Re: I need help. (long)

["John S. Denker" <jsd@MONMOUTH.COM>]

David Bowman wrote:
> > claim that the finite size of the sheet
> > has a negligible effect for the geometry chosen?
>
> It probably *does* have a negligible effect on the overall value for
> the interelectrode resistance.  But this does not mean that it has a
> negligible effect on the potential values out in the parts of the
> paper that are distant from the close vicinity of the electrodes, and
> it does not mean that the equipotential curves out there are only
> negligibly distorted from circles.

Bravo.  I didn't even understand the question until
I saw David's answer.

> A variational principle is at work here.

Ah, we get even more than we bargained for.
A nice lesson on variational methods.

The variational approach not the only way to answer
the question, but it is an elegant and incisive way.

> Effectively, the error in the resistance (& field
> energy, & capacitance) is proportional to the *square* of the error
> in the potential used to calculate it.

1) To emphasize David's point:  This is not just an isolated
odd factoid.  There are lots and lots of physics problems
that exhibit this behavior.

A particular case of this is the idea behind the
Hellman-Feynman theorem.

2) The "relaxation methods" we used last summer for numerically
solving Laplace's equation can be considered variational
methods.

3) David explained why the potential error goes like the
square of the field error.  The implicit claim is that
the latter is small, so the former is small squared.  So
to fill in the last detail, let's justify this claim,
which is in fact a valid claim.

3a) Almost (but not quite) all of the resistance occurs
right near the electrodes, when the electrodes are small
compared to the distance between them.  The field is
falling off like 1/r, which is not a super-strong falloff
(compared to, say 1/r^2 or exp(-r)), but it's something.

3b) At typical points far from the electrode pair, the
field is a _destructive_ superposition of two
contributions.  So that makes it even smaller.

3c) Here's another way of saying the same thing that may
be more appealing to students:  We have here a problem
with _mixed_ boundary conditions:  The voltage is
fixed at the electrodes, and the current is fixed
at the boundaries of the sheet.  Mixed boundary
conditions make life difficult.  So here's a useful
trick:  Impose periodic boundary conditions.  That
is, tile the floor of Madison Cube Garden with identical
copies of the resistance paper, with one pair of electrodes
per copy.  In fact, make it endless in both directions,
either by making it toroidal, or by making it plain-old
infinite.  Now at the boundary of each cell, the current
will vanish by symmetry.  (I love variational arguments,
but I love symmetry arguments even more.)  So in the
periodic-boundary system, we no longer have mixed
boundary conditions; all we have is the specified
potential at the electrodes.  You can easily calculate
(directly and/or using Fourier methods) how any given
cell is affected by the far-field of all the other
cells.  It's pretty small compared to the field near
the electrodes.

==> This far-field contribution is not, however, small
compared to the field at typical points in "this" cell
not near the electrodes.  It might make a perturbation
of order 100% on this field, because this field
is also small at that point.  The effect on the
total electrode voltage is small, because 100% of
small is still small.