Maxwell's equations, the Lorentz force law, and conservation of charge. Classical relativistic field theory of charges, currents, and fields.
electromagnetism
Gauss's law (electric)
Electric flux through a closed surface equals enclosed charge divided by ε₀; differential form: ∇·E = ρ/ε₀.
Gauss's law for magnetism
No magnetic monopoles exist; total magnetic flux through any closed surface is zero. Differential form: ∇·B = 0.
Faraday's law of induction
A time-varying magnetic field induces a circulating electric field.
Ampère-Maxwell law
Circulating magnetic field is generated by current density and by displacement current ∂E/∂t.
Lorentz force law
The electromagnetic force on a point charge q moving with velocity v is F = q(E + v × B).
Conservation of electric charge
Electric charge is conserved: ∂ρ/∂t + ∇·J = 0. Derivable from Maxwell's equations; follows from U(1) gauge symmetry (Noether).
Electromagnetic field tensor F^{μν}
Antisymmetric 2-tensor unifying E and B: F^{μν} = ∂^μ A^ν − ∂^ν A^μ. Maxwell's equations become ∂_μ F^{μν} = μ₀ J^ν and ∂_μ *F^{μν} = 0.
Electromagnetic four-potential A^μ
Four-vector (φ/c, A) generating E = −∇φ − ∂_t A and B = ∇×A. Gauge ambiguity A → A + ∂^μ λ fixed by Lorenz, Coulomb, or radiation gauges.
Retarded potentials
Causal solution to Maxwell's equations in the Lorenz gauge: A^μ(x,t) = (μ₀/4π) ∫ J^μ(x', t_r)/|x−x'| d³x' with t_r = t − |x−x'|/c.
Liénard–Wiechert potentials
Exact retarded potentials of a point charge in arbitrary motion. Basis for Larmor radiation, synchrotron spectra, and radiation reaction.
Electromagnetic multipole expansion
Systematic expansion of 1/|x−x'| in spherical harmonics: fields far from a bounded source decompose into monopole, dipole, quadrupole, ……
Radiation reaction (Abraham–Lorentz force)
Self-force on an accelerating charge from its own radiation: F_rad = (μ₀ q² / 6π c) ȧ (non-relativistic). Abraham–Lorentz–Dirac form is…
Jefimenko's equations
Retarded expressions for E(r,t) and B(r,t) as explicit integrals of ρ and J at retarded time; causal form of Maxwell solutions.
Liénard–Wiechert potentials
Retarded potentials of point charge: φ = q/(4πε₀)/(R-β·R)|_ret, A = v φ/c². Basis of radiation from accelerated charges.
Larmor radiation formula
P = q² a² /(6πε₀ c³) non-relativistic; relativistic: P = (q² γ⁶/(6πε₀ c³))(a² - (β×a)²). Synchrotron power scaling.
Abraham–Lorentz radiation reaction
F_rad = (μ₀ q²/(6πc)) ä; runaway/pre-acceleration pathologies cured by Abraham–Lorentz–Dirac or Landau–Lifshitz reduction.
Thomson scattering
Low-energy (ω ≪ m_e c²) EM scattering off free electron: dσ/dΩ = r_e² (1+cos²θ)/2; σ_T = (8π/3)r_e² = 6.65×10⁻²⁹ m².
Rayleigh scattering (ω⁴ law)
σ ∝ ω⁴/(ω₀²-ω²)² for bound electrons; explains blue sky and red sunsets. Induced dipole oscillator picture.
Mie scattering
Exact multipole solution for EM scattering by homogeneous sphere of arbitrary size; Mie resonances; forward lobe for ka ≫ 1.
EM multipole expansion
Radiation fields expanded in electric/magnetic multipoles (E1, M1, E2...); selection rules for atomic transitions and antenna patterns.
Synchrotron radiation
Relativistic circular-motion radiation; power ∝ γ⁴, critical frequency ω_c = (3/2) γ³ c/ρ; forward-beamed into 1/γ cone.
Bremsstrahlung
Radiation from charged particle decelerated in Coulomb field; X-ray tubes, astrophysical hot plasmas (thermal bremsstrahlung, free-free).
Maxwell stress tensor
T_ij = ε₀(E_i E_j - ½δ_ij E²) + (1/μ₀)(B_i B_j - ½δ_ij B²); gives EM momentum flux and radiation pressure.
Poynting theorem
∂u/∂t + ∇·S = -J·E with S = E×H/μ₀; energy conservation for EM field plus matter.
Casimir force
F/A = -π²ℏc/(240 d⁴) between parallel conducting plates from vacuum zero-point modes; measured to %-level (Lamoreaux 1997).
Optical theorem (scattering)
Total cross-section σ_tot = (4π/k) Im f(0): imaginary part of forward scattering amplitude from unitarity; holds in QM too.
Gyromagnetic ratio and Larmor precession
γ = g q/(2m); μ = γ L; spin precesses at ω_L = γB. Basis of NMR, ESR, MRI.
Hall effect (classical)
Transverse voltage V_H = I B/(n q t) in current-carrying conductor in magnetic field; distinguishes electron vs hole carriers.
Drude model of conductivity
σ = n e² τ/m; ω-dependent σ(ω) = σ₀/(1-iωτ); plasma frequency ω_p² = ne²/(ε₀ m); foundation of metallic optics.
London equations (superconductors)
∂J/∂t = (n_s e²/m)E and ∇×J = -(n_s e²/m)B; exponential field expulsion λ_L = √(m/μ₀ n_s e²); Meissner effect.
Optical activity and circular birefringence
Chiral media have different n_L, n_R; plane polarization rotates by α = (πℓ/λ)(n_L-n_R). Faraday effect is magnetically induced version.
Electromagnetic duality (E↔B)
Vacuum Maxwell symmetric under (E,B) → (B,-E); breaking requires magnetic monopoles; Dirac quantization eg/ℏc ∈ ℤ/2.
Waveguide modes (TE/TM/TEM)
Rectangular waveguide: TE_mn, TM_mn with cutoff ω_c = c√((mπ/a)²+(nπ/b)²); coax supports TEM; used in radar, microwave engineering.
Antenna radiation and reciprocity
Half-wave dipole, Yagi arrays; directivity D, gain G; Lorentz reciprocity: transmit/receive patterns equal.
Plasma oscillation and ω_p
Electron density oscillates at ω_p = √(ne²/ε₀ m_e); EM waves reflected below ω_p; ionospheric radio propagation.