(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For the potentials:

[tex] V(\vec{r}, t) = ct [/tex]

[tex] \vec{A}(\vec{r}, t) = -\frac{K}{c} x \^x[/tex]

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 conditionsufficientfor the two gauges not to be mutually exclusive?

2. Relevant equations

For the Coulomb gauge:

[tex]\bigtriangledown \cdot \vec{A} = 0[/tex]

For the Lorentz gauge:

[tex] \bigtriangledown \cdot \vec{A} = -\mu_0 \epsilon_0 \frac{\partial V}{\partial t}[/tex]

Also:

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

[tex] (\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}[/tex]

Which contain all the information in Maxwell's equations.

3. The attempt at a solution

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

[tex] K = c^2 \mu_0 \epsilon_0[/tex]

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?

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# Homework Help: Lorentz and Coulomb gauges

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