Surface charge and surface current densities at material boundaries

In electromagnetism, materials with a polarization density P→ or a magnetization density M→ are known to exhibit a bound surface charge density σb=P→·n̂ or a surface current density κ→b=M→×n̂, respectively, where n̂ is the unit vector perpendicular to the material boundary surface, directed outward. These expressions can be obtained from volume integrations for the electric potential V, or the magnetic vector potential A→, in which the integrals are restricted to the material volumes delimited by their respective boundaries. In that case, applying the divergence theorem leads to surface integrals on material boundaries and to the above-mentioned surface quantities. In this paper, a simple derivation is presented, which shows that both σb and κ→b are included in the expressions for the volume charge or current densities, provided that the divergence and curl operators are evaluated at the boundary so as to account for discontinuities at interfaces.