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Re: A funny capacitor.



At 10:21 PM 3/1/01 -0500, Ludwik Kowalski wrote:
Referring to this:

Q1=C11*V1 + C12*V2 + C13*V3
Q2=C21*V1 + C22*V2 + C23*V3 (Equations 2)
Q3=C31*V1 + C32*V2 + C33*V3

John wrote:

> You will discover the following facts, which will soon come in handy:
>
> Gauge invariance requires Cij to be symmetric:
> Cij = Cji (equation 3)
>
> Charge conservation requires each row (and/or column) to sum to zero
> SUM_j Cij = 0 for all i (equation 4)
> assuming (!) the sum runs over all relevant objects.
...

2) I am not sure I understand the last line in the above quote.

As an illustration: If you are doing experiments inside a Faraday cage,
objects outside the cage are not relevant and need not be summed over.

This is important, because a law that holds only if you sum over all
objects in the universe wouldn't be very useful -- you would never be able
to apply it.

> How does the equation 4 follow from the law of
conservation of charges?

The total charge in the world is Q. If Q = const then
(partial Q) / (partial Vj) = 0
but
(partial Q) / (partial Vj) = SUM_i Cij
which follows immediately from equation (2).

Hint: (partial Vj) / (partial Vk) = delta_jk

=======================
Equation 3 reminds of
Newton's third law. The influence of object 2 on object 1
is as strong as the influence of object 1 on object 2.

Huh???? N3 has to do with equal and opposite vectors. Equation 3 doesn't
deal in vectors, and it doesn't express an equal-and-opposite relationship.