Re: funny capacitor

["John S. Denker" <jsd@MONMOUTH.COM>]

At 01:30 PM 3/5/01 -0500, Bob Sciamanda wrote:
> The trivial case of a single conductor: Q = C V is useful.  C is a
> constant of geometry, even for various values of Q and V.  IOW one need
> not involve the the "sphere of charge at infinity" - the equations are
> still valid and useful.

When I see the scalar equation
        Q = C V
I assume it is shorthand for
        Q = C delta V
          = C (V1 - V2)

where V1 and V2 are the voltages on the two terminals of the
capacitor.  That is a fine equation, valid and useful.  The expression
(V1-V2) is manifestly gauge invariant.

If somebody wants to interpret
        Q = C V
in terms of some "absolute" voltage V, then the equation does not look
valid to me.  It begs the question of how to measure V.  All the voltmeters
I've ever seen have *two* leads, and can only measure a voltage *difference*.

================

If somebody wants to interpret
        Q = C V
as a matrix equation
        Qi = Cij Vj
then in the case Bob mentioned Q is a 1x2 matrix (one row of two columns)
and V is a matrix with components V1 and V2 as above.

It's fairly obvious that it will be fairly hard to invert this 1x2 matrix
to find V1 and V2 as a function of Q.  We can find V1-V2 as a function of
Q, but that's quite a departure from the claims that A.F.Kip reportedly
made about the NxN capacitance matrix.

Now that we have focused attention on what is actually possible, namely
finding voltage DIFFERENCES as a function of Q, we can discuss the general
case.

Here's a procedure that seems to make sense:

As always, write Qi = Cij Vj and find the Cij by solving Laplace's equation.

Form C' = C + x I, where I is the identity matrix and x is a small
regularizer.  C is singular, but C' is not.

Invert C' to form B'.  Actually we should write this as B'(x) since it
depends on x.  Choose one of the objects to be the gauge, i.e. we will
arbitrarily choose to hook the black lead of our voltmeter there.  Make a
note of the matrix elements on the corresponding row of B', and subtract
that from all the rows (including that row itself, zeroing it out).

Finally, take the limit as x goes to zero.
        B'' = lim B'(x) as x -> 0
The amusing thing is that this limit exists.  B'' is a much nicer looking
matrix than B'.  It is much better behaved if we make small changes in Q in
the equation
        Vi = B''ij Qj

I worked out an example of this and added it near the bottom of
  http://www.monmouth.com/~jsd/physics/singular-inverse.xls

I suspect this must be a standard technique for forming stabilized
quasi-inverses of a singular matrix, but I've never seen it before.  If
anybody knows of a general discussion, please let me know.