# Top 10 Electromagnetics Books

## Classical Electrodynamics

### Table Content of  Classical Electrodynamics

Introduction and Survey 1
1.1 Maxwell Equations in Vacuum, Fields, and Sources 2
1.2 Inverse Square Law, or the Mass of the Photon 5
1.3 Linear Superposition 9
1.4 Maxwell Equations in Macroscopic Media 13
1.5 Boundary Conditions at Interfaces Between Different Media 16
1.6 Some Remarks on Idealizations in Electromagnetism 19
Chapter 1 / Introduction to Electrostatics 24
1.1 Coulomb's Law 24
1.2 Electric Field 24
1.3 Gauss's Law 27
1.4 Differential Form of Gauss's Law 28
1.5 Another Equation of Electrostatics and the Scalar Potential 29
1.6 Surface Distributions of Charges and Dipoles and Discontinuities
in the Electric Field and Potential 31
1.7 Poisson and Laplace Equations 34
1.8 Green's Theorem 35
1.9 Uniqueness of the Solution with Dirichlet or Neumann Boundary
Conditions 37
1.10 Formal Solution of Electrostatic Boundary-Value Problem
with Green Function 38
1.11 Electrostatic Potential Energy and Energy Density; Capacitance 40
1.12 Variational Approach to the Solution of the Laplace and Poisson
Equations 43
1.13 Relaxation Method for Two-Dimensional Electrostatic Problems 47
Problems 50
Chapter 2 / Boundary- Value Problems in Electrostatics: I 57
2.1 Method of Images 57
2.2 Point Charge in the Presence of a Grounded Conducting Sphere 58
2.3 Point Charge in the Presence of a Charged, Insulated, Conducting
Sphere 60
2.4 Point Charge Near a Conducting Sphere at Fixed Potential 61
2.5 Conducting Sphere in a Uniform Electric Field by Method
of Images 62
2.6 Green Function for the Sphere; General Solution
for the Potential 64
2.7 Conducting Sphere with Hemispheres at Different Potentials 65
2.8 Orthogonal Functions and Expansions 67
2.9 Separation of Variables; Laplace Equation in Rectangular
Coordinates 70
2.10 A Two-Dimensional Potential Problem; Summation
of Fourier Series 72
2.11 Fields and Charge Densities in Two-Dimensional Corners
and Along Edges 75
2.12 Introduction to Finite Element Analysis for Electrostatics 79
Problems 85
Chapter 3 / Boundary- Value Problems in Electrostatics: II 95
3.1 Laplace Equation in Spherical Coordinates 95
3.2 Legendre Equation and Legendre Polynomials 96
3.3 Boundary-Value Problems with Azimuthal Symmetry 101
3.4 Behavior of Fields in a Conical Hole or Near a Sharp Point 104
3.5 Associated Legendre Functions and the Spherical Harmonics
Ylm@, Ð¤) Ð®7
3.6 Addition Theorem for Spherical Harmonics 110
3.7 Laplace Equation in Cylindrical Coordinates; Bessel Functions 111
3.8 Boundary-Value Problems in Cylindrical Coordinates 117
3.9 Expansion of Green Functions in Spherical Coordinates 119
3.10 Solution of Potential Problems with the Spherical Green Function
Expansion 112
3.11 Expansion of Green Functions in Cylindrical Coordinates 125
3.12 Eigenfunction Expansions for Green Functions 127
3.13 Mixed Boundary Conditions, Conducting Plane with a Circular
Hole 129
Problems 135
Chapter 4 / Multipoles, Electrostatics of Macroscopic Media,
Dielectrics 145
4.1 Multipole Expansion 145
4.2 Multipole Expansion of the Energy of a Charge Distribution
in an External Field 150
4.3 Elementary Treatment of Electrostatics with Ponderable Media 151
4.4 Boundary-Value Problems with Dielectrics 154
4.5 Molecular Polarizability and Electric Susceptibility 159
4.6 Models for Electric Polarizability 162
4.7 Electrostatic Energy in Dielectric Media 165
Problems 169
Chapter 5 / Magnetostatics, Faraday's Law, Quasi-Static Fields 174
5.1 Introduction and Definitions 174
5.2 Biot and Savart Law 175
5.3 Differential Equations of Magnetostatics and Ampere's Law 178
5.4 Vector Potential 180
5.5 Vector Potential and Magnetic Induction for a Circular Current
Loop 181
5.6 Magnetic Fields of a Localized Current Distribution, Magnetic
Moment 184
5.7 Force and Torque on and Energy of a Localized Current Distribution
in an External Magnetic Induction 188
5.8 Macroscopic Equations, Boundary Conditions on Ð’ and H 191
5.9 Methods of Solving Boundary-Value Problems
in Magnetostatics 194
5.10 Uniformly Magnetized Sphere 198
5.11 Magnetized Sphere in an External Field; Permanent Magnets 199
5.12 Magnetic Shielding, Spherical Shell of Permeable Material
in a Uniform Field 201
