Classical Electrodynamics · Problem 6.23

Plane Electromagnetic Waves and Wave Propagation — John David Jackson

Physics — E&M · hertz-vector-gauge-transformation

Problem

The wave equations (6.168) for the Hertz vectors contain arbitrary source terms involving the functions

\mathbf{V}

and

\xi

. Consider the gauge transformations

\mathbf{\Pi}'_e = \mathbf{\Pi}_e + \mu_0 \nabla \times \mathbf{G} - \nabla g; \quad \mathbf{\Pi}'_m = \mathbf{\Pi}_m - \mu\frac{\partial \mathbf{G}}{\partial t}

where

\mathbf{G}

and

g

are well-behaved functions of space and time. (a) Show that, if

\mathbf{G}

and

g

satisfy the wave equations

\left(\mu\epsilon\frac{\partial^2}{\partial t^2} - \nabla^2\right)\begin{Bmatrix}\mathbf{G}\\g\end{Bmatrix} = \begin{Bmatrix}\dfrac{1}{\mu}(\mathbf{V} + \nabla\xi)\\0\end{Bmatrix}

then the new polarization potentials

\mathbf{\Pi}'_e

and

\mathbf{\Pi}'_m

satisfy (6.168) with vanishing

\mathbf{V}

and

\xi

. (b) Show that the gauge transformation on the Hertz vectors is equivalent to a gauge transformation on

\mathbf{A}

and

\Phi

. What is the gauge function

\Lambda

of (6.19) in terms of

\mathbf{G}

and

g