Boundary conditions in an integral approach to scattering

Scattering of electromagnetic radiation by an object of arbitrary shape or a structured surface, infinite in extent, is considered. When radiation is incident on an interface separating vacuum from a material medium, a current density is induced in the bulk and a surface current density may appear on the boundary surface. The electromagnetic field is then the sum of the incident field and the field generated by the current densities. This concept leads to expressions for the electric and magnetic fields that can easily be shown to be exact integrals of Maxwell's equations both in the vacuum and in the medium. At the boundary surface, the electric and magnetic fields must be discontinuous, with the discontinuity determined by the surface charge and current densities. This is usually referred to as boundary conditions for Maxwell's equations. We show that the integrals for the electric and magnetic fields automatically satisfy these boundary conditions, no matter the origin of the current densities.