Is the fact that there could be no E-field inside a conductor purely experimental? I don't see any way to apply Maxwell's equations to prove this fact.
Yeah, this makes sense. Eventhough in order to prove that there is no E-field inside the conductor we are assuming that there is no current inside the conductor, the assumption of there being no current is more pleasing.If there is an E field in a region with mobile charge carriers then there will be a current. So after all currents have died away, either the E field must be cancelled or the medium will have run out of charge carriers. Does that help?
No, I meant to say that in general when we say "There is no E-field inside a conductor," what does "inside" actually mean?If you mean inside a void in a conductor - sure, E can be non-zero in there.