5.13 Effect of a Circular Hole in a Perfectly Conducting Plane
with an Asymptotically Uniform Tangential Magnetic Field
on One Side 203
5.14 Numerical Methods for Two-Dimensional Magnetic Fields 206
5.15 Faraday's Law of Induction 208
5.16 Energy in the Magnetic Field 212
5.17 Energy and Self- and Mutual Inductances 215
5.18 Quasi-Static Magnetic Fields in Conductors; Eddy Currents; Magnetic
Diffusion 218
Problems 225
Chapter 6 / Maxwell Equations, Macroscopic Electromagnetism,
Conservation Laws 237
6.1 Maxwell's Displacement Current; Maxwell Equations 237
6.2 Vector and Scalar Potentials 239
6.3 Gauge Transformations, Lorenz Gauge, Coulomb Gauge 240
6.4 Green Functions for the Wave Equation 243
6.5 Retarded Solutions for the Fields: Jefimenko's Generalizations
of the Coulomb and Biot-Savart Laws; Heaviside-Feynman
Expressions for Fields of Point Charge 246
6.6 Derivation of the Equations of Macroscopic Electromagnetism 248
6.7 Poynting's Theorem and Conservation of Energy and Momentum
for a System of Charged Particles and Electromagnetic Fields 258
6.8 Poynting's Theorem in Linear Dissipative Media with Losses 262
6.9 Poynting's Theorem for Harmonic Fields; Field Definitions
6.10 Transformation Properties of Electromagnetic Fields and Sources
Under Rotations, Spatial Reflections, and Time Reversal 267
6.11 On the Question of Magnetic Monopoles 273
6.12 Discussion of the Dirac Quantization Condition 275
6.13 Polarization Potentials (Hertz Vectors) 280
Problems 283
Chapter 7 / Plane Electromagnetic Waves and Wave Propagation 295
11 Plane Waves in a Nonconducting Medium 295
7.2 Linear and Circular Polarization; Stokes Parameters 299
7.3 Reflection and Refraction of Electromagnetic Waves at a Plane
Interface Between Two Dielectrics 302
7.4 Polarization by Reflection, Total Internal Reflection; Goos-Hanchen
Effect 306
7.5 Frequency Dispersion Characteristics of Dielectrics, Conductors,
and Plasmas 309
7.6 Simplified Model of Propagation in the Ionosphere
and Magnetosphere 316
7.7 Magnetohydrodynamic Waves 319
7.8 Superposition of Waves in One Dimension; Group Velocity 322
7.9 Illustration of the Spreading of a Pulse As It Propagates in a Dispersive
Medium 326
7.10 Causality in the Connection Between D and E; Kramers-Kronig
Relations 330
7.11 Arrival of a Signal After Propagation Through a Dispersive
Medium 335
Problems 340
Chapter 8 / Waveguides, Resonant Cavities, and Optical Fibers 352
8.1 Fields at the Surface of and Within a Conductor 352
8.2 Cylindrical Cavities and Waveguides 356
8.3 Waveguides 359
8.4 Modes in a Rectangular Waveguide 361
8.5 Energy Flow and Attenuation in Waveguides 363
8.6 Perturbation of Boundary Conditions 366
8.7 Resonant Cavities 368
8.8 Power Losses in a Cavity; Q of a Cavity 371
8.9 Earth and Ionosphere as a Resonant Cavity:
Schumann Resonances 374
8.10 Multimode Propagation in Optical Fibers 378
8.11 Modes in Dielectric Waveguides 385
8.12 Expansion in Normal Modes; Fields Generated by a Localized
Source in a Hollow Metallic Guide 389
Problems 396
9.1 Fields and Radiation of a Localized Oscillating Source 407
9.2 Electric Dipole Fields and Radiation 410
9.3 Magnetic Dipole and Electric Quadrupole Fields 413
9.4 Center-Fed Linear Antenna 416
9.5 Multipole Expansion for Localized Source or Aperture
in Waveguide 419
9.6 Spherical Wave Solutions of the Scalar Wave Equation 425
9.7 Multipole Expansion of the Electromagnetic Fields 429
9.8 Properties of Multipole Fields, Energy and Angular Momentum
9.9 Angular Distribution of Multipole Radiation 437
9.10 Sources of Multipole Radiation; Multipole Moments 439
9.11 Multipole Radiation in Atoms and Nuclei 442
9.12 Multipole Radiation from a Linear, Center-Fed Antenna 444
Problems 449
Chapter 10 / Scattering and Diffraction 456
10.1 Scattering at Long Wavelengths 456
10.2 Perturbation Theory of Scattering, Rayleigh's Explanation
of the Blue Sky, Scattering by Gases and Liquids, Attenuation
in Optical Fibers 462
10.3 Spherical Wave Expansion of a Vector Plane Wave 471
10.4 Scattering of Electromagnetic Waves by a Sphere 473
10.5 Scalar Diffraction Theory 478
10.6 Vector Equivalents of the Kirchhoff Integral 482
10.7 Vectorial Diffraction Theory 485
10.8 Babinet's Principle of Complementary Screens 488
10.9 Diffraction by a Circular Aperture; Remarks on Small
