Prove that, in a uniform magnetic field, the net force on a current loop carrying current I is zero.
Green's theorem, and magnetic force equation
The Attempt at a Solution
My main question is whether I can use Green's theorem to prove this result... it seems rather too simple if I can!
Since F=∫I(dLxB) integrated around the whole loop, let's say the B=(Bx)i+(By)j+(Bz)k and let's parameterize the closed loops as the path L: x(t)i+y(t)j+z(t)k. Then dL= dx i+ dy j + dz k. And dL x B = (Bz*dy - By*dz)i+(Bz*dx-Bx*dz)j+(Bydx-Bxdy)k.
Can I simply break up the integral expression for F into three closed-loop line integrals for Fx, Fy, and Fz respectively? And then can I apply Green's theorem and (using the statement that it's a uniform B-field), I know the derivatives of any component of the magnetic field must be zero... so the line integrals all turn into double integrals over some region, but with 0 as the integrand. Thus the net force in all directions is 0!
Is that allowed? The question called for me to integrate directly...