Maxwell equations in matter

In macroscopic matter with free charge density $\rho$ and free current density $J$,The electric displacement $D$ and magnetic field strength $H$ encode the material response.

Constitutive relations specify how matter responds to fields. A homogeneous isotropic linear electromagnetic medium obeys $\mathbf{D}=\epsilon \mathbf{E}$ and $\mathbf{B}=\mu \mathbf{H}$.

Electromagnetic permeability relates magnetic flux density to magnetic field strength through $\mathbf{B}=\mu \mathbf{H}$ in an isotropic linear medium.

A linear anisotropic medium has constitutive relations $D_i={\epsilon }_{ij}{E}_j$ and $B_i={\mu }_{ij}{H}_j$. Symmetric, time-independent tensors make the field-energy density a quadratic form.