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*From*: John Denker <jsd@MONMOUTH.COM>*Date*: Tue, 6 Feb 2001 08:49:06 -0500

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.

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