous and stimulated emissions . . . . . . . . . . . . . . . . . . 325
10.13. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
Chapter 11. Reflection, Interference, Diffraction and Diffusion . . . . . . . 337
11.1. General laws of reflection and refraction . . . . . . . . . . . . . . . . . . 337
11.2. Reflection and refraction on the interface of two dielectrics . . . . . . 340
11.3. Total reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
