Re: two charged spheres

[John Denker <jsd@MONMOUTH.COM>]

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.