Re: funny capacitor

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

At 12:25 PM 3/9/01 -0500, Ludwik Kowalski wrote:

> On one hand I believe in one and only one reproducible
> solution (funny capacitor with V1=-50, V2=+50 and
> V3=+20) on the other I am facing a situation in which
> the mathematics contradicts me. Is this an artificially
> created dilemma due to different interpretations of the
> meaning of the phrase "one and only one?"

That's the right question, I think.

I think the answer hinges on what one means by "the mathematics".

If we let Cij denote the *full* capacitance matrix, then the mathematics is
fully gauge invariant and charge conserving, and the matrix is non-invertible.

It may be your _intention_ to measure V1, V2, and V3 relative to V4, but
you haven't yet communicated that intention to the mathematics.  It doesn't
know, and it doesn't need to know.  It can calculate the Qi from the four
Vj in a gauge-independent way.  This is the Q(V) problem.

The inverse problem V(Q) is not well posed, and it does not have a unique
solution.  V does not depend on the four Q values.  It depends on 3 Q
values and one gauge.

If you want to express your gauge-dependent intentions mathematically, you
need to form the appropriate *diminished* capacitance matrix.  This
expresses the Q(delta_V) problem.  The diminished capacitance matrix can be
inverted to give delta_V(Q).

Moral of the story:  Ask for what you want.  Don't ask for what you don't
want.  Don't command the mathematics to give you V when what you really
want is delta_V.