infinite grid of resistors

Hello,

I have a question about the infinite plane of resistors mentioned in the following FAQ entry:
http://rec-puzzles.org/new/sol.pl/physics/resistors

Given a grid of one ohm resistors, what is the resistance across 2 diagonally adjacent nodes? The FAQ entry gives a formula but does not derive it. The derivation is what I am interested in.

I have read some postings from people hinting that there is a simple 'trick' to solve this problem without doing any integrals. Anyone have any idea?

The math-intensive answer is here:

Possibly the simple 'trick' gives a different answer than the math-intensive one.

The infinite grid of resistors was one of my favorite problems. Unfortunately, that was 50 years ago. I seem to recall the use of Norton's theorem to solve. Inject a positive current, calculate the drop across two resistors, then inject a negative current, calculate that drop and sum the voltage drops appropriately. The answer, using this approach does not agree with the frooha page.

Checking the web, I found: http://www.egr.up.edu/contrib/oster/ajp.pdf a paper by Osterberg and Inan titled: Impedance between adjacent nodes of infinite uniform D-dimensional resistive lattices. Their answer agrees with mine. They provide references as well.

With one answer giving a transcendental number and the other a simple fraction, I must print the frooha page and see if I can find truth.

I never quite trusted the simple answer on the grounds that it was unclear how an injected current going to infinity could ever return to the source.

John Bailey

No, I heard someone say there was a trick for the diagonally adjacent node. But, I'm beginning to suspect it doesn't exist.

Thanks for a thoroughly amusing problem with good opportunities for learning. If you or anyone else is as fascinated with the problem as I now am, the page: http://www.av8n.com/physics/laplace.html provides several examples of using a spreadsheet for solving the potential flow problems with a discrete mesh approximation. It requires some physics reasoning to relate the potential flow solution to the resistive mesh problem. It is easily seen that the calculated discrete case approximation is applicable to the resistive mesh. At this point I use a 60x60 grid and am experimenting with boundary conditions to improve on 0.01% accuracy.

If symbolic solutions are more to your taste, realize that the classic methods which solve the continuous resistive sheet case are (1) separation of variables and (2) analytic functions of complex variables. Both the Atkinson and Frooha papers are based on the principle of mapping the discrete case (resistor mesh) to the continuous case (2D resistive sheet) and solving the continuous case. Neither explains the logic that supports this assumption. So far, I have found nothing in the symbolicly solved classic method which would yield a number involving pi for the adjacent diagonal location. John Bailey

See my easy and foolproof solution: http://www.youtube.com/watch?v=v1YrANSmOGY