I have always found it difficult to explain the field in a capacitor using
Gauss' law by itself. If I were speaking I would emphasize "by itself."
Here is why...
Assume we're using a right cylinder for the gaussian surface.
Assume the gaussian cylinder has "end caps" parallel to the capacitor
plates.
Assume the plates are large enough and we are far enough from the ends that
all field lines are perpendicular to the end caps and parallel to the curved
surfaces. This means there is never any flux through the curved wall; i.e.
all flux goes through one or both ends.
Assume the capacitor plates are isolated in space; i.e. we don't have to
worry about other charges in the universe because they are sufficiently far
away to be of no concern.
Assume the plates are conductors of finite thickness.
Assume the plates have equal/opposite charge.
Now... where will we put the end caps of our Gaussian surface, and what will
the result tell us?
(1) Put one end cap within one plate and one end cap within the other plate.
Static E inside a conductor is zero, there is no net flux, there is no net
charge in our cylinder. This tells us the surface charge densities on the
inner surfaces of the two plates are equal/opposite. It does not tell us
whether there is or isn't any charge density on the outer surfaces of the
plates.
(2) Put one end cap outide the the negative plate and the other end outside
the positive plate. If the two plates are equally charged and have equal
density (which we'll assume from symmetry) this doesn't tell us anything.
We assume net q inside is zero, but cannot tell whether the flux through the
end near the negative plate is zero or negative, and we can't tell whether
the flux through the end near the positive plate is positive or zero.
Therefore this does not clear up the question of whether there is any charge
density on the outer surfaces of the plates.
(3) Put one end outside the capacitor and one end inside the nearest plate.
If there is +/- charge densisty on the outer surface, then there is a +/-
flux through the outside end cap, but none through the cap inside the
conductor. But there could be zero flux through each end and no charge on
the outer surface. This test does not tell us which we have.
(4) Put one end outside the capacitor and one end inside the farthest plate.
We already know the inner charge densities are equal/opposite, but we don't
know the outer density. This test will come out okay either way. The flux
through the outside cap might be zero (in which case there is no outer
charge density) or it might be something (in which case there is outer
charge density).
(5) Put one end inside the gap and one end within one of the plates. This
gets interesting. In hindsight we see we will get the wrong answer for the
field between the two plates if we assume the charge density on the surface
contained within the gaussian surface is Q/2A where A is the surface area of
just the inner surface, but Q is the total charge. In order to account for
the electric field in the gap that is created by the charges on the
capacitor plate that is NOT currently contained on our gaussian surface, we
have to assume the charge on the plate which contains the end of our
cylinder has all its charge on the inner side so that the charge density on
the inner surface is Q/A rather than Q/2A.
Therefore, once we reason there is no charge on the outer surfaces, then
Gauss' Law can give the correct expression for the field inside the gap by
putting one end cap of the gaussian surface in the gap and one end cap
within one of the plates. But until we reason that all charges migrate to
the inner surfaces, Gauss' Law will not solve this problem, or at least
solves it in a round-about way. We have to realize the electric field in
the gap is determined by both plates, so that when we draw a gaussian
surface with one end cap in the gap and the other end cap within one plate,
that the charge density on the plate surface within our gaussian surface is
determined by both plates. It is not the same as it would be if the other
plate were not there.
I think this must be very confusing for students because it is something
that I have had to struggle to keep straight for the whole time I have tried
to teach this.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817