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The Coulomb Gauge:

$\nabla \cdot A=0\\$

The Lorenz Gauge:

$ \nabla \cdot A= { \mu }_{ 0 }{ \epsilon }_{ 0 }\frac { \partial V }{ \partial t }$

Can both of these gauges be satisfied for some potential?

For example, the potentials:

$V(\vec { r } ,t)\quad =\quad 0\\ \vec { A } (\vec { r } ,t)\quad =\quad \begin{cases} \hat { j } { A }_{ 0 }cos(kx-\omega t)\quad ,\quad x>0 \\ \hat { j } { A }_{ 0 }cos(kx+\omega t)\quad ,\quad x<0 \end{cases}$

These potentials seem to satisfy both gauges. I'm unsure whether this is correct.

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    $\begingroup$ it seems to me that you already answered your own question. $\endgroup$ – AccidentalFourierTransform May 2 '17 at 20:15
  • $\begingroup$ Ahh thanks. For some reason, I thought that gauges were mutually exclusive. $\endgroup$ – Rahul Chowdhury May 2 '17 at 20:26
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Only in certain simple situation can both gauges be satisfied. $V=0$ is obviously one of them. For a general electric and magnetic field configuration, however, only one gauge can be satisfied.

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