Re: invertible 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.

Yes!  Very useful!
I now see the trick to inverting capacitance matrices.

Start with the complete Cij matrix.  Here is a famous example:

        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

This gives the charge on object (i) in terms of the voltage on object (j).
This matrix has no inverse.  It is singular.  It has to be.

But there is a way to invert "something".  The trick has two parts:
 1) The matrix must no longer express charge conservation.
    That is, we shall keep only three of the four rows.
    We shall calculate the charge on three objects using the matrix,
    and attribute the "rest of the charge", whatever that may be,
    to the fourth object.
 2) The matrix must no longer express gauge invariance.
    That is, we shall set one of the four voltages to zero.
    That means we can throw away one of the columns.
    It also means that the inverse matrix will only be responsible
    for calculating three voltages, not four.

Note that the missing row and the missing column do NOT need to intersect
on the diagonal.  We can set one object to be the gauge reference (V=0) and
select some *other* object to be the sump for the "rest of the charge".

So let's drop row 2 and column 3 in the example.  We are left with the
following peculiar matrix which gives the charge on objects 1, 3, and 4 in
terms of the voltages on objects 1, 2, and 4:

                 2.86   -0.46   -1.25
                -1.15   -1.15   -0.96
                -1.25   -1.25    3.46

This has a perfectly fine inverse:

                 0.30   -0.18    0.06
                -0.30   -0.48   -0.24
                 0.00   -0.24    0.22

which can be used to calculate the voltages on objects 1, 2, and 4 in terms
of the charges on objects 1, 3, and 4.

If you want to play with this, I added it near the bottom of
  http://www.monmouth.com/~jsd/physics/singular-inverse.xls

===========

We can now recognize the elementary formula for an ordinary non-funny
2-terminal capacitor
        Q = C V
as a special case of the 2x2 capacitance matrix (for the two capacitor
plates) where we have dropped a row and a column in accordance with the
scheme outlined above.

We have here the ingredients for a notational nightmare, unless we invent
some terminology to indicate whether a certain matrix is
 -- a full capacitance matrix, with explicit gauge invariance and
    explicit charge conservation, or
 -- a diminished capacitance matrix, with broken gauge invariance and
    implicit charge conservation.