Re: Thoughts on series Capacitors

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Joel Rauber mentioned:
> ...
> 1) First a minor comment.  I disagree with a statement David made.  He 
> wrote:  "If the charges on plates B and C of the interior plates were not -Q 
> and +Q respectively then there would be an electric field in the region 
> between the capacitors . . ."
>
> If we look at an ideal case where the two capacitors are well approximated 
> by two infinite parallel plate capacitors far apart and connected by a very 
> thin wire.  We can model the static charge distribution as 4 sheets of 
> charge and the field in  between the two plates is zero, even if the 
> interior charges are +- 1.1 Q.

No.  If plate A has a +Q charge uniformly distributed on it, plate B has a
-1.1*Q charge uniformly distributed on it, plate C has a + 1.1*C uniformly
distributed on it, and plate D has a - Q charge uniformly distributed over
it, then capacitor AB has a net -0.1*Q charge on it while capacitor CD has a
net +0.1 *Q charge on it  This means that there would be field lines going
from the net positively charged CD capacitor to the net negatively charged
AB capacitor.  Using your (nearly) infinite parallel plate approximation
where A is the (actually finite) area of the plates and L is the separation
between plates B and C where L^2 << A and L >> max(AB separation, CD
separation), then these (effectively parallel and uniform) E field lines in
the region between the B and C plates will approximately represent a field of
strength 0.1*Q/([epsilon_0]*A) in the region between plates B and C.  Because
of this field there is a potential difference of about
0.1*Q*L/([epsilon_0]*A) between plates B and C.  Since these plates are
connected by a conducting wire the mobile charge in the metal is free to flow
along this wire in response to the field/potential difference.  In this case
plate B attracts positive charge and plate attracts negative charge. There is
thus a tendency for the excess charges to try to equalize via the wire.  If
there were no dissipation mechanism then once the charges equalize they would
tend to continue to flow (er, or move) due to inertial (inductance) effects
and this would cause a reversed imbalance to develop.  The excess charges
would then continue to oscillate across the wire and alternately
overcharging/undercharging plates B and C.  Dissipation effects guarantee
that the charge eventually settles down into an equilibrium configuration of
minimal potential energy.  This minimum corresponds to a situation where
there is charge neutrality on both capacitors AB and CD.  (Of course, in
describing the above senario I implicitly and incorrectly assumed that the
charges on plates A and D remained constant during the equilibration
process.  Although including charge fluctuations on plates A and D through
the battery during the equilibration process would be correct, the neclect of
them doesn't change the result that the final minimal potential energy
configuration is the one where each separate capacitor has no net charge, and
this zeros out the field (except for stray fringe effects) in the interior
region between plates B and C.)

> 2) I rather like John's response.  Let's assume we are not dealing with an 
> extreme geometry case; but rather a case where the textbook rule applies to 
> very good approximation.  I think the rule hinges on two points.  One must 
> argue that all field lines originating on plate A terminate on plate B. 
> (This is what gets violated in extreme geometry cases).  For situations 
> like (1) above this isn't too hard to justify, because in a single capacitor 
> one has already talked about this in the usual "what is a capacitor 
> discussion".  Secondly, you need to invoke Gauss' Law, put surfaces around 
> each of plates A and B; the field line flux is the same (in magnitude, there 
> is a sign difference of course), therefore the magnitudes of the net charge 
> on plates A and B are the same.

Notice that this requirement of no net excess electric flux external to each
of the capacitors is equivalent to a requirement of a vanishing field in the
region between plates B and C.  The reason why even a single capacitor would
want to be charged in such a way as to keep it overall charge-neutral is that
if this condition did not hold then the net electric flux emanating from the
capacitor would result in an electric field external to the capacitor and
this would raise the potential energy of the system above the minimal value
that occurs when nearly all the electric flux (and field) is confined to the
relatively small region between the capacitor plates.

David Bowman
dbowman@gtc.georgetown.ky.us