John wrote:
* * *
I think Martha is on to something critical here. I believe that I can
show that the potential difference of any two isolated conductors carrying
charges Q1 and Q2 will be given by
Q1 - Q2 Q1 + Q2
delta V = ------- + -------
2 Cb 2 Cc
where we might call Cb the "balanced mode capacitance" and Cc the "common
mode capacitance." Cb corresponds to the usual definition of capacitance;
it gives the ratio of charge *separation* to associated potential
difference. Cc, on the other hand, gives the ratio of *total* charge to
the associated potential difference. Both Cb and Cc depend *only* on
geometry.
For two symmetric parallel plates Cc -> infinity (i.e., there is zero
"common mode potential difference") while for two concentric spherical
shells Cc = Cb. I *think* I can show that, in general, |Cc| >= |Cb|. One
needs to be particularly careful with the sign of Cc as it depends
critically upon which particular conductor takes on the larger potential
when both carry the same charge.
.. . .
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A. John Mallinckrodt email: mallinckrodt@csupomona.edu
* * *
John,
This seems to be a special case of the matrix equation relating
the potentials and charges on a system of N separated conductors:
|V| = P*|Q|
where |V| is the column vector of N potentials, Q is the column
vector of charges, and P is the matrix of "coefficients of potential".
The P_i,j are functions only of the geometry. In your case N=2.
The relation can obviously be turned into:
|Q| = C*|V| , where the C_i,j are the "coefficients of capacitance".
These relations have gone out of fashion in most recent textbooks
(only the "capacitor" of two equally and oppositely charged
conductors is typically now treated). An exception is
"Electromagnetic Fields", Roald K. WAngsness
(Also, one of the earlier editions of Corson & Lorrain).