Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 36D / b

The magnetization $M$ is magnetic dipole moment per unit volume. It produces bound volume and surface current densityStarting from the vector potential of magnetic dipoles and integrating by parts giveswhich has precisely the form generated by those bound currents.

Write the cylinder radius as $R$ and use cylindrical radius $\rho$. Ampère's law for the free axial current gives, inside the material,At $\rho =R$ this is tangential to the cylinder. Since $n=\hat{r}$,andFor $\mu >{\mu }_0$, the bound surface current therefore runs opposite to the free wire current.