Magnetization

Remarks

The magnetization is a macroscopic average. $M(x)$ is the sum of the atomic dipole moments in a region around $x$, divided by the volume of that region. The region is small at the scale of the body but large at the atomic scale.

The magnetic field of a magnetized body equals the field of two bound currents. They are the current density $J_b = \nabla \times M$ in its interior and the surface current density $K_b = M \times n$ on its boundary, where $n$ is the outward unit normal. Here $\nabla \times$ is the curl. The magnetizing field is defined by $H = B / \mu_0 - M$, where $\mu_0 > 0$ is the vacuum permeability.

Example: a ball of radius $R$ with constant magnetization $M_0$ has $J_b = 0$, since the curl of a constant field is zero. Its total dipole moment is $\frac{4}{3} \pi R^3 M_0$. The field inside the ball is the constant field $B = \frac{2}{3} \mu_0 M_0$.

Non-example: a copper wire carrying a current produces a magnetic field, but its magnetization is $0$. Its field comes from the free current of moving conduction electrons and not from aligned dipole moments of the material.