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Q: Infinite grid of resistors ( No Answer,   4 Comments )
Subject: Infinite grid of resistors
Category: Science > Physics
Asked by: racecar-ga
List Price: $2.00
Posted: 29 Oct 2002 16:18 PST
Expires: 28 Nov 2002 16:18 PST
Question ID: 92564
Well known problem:
Given an infinite grid of 1 ohm risistors, what is the resistance
across a single diagonal?

Well known answer:
2/pi ohms.

Where can I find a full solution to this problem?
There is no answer at this time.

Subject: Re: Infinite grid of resistors
From: scodad-ga on 31 Oct 2002 05:01 PST
Lets call the resistance of the grid Req. Since the grid is infinite
you could "break off" a segment, and the resistance of the remaining
grid will still be Req. The remaining segment is in parallel with one
of the 1 ohm resistors that has been "broken off."  The equation for
reisistors R1 and R2 in parallel is: Rp = R1*R2/(R1 + R2) . There is
still a one ohm resistor from the broken-off section that is in series
with Req.

The equivalent resistance Req  can be calculated using the equation:

Req = 1 + (Req * 1)/(1+Req)
Req^2 - Req - 1 = 0
Use quadratic equation to solve
Subject: Re: Infinite grid of resistors
From: racecar-ga on 31 Oct 2002 18:00 PST
To clarify: the grid in question is infinite in 2 dimensions, not just
1.  The method mentioned by scodad works fine for an infinite "ladder"
of resistors, but since the answer has a pi in it, there is no point
in looking for a solution by such a simple algebraic method in the
case of an infinite (in 2 dimensions) grid.
Subject: Re: Infinite grid of resistors
From: iota-ga on 03 Nov 2002 22:53 PST
By full solution, do you mean across more than one node?  If so, the answer is:
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)

the derivation of this can be found among the following
 (with embedded links)

Other links...
 (with embedded links)

Comments on a three dimensional grid...
Subject: Re: Infinite grid of resistors
From: nokay-ga on 01 May 2004 00:45 PDT
A current link showing the integral solution for one 2D case

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