A question about resistance

laurelelizabeth

The question is "On this infinite grid of ideal one-ohm resistors, what is the equivalent resistance between the two marked nodes?"

n here's the picture: http://xkcd.com/356/

mgb_phys

Arghhhh don't bring that here!!!!!!!

Interestingly the answer doesn't seem to be independant of the distance between the two test points.
One of the odd features of conductive anti-static mats is that the resistance between any two points is the same (the matts are assumed to be infinite) - it must be because of the discrete number of paths in the resistor case

chroot

laurelelizabeth,

Are you asking us to solve it? It's actually really easy; just use superposition.

- Warren

Kittel Knight

Really easy?
It seems you can solve it in 4 or 5 lines!
...I'd like to see how...

Kittel Knight

Could you, Warren? please?

Kittel Knight

Hey Warren, wake up !

kudoushinichi88

Let me try... Total resistance = 1 ohm?

Kittel Knight

I think it is wrong, but, in fact, I don't care about the number - I just wanna know HOW TO GET IT.
Could you please explain your method?

Chroot had told us it could be done easily, by using superposition.

mgb_phys

The case for an exact diagonal is quite easy.
R(n,n) = 2/pi * Sum{k=1...n: 1/(2*k-1)}
R(1,1) = 2/pi
R(2,2) = 2/pi * (1 + 1/3)
R(3,3) = 2/pi * (1 + 1/3 + 1/5)

For a "2 accross 1, up 2" path as in the cartoon I'm not sure, but it must lie between 2/pi and 1.33*2/pi

There is a page of references for the calculation - you will probbaly need an academic subscription to get the text http://www.physics.thetangentbundle....istive_lattice

kudoushinichi88

Lol~ my answer was a wild guess. Sorry.

I think you'd better conduct a lab experiment and see the results...

mgb_phys

You can write something in python/mathematica to try a 100,100 array and get a good approximation. Then fiddle around with 2/pi and a constant to work out the 'correct' answer.

kudoushinichi88

I don't get it at all. How did you come up with that? I am not familiar with those math at all...

mgb_phys

Derivation stolen from http://www.geocities.com/frooha/grid/node2.html

kudoushinichi88

Oh my... I never thought this thing is really that complicated...

Kittel Knight

Thanks a lot, mgb_phys !!!


And it is not that easy Warren had said...

Well, who knows, maybe CHROOT could explain it in a other way, using superpostition...

laurelelizabeth

Whoosh that all went over my head :P I hope that I didn't annoy to many people by posting that there