Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Charged disk; was electrostatic ...



For the general case of N conductors, the charge on the jth one is given
by:

Qj = 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.


Bob Sciamanda
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "Ludwik Kowalski" <KowalskiL@MAIL.MONTCLAIR.EDU>
To: <PHYS-L@lists.nau.edu>
Sent: Tuesday, February 06, 2001 04:33 PM
Subject: Re: Charged disk; was electrostatic ...


My brain is not comfortable with the term "mutual capacitance"
in the context of two metallic objects carrying charges of identical
sign. John D. used this term, as shown after this message.

When signs are opposite then mutual capacitance is represented
by the flux originating on +object and terminating on -object.
The larger the flux, for a give Q, the larger C. But two positive
or negative spheres are not always directly linked by field lines,
even when they are very close; they are linked indirectly ("via
infinity").

Suppose charges are large, or spheres are not too far away
(say one radius or so). In that case we always have "mutually
induced polarization". But this by itself does not lead to field
lines going from one sphere to another. Direct linking appears
only when one charge is much larger than another. Is "mutual
capacitance" defined in terms of direct flux due to induced
polarization?

I have no trouble with the term "self-capacitance", also used
by John D. Suppose we have several metallic objects which
are positively charged. They are not linked directly by field
lines. The self-capacitance of each positive metallic object
(in farads) is equal to the number of coulombs it needs to
change its potential (with respect to V=0 at infinity) by one
volt. Would it be correct to say that a "mutual capacitance",
between any such objects, is zero, unless there is a direct
flux between them?

Note that the amount of direct flux, between two positive
spheres, for example, increases with the Q1/Q2 ratios. Does
this mean that the "mutual capacitance" depends not only
on the geometry but on charges as well?
Ludwik Kowalski

John Denker wrote:

At 02:57 PM 2/5/01 -0500, Bob Sciamanda wrote:
Separate spheres widely.
Connect them with a long wire.
Charge the system.
Remove the wire.
Now you have two spheres at the same potential and sufficiently
isolated from each other to allow a closed form calculation

This is a nice way to think about the problem. It makes it easy to
see some important physics.

Some comments and refinements:

1a) It might help to use a very _thin_ connecting wire. Otherwise
there will be a charge on the wire, which will induce charges on the
spheres. The capacitance per unit length depends inversely on the
logarithm of the wire's radius. The logarithm makes it painful to
achieve a really low capacitance, but it's possible in theory.

1b) Another option would be to do without a wire altogether.
We could have somebody with an ion gun deliver to each sphere
whatever charge is necessary to give it the desired potential.

2) Obviously we are assuming size-scales and voltage-scales large
enough that the quantum of charge is unimportant.

3) Treating the two spheres as "isolated" is an approximation that
gets better and better as the separation L increases.

Now if we want to be really professional about this, we should try
to make a _controlled_ approximation. So we want to get a bound
on the error.

For one sphere in isolation, its charge versus voltage relationship is
called its self-capacitance, also known as its
capacitance-to-infinity.
The same goes for the other sphere in isolation. When we have the
two spheres together, in addition to the two self-capacitances we must
also account for their mutual capacitance. That is, suppose we charge
sphere A with the ion gun, temporarily ignoring sphere B. Then we
charge sphere B, temporarily ignoring sphere A. But then when we
go back and check on sphere A, its voltage changed while we
were charging sphere B, because of the mutual capacitance.

If we now adjust the charge on sphere A, this changes the voltage
on sphere B, so we have to adjust that, and so on _ad infinitum_.
It's good that this is a convergent series. Even better, it is an
alternating series, so truncating the series causes an error that is
no bigger than the first term dropped. The formula Bob gave
results from approximating this series by its first term.

The magnitude of the mutual capacitance can be bounded by the
parallel-plate formula, something that (at worst) scales like R1^2 /
(L-R1-R2); for large L this is small compared to the
self-capacitance which scales like R1.
====

3b) An amusing corollary concerns the case where a small sphere
is rather near a large sphere, so that the mutual capacitance is
significant compared to the small sphere's self-capacitance. In
that case the small sphere gets markedly less charge than the
independent-isolated-spheres model would suggest.

This makes sense. Imagine charging the large sphere first. Then
the as-yet-uncharged small sphere is already sitting at almost the
full potential; it needs very little charge to bring it up the rest
of the way.