Re: funny capacitor

[Bob Sciamanda <trebor@VELOCITY.NET>]

John,
The "full" (your term) capacitance matrix Cij which you are defining
restricts the validity of the equation Qi=SUM CijVj to states with the
same total charge, SUM Qi.  In fact,  the values of the Cij depend on the
value of that total charge.  There is no one set of Cij that is a property
only of geometry and the choice of voltage reference point.

This is not very useful.  EG: If I have an isolated system of two
separated conductors, I would like an equation Qi=SUM Cij Vj which defines
the Cij as properties of the system geometry (and the gauge) and are
independent of the total system charge, so that I can consider cases in
which Q1 = -Q2, or Q1=2*Q2, or Q1 = -3*Q2, or etc, using the same set of
Cij.  I want to use the same Cij to describe all various ways of
depositing charges on these conductors, holding fixed only the geometry
and the voltage reference space-point.  This is the way all texts familiar
to me treat Qi = SUM CijVj.  The Vj are independent variables, and all
possibilities are included.  Adding the constraint of fixed total charge
is doing a very different, and severely restricted, problem.

Bob Sciamanda
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "John S. Denker" <jsd@MONMOUTH.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Tuesday, March 06, 2001 09:00 AM
Subject: Re: funny capacitor


> At 04:58 AM 3/6/01 -0500, Bob Sciamanda wrote:
> > Qi = SUM Cij Vj is invertible because it is NOT gauge invariant
>
> I suspect we have a communication problem because Cij means different
> things to different people:  if it means the *full* capacitance matrix
> that's one thing, and if it means the *diminished* capacitance matrix
> that's another thing.
>
> > In practice one can invert the Cij of a particular geometry
>
> In practice, the following is the *full* capacitance matrix Cij for a
> particular geometry:
>          2.86,  -0.460, -1.15,  -1.25
>         -0.460,  2.86,  -1.15,  -1.25
>         -1.15,  -1.15,   3.26,  -0.96
>         -1.25,  -1.25,  -0.96,   3.46
>
> I would be very impressed if anybody can invert this, where by "invert"
I
> mean exhibit a matrix B such that B C = I, the identity matrix.
>
> OTOH, starting from the *full* capacitance matrix it is possible to form
a
> number of different *diminished* capacitance matrices, and these
> *diminished* capacitance matrices are invertible.
>
> The *full* capacitance matrix is gauge invariant and manifestly
> charge-conserving.  The *diminished* capacitance matrix is neither.