Gauge conditions concerning vector potential and potential

[Lindsayyyy]

Hi everyone

Homework Statement

Give is a generall gauge transformation $$\Phi \rightarrow \Phi ' =\Phi -\frac {\partial \chi}{\partial t}$$
and
$$\vec A \rightarrow \vec A' = \vec A + \nabla \chi$$

first task for now is the following: How do I have to choose Chi in order to fulfill the lorenz gauge condition.

Homework Equations

$${\rm div} \vec A + \frac{1}{c^2} \frac{\partial}{\partial t}\phi = 0$$

The Attempt at a Solution

FIrst of all I'm not even sure if I have to discuss phi and A as if they are linked to each other or not. But let's take a look at my A

I tried to use the divergence on my A'

$$div \vec A' = div \vec A + div \nabla \chi$$ then I use the Lorenz gauge condition for div a and I finally get

$$\nabla ^2 \chi +\mu_0 \epsilon_0 \frac {\partial^2 \chi}{\partial t^2}=0$$

Is this the right approach ? I'm stuck here though I don't know how I have to choose my chi now and I still haven't taken a look at my potential.

Thanks for your help in advance.

 

[dextercioby]

That's correct. Lorenz gauge implies that the parameter is a solution of the free wave equation.

 

[Lindsayyyy]

thanks for the quick reply. Do I have to do something else with my potential phi or is the task done with that?