Re: How many volts ?

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

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. 

At any rate, I'm going to look for references on this one.  It's probably
already written up somewhere, but if not (and *if true*) it might make a
nice phys-l inspired note to AJP or the Physics Teacher. 

John

On Tue, 25 Mar 1997, Martha Takats wrote:

> Here is a related idea which may or may not be relevant!
> For an "ideal" parallel-plate capacitor, i.e. with plate separation much 
> less than the dimensions of the plates, I maintain that the potential 
> difference between the plates equals |q1-q2|/2C. For example, if q1=Q 
> and q2=-Q (the usual case), V=Q/C. If q1=q2 (same magnitude and sign), 
> V=0.
> This relationship is not true for other geometries. A counter-example is 
> a capacitor consisting of two concentric spherical shells. The potential 
> difference depends on the electric field in the space between the 
> shells, which depends ONLY on the charge on the inner shell. (Same goes 
> for coaxial cylinders).
> This is probably why we don't deal much with cases where q1 and q2 are 
> not equal and opposite: we can't use a general equation involving only 
> the capacitance C rather than details of the geometry.

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