Re: let's outgrow puzzles

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

> Yes, Herb; you got it right. My next problem for you is to consider the
> infinite square array of resistors each having resistance R. What is the
> resistance between junctions across a square diagonal?
>
> Hint: This problem might not be solved on an examination, but it might
> yield to work done while in transcendental meditation.
>
> Leigh

I don't want to give away the answer to Leigh's assignment for Herb, but
I want to mention that solving this problem for the resistance between
any two nodal points in the lattice can be an excellent excercise in the
application of the power of Fourier analysis and the use of recursion.
(However, I also agree that it makes for a lousy test question for an
in-class time-limited test).

Does anyone want to check my result that the resistance between one nodal
point and a second one such that the second one is translated 5 lattice
units along one nearest neighbor direction *and* translated by 2 units in
the other orthogonal direction w.r.t the first nodal point is:
R_eff = (97/2 - 2236/(15*[pi]))*r
where r is the the uniform resistance of each nearest neighbor resistor?
I hope not too many people object to me showing off here.

Leigh, have you ever worked out the cases for other
nearest-neighbor-bond-connected lattices which have uniform resistor
bonds?  An interesting result in 2 dimensions (square, honeycomb, and
triangular lattices) is that the effective resistance diverges
logarithmically when the distance between the nodal connection points
diverges toward infinity.  The asymptotic behavior becomes *isotropic* on
sufficiently large length scales.  In 3 dimensions, though, the
resistance is *finite* for infinitely separated nodal connections.  In
fact, for the simple cubic lattice the resistance asymptotically
approaches R_eff --> 0.50546201973*r as the connection points diverge to
infinity in any direction.  BTW, the answer for Herb's problem (2nd
neighbor distance) *but* for a *triangular lattice* rather than a square
one (like Leigh asked about) is R_eff = (2*sqrt(3)/[pi] - 2/3)*r.

David Bowman
David_Bowman@georgetowncollege.edu