Apertures 490
10.10 Scattering in the Short-Wavelength Limit 495
10.11 Optical Theorem and Related Matters 500
Problems 507
Chapter 11 / Special Theory of Relativity 514
11.1 The Situation Before 1900, Einstein's Two Postulates 515
11.2 Some Recent Experiments 518
11.3 Lorentz Transformations and Basic Kinematic Results of Special
Relativity 524
11.4 Addition of Velocities; 4-Velocity 530
11.5 Relativistic Momentum and Energy of a Particle 533
11.6 Mathematical Properties of the Space-Time of Special
Relativity 539
11.7 Matrix Representation of Lorentz Transformations, Infinitesimal
Generators 543
11.8 Thomas Precession 548
11.9 Invariance of Electric Charge; Covariance of Electrodynamics 553
11.10 Transformation of Electromagnetic Fields 558
11.11 Relativistic Equation of Motion for Spin in Uniform or Slowly Varying
External Fields 561
11.12 Note on Notation and Units in Relativistic Kinematics 565
Problems 568
Chapter 12 / Dynamics of Relativistic Particles
and Electromagnetic Fields 579
12.1 Lagrangian and Hamiltonian for a Relativistic Charged Particle
in External Electromagnetic Fields 579
12.2 Motion in a Uniform, Static Magnetic Field 585
12.3 Motion in Combined, Uniform, Static Electric and Magnetic
Fields 586
12.4 Particle Drifts in Nonuniform, Static Magnetic Fields 588
12.5 Adiabatic Invariance of Flux Through Orbit of Particle 592
12.6 Lowest Order Relativistic Corrections to the Lagrangian for Interacting
Charged Particles: The Darwin Lagrangian 596
12.7 Lagrangian for the Electromagnetic Field 598
12.8 Proca Lagrangian; Photon Mass Effects 600
12.9 Effective "Photon" Mass in Superconductivity; London Penetration
Depth 603
12.10 Canonical and Symmetric Stress Tensors; Conservation Laws 605
12.11 Solution of the Wave Equation in Covariant Form; Invariant Green
Functions 612
Problems 617
Chapter 13 / Collisions, Energy Loss, and Scattering of Charged Particles,
13.1 Energy Transfer in Coulomb Collision Between Heavy Incident Particle
and Free Electron; Energy Loss in Hard Collisions 625
13.2 Energy Loss from Soft Collisions; Total Energy Loss 627
13.3 Density Effect in Collisional Energy Loss 631
13.5 Elastic Scattering of Fast Charged Particles by Atoms 640
13.6 Mean Square Angle of Scattering; Angular Distribution of Multiple
Scattering 643
Problems 655
Chapter 14 / Radiation by Moving Charges 661
14.1 Lienard-Wiechert Potentials and Fields for a Point Charge 661
14.2 Total Power Radiated by an Accelerated Charge: Larmor's Formula
and Its Relativistic Generalization 665
14.3 Angular Distribution of Radiation Emitted by an Accelerated
Charge 668
14.4 Radiation Emitted by a Charge in Arbitrary, Extremely Relativistic
Motion 671
14.5 Distribution in Frequency and Angle of Energy Radiated
by Accelerated Charges: Basic Results 673
14.6 Frequency Spectrum of Radiation Emitted by a Relativistic Charged
Particle in Instantaneously Circular Motion 676
14.7 Undulators and Wigglers for Synchrotron Light Sources 683
14.8 Thomson Scattering of Radiation 694
Problems 698
Chapter 15 / Bremsstrahlung, Method of Virtual Quanta,
15.1 Radiation Emitted During Collisions 709
15.2 Bremsstrahlung in Coulomb Collisions 714
15.3 Screening Effects; Relativistic Radiative Energy Loss 721
15.4 Weizsacker-Williams Method of Virtual Quanta 724
15.5 Bremsstrahlung as the Scattering of Virtual Quanta 729
15.6 Radiation Emitted During Beta Decay 730
15.7 Radiation Emitted During Orbital Electron Capture: Disappearance
of Charge and Magnetic Moment 732
Problems 737
Chapter 16 / Radiation Damping, Classical Models
of Charged Particles 745
16.1 Introductory Considerations 745
16.2 Radiative Reaction Force from Conservation of Energy 747
16.3 Abraham-Lorentz Evaluation of the Self-Force 750
16.4 Relativistic Covariance; Stability and Poincare Stresses 755
16.5 Covariant Definitions of Electromagnetic Energy
and Momentum 757
16.6 Covariant Stable Charged Particle 759
16.8 Scattering and Absorption of Radiation by an Oscillator 766
Problems 769
Appendix on Units and Dimensions 775
1 Units and Dimensions, Basic Units and Derived Units 775
2 Electromagnetic Units and Equations 777
3 Various Systems of Electromagnetic Units 779
4 Conversion of Equations and Amounts Between SI Units
and Gaussian Units 782
Bibliography 785
Index 791

