extended to the case where the charge is not along any axis of the
cube,^11 and based on additivity we then have a proof that the flux
due to an outside charge is always zero.
No charge on the interior of a conductor example 38
I asserted on p. 543 that for a perfect conductor in equilibrium, ex-
cess charge is found only at the surface, never in the interior. This
can be proved using Gauss’s theorem. Suppose that a chargeq
existed at some point in the interior, and it was in stable equilib-
rium. For concreteness, let’s sayqis positive. If its equilibrium
is to be stable, then we need an electric field everywhere around
it that points inward like a pincushion, so that if the charge were
to be perturbed slightly, the field would bring it back to its equi-
librium position. Since Newton’s third law forbids objects from
making forces on themselves, this field would have to be the field
contributed by all the other charges, not byqitself. But this is im-
possible, because this kind of inward-pointing pincushion pattern
would have a nonzero (negative) flux through the pincushion, but
Gauss’s theorem says we can’t have flux from outside charges.(^11) The math gets messy for the off-axis case. This part of the proof can be
completed more easily and transparently using the techniques of section 10.7,
and that is exactly we’ll do in example 40 on page 653.
Section 10.6 Fields by Gauss’ law 645