(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider a segment of a toroidal (doughnut-shaped) resistor with a horizontal cross-section (see attachment for the figure). Show that the resistance between the flat ends having a circular cross-section is given by

R = [itex] \frac{\phi_o}{σπ(√b-√a)^2} [/itex]

2. Relevant equations

Laplace's equation: [itex]\nabla^2\phi = 0[/itex].

E-field in terms of the potential: E=[itex]-\nabla\phi[/itex]

(both for cylindrical coordinates)

I = ∫[itex]J\cdot dA[/itex]

J = σE

R = [itex]\frac{\phi}{I}[/itex]

3. The attempt at a solution

From Laplace's equation we know that the potential will only vary with the angle

[itex]\phi[/itex] and it will vary linearly : [itex]\phi = k_1\phi+k_2[/itex]

due to BC [itex]\phi(0) = 0 = k_2[/itex] and [itex]\phi(\phi_o) = k_1\phi_o = V_o[/itex]

the E-field is [itex]\frac{-k_1}{r}\phi-direction[/itex]

Using the equation for the current: I = ∫[itex]-σk_1∫\frac{1}{r}drdz[/itex] where the dz portion is the height of the strip and the integrand goes from a to b (the change of radius).

Where I'm having issue is setting up the height of the strip. If you look at the attached picture I've drawn out what I think should be the height, but the integral gets pretty complex and I get some complex numbers when I apply the limits. If someone would please show me my mistake, I would be most grateful.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Resistance of a toroidal conductor

**Physics Forums | Science Articles, Homework Help, Discussion**