What is the total force on a semicircular wire in a uniform magnetic field?

[bakin]

Homework Statement

A very thin wire, which follows a semicircular curve C of radius R, lies in the upper half of the x-y plane with its center at the origin. There is a constant current I flowing counter clockwise, starting upward from the end of the wire on the positive x-axis and ending downward at the end on the negative x axis. The wire is in a uniform magnetic field, which has magnitude Bo and direction parallel to the z axis in the positive z direction. Determine a symbolic answer in unit vector notation for the total force on the wire due to the magnetic field. Ignore the forces on the leads that carry the current into the wire at the right end and out of the wire at the left end.

Solution check: The numerical value with I = 2.00A, Bo = 3.00T, and R = 4.00m is 48.0j N.

Homework Equations

dF_b= i dLxB

The Attempt at a Solution

Because it's curved, I don't think you can use F_b = iLBsin. Instead, the cross product version above has to be used. I read some lecture notes here: http://www.wfu.edu/~matthews/courses/phy114/ppt/Ch29-Magnetic_Fields.ppt
And it says that you can just take the length from one endpoint to the other, and use that as your dL. Using that works, because (2)(3)(2x4) = 48, but I don't understand why. Can anybody help clarify? Thanks!

 

[Doc Al]

Because it's curved, I don't think you can use F_b = iLBsin. Instead, the cross product version above has to be used.

The magnitude of the cross product i dL X B = i dL B sinθ. (What's θ?)

I read some lecture notes here: http://www.wfu.edu/~matthews/courses/phy114/ppt/Ch29-Magnetic_Fields.ppt
And it says that you can just take the length from one endpoint to the other, and use that as your dL.

Where does it say that? Set up the integral of dF over the length of the semicircle. (It's an easy integral.) Which way does dF point at each position along the arc?

 

[bakin]

The magnitude of the cross product i dL X B = i dL B sinθ. (What's θ?)

Where does it say that? Set up the integral of dF over the length of the semicircle. (It's an easy integral.) Which way does dF point at each position along the arc?

Soooo helpful. I was able to figure it out after thinking about your questions a little bit. Thanks for helping me understand!

 

[Doc Al]

Excellent. (Glad it helped.)