Potential Due to a Magnetized Object

Magnetization is the magnetic dipole moment per unit volume that occurs when magnetic materials are placed in an external magnetic field. In paramagnets and ferromagnets, dipoles align with the field direction, while in diamagnets they align opposite to it. This state of magnetic polarization plays a similar role to polarization in electrostatics, enabling the material to generate its own magnetic field.

In a uniformly magnetized material, equal and opposite currents in adjacent current loops cancel internally, leaving a net surface-bound current along the boundary. When magnetization is non-uniform, currents no longer cancel completely, creating a net volume-bound current inside the material. The volume-bound current equals the curl of the magnetization vector, while surface current is the cross product of magnetization and the boundary's unit vector.

Surface-bound current density represents the current per unit thickness at the material's boundary. By substituting the dipole moment into the magnetization expression, the surface current density equals the magnetization. This relationship can be expressed in vector form since the surface current exists only across the boundary of the magnetized material.

The vector potential created by a magnetized material is determined solely from bound currents and equals the sum of potentials produced by surface and volume-bound currents. This approach mirrors electrostatic polarization, where the field of a polarized object results from bound volume and surface charges. The magnetic vector potential provides an alternative method to calculate fields from magnetized materials.

In a uniform external magnetic field, magnetization remains constant throughout the material, so adjacent current loops maintain equal and opposite currents that cancel completely. This cancellation eliminates volume-bound currents, leaving only surface-bound current at the material's boundary. Non-uniform fields prevent this cancellation, allowing volume currents to persist inside the material.

Like any steady current, bound volume current obeys current conservation laws. The divergence of the bound volume current is zero because the volume current equals the curl of the magnetization vector, and the divergence of any curl is mathematically zero. This ensures that charge is conserved throughout the magnetized material.

Paramagnets and ferromagnets exhibit magnetization with dipoles aligning in the direction of the applied external magnetic field. Diamagnets show opposite behavior, with dipoles aligning against the field direction. Despite these directional differences, all three material types undergo the same magnetization process—a state of magnetic polarization defined as dipole moment per unit volume.