Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 6D / a / Solution

In SI units the Ampère-Maxwell equation is $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0{\partial }_t\mathbf{E}$. Since $\mathbf{E}=0$, the displacement-current term vanishes and $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Apply Stokes theorem to an oriented surface $S$ with positively oriented boundary $C$:Hence the integral form of Ampère's law isHere ${\mu }_0$ is vacuum magnetic permeability, $\mathbf{J}$ is current density, and positive enclosed electric current is defined by the orientation of $S$.