Advanced Classical Electrodynamics

Contents......Page 10
Preface......Page 8
1.1.1 Integral and Differential Forms of the Maxwell Equations......Page 16
1.1.2 Relation of the Differential to the Integral Form......Page 19
1.1.3 Dirac δ Function......Page 24
1.2.1 Transverse and Longitudinal Components of a Vector Field......Page 26
1.2.2 Vector and Scalar Potentials......Page 28
1.2.3 Lorenz Gauge......Page 31
1.2.5 Coulomb Gauge......Page 35
1.3.1 Electric and Magnetic Field Energies......Page 38
1.3.2 Maxwell Stress Tensor......Page 40
1.4 Exercises......Page 42
2.2 Green Function of the Harmonic Oscillator......Page 44
2.3.1 Poisson and Laplace Equations in Electrostatics......Page 48
2.3.2 Laplace Equation in Spherical Coordinates......Page 49
2.3.3 Legendre Functions and Spherical Harmonics......Page 52
2.3.4 Expansion of the Green Function in Spherical Coordinates......Page 55
2.3.5 Multipole Expansion of Charge Distributions......Page 60
2.3.6 Addition Theorem for Spherical Harmonics......Page 62
2.3.7 Multipole Expansion in Cartesian Coordinates......Page 63
2.3.8 Multipole Expansion in an External Field......Page 65
2.4.1 Laplace Equation in Cylindrical Coordinates......Page 67
2.4.2 Cylindrical Coordinates and Bessel Functions......Page 68
2.4.3 Orthogonality Properties of Bessel Functions......Page 71
2.4.4 Expansion of the Green Function in Cylindrical Coordinates......Page 75
2.5.1 Eigenfunction Expansions for the Green Function......Page 78
2.5.2 Application to Electrostatics in Spherical Coordinates......Page 80
2.6 Summary: Green Function of Electrostatics......Page 82
2.7 Exercises......Page 83
3.1.1 Differential Equations of Electrostatics......Page 86
3.1.2 Boundary-Value Problems: One-Dimensional Analogy......Page 87
3.1.3 Green’s Theorem: Dirichlet and Neumann Green Functions......Page 88
3.1.4 Boundary-Value Problems and Laplace Equation......Page 91
3.2.1 Definition of the Functional Derivative......Page 93
3.2.2 Variational Principle and Euler–Lagrange Equations......Page 95
3.2.3 Variations with Constraints......Page 97
3.2.4 Second Functional Derivative and Hilbert Space......Page 98
3.2.5 Variational Calculation of the Capacitance of Plates......Page 99
3.2.6 Variational Calculation for Coaxial Cylinders......Page 103
3.2.7 Exact Integration of the Capacitance of Coaxial Cylinders......Page 106
3.3.1 Coordinate Systems and Special Functions......Page 112
3.3.2 Laplace Equation in a Rectangular Parallelepiped......Page 116
3.3.3 Laplace Equation in a Two-Dimensional Rectangle......Page 121
3.3.4 Boundary Conditions on the Finite Part of a Long Strip......Page 123
3.3.5 Cauchy’s Residue Theorem: A Small Digression......Page 125
3.3.6 Boundary Conditions on the Infinite Part of a Long Strip......Page 129
3.3.7 Cylinders and Zeros of Bessel Functions......Page 134
3.4.1 Dirichlet Green Function for Spherical Shells......Page 139
3.4.2 Boundary Condition of Dipole Symmetry......Page 140
3.4.3 Verification Using Series Expansion......Page 141
3.5.2 Sources and Fields in a Spherical Shell......Page 142
3.5.3 Induced Charge Distributions on the Boundaries......Page 145
3.6 Exercises......Page 146
4.1.1 Integral Representation of the Green Function......Page 152
4.1.2 Why So Many Green Functions?......Page 156
4.1.3 Retarded and Advanced Green Function......Page 157
4.1.4 Feynman Contour Green Function......Page 162
4.1.5 Summary: Green Functions of Electrodynamics......Page 168
4.2.1 Potentials and Sources......Page 169
4.2.2 Cancellation of the Instantaneous Term......Page 171
4.2.3 Longitudinal Electric Field as a Retarded Integral......Page 173
