The two-body multipole problem of electrodynamics

A 2-body system composed of two objects having arbitrary distributions of charge and current is discussed. An expression for the velocity dependent potential between these two objects has been obtained in the non-relativistic approximation. This potential consists of two parts viz., a velocity independent scalar potential Φsub(eff) and another part which is linearly dependent on the relative velocity between the objects. The second part naturally suggests a vector potential Asub(eff). The potentials have been expanded into multipole terms. It has been found that Φsub(eff) is a sum of two components viz. Φsub(EE) and Φsub(MM) such that each multipole term in Φsub(EE) represents an interaction between the electric multipoles of the two systems, each term in Φsub(MM) represents an interaction between their magnetic multipoles whereas each term in Asub(eff) represents an interaction between an electric multipole of one and a magnetic multipole of the other. The results have been applied to the interaction between an electric dipole and a magnetic dipole. The symmetry among the multipole terms in Asub(eff) suggests vanishing vector potential between two identical objects. A corollary of this appears to be absence of spin orbit interaction between two identical particles in the same spin state. (author). 12 refs