electromagnetism - Electromagnetic braking hypothetical situation?

Let's say there is a conducting rod in space - with a constant external magnetic field. If you spin the rod - will electromagnetic braking make the rod stop eventually? I feel like there is a change in flux through the rod when it rotates - and due to Faraday's law and Lenz's law eddy currents will be induced to oppose the motion of the rod

Answer

There is only one case where isn't induced a current in the rod. One have to move the rod parallel to the magnetic field. In any other case and especially in any case of rotation the magnetic field will move the electrons inside the rod.

For example, if the rotating axis is perpendicular to the rods symmetry axis, there are two pure cases.

(1) The rotating axis is aligned parallel to the magnetic field. Free available electrons will be distributed more to the ends of the rod or more to the centre of the rod. This depends from how the direction of the magnetic field and the direction of the rods rotation are related. Since the rod isn't an closed electric circuit this happens only once and after it does not effect the rotation any more.

(2) The rotating axis is aligned perpendicular to the magnetic field. Electrons are moved periodically parallel to the axis of rotation. In case (2) electrons get accelerated periodicaly and this will be accompanied by electromagnetic radiation. The energy for this radiation has to come or from the kinetic energy of the rotating rod or from a cooling down rod. Cooling may be happens but it is not the dominant process.

Last not least I have to explain why the electrons will be moved. First at all think about the Lorentz force $ \vec F = q \vec v \times \vec B $. This vector cross product can be rewritten for orthogonal vectors to $ q \vec v = \dfrac {(\vec B \times \vec F)}{\|\vec {B}\|^2} $. This corresponds with what happens in an electric generator. In case (1) it will be an DC generator, in case (2) an AC generator. To be precise, in case (1) it is a homopolar generator, ie a special DC generator.

Second one can ask why the electrons get moved in this way under the influence of a magnetic field. One has te remember that attraction or repulsion happens only between charged bodies or between magnetic dipoles. It is nice for us that electrons have magnetic dipole moment and it is spinning too. The electrons magnetic moment will be aligned under the influence of a magnetic field. This isn't spectacular. More interesting is it if the electron moves in a magnetic field (not parallel to the field). The electrons magnetic moment will be aligned and this is accompanied by a gyroscopic phenomena. Gyroscopic effect means that every rotating body act again the force which try to align it. Because the electron has not a magnetic moment only but spins too the moving electron will be deflected. Since any deflection is an acceleration, a photon emission takes place and this emission is directed against the gyroscopic effect. The direction of the magnetic moment of the electron more or less fall back in his previous orientation.

Periodical displacement of the electrons and an associated EM radiation, this is the full picture we see in case (2). The rod will come to a stop.

At the end proof the described movement of the electrons with the Right-hand rule. Don't worry about right or left hand. In your case it is more important to follow the orthogonality of the three vectors magnetic field $ \vec B $, movement of the rod, represented by $ \vec F $ and the resulting current, represented by $ q \vec v $.

If anyone is surprised that such a hand rule is to use, this is because the spin of the electron and its magnetic moment are clearly coupled. If this were not so in nature, then we could not use generators and motors.