The Kirchhoff gauge

TL;DR

This paper analyzes the Kirchhoff gauge in classical electrodynamics, showing that despite its unphysical scalar potential, the resulting electric and magnetic fields match the standard retarded solutions, highlighting its relation to velocity gauges.

Contribution

It provides explicit solutions for the scalar and vector potentials in the Kirchhoff gauge and clarifies their physical implications and relation to retarded fields.

Findings

Scalar potential satisfies an elliptical equation

Electric and magnetic fields match retarded solutions

Kirchhoff gauge is a type of velocity gauge

Abstract

We discuss the Kirchhoff gauge in classical electrodynamics. In this gauge the scalar potential satisfies an elliptical equation and the vector potential satisfies a wave equation with a nonlocal source. We find the solutions of both equations and show that, despite of the unphysical character of the scalar potential, the electric and magnetic fields obtained from the scalar and vector potentials are given by their well-known retarded expressions. We note that the Kirchhoff gauge pertains to the class of gauges known as the velocity gauge.