Re: funny capacitor

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

At 11:46 AM 3/10/01 -0500, Ludwik Kowalski wrote:
> We all know that the term potential, like the term  elevation,
> has no unique meaning unless a reference level is defined.

Right.

> .... the matrix of coefficients, Cij, is singular. Clearly
> something was wrong in the previous step; that is when
> the Laplace method was used to solve the Q(V) problem
> four times.

It's not at all obvious that there is anything wrong here.  Singular is not
the same as evil.

This procedure, namely turning the crank on Laplace's equation, is an
exceedingly reasonable way of handling the Q(V) problem.

> We have no right to declare that V4=1 because the enclosure
> is our "sea level reference";  it must remain zero no matter
> what else we are doing. Right?

It depends on what we mean by "we".  I have every right to set V4=1 if I
want.  You don't have to if you don't want to (that's YOUR freedom) but I
can if I want to (that's MY freedom).

I won't give up my gauge invariance until they pry it from my cold, dead
fingers.

> We need a method of calculating Cij which does not introduce
> the ambiguity. Do you agree?

I would rather say that we need a way to calculate the
physically-meaningful quantities.  We can allow V to remain ambiguous, as
long as our physically-meaningful results don't depend on absolute
V.  Ambiguity in V is not _per se_ a problem that needs solving.  It's like
elevation, or like relativity:  any reported velocity is ambiguous, unless
we know what it is relative to.

Specifically, the Q values are physically meaningful, the delta_V values
are physically meaningful, but the absolute V values are not physically
meaningful.  Therefore
  a) We need a way to calculate Q(delta_V).  We have that.  No problem.
  b) We need a way to calculate delta_V(Q).  We have that.  No problem.
  c) We don't need a way to calculate V(Q).  We don't have that.  No problem.
  d) We don't need a way to calculate Q(V).  Actually we happen to have
that, even though it's not strictly needed.  No problem.

>  By using this method we would
> not be forced to use tricks to solve V(Q) problems.

Calculating the absolute V(Q) is not just tricky -- it's impossible.

> What is the
> correct way of calculating nine (not sixteen) Cij coefficients
> for the three equations with three unknowns? I suppose the
> 3 by 3 matrix of such coefficients would not be singular.

It suffices to calculate the 4x4 full capacitance matrix, and then throw
away the 4th row and 4th column.  This results in the 3x3 diminished
capacitance matrix, which applies to the (V1, V2, V3, 0) subspace.

There exist other approaches that will arrive at the same results.

The alert reader will notice the possibility of speeding up the calculation
by a factor of 4/3 by not bothering to calculate the 4th column of the full
capacitance matrix.  IMHO this optimization is for experts only, and is NOT
recommended for students, because the 4th column provides a valuable check
on the performance of the Laplace-equation solver.

In all cases, calculating the 4th row is extremely cheap, so you might as
well do it, even if you plan to throw it away.  Before throwing it
away  you should use it as a valuable check on the number-crunching.