### Table Content of  Classical Electrodynamics for UnderGraduates

1 MATRICES 5
1.1 Einstein Summation Convention ................ 5
1.2 Coupled Equations and Matrices ................ 6
1.3 Determinants and Inverse .................... 8
1.4 Solution of Coupled Equations . . . . . . . . . . . . . . . . . 11
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 VECTORS 19
2.1 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Vector Product . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Triple and Mixed Products . . . . . . . . . . . . . . . . . . . 25
2.5 Div, Grad and Curl (differential calculus for vectors) . . . . . 26
2.6 Integrals of Div, Grad and Curl . . . . . . . . . . . . . . . . . 31
2.6.1 Fundamental Theorem of Gradients . . . . . . . . . . 32
2.6.2 Gauss’ theorem (Fundamental theorem of Divergence) 34
2.6.3 Stokes’ theorem (Fundamental theorem of curl) . . . . 35
2.7 Potential Theory . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . 37
2.8.1 Plane Cartesian (Rectangular) Coordinates . . . . . . 37
2.8.2 Three dimensional Cartesian Coordinates . . . . . . . 38
2.8.3 Plane (2-dimensional) Polar Coordinates . . . . . . . 38
2.8.4 Spherical (3-dimensional) Polar Coordinates . . . . . 40
2.8.5 Cylindrical (3-dimensional) Polar Coordinates . . . . 41
2.8.6 Div, Grad and Curl in Curvilinear Coordinates . . . . 43
2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.11 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.12 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.13 Figure captions for chapter 2 . . . . . . . . . . . . . . . . . . 51
3 MAXWELL’S EQUATIONS 53
3.1 Maxwell’s equations in differential form . . . . . . . . . . . . 54
3.2 Maxwell’s equations in integral form . . . . . . . . . . . . . . 56
3.3 Charge Conservation . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . 58
3.5 Scalar and Vector Potential . . . . . . . . . . . . . . . . . . . 60
4 ELECTROSTATICS 63
4.1 Equations for electrostatics . . . . . . . . . . . . . . . . . . . 63
4.2 Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Electric Scalar potential . . . . . . . . . . . . . . . . . . . . . 68
4.4 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.1 Arbitrariness of zero point of potential energy . . . . . 74
4.4.2 Work done in assembling a system of charges . . . . . 74
4.5 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . 76
5 Magnetostatics 77
5.1 Equation for Magnetostatics . . . . . . . . . . . . . . . . . . . 77
5.1.1 Equations from Amp`er ́e’s Law . . . . . . . . . . . . . 78
5.1.2 Equations from Gauss’ Law . . . . . . . . . . . . . . . 78
5.2 Magnetic Field from the Biot-Savart Law . . . . . . . . . . . 79
5.3 Magnetic Field from Amp`er ́e’s Law . . . . . . . . . . . . . . . 81
5.4 Magnetic Field from Vector Potential . . . . . . . . . . . . . . 81
5.5 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 ELECTRO- AND MAGNETOSTATICS IN MATTER 83
6.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Maxwell’s Equations in Matter . . . . . . . . . . . . . . . . . 85
6.2.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . 86
6.2.3 Summary of Maxwell’s Equations . . . . . . . . . . . . 88
6.3 Further Dimennsional of Electrostatics . . . . . . . . . . . . . 89
6.3.1 Dipoles in Electric Field . . . . . . . . . . . . . . . . . 89
6.3.2 Energy Stored in a Dielectric . . . . . . . . . . . . . . 90
6.3.3 Potential of a Polarized Dielectric . . . . . . . . . . . 91
7 ELECTRODYNAMICS AND MAGNETODYNAMICS 93
7.0.4 Faradays’s Law of Induction . . . . . . . . . . . . . . . 93
7.0.5 Analogy between Faraday field and Magnetostatics . . 96
7.1 Ohm’s Law and Electrostatic Force . . . . . . . . . . . . . . . 97
8 MAGNETOSTATICS 101
9 ELECTRO- & MAGNETOSTATICS IN MATTER 103
10 ELECTRODYNAMICS AND MAGNETODYNAMICS 105
11 ELECTROMAGNETIC WAVES 107
12 SPECIAL RELATIVITY 109

## Classical Electrodynamics

### Table Content of  Classical Electrodynamics

I Electrostatics

I Introduction and Fundamental Concepts

2 Green's Theorems

3 Orthogonal Functions and Multipole Expansion: Mathematical Supplement

4 Elementary Considerations on Function Theory: Mathematical Supplement

II Macroscopic Electrostatics

5 The Field Equations for Space Filled with Matter

6 Simple Dielectrics and the Susceptibility

7 Electrostatic Energy and Forces in a Dielectric

III Magnetostatics

8 Foundations of Magnetostatics

9 The Vector Potential

10 Magnetic Moment

11 The Magnetic Field in Matter

IV Electrodynamics

13 Maxwell's Equations

14 Quasi-Stationary Currents and Current Circuits

15 Electromagnetic Waves in Vacuum

16 Electromagnetic Waves in Matter

17 Index of Reflection and Refraction

18 Wave Guides and Resonant Cavities

19 Light Waves

20 Moving Charges in Vacuum

21 The Hertzian Dipole

22 Covariant Formulation of Electrodynamics

23 Relativistic-Covariant Lagrangian Formalism

24 Systems of Units in Electrodynamics: Supplement

25 About the History of Electrodynamics

Index

## Electromagnetics of Continuous Media

### Table Content of  Electromagnetics of Continuous Media

I. ELECTROSTATICS OF CONDUCTORS §1. The electrostatic field of conductors §2. The energy of the electrostatic field of conductors §3. Methods of solving problems in electrostatics §4. A conducting ellipsoid §5. The forces on a conductor

II. ELECTROSTATICS OF DIELECTRICS §6. The electric field in dielectrics §7. The permittivity §8. A dielectric ellipsoid §9. The permittivity of a mixture §10. Thermodynamic relations for dielectrics in an electric field §11. The total free energy of a dielectric §12. Electrostriction of isotropic dielectrics §13. Dielectric properties of crystals §14. The sign of the dielectric susceptibility §15. Electric forces in a fluid dielectric §16. Electric forces in solids §17. Piezoelectrics §18. Thermodynamic inequalities §19. Ferroelectrics §20. Improper ferroelectrics

III. STEADY CURRENT §21. The current density and the conductivity 86 §22. The Hall effect 90 §23. The contact potential 92 §24. The galvanic cell 94 §25. Electrocapillarity 96 §26. Thermoelectric phenomena 97 §27. Thermogalvanomagnetic phenomena 101 §28. Diffusion phenomena 102

