Infinite cubic grid of resisrors

Can we find net resistance between the body diagonal points of a infinite cubic lattice

?

The objective is to find net resistance between A & B of the given cube which is part of infinite grid. Let the resistance between any two adjacent vertices is R

Note by Pranjal Prashant
2 years, 7 months ago

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Oh my.

Josh Silverman Staff - 2 years, 7 months ago

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Yes, I know about 2 d version for resistors and fourier series for getting across square diagonal 2r/pi.[you know that Ishan :D ] But this is quite different fom those, much more difficult in getting the approach. You should share this so that someone reaches at the final ansWeR.

Pranjal Prashant - 2 years, 7 months ago

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Here is the 2-d version of the problem.

Calvin Lin Staff - 2 years, 7 months ago

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It must be R/3

Jatin Chauhan - 2 years ago

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Solution ?

Rajdeep Dhingra - 2 years ago

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I asked for diagonally opposite points. Not adjacent ones

Pranjal Prashant - 1 year, 11 months ago

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A simpler version would be two find the equivalent resistance between two diagonally opposite points in an infinite grid of resistors, which is still quite difficult. See the page I have linked Infinite grid of resistors

Ishan Tarunesh - 2 years, 7 months ago

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Total resistance (across points A and B) = ( 5/6 ) * R
I can't post a picture of my solution here

Vincent Miller Moral - 2 years, 7 months ago

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Incorrect, This answer holds if there was a single cube of resistances rather than an infinite one.

Ishan Tarunesh - 2 years, 7 months ago

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but infinite series of resistors just push the "single cube value" to an exact number so my idea is that effective resistance is near (5/6)R (like 0.85R or 0.9R max but not less than (5/6)R). Please notify me if there is a mistake in my assumption.

Yeshas Bharadwaj - 2 years, 7 months ago

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@Yeshas Bharadwaj Now what significance pushing values is of, the question is not an mcq, I want to know how it can be done, and I am sure that it is solvable. {although answer would not be beautiful perhaps}.

Pranjal Prashant - 2 years, 7 months ago

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@Pranjal Prashant got it but I dont know how to solve the way you told and yes the answer sure will be weird

Yeshas Bharadwaj - 2 years, 7 months ago

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@Yeshas Bharadwaj NO, IT IS NOT SO. In the fourier series , steps of integration are nasty, but answer is simply 2r/ \(\pi\)

Pranjal Prashant - 2 years, 7 months ago

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@Pranjal Prashant thanks for the info I will surely try to get the steps for your answer.

Yeshas Bharadwaj - 2 years, 7 months ago

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@Yeshas Bharadwaj I could not understand "push the single cube value to an exact number". Also think of 2-D grid of infinite resistors in which we have to find the equivalent resistance between adjacent points which everyone knows to be R/2. What will be your argument in this one? Just because other resistances are present does not mean that the net would be greater than 5R/6 because resistances in parallel decrease the value

Ishan Tarunesh - 2 years, 7 months ago

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Yes it is haha, I considered only a 'cube'.

Vincent Miller Moral - 2 years, 7 months ago

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