Note on Jackson's formalism of gauge transformation

Abstract

An outline is given of how Jackson may have obtained the inhomogeneous wave equations for the auxiliary functions $\Psi$ and $\bf V$ in his influential 2002 AJP paper on the transformation from the Lorenz gauge to other electromagnetic gauges. It clarifies the roles of these functions in the calculation of the Coulomb-gauge vector potential ${\bf A}_C$ by showing that while ${\bf A}_C$ is given directly $\nabla\times{\bf V}$, only the subtraction of $\nabla\Psi$ from the Lorenz-gauge vector potential ${\bf A}_L$ yields ${\bf A}_C$.