Coulomb gauge

The Coulomb gauge of the magnetic potential $⃗ 𝐀$ is defined by the gauge fixing condition

⃗ \nabla \cdot ⃗ 𝐀 = 0

Proof of gauge validity

Let $⃗ 𝐁 = ⃗ \nabla \times ⃗ 𝐀$ and let $𝑓$ solve Poisson's equation

\nabla 2 𝑓 = - ⃗ \nabla \cdot ⃗ 𝐀

i.e. $𝑓 = - 𝐺 * (⃗ \nabla \cdot ⃗ 𝐀)$ where $𝐺$ is the Green's function for the Laplacian. Then letting $⃗ 𝐀' = ⃗ 𝐀 + ⃗ \nabla 𝑓$ we have $⃗ \nabla \times ⃗ 𝐀' = ⃗ 𝐁$ and

⃗ \nabla \cdot ⃗ 𝐀' = ⃗ \nabla \cdot (⃗ 𝐀 + ⃗ \nabla 𝑓) = ⃗ \nabla \cdot ⃗ 𝐀 + \nabla 2 𝑓 = 0

as required.

The main advantage of this gauge is that Ampère's circuital law is simplified using

⃗ \nabla \times ⃗ 𝐁 = ⃗ \nabla \times ⃗ \nabla \times ⃗ 𝐀 = ⃗ \nabla (⃗ \nabla \cdot ⃗ 𝐀) - \nabla 2 ⃗ 𝐀 = - \nabla 2 ⃗ 𝐀

see e.g. Magnetostatics MOC.

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