Let's start with something we actually know and trust:
conservation of charge. This is a direct corollary of
the Maxwell equations. Specifically:
∇-j = − (∂/∂t) q [1]
and since we are interested in the DC limit this reduces to
∇-j = 0 [2]
For a resistive material, we need Ohm's law. Neglecting
small magnetic effects, the microscopic continuum version is
E = (resistivity) j [3]
Even if the material is non-Ohmic, equation [3] serves as a
first-order approximation.
The overall goal is to find the charge density ρ, so we are
tempted to take the divergence of E. Then we can find ρ
via the Maxwell equation:
∇·E = (1/є0) ρ [4]
Note that today ρ stands for charge per unit volume (not
resistivity). Taking the divergence of equation [3] gives
us
∇·E = ∇(resistivity) · j + (resistivity) ∇-j [5]
where the last term vanishes in accordance with equation [2[.
Putting it all together we find
ρ = є0 j · ∇(resistivity) [6]
This tells us all sorts of interesting things:
*) If the resistivity is zero, the charge density is zero.
*) If the current is zero, the charge density is zero.
*) If the resistivity of the material is spatially uniform,
the charge density is zero
-- even if the wire is non-straight
-- even if the wire is non-circular
-- even if the cross-sectional shape is changing in weird ways
-- even if the cross-sectional area is changing in weird ways
*) Only the variation in resistivity /in the direction of the current/
is relevant. If there is a high-resistivity region in parallel
with a low-resistivity region, this does not give rise to any
charge density.
In the real world, almost any material will have point-to-
point variations in resistivity. Even if the material has
the same composition, if there is a large change in cross-
sectional area there will be a change in the amount of I^2 R
heating, and this can produce a change in resistivity. Note
that for a good metallic conductor, resistivity is directly
proportional to absolute temperature.
I leave it as an exercise to the reader to go back to
equation [3] and stick in the self-induced Hall voltage
that Bob Sciamanda mentioned.
============================================
Here are some remarks about the process that led to equation [6].
1) There is probably more than one right answer, depending on
what assumptions you want to make, but treating the charge
as a continuous fluid and treating the material as approximately
Ohmic is certainly within the reasonable envelope, and consistent
with the level of detail used in the book.
2) 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.»
However....
a) AFAICT, Gauss's law has got nothing to do with it.
b) The interior charge is not necessarily zero. The RHS
of equation [6] is a lot more interesting than zero.
c) My derivation required invoking two concepts that the
book did not even hint at in this context, namely Ohm's
law and conservation of charge.
I mention this because it makes this something of an "AHA
problem". Once you realize that the appropriate starting
point is conservation of current and not Gauss's law, the
calculation is straightforward. Conversely, the longer you
spend fussing with Gauss's law, the worse off you will be.
Also, the longer you spend trying to prove that ρ=0 the worse
off you will be.
Up to this point I have told the story in reverse, giving
the answer and then explaining where it come from. There
are sound pedagogical reasons for this, especially in the
introductory class. However, among friends, among experts,
I can tell what really happened, in order:
1) I read the book.
2) I read the passage in question:
«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.»
I knew that couldn't possibly be true. This is the hardest step,
because you can't afford to disbelieve everything you read. Life
is too short. So it takes some judgment, some well-developed
sense of smell, to decide when a statement smells of rat.
3) I did some numerical simulations. I thought it would be
easy to set up situations where the interior charge was
strongly nonzero, but this turned out to be trickier than
I expected. Almost anything I did using a uniform material
resulted in zero interior charge. So the claim was closer to
being true than I expected it to be. Still, I knew it couldn't
possibly be true in general.
4) I thought about some screwy geometries, involving sudden
changes of cross-section plus blind alleys et cetera, where
the E-field is wildly non-uniform ... yet there was still no
interior charge. I started wondering what sort of property
the E-field could have such that ∇·E would be zero almost
everywhere ... and then AHA ... ∇·E looks like part of a
conservation law, so if E is proportional to current then
....... figuring out the rest took about a millisecond.
5) Combining points (2) and (3): The anybody who has decent
simulation tools has a huuuuge advantage over anybody who
doesn't. You can afford to be more skeptical. That is, you
can check more things if you can check them quickly and easily.
This is relevant to the recent discussion of using or not
using calculators. To be sure, there are plenty of people
who use the computer instead of thinking, but it doesn't
have to be that way. The computer is a tool, and folks
should learn to use the tool intelligently. In general
there is (at least!) a three-legged stool, consisting of
intuition plus computation plus formal mathematical analysis.
In today's example, compare the advantages and limitations
of the various legs:
-- The computer is never going to cough
up the desired answer (equation [6]).
-- Nobody is going to solve the Maxwell equations
analytically in complicated geometries.
++ The computer can relatively quickly tell me
where I should look -- or shouldn't look -- to
find the answer.
++ Once I know what I'm looking for, the analysis
is easy. I can do it on half a post-it note.
++ Simulation and visualization provide
some insights.
++ Analysis provides additional, different
insights.
++++ And so on, iteratively. The analysis and the intuition
guide the computation, and conversely, in all combinations.
===============================
This is hard to teach. "AHA problems" in general are hard
to use in a classroom setting.
-- If there's too much of a hint, it spoils the fun.
-- If there's not enough of a hint, nobody knows how to get
started.
-- If the problem has ever been assigned before, folks can
google the answer, which spoils the fun.
-- In class, the first guy who blurts out the answer spoils
it for the others.
-- OTOH there needs to be a statute of limitations. If you
haven't figured it out after a certain time, looking up the
answer is better than never seeing the answer at all. In
such a case you owe it to yourself to ask, what do I need
to do differently so that the next time something like this
comes along, I will be able to figure it out?