Why Does the First Term in the Force Equation Between Current Loops Cancel Out?

[klawlor419]

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?

 

[qbert]

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 \nabla f \cdot d\mathbf{r} = df and you're integrating
over a loop that starts and end at the same place.

 

[klawlor419]

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

 

[jsholliday7]

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

 

[klawlor419]

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

there's 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.

 

[alva]

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?

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.

 

[Dale]

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.