Conservation of charge at an interface

In a continuous medium, the charge density and the current density are related by the continuity equation, expressing conservation of charge. When an interface separates two continuous media, we can also have a surface charge density σ and a surface current density i on the interface surface. We consider an interface of arbitrary shape, and derive the continuity equation relating σ and i from the principle of conservation of charge. The result can be expressed in terms of the surface divergence of i on the surface, familiar from tensor analysis. In an independent approach, we derive the same equation from Maxwell’s equations at the boundary.