classical-electrodynamics

name	classical_electrodynamics
description	Use when solving problems involving Maxwell's equations, electrostatics, magnetostatics, electromagnetic waves, radiation, or relativistic electrodynamics.

Classical Electrodynamics

Apply this skill when the problem involves electric and magnetic fields, charges and currents, electromagnetic wave propagation, radiation from accelerating charges, or relativistic formulations of electrodynamics.

Goal

Solve electromagnetic problems using Maxwell's equations, boundary conditions, and standard techniques (method of images, multipole expansion, Green's functions, retarded potentials).

Scope

• Electrostatics: Coulomb's law, Gauss's law, Poisson/Laplace equations, boundary value problems, method of images, multipole expansion
• Magnetostatics: Biot-Savart law, Ampere's law, magnetic vector potential, magnetic multipoles
• Electromagnetic waves: plane waves, polarization, reflection/refraction, waveguides, cavities
• Radiation: Larmor formula, retarded potentials, Lienard-Wiechert fields, dipole radiation, synchrotron radiation
• Relativistic formulation: field tensor F^{mu nu}, covariant Maxwell equations, Lorentz transformations of fields
• Electromagnetic energy and momentum: Poynting vector, Maxwell stress tensor, radiation pressure

Inputs

• charge_current_distribution: Source charges and/or currents (static or time-dependent)
• geometry: Boundary conditions, conductor shapes, dielectric interfaces
• medium_properties: Permittivity, permeability, conductivity (if not vacuum)

Outputs

• fields: Electric and magnetic field solutions (E, B or potentials phi, A)
• energy_momentum: Energy density, Poynting vector, radiated power
• multipole_moments: Electric and magnetic multipole moments (if applicable)

Workflow

1. Identify the problem type.

   • Static vs time-dependent.
   • Free space vs boundaries/media.
   • Source-driven vs eigenmode problem.

2. Choose the appropriate method.

   • High symmetry (spherical, cylindrical, planar): use symmetry-adapted coordinates and separation of variables.
   • Conductors: method of images when applicable.
   • Arbitrary charge distributions: multipole expansion for far-field behavior.
   • Time-dependent sources: retarded Green's function / Lienard-Wiechert potentials.
   • Waveguides/cavities: eigenmode decomposition (TE, TM, TEM modes).

3. Solve the equations.

   • Apply boundary conditions: tangential E continuous, normal D discontinuous by sigma_free, etc.
   • For radiation problems, compute fields in the radiation zone (far field) where 1/r terms dominate.

4. Compute derived quantities.

   • Poynting vector: S = (1/mu_0) E x B.
   • Radiated power: P = integral S . dA over a closed surface.
   • Larmor formula for non-relativistic radiation: P = q^2 a^2 / (6 pi epsilon_0 c^3).

5. Verify.

   • Check Maxwell's equations are satisfied.
   • Verify boundary conditions.
   • Check limiting cases (far field, near field, static limit).

Quality Checks

• Divergence of B must be zero everywhere.
• In source-free regions, divergence of E must be zero (or equal to rho/epsilon_0 with sources).
• Energy conservation: rate of change of field energy + Poynting flux = - J . E (work done on charges).
• Far-field radiation must fall off as 1/r, with power going as 1/r^2.

Constraints

• Always specify the gauge choice when using potentials (Coulomb gauge, Lorenz gauge, etc.).
• Do not mix SI and Gaussian units; state the unit system explicitly.
• Method of images applies only when the geometry allows exact image placement; do not use it for arbitrary shapes.
• Retarded potentials, not instantaneous ones, must be used for time-dependent sources.