Coulomb Gauge
Fixing the Gauge Freedom of the Potentials
Maxwell's equations leave the electromagnetic potentials underdetermined: the physical fields E and B are unchanged if the vector potential is shifted by the gradient of any scalar function λ and the scalar potential is shifted by the negative time derivative of that same function. This freedom, called gauge invariance, lets engineers and physicists impose one extra scalar condition to simplify the field equations without altering any measurable result. The Coulomb gauge is the choice that demands the vector potential carry no divergence, anchoring its longitudinal degree of freedom to zero and leaving only the rotational part.
The immediate payoff is that the equation for the scalar potential collapses to a Poisson equation identical in form to electrostatics, so φ is determined instantaneously across all space by the present charge distribution. The vector potential then absorbs all the time-retarded, wave-like behavior of the field. Crucially, the current that drives A is not the full current but its transverse part, obtained by stripping away the longitudinal piece tied to charge conservation. This split, rooted in the Helmholtz theorem, is what makes the gauge so clean for describing radiation: in a region with no free charge, φ can be set to zero and the entire field reduces to a transverse A.
For practical RF and microwave work the gauge is mostly a calculational and conceptual tool rather than something measured on the bench. It underpins how antenna near-field and far-field decompositions are derived, how the photon is treated as a transverse excitation in quantum optics, and how molecular and atomic light-matter coupling is written in the dipole approximation. Where retarded, manifestly causal radiation expressions are needed, designers instead reach for the Lorenz gauge, whose symmetric wave equations feed directly into the retarded potential integrals used for antenna radiation.
Governing Equations
∇·A = 0
Scalar potential (instantaneous Poisson):
∇2φ = −ρ / ε0
Vector potential (transverse-current wave equation):
∇2A − μ0ε0 ∂2A/∂t2 = −μ0JT
Helmholtz current split:
J = JL + JT, ∇×JL = 0, ∇·JT = 0
Where A = vector potential, φ = scalar potential, ρ = charge density, ε0 ≈ 8.854 × 10−12 F/m, μ0 ≈ 4π × 10−7 H/m, and JT = transverse current. A gauge transformation with scalar λ satisfying ∇2λ = 0 preserves the condition.
Coulomb Gauge vs. Lorenz Gauge
| Property | Coulomb gauge | Lorenz gauge | Practical advantage |
|---|---|---|---|
| Constraint | ∇·A = 0 | ∇·A + μ0ε0∂φ/∂t = 0 | Lorenz: symmetric form |
| φ equation | Poisson (instantaneous) | Wave equation (retarded) | Coulomb: simpler φ |
| A equation | Driven by JT only | Driven by full J | Lorenz: decoupled potentials |
| Lorentz covariance | No | Yes | Lorenz: relativistic / QED |
| Causality of fields | Preserved (cancellation) | Manifest | Lorenz: explicit retardation |
| Typical use | Bound states, field quantization | Antenna radiation, retarded potentials | Choose per problem |
Frequently Asked Questions
How does the Coulomb gauge differ from the Lorenz gauge?
The Coulomb gauge sets ∇·A = 0, so φ obeys an instantaneous Poisson equation like electrostatics and A is driven only by the transverse current. The Lorenz gauge sets ∇·A + μ0ε0∂φ/∂t = 0, giving symmetric, decoupled wave equations for both potentials. The Lorenz form is Lorentz covariant and preferred for radiation and retarded-potential work, while the Coulomb form simplifies bound-state and non-relativistic analysis.
What makes the Coulomb gauge also the transverse or radiation gauge?
Forcing ∇·A = 0 removes the longitudinal component of the vector potential and keeps only the divergence-free transverse part, per the Helmholtz decomposition. Since freely propagating waves and radiation fields are transverse, the gauge earns the names transverse gauge and radiation gauge. In source-free regions φ vanishes and the field is described entirely by the transverse A, which is why it is the standard gauge for quantizing the radiation field.
Does the instantaneous Coulomb potential violate causality?
No. The scalar potential responds instantaneously to charge changes, which looks acausal, but the measurable fields E and B stay causal. The faster-than-light pieces in the scalar-potential term and in the longitudinal vector-potential contribution cancel exactly when E is assembled, so no signal exceeds c. The instantaneous behavior is a gauge artifact with no physical consequence.