Integral equation for scattering by a dielectric

The determination of the scattered and transmitted transient electromagnetic waves produced by a uniform dielectric body is reduced to the solution of a singular integral equation of the first kind for one tangential vector field defined on the surface. All derivations are carried out within the heuristic approach to Green functions and delta functions. The electric and magnetic fields are expressed in terms of the sources, initial values, and the boundary values by means of the Green function for the scalar wave equation. The appropriate integral equation is derived, and the integrals for the scattered and transmitted fields are given. The simpler problem of scattering of scalar waves is developed first. Formulas for the scattering of monochromatic fields are also given in the scalar and electromagnetic cases when transmitted fields do not vanish.