Consider a uniform magnetic field. We know that if we have a conductive loop rotating in this magnetic field then we get a current running through that loop due to Faraday's Law, which in turn exerts a torque on the loop via the Lorentz force.
We can explain the effect of the magnetic field on a loop by observing that the flux intersecting the loop changes as it rotates, because the component of the area perpendicular to the magnetic field changes.
However, now consider a conducting sphere rotating in the magnetic field. On one hand, looking at the sphere as a whole, the flux through the sphere never changes, so one would expect there to be no effect on the sphere. But on the other hand, if I divide the sphere into lots of circular loops then there would be an effect on each of those loops, which I imagine might add up to some net effect. Probably something to do with eddy currents.
However, my electromagnetism is quite rusty. I do not know whether it is valid to treat a conducting sphere as many independent loops and integrate over it. The fact that all these loops intersect (if I choose them as great circles which share an axis of rotation) might make this choice of integration invalid. But I don't know how to justify whether it is valid or not. I would like some sort of proof or explanation one way or the other (unless, that is, there exists a better way to tackle the problem which makes this loops business redundant).
Ultimately, I wish to be able to calculate the torque exerted on a rotating conducting sphere in a uniform magnetic field. Bonus points if you can do this for arbitrarily oriented magnetic fields.