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Re: mutual capacitance

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

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.
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.