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. 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?
a couple of things. 1. s can't be pulled out of the integrals. 2. look at this term: $$ \mathbf{F}_{12}=\frac{\mu_0 I_1 I_2}{4\pi}\oint\oint d\mathbf{l_2}\left( d\mathbf{l_1}\cdot \frac{\hat{s}}{s^2}\right) $$ then realize that if we do the integral over loop 1 first, we have $$ \oint d\mathbf{l_1}\cdot \frac{\hat{s}}{s^2} = \oint d_1\frac{1}{s} = 0. $$ where the subscript on 1 means i'm treating all the terms associated with loop 1 as variables and freezing all terms associated with loop 2. Just remember [itex] \nabla f \cdot d\mathbf{r} = df [/itex] and you're integrating over a loop that starts and end at the same place.
please could you explain why the integral over loop 1 is zero? I'm struggling to see your method.. :/
Theres an important identity involving one of the terms. Once you use that identity, you can deduce by divergence theorem that the entire term for loop 1 vanishes.
So if there is a loop current in a start say 10 light years away and I switch on a loop current in the earth now the Newtons third law holds true. Please, prove it with integrals and divergence theorem. By the way I did not see any t (time) variable in your formulae.
Alva, the Biot Savart law uses the magnetostatic approximation. It doesn't apply in the far field and there is no time. This is a standard approximation, but it is an approximation.