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My understanding is that that the claim that non-zero net charge only
resides on the surfaces is based on a Gauss' Law arguement that hinges
crucially on the idea that the E field is zero to the interior of a
conductor. The E field is of course manifestly *not* zero to the interior
of the conductor where we have non-zero current. Consequently I see no
fundamental reason why one can not have net charge as well as non-zero
current to the interior of a wire.