Non-relativistic scattering in the presence of moving objects: The Mie problem for a moving sphere

Recently non-relativistic boundary conditions based on the Lorentz force formulas have been introduced. It was shown that to the first order in the relative velocity v/c the results are in agreement with the exact relativistic formalism. Specific boundary value problems have been solved to get concrete results and demonstrate the feasibility of implementing the formalism. These included examples involving plane and cylindrical interfaces. Presently the velocity-dependent Mie problem, viz. scattering of a plane wave by a moving sphere, is investigated. The sphere is assumed to move in a material medium without mechanically affecting the medium. The analysis follows closely the solution for the cylindrical case, given before. The mathematics here (involving spherical vector waves and harmonics) is more complicated, and therefore sufficient detail and references are provided. The interesting feature emerging from the present analysis is that the velocity-dependent effects induce higher order multipoles, which are not present in the classical Mie solution for scattering by a sphere at rest. The formalism is sufficiently general to deal with arbitrary moving objects.