Bound charge and bound current

Polarization and magnetization produce the effective sourcesDefining $D={ϵ}_{0}E+P$ and $H=B/{\mu }_0-M$ moves these sources into the macroscopic fields.

The current associated with electric polarization and magnetization in matter. If the magnetic response is neglected, it is ${\mathbf{J}}_b={\partial }_t\mathbf{P}$. It is distinguished from a free conduction current in the Maxwell equations in matter.

Electric polarization is electric dipole moment per unit volume. It produces bound volume and surface chargeAcross an interface, the net bound sheet charge is $({\mathbf{P}}_1-{\mathbf{P}}_2)\cdot \mathbf{n}$, where $\mathbf{n}$ points from medium one to medium two.

The electric displacement is $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$ and obeys $\nabla \cdot \mathbf{D}={\rho }_{\mathrm{free}}$. In a linear isotropic dielectric, $\mathbf{D}=ϵ\mathbf{E}$.

Magnetization is magnetic dipole moment per unit volume. It produces bound volume and surface currentsAcross an interface, the net bound sheet current is $({\mathbf{M}}_1-{\mathbf{M}}_2)\times \mathbf{n}$.

A relativistic magnetization current has the form $J_{\mathrm{mag}}^{\mu }={\partial }_{\nu }{M}^{\nu \mu }$ for an antisymmetric polarization-magnetization tensor. It is identically conserved because commuting derivatives contract an antisymmetric tensor.

The magnetic field intensity is $\mathbf{H}=\mathbf{B}/{\mu }_0-\mathbf{M}$ and obeys $\nabla \times \mathbf{H}={\mathbf{J}}_{\mathrm{free}}$ in magnetostatics. In a linear isotropic medium, $\mathbf{B}=\mu \mathbf{H}$.