EM-wave equation in conductors with source terms

The traditional modified Maxwell’s equations to express em wave inside conductors that I have come across are:

$$
nablacdotmathbf E = 0
nablacdotmathbf B = 0
nablatimesmathbf E = -frac{dmathbf B}{dt}
nablatimesmathbf B = musigmamathbf E+muepsilonfrac{partialmathbf E}{partial t}$$

where use of $mathbf J=sigma mathbf E$ has been made

This makes sense, inside a conductor wherever there is an electric field, there is a corresponding current density due to free charge throughout

However I am not sure of the physical significance of setting the divergence of $mathbf E$ to be zero?

Why is this done, and for that matter in the general wave equation in free space why does this also occur? As an EM wave needs to be generated by a source (the free space wave equation I’m guessing is to show the field itself behaves like a wave in general).

But for inside conductors, what would adding that the $nablacdotmathbf E = rho/epsilon$ actually physically mean, and what is the difference between the two?

I have had a good go at solving for the potentials $phi$ and $mathbf A$ with source terms.

Solving for $mathbf A$ is pretty straightforward, providing the gauge choice is

$$nablacdotmathbf A – musigmaphi – muepsilonfrac{partial phi}{partial t} = 0 $$

and the equation for the magnetic vector potential I get is:

$$
– nabla^2mathbf A+musigmafrac{dmathbf A}{dt} + muepsilonfrac{d^2mathbf A}{dt^2}=0
$$
(correct me if I’m wrong)

The standard potential formulation equation for $phi$ is initially unchanged, however adding the gauge condition mentioned earlier you get a relatively complicated equation.

Another idea to use (most likely useless):

However is it valid to substitute $rho/epsilon$ for $frac{sigmamathbf E}{epsilonmathbf v}$ where $mathbf v$ is the velocity field, then exchange $mathbf E$ for the potentials $mathbf A$ and $phi$, (as obviously $mathbf J = rhomathbf v = sigmamathbf E$)?

Edit: doesnt setting p to be zero contradict the statement that $$ J = sigma E $$ as $$ J = rho * V $$ so must conclude that E =0