January 17, 2006

Consider a spherical cow, covered in resistors

(Anybody who didn't take an A-level in Physics may as well switch off now. Any readers who sat next to me throughout A-level Physics lessons may wish to consider that it might be their fault I wasn't paying enough attention to be able to work this out. ;-) )

This question's been bugging me for days. I did have an answer, but after a spot of googling showed me I was wrong, I realised I was talking rubbish.

Imagine, if you will, a flat square grid of points, such that every point is connected to the 4 adjacent points by a 1Ω resistor. The grid extends infinitely in all directions. What is the resistance between 2 points on the grid a knight's move apart? (e.g. 2 squares up and 1 across, WLOG) Or, for a simpler starter question, what about between 2 points diagonally adjacent? Or even just 2 adjacent points? (Is that as simple as just 1Ω? My hunch is no.)

Maybe I've missed something while googling for answers to this, but if anyone has a good solution I'd be very interested to hear it! Just to clarify, I have some numerical answers from googling, but I'm far more interested in how they're arrived at than what the actual numbers are.

[Update: Chris has considerably better google-fu than me, and found a solution for the diagonal case. It contains hard maths that I can't really justify spending the time to get my head around in the middle of exam season, alas. There's a paper with even more on this too. Eek!]

Posted by James at 22:59 | Comments (0) | TrackBack
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