I am working on a problem involving force between two loops of current. The problem is to prove the for any arbitrary loops of current, Newtons third law holds true.(adsbygoogle = window.adsbygoogle || []).push({});

I understand the basics of the approach but I am having trouble seeing why a term goes to zero. The basic setup is to use the Biot-Savart law to predict the field of a segment of the current loop then to use the Lorentz force law to predict the force acting on a segment of the second loop.

This has the form,

$$d\mathbf{F}_{12}=\frac{\mu_0 I_1 I_2}{4\pi s^3}(d\mathbf{l_1}\times(d\mathbf{l_2}\times \mathbf{s}))$$

Which when simplified by the triple-product and integrated gives the following form,

$$ \mathbf{F}_{12}=\frac{\mu_0 I_1 I_2}{4\pi s^2}(\oint\oint d\mathbf{l_2}(d\mathbf{l_1}\cdot \hat{s}) -\oint\oint\hat{s}(d\mathbf{l_1}\cdot d\mathbf{l_2}))$$

Its easy enough to see right from here that F12=-F21 just by the fact that you pick up a minus sign by switching the separation vector. So in that sense the problem is solved, at least from what I see right now.

However I was looking in the Griffiths EM and apparently the first term on the left cancels out somehow and I can't figure why. (Problem 5.49) Any suggestions?

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# Force between Current Loops

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