Re: question about series capacitors

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

Carl 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?
>
> If the isolated circuit consisting of plates B and C and the wire between
> them is initially uncharged, then the sum of the charges presumably has to
> remain zero. But why couldn't I get -1.1Q and +1.1Q on these two plates,
> say? Why *exactly* -/+Q? Is it always exact: what if plates A and B have
> different shapes? Or what if I imagine distorting the wire between plates B
> and C, so that B and C are both portions of some larger object, say the two
> ends of a solid rectangular block, or even a sphere? Surely at some point
> the answer will no longer be -/+Q. At what point - in other words, what
> assumptions go into the usual derivation? I've looked in several textbooks
> and it's presented as though -/+Q is patently obvious. Carl

The key idea here is that the system of charges will redistribute itself in
such a way as to minimize the resulting electrostatic potential energy of
the system subject to the external constraint imposed by the battery
potential.  This is a consequence of the 2nd law.  The system tries to
dissipate as much of its energy as possible as (dare I say it) thermal
energy which ultimately spreads throughout the universe raising its entropy
as it goes.  This means that only the irreducibly minimal energy remains in
the electrostatic field configuration after the charges are through moving
into their equilibrium positions.  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 and the mobile charges in
the connecting conductor wire would feel an unbalanced restoring force which
would tend to accelerate them toward the equilbrium configuration.  If there
was no dissipative damping the interior charges would slosh back and forth at
a frequency given by the effective inductance and capacitance of the sloshing
mode.  The charges will move around dissipating energy until they are in the
minimum potential energy configuration.  This requires that both surfaces B
and C be at the same potential with no electric field between them.  The
energy tied up in an electrostatic field is proportional to the volume
integral of the square of the field strength.  The greater the field strength
and the greater the volume it occupies the greater the energy in it.  If the
interior charges are partitioned as -Q and + Q then all of the electric
field is effectively confined to the interior of each capacitor and its
fringe area.  This minimizes the potential energy of the system and balances
the forces on the mobile charges so they stay put there.  If the geometry of
the setup is distorted so the distance between B and C was comparable to or
less than the capacitor gaps which would then have a huge fringe field
volume, and if the geometry of the system significantly broke any left-right
symmetry that the system might possess, then the values of the charges would
not necessarily split up into exactly -Q and +Q on plates B and C but
would also be some partial charge distributed along the surface of the
conducting connector between them.  To solve for the final charge distribution
in this case would entail, in general, solving Laplace's equation over all
space with the requirement the the potential gradient at all metallic
surfaces be locally perpendicular to the surface.  This will zero out the
electric field in the interior of the metal.

David Bowman
dbowman@gtc.georgetown.ky.us