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*From*: John Denker <jsd@MONMOUTH.COM>*Date*: Tue, 6 Feb 2001 23:10:42 -0500

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.

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