Let's say you had a wave from a distant source hitting a sphere/ball inside a shell, both made of a good conductor, with the charge on the center sphere/ball held constant by some means, as in the below set-up:
Would the wave have any effect on (my questions and my attempt at answering them):
The E field inside the shell? I'd guess not due to Gauss's law - the charge inside the shell is fixed, so the E field would have to be fixed too.
The B field inside the shell? Not sure, I think Ampere's is the most relevant here: the E field would be fixed, but the wave would create J on the surface of the exterior shell. I'm not sure if that'd affect B inside it though.
The magnetic vector potential A inside the shell? Possibly...depends on the answer to (2).
The scalar electric potential $\phi$ inside the shell? Depends, if the E field is static, but if the A field is not, then $\phi$ would change inside the shell: $\textbf{E}=-\nabla \phi-\frac{\partial \textbf{A}}{\partial t}$
