Multipole Expansion of Relativistic Electric Potential

Abstract

The concept of relativistic multipole expansion is applied to determine the multipole moments of the electric potential of a charged system moving at constant velocity with respect to a stationary observer. A system of discrete point charges and a charged plate of uniform surface charge density were considered for analysis. For relativistic considerations, two reference frames: namely S and Sˊ were chosen. Sˊ, which contains the charged system, is moving with constant velocity relative to frame S, to which stationary observer is attached. For easy comparison, both multipole expansions are expressed in terms of the coordinates of the Sˊ frame. The two observers in S and Sˊ, in general, calculate different expressions of multipole expansion for the same charged system because of Lorentz contraction effect. The multipole terms of the potential evaluated by the observer in S is time dependent. This shows that there are also magnetic effects, which are not observed by an observer in Sˊ.

Keywords: Charge Density; Electric Potential; Magnetic Vector Potential; Multipole Expansion