PH 622,3: Classical Electrodynamics

W, S 2021 (D. Belitz)

Chapter 0: Relativistic Mechanics

$1: The axioms of Mechanics
1.1 One free point particle
1.2 Potentials

$2: The Euler-Lagrange equations
2.1 Three classic problems
2.2 The fundamental lemma of the calculus of variations
2.3 The Euler-Lagrange equations
2.4 Variational problems with a constraint
2.5 Euler-Lagrange equations for fields

$3: Relativistic Mechanics
3.1 Newton's first law
3.2 Newton's second law
3.3 Example: Einstein's law of falling bodies

Chapter 1: Maxwell’s Equations

$1: The variational principle of classical electrodynamics
1.1 The Maxwellian action
1.2 Euler-Lagrange equations for fields
1.3 The field equations

$2: Conservation laws, and gauge invariance
2.1 The continuity equation for the 4-current
2.2 The energy-momentum tensor
2.3 The continuity equation for the energy-momentum tensor
2.4 Gauge invariance

$3: Electric and magnetic fields
3.1 The field tensor in terms of Euclidian vector fields
3.2 Maxwell’s equations
3.3 Discussion of Maxwell’s equations
[3.4 Relation between fields and potentials]
[3.5 Charges in electromagnetic fields]
[3.5a Digression: Potentials in Mechanics, and Newton's Second Law]
3.6 Poynting’s theorem

$3a: Point charges subject to electromagnetic fields
3a.1: Einsteinian mechanics
3a.2: Potentials, and Newton's Second Law
3a.3: The motion of charges in electromagnetic fields

$4: Lorentz transformations of the fields
4.1 Physical interpretation of a Lorentz boost
4.2 Transformation of E and B under a Lorentz boost
4.3 Lorentz invariants

$5: The superposition principle of Maxwell theory
5.1 Real solutions
5.2 Complex solutions

Chapter 2: Static Solutions of Maxwell’s Equations

$1: Poisson's equation
1.1 Electrostatics
1.2 Magnetostatics

$2: Solutions of Poisson's equation
2.1 The general solution of Poisson's equation
2.2 The Coulomb potential
2.3 Poisson's formula
2.4 The field of a uniformly moving charge
2.5 The electrostatic interaction
2.6 The law of Biot and Savart
2.7 The magnetostatic interaction

$3: Multipole expansion for static fields
3.1 The electric dipole moment
3.2 Legendre functions, and spherical harmonics
3.3 Separation of the Laplace operator in spherical coordinates
3.4 Expansion of harmonic functions in spherical harmonics
3.5 Multipole expansion of the electrostatic potential
3.6 Multipole expansion of the elecrostatic interaction
3.7 The magnetic moment
3.8 The magnetostatic energy of a current distribution
3.9 The energy of dipoles in external fields

Chapter 3: Electromagnetic Waves in Vacuum

$1: Plane electromagnetic waves
1.1 The wave equation
1.2 Plane waves
1.3 The orientation of the fields
1.4 Monochromatic plane waves
1.5 Polarization of electromagnetic waves
1.6 The 4-wave vector, and the Doppler effect

$2: The wave equation as an initial value problem
2.1 The wave equation in Fourier space
2.2 The general solution of the wave equation

Chapter 4: Electromagnetic Radiation

$1: Review of potentials and gauges
1.1 Potentials and fields
1.2 Gauge conventions

$2: Green's functions in the Lorentz gauge
2.1 The concept of a Green's function
2.2 Green's functions for the wave equation
2.3 The retarded potentials

$3: Radiation by time-dependent sources
3.1 Asymptotic potentials and fields
3.2 The radiated power
3.3 Radiation by an accelerated charged point particle
3.4 Dipole radiation

$4: Spectral distribution of radiated energy
4.1 Retarded potentials in frequency space
4.2 Asymptotic potentials and field
4.3 The spectral distribution of the radiated energy
4.4 Spectral distribution for dipole radiation
4.5 Example: Radiation by a damped harmonic oscillator

$5: Cerenkov radiation
5.1 The time-Wigner function, and the macroscopic power spectrum
5.2 Cerenkov radiation

$6: Synchrotron radiation
6.1 Relativistic motion of a charged particle in a homogeneous B-field
6.2 The power spectrum of synchrotron radiation
6.3 Qualitative explanation of the main features
6.4 The polarization of synchrotron radiation

$7: Scattering of light by small obstacles
7.1 The scattering cross section
7.2 Thomson scattering
7.3 Scattering by a bound charge

Chapter 5: Some Aspects of the Electrodynamics of Continuous Media

$1: Maxwell equations for a dielectric medium
1.1 Electrostatics of dieletrics
1.2 Magnetostatics
1.3 Summary of static Maxwell equations in a dielectric medium
1.4 Generalization to dynamics

$2: Introduction to the theory of causal functions
2.1 Causal functions
2.2 Spectrum and reactive part
2.3 Hilbert-Stieltjes transforms
2.4 Simple examples of Hilbert-Stieltjes transforms
2.5 Spectral representation, and Kramers-Kronig relations
2.6 Application: The dielectric function