[[Electric and magnetic potentials]]
# Coulomb gauge

The **Coulomb gauge** of the [[Electric and magnetic potentials|magnetic potential]] $\vab A$ is defined by the gauge fixing condition
$$
\begin{align*}
\vab{\nabla}\cdot\vab A = 0
\end{align*}
$$

> [!check]- Proof of gauge validity
> Let $\vab B = \vab{\nabla}\times\vab A$ and let $f$ solve [[Poisson's equation]]
> $$
> \begin{align*}
> \nabla^2 f = - \vab{\nabla} \cdot \vab A
> \end{align*}
> $$
> i.e. $f = - G * (\vab{\nabla} \cdot\vab A)$ where $G$ is the [[Green's function for the Laplacian]].
> Then letting $\vab A' = \vab A + \vab{\nabla}f$ we have $\vab{\nabla}\times\vab A' = \vab B$ and
> $$
> \begin{align*}
> \vab{\nabla}\cdot \vab A' = \vab{\nabla} \cdot (\vab A + \vab{\nabla}f) = \vab{\nabla} \cdot \vab A + \nabla^2 f = 0
> \end{align*}
> $$
> as required.
> <span class="QED"/>

The main advantage of this gauge is that [[Ampère's circuital law]] is simplified using
$$
\begin{align*}
\vab{\nabla}\times\vab B = \vab{\nabla}\times \vab{\nabla}\times\vab A = \vab{\nabla}(\vab{\nabla}\cdot\vab A) - \nabla^2 \vab A = -\nabla^2 \vab A
\end{align*}
$$
see e.g. [[Magnetostatics MOC]].


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