Coulombgauge

The Coulomb gauge is a condition used in electromagnetism to fix the gauge freedom of the vector potential A. It is defined by the divergence-free requirement ∇·A = 0. The scalar potential φ remains and is determined by the charge distribution through Poisson’s equation ∇^2 φ = -ρ/ε0, so φ is an instantaneous function of the charge density ρ.

In this gauge, Maxwell’s equations separate into two parts. The scalar potential is set by the charge density at all times, while the vector potential A obeys a wave equation driven by the transverse part of the current J. Specifically, ∇^2 A − μ0ε0 ∂^2A/∂t^2 = −μ0 J_T, where J_T is the solenoidal (divergence-free) component of J. The full current J can be decomposed into transverse and longitudinal parts, with the longitudinal part effectively accounted for by φ. The result is that the Coulomb interaction is contained in φ, while radiation and propagation are carried by A.

Gauge transformations of the form A → A + ∇χ and φ → φ − ∂χ/∂t preserve the physics, but in the Coulomb gauge χ is constrained to keep ∇·A = 0 (i.e., the transformed potential must also satisfy the gauge condition).

The Coulomb gauge is not Lorentz invariant, so it is most convenient in nonrelativistic or quasi-static contexts and in quantum electrodynamics when separating instantaneous Coulomb interactions from transverse radiation. It is widely used in atomic, molecular, and condensed-mmatter physics, as well as in certain formulations of cavity and plasma physics.