Pre-Quantum Electrodynamics

In para/diamagnets: when \({\bf B}\) is removed, \({\bf M}\) disappears. For not too strong fields, proportionality. Custom (slightly different than for dielectrics): {\bf magnetic susceptibility} \(\chi_m\) defined as

(and not \({\bf M} = \frac{1}{\mu_0} \chi_m {\bf B}\) had the electrostatics parallel been followed historically). Materials that obey this are called linear media. Then, \[ {\bf B} = \mu_0 ({\bf H} + {\bf M}) = \mu_0 (1 + \chi_m) {\bf H} \label{Gr(6.30)} \] and thus \[ {\bf B} = \mu {\bf H}, \hspace{1cm} \mu \equiv \mu_0 (1 + \chi_m) \label{Gr(6.31)} \] where \(\mu\) is called the {\bf permeability} of the material.

Again, although \({\bf M}\) and \({\bf H}\) are proportional to \({\bf B}\), it doesn't follow that their divergence vanishes. At the boundary between two different media, they can have nonzero divergence.

In linear medium: bound volume current proportional to free current, \[ {\bf J}_b = {\boldsymbol \nabla} \times {\bf M} = {\boldsymbol \nabla} \times (\chi_m {\bf H}) = \chi_m {\bf J}_f. \label{Gr(6.33)} \]