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): 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,

\[ {\bf B} = \mu_0 ({\bf H} + {\bf M}) = \mu_0 (1 + \chi_m) {\bf H} \tag{BHM}\label{BHM} \] and thus

\[ {\bf B} = \mu {\bf H}, \hspace{1cm} \mu \equiv \mu_0 (1 + \chi_m) \tag{BH}\label{BH} \] 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,

\[ {\bf J}_b = {\boldsymbol \nabla} \times {\bf M} = {\boldsymbol \nabla} \times (\chi_m {\bf H}) = \chi_m {\bf J}_f. \tag{JbchimJf}\label{JbchimJf} \]