How Do You Calculate the Magnetic Moment of a Current Loop?

[iknowone]

I cannot figure out how to solve this problem:

The calculation of the magnetic moment of a current loop leads to the line integral (around a closed loop)

INT r x dr

a) Integrate around the perimeter of a current loop (in the xy plane) and show that the scalar magnitude of this line integral is twice the area of the enclosed surface.



b) The perimeter of an ellipse is described by r = x_hat*(a*cos(t)) + y_hat*(b*sin(t)), Using part (a) show that the area of the ellipse is pi*a*b.

notation (x_hat, y_hat are unit vectors in the x and y directions respectively)

Thank you for your help.

 

[Meir Achuz]

A) You can show this most easily by a sketch of the loop.
rXdr is seen to be twice the area of a triangle with one corner at the center of the loop and the opposite leg on the circumference. Doing the integral means adding up all these areas and getting 2S. It can also be done using vector calculus, but that is a bit tricky.

B) For b, just take the cross product of the given r with dr (the differential of the given r) and do the integral over t from 0 to 2 pi.

 

[Fjolvar]

I need help with this same problem so rather than create a new thread I figured I'd just respond to this one. I've tried many ways to approach this problem including the use of Stoke's theorem to rewrite the line integral as a surface integral, but I still can't get it. For part a to show that the scalar magnitude of the line integral is twice the area of the enclosed surface, does that involve using 2pi for the perimeter of the loop? Any help with this problem would be immensely appreciated. Thank you.

 

[Fjolvar]

Also, I'm trying to solve part A using vector calculus. I have a midterm tomorrow and I have a strong feeling this will be on there.. :(

 

[Fjolvar]

A hint to get this problem started or set up would suffice!

 

[diazona]

Did you try what Meir Achuz suggested?

Other than that, I'd imagine it's easiest to do part (a) for a rectangle. Then you can build up any shape out of infinitesimal rectangles.

Or if you're trying to use theorems of vector calculus, have a look at theorem[/url] (it's a special case of Stokes' theorem, but probably a little less mysterious how to apply it here).

 

[Fjolvar]

I had Green's theorem in mind, but I'm stuck on the part where I need to show that the line integral is 2 times the surface integral..
unless I'm understanding the problem wrong.

 

[diazona]

Did you happen to look at the part at the bottom of the wiki article entitled "Area calculation"? I think that's just the sort of thing you're trying to do here.