Electromagnetism

Potentials and Gauge Freedom

Six field components traded for four potentials. The redundancy left over is the gauge freedom.

Updated July 2026

Maxwell's equations are six coupled equations for six field components. Two of them, the ones with no sources, are constraints rather than dynamics: they say the magnetic field has no divergence and the electric field's curl is tied to a changing magnetic field. Those two can be solved once and for all by writing the fields in terms of potentials, which is both a simplification and, as it turns out, a source of a subtle kind of freedom.

Solving two equations for free

Since $\nabla \cdot B = 0$ everywhere, $B$ is the curl of something:

B = \nabla \times A .

Put that into Faraday's law and it forces $\nabla \times (E + \partial_{t} A) = 0$ , so the combination in the bracket is a gradient:

E = - \nabla φ - \frac{\partial A}{\partial t} .

Four numbers now, the scalar $φ$ and the vector $A$ , in place of six, and the two source-free Maxwell equations are satisfied automatically by construction.

The freedom

The potentials are not unique. The same fields come from

A \to A + \nabla λ, φ \to φ - \frac{\partial λ}{\partial t},

for any scalar function $λ (r, t)$ , because the curl of a gradient and the mismatched time derivative cancel exactly. This is a gauge transformation, and the fact that no measurement can tell the two sets of potentials apart is gauge freedom.

Definition

Fixing a gauge

Choosing a condition that pins down $λ$ is called fixing the gauge. It removes the redundancy without changing any field. The two common choices are the Coulomb gauge, $\nabla \cdot A = 0$ , and the Lorenz gauge, $\nabla \cdot A + μ_{0} ε_{0} \partial_{t} φ = 0$ .

Why the Lorenz gauge is worth it

Substitute the potentials into the two Maxwell equations that do have sources, and in the Lorenz gauge the mess uncouples into two identical wave equations:

□ φ = - \frac{ρ}{ε _{0}}, □ A = - μ_{0} J,

where $□ = \nabla^{2} - \frac{1}{c ^{2}} \partial_{t}^{2}$ . Each potential obeys a wave equation driven by its own source, the charge for $φ$ and the current for $A$ . Nothing could be cleaner, and the solutions are the retarded potentials, fields that arrive at a distance a time $r / c$ after the source moved.

The redundancy is not a defect to be tidied away and forgotten. Insisting that physics should not depend on the choice of $λ$ is, in more advanced treatments, exactly what forces the existence of the electromagnetic field in the first place. Here it is enough to notice that the freedom is real, that fixing it is a convenience rather than a physical statement, and that the right fix turns six coupled equations into two wave equations.