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## Homework Statement

Derive equation 8.43 (given below). The equation shows the internal energy stored in a system of two current loops, one of radius b with current Ib and one of radius a with current Ia. The currents are spinning in the same direction, the loops are parallel, and they share the same axis. The distance between them is h.

Hint: Build the two currents up from zero to their final values.

## Homework Equations

##U = \frac{1}{2} L_{a}I_{a}^2 + \frac{1}{2} L_{b}I_{b}^2 + MI_{a}I_{b} ##

M is also given:

## M = \frac{\mu_{0} \pi b^2 a^2}{2(h^2+b^2)^{3/2}} ##

An earlier part of the question required that I calculate B on the larger wire due to the smaller (which I assumed to be a dipole. I know I'm right here:

##B = \frac{\mu_{0} I_{a} a^2}{8(h^2+b^2)^{5/2}} (2h^2-b^2)\hat{r} - 3hb\hat{z}##

## The Attempt at a Solution

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The first two terms seem simple enough to me - the energy stored in a loop of current of radius a is ##U = \frac{1}{2} L_{a}I_{a}^2 ##. But where the third term comes from, I haven't the slightest of clues. I did find the B-field due to the smaller loop (which I assumed to be the dipole) at the larger loop. This was for an earlier part of the problem which I know I got right.