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Re: funny capacitor



WHAT WAS THE LAST PART OF THIS THREAD ABOUT?
It seems to me that the meaning of Cij and Bij coefficients,
probably obvious to many, should be discussed before
addressing the issues of inversion. The mathematical
definitions of coefficients (for a three body system) are:

Q1=C12*V1 + C12*V2 + C13*V3
Q2=C21*V1 + C22*V2 + C23*V3 Equation (2)
Q3=C31*V1 + C32*V2 + C33*V3

and

V1=B12*Q1 + B12*Q2 + B13*Q3
V2=B21*Q1 + B22*Q2 + B23*Q3 Equation (1)
V3=B31*Q1 + B32*Q2 + B33*Q3

Equation (1) tells me what happens to potentials of floating
metallic objects. Each of them is given a net charge and we
want to know the potentials which result from this. As usual,
potentials are defined in terms of work per unit charge necessary
to bring an additional small probe charge from infinity, where
V4 is assumed to be zero. Assigning zero to infinity is convenient
but not necessary. (With this assignment V=kQ/r, with another
assignment the relation would be V=k*Q/r + A, where A is a
nonzero constant. )

So what is the meaning of equation (1)? It tells us, for
example, that the potential on object #2, V2, depends not
only on the charge it receives, Q2, but also on charges
received by other objects, Q1 and Q3, according to

V2=B21*Q1 + B22*Q2 + B23*Q3

The values of B21 and B23 become small, in comparison
with B22, when objects 1 and 3 are moved further and
further away from object 2. B12 shows how strongly the
charge on the object #1 influences potential on the object #2.
Note that the unit of a B coefficient is volt/coulomb=1/Farad.

Let us now turn to equation (2). It tells us how much charge
should objects receive in order to maintain imposed potentials.
A charge Q2, for example, depends not only on its own
potential, V2, but also on potential on other objects, V1 and V3.

Q2=C21*V1 + C22*V2 + C23*V3

The dependence of Q2 on V4 is implicit; the numerical values
of V1, V2 and V3 would change if V4 were not zero. Note that
the unit of Cij is coulomb/volt=F. This observation about units
does not mean that Cij and Bij coefficients are reciprocal of each
other. If they were reciprocal then a larger value of Cij would be
associated with a smaller value of Bij, and vice versa. But this
does not happen; both C12 and B12, for example, become
smaller when the object 1 is moved further away from the
object 2.

The illusion of reversibility can easily be reinforced when the
matrix form of Q=C*V is compared with the formula defining
a common capacitance c, (q=c*v). We do this in terms of the
absolute value of q and v. Note that in this formula v is the
potential difference between objects 1 and 2; it is not determined
by the amount of work involved in bringing an additional small
charge from a third object (infinity) to a plate. Also note that the
q=c*v is inversible, (v=b*c, where b=1/c) so that b is small
when c is large, and vice versa. The parameter b has no special
name. This can be contrasted with distinct names; resistance and
conductance, for r and 1/r (ohms versus siemens).

John exposed the textbook misconception according to which
Bij can be calculated from Cij, or vice versa, by inverting the
corresponding matrix equation. Was he the first one to do this?
It seems to me that he did more than exposing a misconception;
he showed how the voltage-influence coefficients Cij can be
calculated from the charge-influence coefficients, Bij. These
are meaningful parameters and one should be able to show
a correct relation between them. Is the "x regularizer" method
invented by John valid? Is it the only method (or the best
method) for establishing a numerical relation between Cij and
Bij? Can it be demonstrated that each Cij coefficient in the first
equation is directly proportional to the corresponding coefficient
in the second equation 2? I am trying to sell you square Farads.

In trying to be scientific we should submit John's theory to
experimental tests. Independent measurements of Cij and Bij
should not be very difficult; many schools have the necessary
equipment to measure the influence coefficients at the level of
5% or better. Would experimental data support the prediction
(for example about Bij when Cij are given for a funny capacitor)
or not? That could be a good research project, I suppose.
Ludwik Kowalski