Re: funny capacitor

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

At 02:02 PM 3/6/01 -0500, Bob Sciamanda wrote:

> Whether or not conservation of charge is a property of the universe, it
> need not be a property of a system which I choose to study.

Agreed.

> The non-invertibility arises only when one insists that
> only states with the same total system charge be considered in Qi = SUM
> Cij Vj  (the Vj's are then no longer independent variables).  These
> authors do not impose this constraint.  (I know of only one person who
> does :)

My Cij calculations do not insist on anything, do not assume anything, and
do not impose any constraints.

Let's just calculate according to the laws of physics.  We can calculate
using Laplace's equation the ratio
        delta Qi / delta Vj             (all Vk constant except Vj)
and in accordance with convention we call this Cij.

  (In this formulation the Vj are most certainly independent variables.)

Now we observe (not impose) the fact that if (!) we happen to sum over
"enough" objects, the sum_over_i of Cij comes out to be zero.

  (It is the Qi that sometimes (!) turn out to be non-independent.)

Anybody who calculates Cij values must either take care NOT to calculate
the charge on all relevant objects, or risks accidentally constructing a
Cij matrix that is singular.



> Cij which depend only on geometry
> and the choice of Voltage reference point

The foregoing method of calculation / definition is manifestly
gauge-invariant.  AFAICT the Cij value depends on geometry and *not* on any
choice of gauge.  Can somebody provide an example where a Cij value does
depend on the gauge?