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runnergirl
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Homework Statement
A toroid of circular cross-section of radius a and mean radius r_{m}. Show that the inductance of this coil is given by L = μ_{r}μ_{o}N^{2}[r_{m}-(r_{m}^2-a^{2})^{1/2}].
Homework Equations
The B-field of a toroid: B = μNI/2πr (phi-direction)
The Attempt at a Solution
Total flux: N∫Bds where ds in the phi direction in spherical coordinates is: r_{t}dr_{t}dθ. In this case we need to solve for r which can not be assumed to be approximated by the mean radius r_{m}. Given that the vectors can vary as a function of θ within the toroid, I used the law of cosines to solve for r = √r_{m}^2+r_{t}^2 - 2r_{m}r_{t}cosθ and then integrated from 0 to a for the radial component and 0 to 2π for the θ component. The issue I'm having is that the integral isn't coming out and I'm not sure where I went wrong. Any help would be much appreciated, thank you.