The rigorous proof is almost obvious. By Gauss' law, a non-zero net charge within a volume V in a conductor produces non-zero electric flux through the surface of the volume (not to be confused with the surface of conductor itself!). For the flux to be non-zero, there must be a non-zero field at least somewhere on the surface, that is, within the conductor, which contradicts its definition for the case of internal equilibrium (absence of currents). This completes the proof. In case of applied voltage and resulting current, the proof does not work since the field is allowed (and even required!) in this case. An interesting detail: in case of DC, current may be internal, not only on the surface of the wire, without any net charge in a volume - the current is carried by free electrons in the conduction band. So there may be a current without any net charge within V, but there cannot be any net charge there without current.
Moses Fayngold,
NJIT
On Sunday, December 29, 2013 2:59 PM, John Denker <jsd@av8n.com> wrote:
Once upon a time, in a galaxy not too far away, there was a student
who actually read the textbook. (Hey, it happens sometimes.)
Student: Several chapters ago, the book said:
«A rigorous proof that all the excess charge goes to the
surface of a metal conductor requires Gauss’s law, which
we will study in a later chapter.»
Teacher: Yeah, so?
Student: Now that we've studied Gauss's law, I still don't see
how to use it to prove there is no charge inside a wire.
Teacher: But you know the charge has to be zero, because the
book says so about 10 times.
Student: Yeah, but repetition is not the same as proof.
Repetition convinces me the topic is relevant and
important, but I still don't see a proof. I skimmed
the entire book. I looked in the index. And the whole
schmear is searchable via Google Books. The most I
could find was a proof that applies to a long straight
wire with uniform circular cross-section in the DC limit
... and I don't see how that applies to practical circuits.
Is it even true? Is it really impossible for there to be
charge running around inside a metal wire? And what
does
Gauss's law have to do with it?
Teacher: ____________________________
So ... Can anybody fill in the blank, to finish this dialog?
What's the right answer?
To speed up the discussion, let's get some easy cases out of
the way:
-- We are treating the charge as a continuous fluid. We are
not interested in discrete atoms or quantized electrons.
This is consistent with the level of detail used throughout
the relevant chapters of the book.
-- If the current is identically zero, the interior charge
is zero.
-- If the resistivity is identically zero, the interior
charge is
zero.
-- Let's focus attention on the DC limit, because for AC
circuits, the interior charge is almost *never* zero.
So, what's the right answer? Is there a «rigorous proof» that
has any relevance to real-world circuits?