IV. STATIC MAGNETIC FIELD §29. Static magnetic field 105 §30. The magnetic field of a steady current 107 §31. Thermodynamic relations in a magnetic field 113 §32. The total free energy of a magnetic substance 116 §33. The energy of a system of currents 118 §34. The self-inductance of linear conductors 121 §35. Forces in a magnetic field 126 §36. Gyromagnetic phenomena 129

V. FERROMAGNETISM AND ANTIFERROMAGNETISM §37. Magnetic symmetry of crystals 130 §38. Magnetic classes and space groups 132 §39. Ferromagnets near the Curie point 135 §40. The magnetic anisotropy energy 138 §41. The magnetization curve of ferromagnets 141 §42. Magnetostriction of ferromagnets 144 §43. The surface tension of a domain wall 147 §44. The domain structure of ferromagnets 153 §45. Single-domain particles 157 §46. Orientational transitions 159 §47. Fluctuations in ferromagnets 162 §48. Antiferromagnets near the Curie point 166 §49. The bicritical point for an antiferromagnet 170 §50. Weak ferromagnetism 172 §51. Piezomagnetism and the magnetoelectric effect of 176 §52. Helicoidal magnetic structures 178

VI. SUPERCONDUCTIVITY §53. The magnetic properties of superconductors 180 §54. The superconductivity current 182 §55. The critical field 185 §56. The intermediate state 189 §57. Structure of the intermediate state 194

VII. QUASI-STATIC ELECTROMAGNETIC FIELD §58. Equations of the quasi-static field 199 §59. Depth of penetration of a magnetic field into a conductor 201 §60. The skin effect 208 §61. The complex resistance of 210 §62. Capacitance in a quasi-steady current circuit 214 §63. The motion of a conductor in a magnetic field 217 §64. Excitation of currents by acceleration 222

VIII. MAGNETOHYDRODYNAMICS §65. The equations of motion for a fluid in a magnetic field 225 §66. Dissipative processes in magnetohydrodynamics 228 §67. The magnetohydrodynamic flow between parallel planes 230 §68. Equilibrium configurations 232 §69. Hydromagnetic waves 235 §70. Conditions at discontinuities 240 §71. Tangential and rotational discontinuities 240 §72. Shock waves 245 §73. Evolutionary shock waves 247 §74. The turbulent dynamo 253

IX. THE ELECTROMAGNETIC WAVE EQUATIONS §75. The field equations in a dielectric in the absence of dispersion 257 §76. The electrodynamics of moving dielectrics 260 §77. The dispersion of the permittivity 264
§78. The permittivity at very high frequencies of 267 §79. The dispersion of the magnetic permeability 268 §80. The field energy in dispersive media 272 §81. The stress tensor in dispersive media 276 §82. The analytical properties of (co)E 279 §83. A plane monochromatic wave 283 §84. Transparent media 286

X. THE PROPAGATION OF ELECTROMAGNETIC WAVES §85. Geometrical optics 290 §86. Reflection and refraction of electromagnetic waves 293 §87. The surface impedance of metals 300 §88. The propagation of waves in an inhomogeneous medium 304 §89. The reciprocity principle 308 §90. Electromagnetic oscillations in hollow resonators 310 §91. The propagation of electromagnetic waves in waveguides 313 §92. The scattering of electromagnetic waves by small particles 319 §93. The absorption of electromagnetic waves by small particles 322 §94. Diffraction by a wedge 323 §95. Diffraction by a plane screen 327

XI. ELECTROMAGNETIC WAVES IN ANISOTROPIC MEDIA §96. The permittivity of crystals 331 §97. A plane wave in an anisotropic medium 333 §98. Optical properties of uniaxial crystals 339 §99. Biaxial crystals 341 §100. Double refraction in an electric field 347 §101. Magnetic-optical effects 347 §102. Mechanical-optical effects 355

XII. SPATIAL DISPERSION §103. Spatial dispersion of 358 §104. Natural optical activity 362 §105. Spatial dispersion in optically inactive media 366 §106. Spatial dispersion near an absorption line 367

XIII. NON-LINEAR OPTICS §107. Frequency transformation in non-linear media 372 §108. The non-linear permittivity 374 §109. Self-focusing 378 §110. Second-harmonic generation 383 §111. Strong electromagnetic waves 388 §112. Stimulated Raman scattering 391

XIV. THE PASSAGE OF FAST PARTICLES THROUGH MATTER §113. Ionization losses by fast particles in the matter: the non-relativistic case 394 §114. Ionization losses by fast particles in the matter: the relativistic case 399 §115. Cherenkov radiation 406 §116. Transition radiation 408

XV. SCATTERING OF ELECTROMAGNETIC WAVES §117. The general theory of scattering in isotropic media 413 §118. The principle of detailed balancing applied to scatter 419 §119. Scattering with a small change of frequency 422
§120. Rayleigh scattering in gases and liquids 428 §121. Critical opalescence 433 §122. Scattering in liquid crystals 435 §123. Scattering in amorphous solids 436

XVI. DIFFRACTION OF X-RAYS IN CRYSTALS §124. The general theory of X-ray diffraction 439 §125. The integral intensity of 445 §126. Diffuse thermal scattering of X-rays 447 §127. The temperature dependence of the diffraction cross-section 449 Appendix 452 Index 455

