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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.