Re: Calculating resistance

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

Ludwik Kowalski wrote:
> How to solve this problem? Find the resistance R between
> two small silver-painted dots separated by a distance L on
> an infinite sheet of a carbon-impregnated paper. The uniform
> thickness of the paper, d, and its conductivity, rho, are given.
> Ignore the contact resistance of silver dots.

OK.  Before we get started, let's inventory what we know.

1) Grand principle: The same equations have the same solutions.

2) Grand principle: Linearity implies superposition.

3) Charge is conserved.  Charge obeys a _local_ conservation
law.  Charge itself is the conserved quantity;  the corresponding
conservative flow is the ordinarly electrical current.

4) Grand principle:  symmetry.  The situation is
symmetric under interchange of the dots (and mirror-
image inversion of the voltage).

5) We seek the steady-state solution.  We inject a steady
current and wait for transients to die down, then measure
the voltage.

6) Electric field is the gradient of the potential.

7) Ohm's law.

8) Gauge invariance.  We can choose whatever gauge
we find convenient.

> My first idea was to think about the rim-to-rim resistance
> when the sheet is circular and when one dot is in the center
> while another "dot" is painted along the circumference.

Basically the right idea.  Let me say it in my own words:

Ignore one dot.  Analyze the other
dot surrounded by a circular counter-electrode.  Take
the limit as the counter-electrode is moved off to infinity.
Superpose the corresponding solution for the other dot.


> The
> radius of the central dot is r1 while the radius of the outer
> "dot" (the radius of the sheet) is r2. This problem is likely
> to be less complicated and I know how I would start
> solving it.

Conservation (item 3) plus stationarity (item 5) plus Ohm's
law in a homogeneous medium (item 7) guarantees that
the field is divergence-free.  Equivalently, also use
the definition of potential (item 6) and conclude that
the potential obeys Laplace's equation.

Can't you think of any solution to this equation in
cylindrical geometry?  Hint:  The same equations have
the same solutions (item 1).

Additional hint:  Using notions of symmetry (item 4)
and conservation (item 3) and gauge invariance (item 8)
we can choose the potential of the counter-electrode
to be zero.

> This problem has probably been solved somewhere.

Probably.  I think some guy named Fourier had something
to say about it.

http://www.google.com/search?q=fourier+analytique+chaleur