Coulomb gauge

The Coulomb gauge is the constraint ∇·A = 0 — the vector potential is required to be divergence-free. Like the Lorenz gauge, it exploits the gauge freedom of electrodynamics (a function χ can be added to V and ∇χ added to A without changing E or B) to pick a particular choice that simplifies the equations.

Under the Coulomb gauge, the scalar potential satisfies the instantaneous Poisson equation ∇²V = −ρ/ε₀ — the same V you know from electrostatics, with no time-derivative term. Whenever the charge density moves, V at every point in space instantly rearranges to the new Poisson solution. This seems to violate causality — V propagates faster than light! — but it is a gauge artefact: the physical fields E and B (which is what you measure) still propagate at c, because the gradient of the instantaneous V exactly cancels the instantaneous part of −∂A/∂t to leave only the retarded, causal E field. The gauge potentials can do unphysical things as long as the observable fields don't.

The vector potential in the Coulomb gauge satisfies the wave equation □A = μ₀J_T, where J_T = J − ε₀ ∂(∇V)/∂t is the transverse (divergence-free) part of the current. All the radiation physics — the part of electromagnetism that actually propagates — is in A; the static Coulomb forces are in V. This makes the Coulomb gauge natural for non-relativistic atomic physics, where the quasi-instantaneous Coulomb interaction between electron and nucleus dominates and radiation is a small correction. It is also commonly used in quantum field theory when working with specific gauge-fixed Hamiltonians. For manifestly Lorentz-covariant work, the Lorenz gauge is preferred; for practical atomic calculations, Coulomb gauge wins.