Re: question about series capacitors

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

On Wed, 10 Sep 1997, Carl E. Mungan wrote:

> Suppose I connect two capacitors in series across a battery. Label the four
> capacitor plates from left to right as A, B, C, and D. Okay, suppose A is
> connected to the positive terminal of the battery, so out goes charge +Q to
> it, and compensating charge off D, leaving -Q on it. My question is: why
> does the charge on B and C have to be -Q and +Q, respectively?
> ...

As you seem to have suspected, it doesn't.  

Consider an extreme geometry like that shown below--i.e., one in which
"plates" B and C are simply the ends of a short piece of wire in the
region between plates A and D.  The wire need not in any way be "centrally
positioned" in the region and the plates A and D need not be planar or
take on any particularly simple form. (It's not *even* necessarily the
case that plates A and D will carry equal and opposite charges due to the
effects of a net charge on the entire circuit or of other nearby charged
objects, but let's assume for the moment that they do.)

              /            
             /                        \
            |                         |
            |                         |---------
   ---------|      -------            /
            |                        |
            A      B?    C?           D
          (+Q)                      (-Q)

In this extreme case, it isn't at all clear what we even mean by "plate B" 
or "plate C" (and it is important to recognize that this ambiguity is
*always* present albeit to a very significantly reduced degree in
practical cases involving real capacitors with large plates that face each
other and are separated by a small distance.)  There is certainly likely
to be some separation of charge on the wire, but you will *not* in general
be able to divide the wire in any way such that one piece carries net
charge +Q and the other -Q.

The best way to see this is by invoking Gauss' Law: Some of the field
lines starting on plate A will end on the nearer portion of the wire and
some will end on plate D.  Unless, *all* of the lines starting on plate A
end on plate B (a situation that is *very, very* closely approximated in
real capacitors), then plate B will not carry a charge of -Q. 

The bottom line is that the textbook rules apply only as an *extremely
good approximation* in the usual case in which the plates of *each*
capacitor are large and close together, in which *different* capacitors
are separated by distances that are large relative to the spacings of
their plates, and in which we only consider the charge delivered to the
plates via the action of the battery (and can negect or ignore any
preexisting charges that might result from violations of neutrality or
external influences.)

John
-----------------------------------------------------------------
 A. John Mallinckrodt      http://www.intranet.csupomona.edu/~ajm
 Professor of Physics    mailto:ajmallinckro@csupomona.edu
 Physics Department       voice:909-869-4054
 Cal Poly Pomona            fax:909-869-5090
 Pomona, CA 91768-4031   office:Building 8, Room 223