Resistance of infinite nested triangles

[Mtnbiker]

Homework Statement

Here is an interesting problem... there is a wire bent in the shape of an equilateral triangle, side length = a and resistivity = rho.

In the center of this triangle is another equilateral triangle (inverted, side = a/2, resistivity = rho) and so on into infinity. What is the overall resistance between points A and B in terms of a and rho?

Homework Equations

R = (rho * length)/area

The Attempt at a Solution

I started by using the equation for resistivity, R = (rho * length)/area, but I wasn't sure if area would apply here. We aren't given any information about the wire beyond the shape and length. So I'm really asking for help in determining a good starting point... I don't know of any other equations that would incorporate rho and length.

 

[tiny-tim]

Welcome to PF!

… In the center of this triangle is another equilateral triangle (inverted, side = a/2, resistivity = rho) and so on into infinity. What is the overall resistance between points A and B in terms of a and rho?
…
I started by using the equation for resistivity, R = (rho * length)/area, but I wasn't sure if area would apply here. We aren't given any information about the wire beyond the shape and length. So I'm really asking for help in determining a good starting point... I don't know of any other equations that would incorporate rho and length.

Hi Mtnbiker! Welcome to PF!

I don't think you can solve this on the information given.

I suggest you say "let the resistance be R/a times length", and carry on from there.

 

[Mtnbiker]

Hi Mtnbiker! Welcome to PF!

I don't think you can solve this on the information given.

I suggest you say "let the resistance be R/a times length", and carry on from there.

Hi... thanks for the welcome, I'm glad to be here.

I agree with you regarding keeping the area incorporated in the answer. However, I'm still struggling with what exactly would the length be (first triangle is 3a, second triangle is 3a/2, then 3a/4 and so on...). There is a point of diminishing returns, so I need to find that point.

Thanks for the input!

 

[tiny-tim]

Hi Mtnbiker!

Yes, you're correct … obviously each triangle has sides half the length of the next one out.

So assume there are n triangles, start from the inside, and work your way outward …

at each stage, get rid of one triangle and calculate the equivalent resistances along the three sides of the next triangle.

 

[Dick]

If the network is infinite you can use self similarity. Call the overall resistance between two vertices on the first inner triangle R. Now you have a simple network with three wires of resistance R and and six of resistance a*rho/2. Solve that for the resistance beween A and B in terms of R. Then realize that the outer triangular network is the same as the inner triangular network, but twice as big. So the resistance from A to B is also just 2R. Equate the two values and solve for R.

 

[Mtnbiker]

Thanks guys!