How the potentials in different gauges yield the same retarded electric and magnetic fields

Abstract

This paper presents a simple and systematic method for showing how the potentials in the Lorentz, Coulomb, Kirchhoff, velocity, and temporal gauges yield the same retarded electric and magnetic fields. The method uses the appropriate dynamical equations for the scalar and vector potentials to obtain two wave equations whose retarded solutions lead to the electric and magnetic fields. The advantage of this method is that it does not use explicit expressions for the potentials in the various gauges, which are generally simple to obtain for the scalar potential but difficult to calculate for the vector potential. The spurious character of the term generated by the scalar potential in the Coulomb, Kirchhoff, and velocity gauges is noted. The nonspurious character of the term generated by the scalar potential in the Lorenz gauge is emphasized.