# Can the Coulomb gauge and the Lorenz gauge be satisfied simultaneously?

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.

• it seems to me that you already answered your own question. Commented May 2, 2017 at 20:15
• Ahh thanks. For some reason, I thought that gauges were mutually exclusive. Commented May 2, 2017 at 20:26
• – Bio
Commented Feb 1 at 6:27

\begin{align} V &= V(\vec{x}); \\ \vec{A} &= \vec{\nabla} \times \vec{F}, \end{align}
Where $$\vec{F}$$ is any vector field, satisfies both gauge conditions.
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.