Cantor Equivalent

In the diagram above, each resistor has a value of 990 ohm. The recursive pattern of branching of the first and third branch continues infinitely.

Calculate the equivalent resistance of this circuit, rounded to the nearest integer.


The answer is 1400.

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1 solution

Arjen Vreugdenhil
Feb 18, 2016

Let R = 990 R = 990 be the value of an individual resistor, and x x the resistance of the entire circuit. The circuit can be analyzed as consisting of two individual resistors, and two copies of itself: Thus we have x = R + ( x R x ) x = R + (x | R | x) where | stands for placing elements in parallel. This translates into the algebraic equation x = R + 1 1 x + 1 R + 1 x x = R + R x 2 R + x ( x R ) ( x + 2 R ) = R x x 2 2 R 2 = 0 x = 2 R = 1400 Ω . x = R + \frac{1}{\frac 1 x + \frac 1 R + \frac 1 x} \\ \therefore\ \ \ x = R + \frac {R x}{2R + x} \\ \therefore\ \ \ (x - R)(x + 2R) = R x \\ \therefore\ \ \ x^2 - 2R^2 = 0 \\ \therefore\ \ \ x = \sqrt 2 R = \boxed{1400}\ \Omega.

Awesome problem!

Pranshu Gaba - 5 years, 3 months ago

where have u mentioned the points at which resultant resistance is to be found ??

Dhruv Aggarwal - 5 years, 1 month ago

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That is pretty obvious from the drawing, I'd say... there are two connectors, one at the top and one at the bottom. Find the resistance between those two.

Arjen Vreugdenhil - 5 years, 1 month ago

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okay but prefer writing points A and B .. makes it easier to comprehend

Dhruv Aggarwal - 5 years, 1 month ago

I had a lot of fun in solving this problem, thank you!

Gabriele Manganelli - 4 years ago

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You're very welcome.

Arjen Vreugdenhil - 4 years ago

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