4.2.4 Coulomb-Gauge Scalar Potential as a Retarded Integral......Page 174
4.3 Exercises......Page 176
5.1.1 General Considerations......Page 178
5.1.2 Wave Equation and Green Functions......Page 179
5.2.1 Helmholtz Equation and Green Function......Page 180
5.2.2 Helmholtz Equation in Spherical Coordinates......Page 182
5.2.3 Radiation Green Function......Page 185
5.3.1 Basic Formulas and Multipole Expansion......Page 189
5.3.2 Asymptotic Limits of Dipole Radiation......Page 192
5.3.3 Exact Expression for the Radiating Dipole......Page 195
5.4.1 Clebsch–Gordan Coefficients: Motivation......Page 196
5.4.2 Vector Additions and Vector Spherical Harmonics......Page 200
5.4.4 Tensor Helmholtz Green Function and Vector Potential......Page 204
5.5.1 Radiated Electric and Magnetic Fields......Page 209
5.5.2 Gauge Condition, Vector and Scalar Potentials......Page 210
5.5.3 Representations of the Vector Spherical Harmonics......Page 212
5.5.4 Poynting Vector of the Radiation......Page 214
5.5.5 Half-Wave Antenna......Page 217
5.5.6 Long-Wavelength Limit of the Dipole Term......Page 223
5.6.1 Moving Charges and Lorenz Gauge......Page 224
5.6.2 Lienard–Wiechert Potentials in Coulomb Gauge......Page 226
5.7 Exercises......Page 229
6.2.1 Macroscopic Equations and Measurements......Page 236
6.2.2 Macroscopic Fields from Microscopic Properties......Page 238
6.2.3 Macroscopic Averaging and Charge Density......Page 243
6.2.4 Macroscopic Averaging and Current Density......Page 244
6.2.5 Phenomenological Maxwell Equations......Page 247
6.2.6 Parameters of the Multipole Expansion......Page 248
6.3 Fourier Decomposition and Maxwell Equations in a Medium......Page 249
6.4.2 Drude Model......Page 253
6.4.3 Dielectric Permittivity and Atomic Polarizability......Page 257
6.4.4 Dielectric Permittivity for Dense Materials......Page 259
6.5.1 Refractive Index and Group Velocity......Page 262
6.5.2 Wave Propagation and Method of Steepest Descent......Page 264
6.6.1 Analyticity and the Kramers–Kronig Relationships......Page 269
6.6.2 Applications of the Kramers–Kronig Relationships......Page 271
6.7 Exercises......Page 273
7.1 Overview......Page 278
7.2.1 General Formalism......Page 279
7.2.2 Boundary Conditions at the Surface......Page 282
7.2.3 Modes in a Rectangular Waveguide......Page 285
7.3.1 Resonant Cylindrical Cavities......Page 294
7.3.2 Resonant Rectangular Cavities......Page 302
7.4 Exercises......Page 306
8.1 Overview......Page 310
8.2.1 Lorentz Boosts......Page 312
8.2.2 Time Dilation and Lorentz Contraction......Page 313
8.2.3 Addition Theorem......Page 314
8.2.4 Generators of the Lorentz Group......Page 315
8.2.5 Representations of the Lorentz Group......Page 318
8.3.1 Maxwell Tensor and Lorentz Transformations......Page 324
8.3.2 Maxwell Stress–Energy Tensor......Page 328
8.3.3 Lorentz Transformation and Biot–Savart Law......Page 330
8.3.4 Relativity and Magnetic Force......Page 331
8.3.5 Covariant Form of the Lienard–Wiechert Potentials......Page 335
8.4.1 Casimir Effect and Quantum Electrodynamics......Page 337
8.4.2 Zero-Point Energy......Page 339
8.4.3 Regularization and Renormalization......Page 341
8.5.1 Potential of a Uniformly Charged Plane......Page 346
8.5.2 Potential of a Uniformly Charged Long Wire......Page 347
8.5.3 Charged Structures in 0.99 and 1.99 Dimensions......Page 349
8.6.1 Abraham–Minkowski Controversy......Page 351
8.6.2 Relativistic Dynamics with Radiative Reaction......Page 353
8.6.3 Electrodynamics in General Relativity......Page 357
8.7 Exercises......Page 362
Bibliography......Page 366
Index......Page 370