Advanced Classical Electrodynamics

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• Basics
  • Integral and Differential Forms of the Maxwell Equations
  • Relation of the Differential to the Integral Form
  • Dirac δ Function
  • Green Function of the Laplacian
• Potentials and Gauges
  • Transverse and Longitudinal Components of a Vector Field
  • Vector and Scalar Potentials
  • Lorenz Gauge
  • “Gauge Always Shoots Twice”
  • Coulomb Gauge
• Poynting Theorem and Maxwell Stress Tensor
  • Electric and Magnetic Field Energies
  • Maxwell Stress Tensor
• Exercises

The following sections are included:

• Overview
• Green Function of the Harmonic Oscillator
• Electrostatic Green Function and Spherical Coordinates
  • Poisson and Laplace Equations in Electrostatics
  • Laplace Equation in Spherical Coordinates
  • Legendre Functions and Spherical Harmonics
  • Expansion of the Green Function in Spherical Coordinates
  • Multipole Expansion of Charge Distributions
  • Addition Theorem for Spherical Harmonics
  • Multipole Expansion in Cartesian Coordinates
  • Multipole Expansion in an External Field
• Electrostatic Green Function and Cylindrical Coordinates
  • Laplace Equation in Cylindrical Coordinates
  • Cylindrical Coordinates and Bessel Functions
  • Orthogonality Properties of Bessel Functions
  • Expansion of the Green Function in Cylindrical Coordinates
• Electrostatic Green Function and Eigenfunction Expansions
  • Eigenfunction Expansions for the Green Function
  • Application to Electrostatics in Spherical Coordinates
• Summary: Green Function of Electrostatics
• Exercises

The following sections are included:

• Overview
  • Differential Equations of Electrostatics
  • Boundary-Value Problems: One-Dimensional Analogy
  • Green’s Theorem: Dirichlet and Neumann Green Functions
  • Boundary-Value Problems and Laplace Equation
• Laplace Equation and Variational Calculations
  • Definition of the Functional Derivative
  • Variational Principle and Euler–Lagrange Equations
  • Variations with Constraints
  • Second Functional Derivative and Hilbert Space
  • Variational Calculation of the Capacitance of Plates
  • Variational Calculation for Coaxial Cylinders
  • Exact Integration of the Capacitance of Coaxial Cylinders
• Laplace Equation and Series Expansions
  • Coordinate Systems and Special Functions
  • Laplace Equation in a Rectangular Parallelepiped
  • Laplace Equation in a Two-Dimensional Rectangle
  • Boundary Conditions on the Finite Part of a Long Strip
  • Cauchy’s Residue Theorem: A Small Digression
  • Boundary Conditions on the Infinite Part of a Long Strip
  • Cylinders and Zeros of Bessel Functions
• Laplace Equation and Dirichlet Green Functions
  • Dirichlet Green Function for Spherical Shells
  • Boundary Condition of Dipole Symmetry
  • Verification Using Series Expansion
• Poisson Equation and Dirichlet Green Functions
  • Source Terms with Boundary Conditions
  • Sources and Fields in a Spherical Shell
  • Induced Charge Distributions on the Boundaries
• Exercises

The following sections are included:

• Green Function for the Wave Equation
  • Integral Representation of the Green Function
  • Why So Many Green Functions?
  • Retarded and Advanced Green Function
  • Feynman Contour Green Function
  • Summary: Green Functions of Electrodynamics
• Action-at-a-Distance and Coulomb Gauge
  • Potentials and Sources
  • Cancellation of the Instantaneous Term
  • Longitudinal Electric Field as a Retarded Integral
  • Coulomb-Gauge Scalar Potential as a Retarded Integral
• Exercises

The following sections are included:

• Overview
  • General Considerations
  • Wave Equation and Green Functions
• Helmholtz Equation
  • Helmholtz Equation and Green Function
  • Helmholtz Equation in Spherical Coordinates
  • Radiation Green Function
• Localized Harmonically Oscillating Sources
  • Basic Formulas and Multipole Expansion
  • Asymptotic Limits of Dipole Radiation
  • Exact Expression for the Radiating Dipole
• Tensor Green Function
  • Clebsch–Gordan Coefficients: Motivation
  • Vector Additions and Vector Spherical Harmonics
  • Scalar Helmholtz Green Function and Scalar Potential
  • Tensor Helmholtz Green Function and Vector Potential
• Radiation and Angular Momenta
  • Radiated Electric and Magnetic Fields
  • Gauge Condition, Vector and Scalar Potentials
  • Representations of the Vector Spherical Harmonics
  • Poynting Vector of the Radiation
  • Half-Wave Antenna
  • Long-Wavelength Limit of the Dipole Term
• Potentials due to Moving Charges in Different Gauges
  • Moving Charges and Lorenz Gauge
  • Liénard–Wiechert Potentials in Coulomb Gauge
• Exercises

The following sections are included:

• Overview
• Microscopic and Macroscopic Equations
  • Macroscopic Equations and Measurements
  • Macroscopic Fields from Microscopic Properties
  • Macroscopic Averaging and Charge Density
  • Macroscopic Averaging and Current Density
  • Phenomenological Maxwell Equations
  • Parameters of the Multipole Expansion
• Fourier Decomposition and Maxwell Equations in a Medium
• Dielectric Permittivity: Various Examples
  • Sellmeier Equation
  • Drude Model
  • Dielectric Permittivity and Atomic Polarizability
  • Dielectric Permittivity for Dense Materials
• Propagation of Plane Waves in a Medium
  • Refractive Index and Group Velocity
  • Wave Propagation and Method of Steepest Descent
• Kramers–Kronig Relationships
  • Analyticity and the Kramers–Kronig Relationships
  • Applications of the Kramers–Kronig Relationships
• Exercises

The following sections are included:

• Overview
• Waveguides
  • General Formalism
  • Boundary Conditions at the Surface
  • Modes in a Rectangular Waveguide
• Resonant Cavities
  • Resonant Cylindrical Cavities
  • Resonant Rectangular Cavities
• Exercises

The following sections are included:

• Overview
• Lorentz Transformations, Generators and Matrices
  • Lorentz Boosts
  • Time Dilation and Lorentz Contraction
  • Addition Theorem
  • Generators of the Lorentz Group
  • Representations of the Lorentz Group
• Relativistic Classical Field Theory
  • Maxwell Tensor and Lorentz Transformations
  • Maxwell Stress–Energy Tensor
  • Lorentz Transformation and Biot–Savart Law
  • Relativity and Magnetic Force
  • Covariant Form of the Liénard–Wiechert Potentials
• Towards Quantum Field Theory
  • Casimir Effect and Quantum Electrodynamics
  • Zero-Point Energy
  • Regularization and Renormalization
• Classical Potentials and Renormalization
  • Potential of a Uniformly Charged Plane
  • Potential of a Uniformly Charged Long Wire
  • Charged Structures in 0.99 and 1.99 Dimensions
• Open Problems in Classical Electromagnetic Theory
  • Abraham–Minkowski Controversy
  • Relativistic Dynamics with Radiative Reaction
  • Electrodynamics in General Relativity
• Exercises

The following sections are included:

• Bibliography
• Index