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Conduing elipsical wire in a magnetic field

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conduing elipsical wire in a magnetic field :)

1. Homework Statement

An elipse shaped wire with a major axis a and minor axis b conducts a current I , and is set on the XY plane.
A const. magnetic field B is present, towards the Y axis (pointing "up")

I need to caculate the moment of force acting upon the elipse.



2. Homework Equations

1) F = ILBsin(alpha) (force acting upon a conduing wire in the presence of a magnetic field)

2) N = r X F (moment of force)


3. The Attempt at a Solution

The force acts towards "inside the page" - spinning the elipse.
Since the force acting upon each element of the eplise is different (and depends on the angle between the wire and the magnetic field), I figured I should somehow sum the cross-products (rXF) of each element throughout the elipse, with integration on "theta", which is the angle between the radius and the major axis as I progress...
The problem I encountred was calculating this angle, of each element, and express it using "theta".
I have reached a very far-fetched expression, using some Wiki help and some friends help, and reached a pretty complex relation. I shall reffer to the wanted angle (again, the angle between the wire element and the magnetic field, which points up) as "alpha":

note: e = c/a
alpha = 90 + "theta" - arctg ((e)^2sin(2theta)/sqrt(1-(e^2)(cos(theta))^2).

No way this thing enters my intergration :).

I'm really lost... maybe there's a much easier way? some symmetry I cannot see?

Thank you very much for reading! I'm sorry if it's messy.... but the question I hope is coherent.

Tomer.
 

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