Magnetic Materials

Matter Responds to B

Just as a dielectric polarises, matter magnetises. The magnetisation $\mathbf{M}$ is the magnetic moment per unit volume, and it is equivalent to bound current densities:

$$\mathbf{K}_b = \mathbf{M} \times \hat{\mathbf{n}}, \qquad \mathbf{J}_b = \nabla \times \mathbf{M}$$

These are real currents, circulating atomic currents that do not transport charge through the material.

To keep Ampère's law usable, magnetostatics defines the auxiliary field

$$\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$$

whose curl counts only free current: $\nabla \times \mathbf{H} = \mathbf{J}_f$. This is the exact counterpart of $\mathbf{D}$ in electrostatics, and it inherits the same warning, since $\mathbf{B}$ is what exerts force.

For linear media $\mathbf{M} = \chi_m \mathbf{H}$ and $\mathbf{B} = \mu_0\mu_r\mathbf{H}$ with $\mu_r = 1 + \chi_m$. Here the parallel with dielectrics breaks in an important way: $\chi_m$ can be negative.

Diamagnetism is universal, an induced response opposing the change, present in every material but usually masked. Paramagnetism comes from permanent moments partially aligning against thermal disorder, following Curie's law $\chi_m \propto 1/T$.

Ferromagnets have moments aligned in domains by exchange coupling. They are non-linear and history-dependent: the $B$-$H$ curve forms a hysteresis loop, whose area is the energy dissipated per cycle. Above the Curie temperature thermal motion destroys the ordering and the material becomes paramagnetic.

Common pitfall: assuming $\mu_r > 1$ because $\varepsilon_r > 1$ always is. Diamagnets have $\mu_r$ slightly below 1 and are pushed out of a field, the reason a superconductor, the perfect diamagnet with $\chi_m = -1$, levitates.