Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 16D / a / Solution

In electrostatics, the relevant Maxwell equations reduce to $\nabla \cdot \mathbf{E}=\rho /{\epsilon }_0$ and $\nabla \times \mathbf{E}=0$. For a volume $V$ with outward unit normal $\mathbf{n}$, integrate the first equation and apply the divergence theorem:This is the integral form of Gauss's law, with $Q_V$ the enclosed electric charge. The zero curl is the accompanying condition allowing an electrostatic potential locally, and globally when the domain permits one.