Advanced Classical Electrodynamics: Green Functions, Regularizations, Multipole Decompositions

by Ulrich D Jentschura

Overview

This textbook introduces advanced classical electrodynamics using modern mathematical techniques, with an emphasis on physical concepts. Connections to field theory and general relativity are highlighted while the book still serves as the basis for a one- or two-semester course on electrodynamics within the graduate curriculum.

Product Details

ISBN-13:	9789813222854
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	06/29/2017
Pages:	372
Product dimensions:	6.70(w) x 9.50(h) x 0.80(d)

Table of Contents

Preface vii

1 Maxwell Equations 1

1.1 Basics 1

1.1.1 Integral and Differential Forms of the Maxwell Equations 1

1.1.2 Relation of the Differential to the Integral Form 4

1.1.3 Dirac δ Function 9

1.1.4 Green Function of the Laplacian 11

1.2 Potentials and Gauges 11

1.2.1 Transverse and Longitudinal Components of a Vector Field 11

1.2.2 Vector and Scalar Potentials 13

1.2.3 Lorenz Gauge 16

1.2.4 "Gauge Always Shoots Twice" 20

1.2.5 Coulomb Gauge 20

1.3 Poynting Theorem and Maxwell Stress Tensor 23

1.3.1 Electric and Magnetic Field Energies 23

1.3.2 Maxwell Stress Tensor 25

1.4 Exercises 27

2 Green Functions of Electrostatics 29

2.1 Overview 29

2.2 Green Function of the Harmonic Oscillator 29

2.3 Electrostatic Green Function and Spherical Coordinates 33

2.3.1 Poisson and Laplace Equations in Electrostatics 33

2.3.2 Laplace Equation in Spherical Coordinates 34

2.3.3 Legendre Functions and Spherical Harmonics 37

2.3.4 Expansion of the Green Function in Spherical Coordinates 40

2.3.5 Multipole Expansion of Charge Distributions 45

2.3.6 Addition Theorem for Spherical Harmonics 47

2.3.7 Multipole Expansion in Cartesian Coordinates 48

2.3.8 Multipole Expansion in au External Field 50

2.4 Electrostatic Green Function and Cylindrical Coordinates 52

2.4.1 Laplace Equation in Cylindrical Coordinates 52

2.4.2 Cylindrical Coordinates and Bessel Functions 53

2.4.3 Orthogonality Properties of Bessel Functions 56

2.4.4 Expansion of the Green Function in Cylindrical Coordinates 60

2.5 Electrostatic Green Function and Eigenfunction Expansions 63

2.5.1 Eigen function Expansions for the Green Function 63

2.5.2 Application to Electrostatics in Spherical Coordinates 65

2.6 Summary: Green Function of Electrostatics 67

2.7 Exercises 68

3 Paradigmatic Calculations in Electrostatics 71

3.1 Overview 71

3.1.1 Differential Equations of Electrostatics 71

3.1.2 Boundary-Value Problems: One-Dimensional Analogy 72

3.1.3 Green's Theorem: Dirichlet and Neumann Green Functions 73

3.1.4 Boundary-Value Problems and Laplace Equation 76

3.2 Laplace Equation and Variational Calculations 78

3.2.1 Definition of the Functional Derivative 78

3.2.2 Variational Principle and Euler-Lagrange Equations 80

3.2.3 Variations with Constraints 82

3.2.4 Second Functional Derivative and Hilbert Space 83

3.2.5 Variational Calculation of the Capacitance of Plates 84

3.2.6 Variational Calculation for Coaxial Cylinders 88

3.2.7 Exact Integration of the Capacitance of Coaxial Cylinders 91

3.3 Laplace Equation and Series Expansions 97

3.3.1 Coordinate Systems and Special Functions 97

3.3.2 Laplace Equation in a Rectangular Parallelepiped 101

3.3.3 Laplace Equation in a Two-Dimensional Rectangle 106

3.3.4 Boundary Conditions on the Finite Part of a Long Strip 108

3.3.5 Cauchy's Residue Theorem: A Small Digression 110

3.3.6 Boundary Conditions on the Infinite Part of a Long Strip 114

3.3.7 Cylinders and Zeros of Bessel Functions 119

3.4 Laplace Equation and Dirichlet Green Functions 124

3.4.1 Dirichlet Green Function for Spherical Shells 124

3.4.2 Boundary Condition of Dipole Symmetry 125

3.4.3 Verification Using Series Expansion 126

3.5 Poisson Equation and Dirichlet Green Functions 127

3.5.1 Source Terms with Boundary Conditions 127

3.5.2 Sources and Fields in a Spherical Shell 127

3.5.3 Induced Charge Distributions on the Boundaries 130

3.6 Exercises 131

4 Green Functions of Electrodynamics 137

4.1 Green Function for the Wave Equation 137

4.1.1 Integral Representation of the Green Function 137

4.1.2 Why So Many Green Functions? 141

