Retardation and relativity: new integrals for electric and magnetic potentials of time-independent charge distributions moving with constant velocity

Abstract

New integrals for electric and magnetic potentials of arbitrary time-independent charge distributions moving with constant velocity are derived. In contrast to the familiar retarded potential integrals, the new integrals express the potentials in terms of the position occupied by the charge distribution at the moment t for which the potentials are being determined. Two different methods are used for the derivation. First, the integrals are derived by converting retarded potential integrals for a moving charge distribution into the corresponding present position integrals. Second, the integrals are derived by applying relativistic Lorentz - Einstein transformations to integrals representing potentials of stationary charge distributions. Two types of integrals are obtained: integrals expressing the potentials in terms of charge density as such, and integrals expressing the potentials in terms of the inhomogeneities in the density of the charge. Illustrative examples on calculating electric and magnetic potentials and fields of moving charge distributions by means of the new integrals are presented.