Torque on a rotating sphere in a uniform magnetic field

Question

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.

Answer 1

What you are asking is extremely difficult to compute.

Moving any conductor in a magnetic field will exert a Lorentz force on the charge carriers, which causes them to move, which produces a current density. However, due to boundaries of the material, this initial current density will cause charges to accumulate on the boundaries of the material, which produces an electric field inside the material, which changes the current density.

To observe electromagnetic induction, you want to look at the steady-state current density. In such a situation, $\nabla\cdot\vec{j}=0$. In a simple wire, this constraint permits a single scenario: the current is constant in all parts of the wire. However, for an extended object (such as your sphere), there is an infinite family of possible solutions which greatly complicates matters. Circular currents, or eddies, satisfy this condition, although the question becomes where to draw the circles and how much current they carry.

If I assume that we have current running in great circles through the axis of rotation, that satisfies $\nabla\cdot\vec{j}=0$. Considering this scenario also allows us to intuit that a rotating sphere of metal should experience a braking torque similar to a rotating ring of metal. However, integrating the sphere in this manner is problematic. In order to correctly treat the volume we would require the rings to become infinitely thin at the poles, which presents problems for the resistance. Additionally, confining the current to flow only along the great circles is artificial.

In general, calculating eddy currents is a hard problem. See, for instance, Method of calculating eddy currents of a conductor, is this correct? The best I can give you is an order of magnitude estimate.

The Lorentz force on the charge carriers is proportional to $q v B$, where $q$ is the charge, $v$ is the velocity and $B$ is the magnetic field. Since our system is undergoing circular motion, $v=\omega R$, with $\omega$ being angular frequency and $R$ being radius.

The current density $j$ is equal to the Lorentz force on the charge carriers divided by the charge and the resistivity $\rho$. Thus we get $j \approx \omega R B / \rho$.

The Lorentz force on each infinitesimal current-carrying volume element $dV$ is proportional to $j B dV$, so $F \approx \omega R B^2 dV/\rho$.

The torque is calculated by the volume integral $\tau = \int r\times F$. Thus, the torque on the object is (very) approximately

$\tau \approx \omega R^2 B^2 V/\rho$

for an object with radius $R$, volume $V$, and resistivity $\rho$ rotating at a frequency $\omega$ in a magnetic field $B$, to within an order of magnitude. Checking the units verifies that the formula yields a torque. This formula probably over-estimates the actual torque, because it assumes all parts of the extended object contribute maximally to the torque, so we can treat it as an upper bound. For a non-spherical object, you might need a more careful treatment of what 'radius' is, as the relevant linear dimension may differ in the two steps.