Re: Calculating resistance

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

Bernard Cleyet wrote:
...
>  The soln. assumes
> infinite in length cylinders.  Therefore, I think the formulae are only
> accurate in three D.  e.g.  long cylinders buried in the earth.

The same equations have the same solution.
The d/dz derivative could vanish because of
translational symmetry (long rods buried in the earth)
or it could vanish because everything is
constrained to lie in the plane (dots on
resistive paper).

=================

> P.s. Harnwell uses complex variables to solve the two dimensional problem
> (only applicable when the third is infinite in length).

Yeah.  Students are often not too pleased with that
solution.  And they've got a point.  Learning that
method of solution in this context is IMHO just an
advanced form of rote learning.  It's nearly impossible
to connect it with anything else they know.  And
there are only a handful of problems that yield to
that technique, so it's not clear what the point is.

Students think the "method" here is to pull a
hypothetical solution out of thin air (or somewhere
else) and then verify it.  That hardly counts as
a "method".  And explaining where this solution
really comes from would take a week.

It's like Obi-Wan's light saber:  An elegant weapon
left over from a bygone age.
http://www.theforce.net/multimedia/sound/characters/ObiWan/civage.wav

Back in the olden days you _needed_ a closed-form
solution, because that was the only way you could
calculate anything.  Nowadays we deal with more
complicated problems, for which there usually is
no closed-form solution.  In such cases a rapidly-
convergent series solution is just the ticket.

====

Consider the real-world version of Ludwik's problem,
where the electrodes are not perfectly circular, but
slightly amoeba-shaped.  As long as the electrodes
are small and reasonably nearly circular, you know
in your bones that it won't matter much.  But you
ain't gonna find the answer by guessing some function
of complex variables.

What you would really do is approximate the dots as
point sources, then estimate a correction term due
to the polarization of each dot in the total field.
If this correction term is small, as it usually would
be, you're done.