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Force between Current Loops 
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#1
Apr1013, 12:04 PM

P: 117

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 BiotSavart 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 tripleproduct 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? 


#2
Apr1013, 02:34 PM

P: 185

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. 


#3
Apr1013, 02:53 PM

P: 117

Ah nice, I see now. Thanks for the trick.



#4
Jan1014, 04:07 PM

P: 1

Force between Current Loops
please could you explain why the integral over loop 1 is zero? I'm struggling to see your method.. :/



#5
Jan1014, 08:00 PM

P: 117

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. 


#6
Jan1114, 04:12 PM

P: 37

Please, prove it with integrals and divergence theorem. By the way I did not see any t (time) variable in your formulae. 


#7
Jan1114, 05:24 PM

Mentor
P: 16,942

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.



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