## Electromagnetic Field Theory

### Table Content of  Electromagnetic Field Theory

1 Classical Electrodynamics 1
1.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Coulomb’s law . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The electrostatic field . . . . . . . . . . . . . . . . . . 2
1.2 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 AmpÃ¨re’s law . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 The magnetostatic field . . . . . . . . . . . . . . . . . 6
1.3 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Equation of continuity . . . . . . . . . . . . . . . . . 9
1.3.2 Maxwell’s displacement current . . . . . . . . . . . . 9
1.3.3 Electromotive force . . . . . . . . . . . . . . . . . . . 10
1.3.4 Faraday’s law of induction . . . . . . . . . . . . . . . 11
1.3.5 Maxwell’s microscopic equations . . . . . . . . . . . 14
1.3.6 Maxwell’s macroscopic equations . . . . . . . . . . . 14
1.4 Electromagnetic Duality . . . . . . . . . . . . . . . . . . . . 15
Example 1.1 Duality of the electromagnetodynamic equations 16
Example 1.2 Maxwell from Dirac-Maxwell equations for a
fixed mixing angle . . . . . . . . . . . . . . . 17
Example 1.3 The complex field six-vector . . . . . . . . 18
Example 1.4 Duality expressed in the complex field six-vector 19
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Electromagnetic Waves 23
2.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.1 The wave equation for E . . . . . . . . . . . . . . . . 24
2.1.2 The wave equation for B . . . . . . . . . . . . . . . . 24
2.1.3 The time-independent wave equation for E . . . . . . 25
2.2 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Telegrapher’s equation . . . . . . . . . . . . . . . . . 27
2.2.2 Waves in conductive media . . . . . . . . . . . . . . . 29
2.3 Observables and averages . . . . . . . . . . . . . . . . . . . . 30
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Electromagnetic Potentials 33
3.1 The electrostatic scalar potential . . . . . . . . . . . . . . . . 33
3.2 The magnetostatic vector potential . . . . . . . . . . . . . . . 34
3.3 The electromagnetic scalar and vector potentials . . . . . . . . 34
3.3.1 Electromagnetic gauges . . . . . . . . . . . . . . . . 36
Lorentz equations for the electromagnetic potentials . 36
Gauge transformations . . . . . . . . . . . . . . . . . 36
3.3.2 Solution of the Lorentz equations for the electromag-
netic potentials . . . . . . . . . . . . . . . . . . . . . 38
The retarded potentials . . . . . . . . . . . . . . . . . 41
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 The Electromagnetic Fields 43
4.1 The magnetic field . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 The electric field . . . . . . . . . . . . . . . . . . . . . . . . 47
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Relativistic Electrodynamics 51
5.1 The special theory of relativity . . . . . . . . . . . . . . . . . 51
5.1.1 The Lorentz transformation . . . . . . . . . . . . . . 52
5.1.2 Lorentz space . . . . . . . . . . . . . . . . . . . . . . 53
Metric tensor . . . . . . . . . . . . . . . . . . . . . . 54
Radius four-vector in contravariant and covariant form 54
Scalar product and norm . . . . . . . . . . . . . . . . 55
Invariant line element and proper time . . . . . . . . . 56
Four-vector fields . . . . . . . . . . . . . . . . . . . . 57
The Lorentz transformation matrix . . . . . . . . . . . 57
The Lorentz group . . . . . . . . . . . . . . . . . . . 58
5.1.3 Minkowski space . . . . . . . . . . . . . . . . . . . . 58
5.2 Covariant classical mechanics . . . . . . . . . . . . . . . . . 61
5.3 Covariant classical electrodynamics . . . . . . . . . . . . . . 62
5.3.1 The four-potential . . . . . . . . . . . . . . . . . . . 62
5.3.2 The LiÃ©nard-Wiechert potentials . . . . . . . . . . . . 63
5.3.3 The electromagnetic field tensor . . . . . . . . . . . . 65
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Interactions of Fields and Particles 69
6.1 Charged Particles in an Electromagnetic Field . . . . . . . . . 69
6.1.1 Covariant equations of motion . . . . . . . . . . . . . 69
Lagrange formalism . . . . . . . . . . . . . . . . . . 69
Hamiltonian formalism . . . . . . . . . . . . . . . . . 72
6.2 Covariant Field Theory . . . . . . . . . . . . . . . . . . . . . 76
6.2.1 Lagrange-Hamilton formalism for fields and interactions 77
The electromagnetic field . . . . . . . . . . . . . . . . 80
Example 6.1 Field energy difference expressed in the field
tensor . . . . . . . . . . . . . . . . . . . . . 81
Other fields . . . . . . . . . . . . . . . . . . . . . . . 84
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7 Interactions of Fields and Matter 87
7.1 Electric polarisation and the electric displacement vector . . . 87
7.1.1 Electric multipole moments . . . . . . . . . . . . . . 87
7.2 Magnetisation and the magnetising field . . . . . . . . . . . . 90
7.3 Energy and momentum . . . . . . . . . . . . . . . . . . . . . 91
7.3.1 The energy theorem in Maxwell’s theory . . . . . . . 92
7.3.2 The momentum theorem in Maxwell’s theory . . . . . 93
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.1 The radiation fields . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 Radiated energy . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.2.1 Monochromatic signals . . . . . . . . . . . . . . . . . 100
8.2.2 Finite bandwidth signals . . . . . . . . . . . . . . . . 100
8.3 Radiation from extended sources . . . . . . . . . . . . . . . . 102
8.3.1 Linear antenna . . . . . . . . . . . . . . . . . . . . . 102
8.4 Multipole radiation . . . . . . . . . . . . . . . . . . . . . . . 104
8.4.1 The Hertz potential . . . . . . . . . . . . . . . . . . . 104
8.4.2 Electric dipole radiation . . . . . . . . . . . . . . . . 108
8.4.3 Magnetic dipole radiation . . . . . . . . . . . . . . . 109
