[Phys-l] defn of Capacitance

["Bob Sciamanda" <trebor@winbeam.com>]

This was originally sent on 2-12.  Apparently lost in the downed server problem.
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John Denker wrote:
> . . .
> 1) In one direction, we have not just a linear equation, but a
> *system* of linear equations:
>       Vi = Pij Qj + G                     [1a]
>       ? Qj = 0                            [1b]
>
> where G expresses gauge invariance and where equation [1b]
> expresses charge neutrality.
>  . . .
> This probably isn't the end of the story.  I need to think
> about this some more.  I've never had to worry about Pij
> before;  Cij has always been sufficient for my needs. . . .

While you're pondering, just  realize that your treatment differs from that 
of all texts I have seen.  All of the texts follow the original treatment of 
Maxwell (Chapter III) by NOT including the additive constatnt G in your [1a] 
and by NOT imposing your equation [1b] .

They honor gauge invariance by allowing the Vi's to be defined from any 
chosen zero reference.  They also leave the system open to describe 
localized situations in which the sum of the Qi's may be anything (not just 
zero).  This only requires that the considered system is distantly removed 
from all other (not included) charges.

Note that this allows complete freedom in imposing Qi's  (or Vi's) when 
considering special situations for the purpose of calculating the Pij's  (or 
Cij's),  since the Cij's and Pij's are determined only by geometry. 
Imposing charge neutrality for the chosen system severly limits these 
choices - and it is not necessary (it doesn't change the Pij's or the 
Cij's).
 _______________________________________________
Bob Sciamanda