Potential Due to a Magnetized Object

When placed in an external magnetic field, a magnetic material shows the alignment of the magnetic dipoles. This state of magnetic polarization, called magnetization, is the magnetic dipole moment per unit volume.

Consider a thin slab of uniformly magnetized material comprising current loops representing the individual dipoles. The equal and opposite currents in adjacent loops cancel each other, leaving a net current along the boundary, called surface bound current.

The surface-bound current density is the current per unit thickness. Substituting the   dipole moment in the magnetization expression shows that it equals the surface current. The surface current can be expressed in vector form since it exists only across the boundary.

When the magnetization is non-uniform, currents in adjacent loops no longer cancel. A net volume-bound current develops inside the material. Vectorially, this current equals the curl of the magnetization vector.

The vector potential created by a magnetized material at a field point can be estimated solely from these bound currents and is the sum of the potentials produced by the surface and volume-bound currents.