Re: mutual capacitance

[John Denker <jsd@MONMOUTH.COM>]

At 05:56 PM 2/6/01 -0500, Bob Sciamanda intended to write:
> For the general case of N conductors, the charge on the jth one is given
> by:
>
> Qi = SUM{Ci,j * Vj}
>
> In this matrix equation Vj is the potential of the jth conductor and the
> Cij (functions of geometry) are called the coefficients of capacitance .
> These are probably what John is thinking of.

Yes, indeedee;  thanks, Bob.

We can turn this around to create an operational definition of the
capacitance matrix elements:

  Cij = Qi / Vj       where object j has nonzero voltage Vj
                      and all other objects are held at Vi=0
                      and we take the obvious limit if Vj is zero also.

Similarly the small-signal capacitance is
  Cij = dQi / dVj     where all objects except object j are
                      held at Vi = constant

And for linear systems, which is what we are talking about, the foregoing
are equivalent.

Physics:  Gauge invariance requires Cij to be symmetric:
        Cij = Cji

Physics:  Charge conservation requires each row (and/or column) to sum to zero
        SUM_j Cij = 0    for all i
if (!) the sum runs over all relevant objects.

Terminology:  The diagonal elements are called the self-capacitances, and
the off-diagonal elements are commonly called (-1) times the mutual
capacitances.  People commonly throw in the (-1) so the mutual capacitance
comes out to be a positive number;  other people leave out the (-1) and
just report all mutual capacitances as negative numbers.  This situation
will snare the unwary in either case.  No matter what English words you
attach to it, the off-diagonal matrix element Cij is a negative number.

===================

You can use the spreadsheet I presented in my previous note to calculate
self-capacitances and mutual capacitances.
    http://www.monmouth.com/~jsd/physics/laplace.xls
In addition to the potential-grid and the |field|-grid, there is a
charge-density-grid (proportional to the Laplacian of the
potential-grid).  So you can put in some objects on the potential-grid at
suitable voltages and observe the induced charges.