8.4.4 Electric quadrupole radiation . . . . . . . . . . . . . . 110
8.5 Radiation from a localised charge in arbitrary motion . . . . . 111
8.5.1 The LiÃ©nard-Wiechert potentials . . . . . . . . . . . . 112
8.5.2 Radiation from an accelerated point charge . . . . . . 114
Example 8.1 The fields from a uniformly moving charge . 121
Example 8.2 The convection potential and the convection
force . . . . . . . . . . . . . . . . . . . . . 123
Radiation for small velocities . . . . . . . . . . . . . 125
8.5.3 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . 127
Example 8.3 Bremsstrahlung for low speeds and short ac-
celeration times . . . . . . . . . . . . . . . . 130
8.5.4 Cyclotron and synchrotron radiation . . . . . . . . . . 132
Cyclotron radiation . . . . . . . . . . . . . . . . . . . 134
Synchrotron radiation . . . . . . . . . . . . . . . . . . 134
Radiation in the general case . . . . . . . . . . . . . . 137
Virtual photons . . . . . . . . . . . . . . . . . . . . . 137
8.5.5 Radiation from charges moving in matter . . . . . . . 139
Vavilov-Cerenk Ë‡ ov radiation . . . . . . . . . . . . . . 142
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
F Formulae 149
F.1 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . 149
F.1.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . 149
Constitutive relations . . . . . . . . . . . . . . . . . . 149
F.1.2 Fields and potentials . . . . . . . . . . . . . . . . . . 149
Vector and scalar potentials . . . . . . . . . . . . . . 149
Lorentz’ gauge condition in vacuum . . . . . . . . . . 150
F.1.3 Force and energy . . . . . . . . . . . . . . . . . . . . 150
Poynting’s vector . . . . . . . . . . . . . . . . . . . . 150
Maxwell’s stress tensor . . . . . . . . . . . . . . . . . 150
F.2 Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . 150
F.2.1 Relationship between the field vectors in a plane wave 150
F.2.2 The far fields from an extended source distribution . . 150
F.2.3 The far fields from an electric dipole . . . . . . . . . . 150
F.2.4 The far fields from a magnetic dipole . . . . . . . . . 151
F.2.5 The far fields from an electric quadrupole . . . . . . . 151
F.2.6 The fields from a point charge in arbitrary motion . . . 151
F.2.7 The fields from a point charge in uniform motion . . . 151
F.3 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . 152
F.3.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . 152
F.3.2 Covariant and contravariant four-vectors . . . . . . . . 152
F.3.3 Lorentz transformation of a four-vector . . . . . . . . 152
F.3.4 Invariant line element . . . . . . . . . . . . . . . . . . 152
F.3.5 Four-velocity . . . . . . . . . . . . . . . . . . . . . . 152
F.3.6 Four-momentum . . . . . . . . . . . . . . . . . . . . 153
F.3.7 Four-current density . . . . . . . . . . . . . . . . . . 153
F.3.8 Four-potential . . . . . . . . . . . . . . . . . . . . . . 153
F.3.9 Field tensor . . . . . . . . . . . . . . . . . . . . . . . 153
F.4 Vector Relations . . . . . . . . . . . . . . . . . . . . . . . . . 153
F.4.1 Spherical polar coordinates . . . . . . . . . . . . . . . 154
Base vectors . . . . . . . . . . . . . . . . . . . . . . 154
Directed line element . . . . . . . . . . . . . . . . . . 154
Solid angle element . . . . . . . . . . . . . . . . . . . 154
Directed area element . . . . . . . . . . . . . . . . . 154
Volume element . . . . . . . . . . . . . . . . . . . . 154
F.4.2 Vector formulae . . . . . . . . . . . . . . . . . . . . . 154
General relations . . . . . . . . . . . . . . . . . . . . 154
Special relations . . . . . . . . . . . . . . . . . . . . 156
Integral relations . . . . . . . . . . . . . . . . . . . . 157
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Appendices 148
M Mathematical Methods 159
M.1 Scalars, Vectors and Tensors . . . . . . . . . . . . . . . . . . 159
M.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 159
Radius vector . . . . . . . . . . . . . . . . . . . . . . 159
M.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Scalar fields . . . . . . . . . . . . . . . . . . . . . . . 161
Vector fields . . . . . . . . . . . . . . . . . . . . . . 161
Tensor fields . . . . . . . . . . . . . . . . . . . . . . 162
Example M.1 Tensors in 3D space . . . . . . . . . . . . 164
M.1.3 Vector algebra . . . . . . . . . . . . . . . . . . . . . 167
Scalar product . . . . . . . . . . . . . . . . . . . . . 167
Example M.2 Inner products in complex vector space . . . 167
Example M.3 Scalar product, norm and metric in Lorentz
space . . . . . . . . . . . . . . . . . . . . . 168
Example M.4 Metric in general relativity . . . . . . . . . 168
Dyadic product . . . . . . . . . . . . . . . . . . . . . 169
Vector product . . . . . . . . . . . . . . . . . . . . . 170
M.1.4 Vector analysis . . . . . . . . . . . . . . . . . . . . . 170
The del operator . . . . . . . . . . . . . . . . . . . . 170
Example M.5 The four-del operator in Lorentz space . . . 171
The gradient . . . . . . . . . . . . . . . . . . . . . . 172
Example M.6 Gradients of scalar functions of relative dis-
tances in 3D . . . . . . . . . . . . . . . . . . 172
The divergence . . . . . . . . . . . . . . . . . . . . . 173
Example M.7 Divergence in 3D . . . . . . . . . . . . . 173
The Laplacian . . . . . . . . . . . . . . . . . . . . . . 173
Example M.8 The Laplacian and the Dirac delta . . . . . 173
The curl . . . . . . . . . . . . . . . . . . . . . . . . . 174
Example M.9 The curl of a gradient . . . . . . . . . . . 174
Example M.10 The divergence of a curl . . . . . . . . . 175
M.2 Analytical Mechanics . . . . . . . . . . . . . . . . . . . . . . 176
M.2.1 Lagrange’s equations . . . . . . . . . . . . . . . . . . 176
M.2.2 Hamilton’s equations . . . . . . . . . . . . . . . . . . 176
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

