Coulomb gauge

The Coulomb gauge is an electrodynamic gauge for Maxwell's equations in which we set the Divergence of the magnetic vector potential to zero:

∇⋅A=0\nabla\cdot \mathbf{A}=0

In this gauge, the electric potential becomes the well known Gauss' law:

∇2V=−ρε0\nabla ^{2}V=- \frac{\rho}{\varepsilon_{0}}

The upside of this gauge is that VV is relatively easy to calculate, since it is given by Poisson's equation and that is a very well known and very well studied equation. The downside is that A\mathbf{A} is a nightmare to calculate. The general solution to Maxwell's equations to this gauge is

V(r,t)=14πε0∫Vρ(r′,t)rdτ′V(\mathbf{r},t)=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}} \frac{\rho(\mathbf{r}',t)}{\mathfrak{r}}d\tau'

for VV. There is no general solution for A\mathbf{A}, which would be the solution to

∇2A−μ0ε0∂2A∂t2=−μ0J+μ0ε0∇(∂V∂t)\nabla ^{2}\mathbf{A}-\mu_{0}\varepsilon_{0}\frac{ \partial ^{2}\mathbf{A} }{ \partial t^{2} } =-\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\nabla\left( \frac{ \partial V }{ \partial t } \right)

a second-order differential equation.