Past exam of the options I course of the University of Oxford / 2007 / J / J1 / Solution

Ampère's law in magnetostatics iswith boundary and surface orientations related by the right-hand rule. Stokes' theorem gives the left side as ${\int }_S(\nabla \times B)\cdot ndS$, proving the integral law from the stated curl equation.

For a circular contour of radius $r$ about the axis, the left side is $2\pi r{B}_{\theta }$. For $r\le a$ the enclosed current is $I{r}^2/a^2$; for $r\ge a$ it is $I$. ThereforeBoth expressions agree at the boundary. At the axis the field is zero, interpreted by continuity even though the angular unit vector is undefined there.