## Electrodynamics and Classical Theory of Fields and Particles

### Table Content of  Electrodynamics and Classical Theory of Fields and Particles

PART I. RELATIVISTIC DESCRIPTION OF FIELDS AND PARTICLES

chapter I Lorentz Transformations 3
1. The Physical Basis of the Lorentz Transformations 3
2. Mathematical Properties of the Lorentz Space 6
Notations 6
Vectors and Scalar Products 8
Basis 11
Complex Lorentz Space 11
3. Properties of the Lorentz Transformations 12
4. Special Lorentz Transformations: Applications 16
5. The Lorentz Group and its Representations 22
A. The Four Parts of the Lorentz Group 22
B. Correspondence with the 2x2 Unimodular Group 23
C. Spinors 25
D. Higher-Order Spinor and Tensor Representations 29
E. Infinitesimal Generators of the Lorentz and
Unimodular Groups 32
F. Complex Lorentz Group 34
6. The Principle of Relativity: Invariance and Covariance 35
7. Tensor and Spinor Fields and Momentum Space Functions 36
8. Analysis. 38
9. Further Developments and Exercises 41
Bibliography for Chapter I 45
chapter II Relativistic Dynamics 47
1. Proper Time Form of the Equations of Motion 48
Particles with Zero Rest Mass 52
Explicit Forms of the Minkowski Force 1Ð¡ 54
Angular Momentum 57
Systems of Colliding Particles 58
2. Lagrangian Form of the Equations of Motion 60
3. Canonical Form of the Equations of Motion 68
4. Electric and Magnetic Moments: Classical Spin 73
5. Further Developments and Exercises 80
Notes and Bibliography for Chapter II 83
chapter in Relativistic Field Theory 85
1. Intuitive Introduction of Fields 85
2. The Electromagnetic Field 88
Basic Equations 88
Lorentz Electrodynamics 92
Covariant form of Maxwell—Lorentz Equations 93
Gauge Transformations 94
Invariants 95
Covariant Form of Maxwell's Equations in Material Media 96
Spinor Form of the Maxwell Equations 97
3. Lagrangian Form of Field Equations 99
(A) Variations with a Fixed Boundary Ð’
(B) Variations Involving a Change of Boundary Ð’ 103
4. Conservation Laws 105
(A) Conservation Laws in Integral Form 105
(B) Conservation Laws in Differential Form 112
(C) Lagrangians Not Invariant Under Translations 115
(D) Explicit Forms of Conserved Quantities 116
5. Canonical Form of the Field Equations 119
6. Lagrangians Involving Higher-Order Derivatives 122
7. Further Developments and Exercises 127
Bibliography for Chapter III 130

PART II INTERACTIONS OF FIELDS AND PARTICLES

chapter iv Equations of Motion and Their Solutions 135
1. Interactions of Fields with External "Currents" 135
2. Interactions of Fields with a Particle 138
3. Interactions Between Fields 142
4. Solutions of Field Equations: Green's Functions 148
5. Further Developments and Exercises 163
Bibliography for Chapter IV 164
1. Radiation Field of a Moving Particle 165
Lienard-Wiechert Potentials 165
The Field Tensor F"" 168
2. Properties of the Radiation Field 170
Null Fields 170
Plane Wave Decomposition of the Radiation Field 172
Energy and Momentum of the Radiation Field 175
3. Canonical Formalism for the Radiation Field in Terms of
the Transverse Vector Potential 177
4. Energy and Momentum Radiated 179
(A) Energy Balance 184
(B) Interacting Fields and Particles 186
(C) Finite Part of Self-Force 187
(D) Mass Renormalization 190
6.. Equations of Motion with Radiation Reaction 195
7. Theory of the Electromagnetic Mass 199
8. Further Developments and Exercises 203
Bibliography for Chapter V 211
chapter vi Action-at-a-Distance Electrodynamics 213
1. The Action Principle of Fokker-Schwarzschild-Tetrode 214
2. Action Principle with Self-Energy 217
3. Mass Renormalization 219
Related Mathematical Books 221
Author Index 225
Subject Index 231

## Introduction to Electrodynamics

### Table Content of  Introduction to Electrodynamics

1 Vector Analysis 1

1.1 Vector Algebra 1

1.1.1 Vector Operations 1

1.1.2 Vector Algebra: Component Form 4

1.1.3 Triple Products 7

1.1.4 Position, Displacement, and Separation Vectors 8

1.1.5 How Vectors Transform 10

1.2 Differential Calculus 13