Formulae for the electrodynamic interaction of point particles with a substrate

A complete expression is derived for the force and energy of electrodynamic (van der Waals) interaction between a point particle and a planar semi-infinite substrate. The new result implicitly includes contributions from charge fluctuations at all frequencies, the role of finite temperature, of retardation due to finite light velocity and terms due to many-body interactions occurring in condensed media. The point particle may be in a vacuum or suspended in a condensed medium.

The required expressions are deduced from Dzyaloshinskii, Lifshitz and Pitaevskii's original formula for the interaction of two flat semi-infinite media; the method is checked by using it to derive an intuitive but complete form for the electrodynamic attraction between two point particles. Evaluation of these formulae for application to measurements of particle adhesion and of deflection of molecular beams can be computed from electromagnetic properties of component materials.