4.1.3 Retarded and Advanced Green Function 142

4.1.4 Feynman Contour Green Function 147

4.1.5 Summary: Green Functions of Electrodynamics 153

4.2 Action-at-a-Distance and Coulomb Gauge 154

4.2.1 Potentials and Sources 154

4.2.2 Cancellation of the Instantaneous Term 156

4.2.3 Longitudinal Electric Field as a Retarded Integral 158

4.2.4 Coulomb-Gauge Scalar Potential as a Retarded Integral 159

4.3 Exercises 161

5 Paradigmatic Calculations in Electrodynamics 163

5.1 Overview 163

5.1.1 General Considerations 163

5.1.2 Wave Equation and Green Functions 164

5.2 Helmholtz Equation 165

5.2.1 Helmholtz Equation and Green Function 165

5.2.2 Helmholtz Equation in Spherical Coordinates 167

5.2.3 Radiation Green Function 170

5.3 Localized Harmonically Oscillating Sources 171

5.3.1 Basic Formulas and Multipole Expansion 174

5.3.2 Asymptotic Limits of Dipole Radiation 177

5.3.3 Exact Expression for the Radiating Dipole 180

5.4 Tensor Green Function 181

5.4.1 Clebsch Gordan Coefficients: Motivation 181

5.4.2 Vector Additions and Vector Spherical Harmonics 185

5.4.3 Scalar Helmholtz Green Function and Scalar Potential 189

5.4.4 Tensor Helmholtz Green Function and Vector Potential 189

5.5 Radiation and Angular Momenta 194

5.5.1 Radiated Electric and Magnetic Fields 194

5.5.2 Gauge Condition, Vector and Scalar Potentials 195

5.5.3 Representations of the Vector Spherical Harmonies 197

5.5.4 Poynting Vector of the Radiation 199

5.5.5 Half-Wave Antenna 202

5.5.6 Long-Wavelength Limit of the Dipole Term 208

5.6 Potentials due to Moving Charges in Different Gauges 209

5.6.1 Moving Charges and Lorenz Gauge 209

5.6.2 Liénard-Wiechert Potentials in Coulomb Gauge 211

5.7 Exercises 214

6 Electrodynamics in Media 221

6.1 Overview 221

6.2 Microscopic and Macroscopic Equations 221

6.2 1 Macroscopic Equations and Measurements 221

6.2.2 Macroscopic Fields from Microscopic Properties 223

6.2.3 Macroscopic Averaging and Charge Density 228

6.2.4 Macroscopic Averaging and Current Density 229

6.2.5 Phenomenological Maxwell Equations 232

6.2.6 Parameters of the Multipole Expansion 233

6.3 Fourier Decomposition and Maxwell Equations in a Medium 234

6.4 Dielectric Permittivity: Various Examples 238

6.4.1 Sellmeier Equation 238

6.4.2 Drude Model 238

6.4.3 Dielectric Permittivity and Atomic Polarizability 242

6.4.4 Dielectric Permittivity for Dense Materials 244

6.5 Propagation of Plane Waves in a Medium 247

6.5.1 Refractive Index and Group Velocity 247

6.5.2 Wave Propagation and Method of Steepest Descent 249

6.6 Kramers-Kronig Relationships 254

6.6.1 Analyticity and the Kramers-Kronig Relationships 254

6.6.2 Applications of the Kramers-Kronig Relationships 256

6.7 Exercises 258

7 Waveguides and Cavities 263

7.1 Overview 263

7.2 Waveguides 264

7.2.1 General Formalism 264

7.2.2 Boundary Conditions at the Surface 267

7.2.3 Modes in a Rectangular Waveguide 270

7.3 Resonant Cavities 279

7.3.1 Resonant Cylindrical Cavities 279

7.3.2 Resonant Rectangular Cavities 287

7.4 Exercises 291

8 Advanced Topics 295

8.1 Overview 295

8.2 Lorentz Transformations, Generators and Matrices 297

8.2.1 Lorentz Boosts 297

8.2.2 Time Dilation and Lorentz Contraction 298

8.2.3 Addition Theorem 299

8.2.4 Generators of the Lorentz Group 300

8.2.5 Representations of the Lorentz Group 303

8.3 Relativistic Classical Field Theory 309

8.3.1 Maxwell Tensor and Lorentz Transformations 309

8.3.2 Maxwell Stress-Energy Tensor 310

8.3.3 Lorentz Transformation and Biot-Savart Law 315

8.3.4 Relativity and Magnetic Force 316

8.3.5 Covariant Form of the Liénard-Wiechert Potentials 320

8.4 Towards Quantum Field Theory 322

8.4.1 Casimir Effect and Quantum Electrodynamics 322

8.4.2 Zero-Point Energy 324

8.4.3 Regularization and Renormalization 326

8.5 Classical Potentials and Renormalization 331

8.5.1 Potential of a Uniformly Charged Plane 331

8.5.2 Potential of a Uniformly Charged Long Wire 332

8.5.3 Charged Structures in 0.99 and 1.99 Dimensions 334

8.6 Open Problems in Classical Electromagnetic Theory 336

8.6.1 Abraham-Minkowski Controversy 336

8.6.2 Relativistic Dynamics with Radiative Reaction 338

8.6.3 Electrodynamics in General Relativity 342

8.7 Exercises 347

Bibliography 351

Index 355