Can Both Lorentz and Coulomb Gauges Be Satisfied Simultaneously?

[ultimateguy]

Homework Statement

For the potentials:

$$V(\vec{r}, t) = ct$$
$$\vec{A}(\vec{r}, t) = -\frac{K}{c} x \^x$$

c being velocity of light in a vacuum, determine the constant K assuming the potentials satisfy the Lorentz gauge.
b) Do these potentials satisfy the Coulomb gauge as well?
c) Show that for a set of potentials the Coulomb and Lorentz gauges can be simultaneously satisfied if V does not vary with time.
d) Is this condition sufficient for the two gauges not to be mutually exclusive?

Homework Equations

For the Coulomb gauge:
$$\bigtriangledown \cdot \vec{A} = 0$$

For the Lorentz gauge:

$$\bigtriangledown \cdot \vec{A} = -\mu_0 \epsilon_0 \frac{\partial V}{\partial t}$$

Also:

$$\bigtriangledown^2 + \frac{\partial}{\partial t} (\bigtriangledown \cdot \vec{A}) = -\frac{1}{\epsilon_0} \rho$$

$$(\bigtriangledown^2 \vec{A} - \mu_0 \epsilon_0 \frac{\partial^2 \vec{A}}{\partial^2 t}) - \bigtriangledown(\bigtriangledown \cdot \vec{A} + \mu_0 \epsilon_0\frac{\partial V}{\partial t}) = -\mu_0 \vec{J}$$

Which contain all the information in Maxwell's equations.

The Attempt at a Solution

I solved the first part, found that the constant K is

$$K = c^2 \mu_0 \epsilon_0$$

My question is, how do I show that these potentials "satisfy" a gauge? Do I just plug the potentials into the condition for the divergence of A or is it something else?

 

[Avodyne]

You plug them into the appropriate "For the ____ gauge" equation, and see if the equation is true or false for those particular potentials.