Thanks for the responses. As to the energy question I was excluding friction, as I think you assumed, so the issue of the motor is also moot, the disk is spinning and it doesn't matter what caused that. So I take it from your answer that the disk would spin indefinitely if the charge is azimuthally symmetric and it would slow down if it isn't, presumably because as the energy is radiated away a force is produced that impedes the spinning of the disk.
Yes.
This also pops up in astronomy. Orbiting bodies create gravitational waves the same way a spinning disk with unbalanced charge would radiate electromagnetically. So gravitationally orbiting systems lose energy over time. Lower-energy orbits actually have a higher frequency (they orbit closer in), but the idea of energy loss through radiation is the same. For the earth and sun, the power loss rate is REALLY low (the frequency is also low, only 1 year
-1), so we can ignore it. But for massive enough objects with orbiting very close to each other (so the frequency is much higher), the power loss rate can be significant enough to create a significant decay in the orbits over time.
One interesting implication here is that photons that are emitted have some angular momentum otherwise angular momentum would not be conserved as the disk slows down. But how can a photon radiating in space have angular momentum?
Pretty easily, as it turns out. First, keep in mind that angular momentum is defined in terms of the momentum
about some point in space. So any object with linear momentum will have nonzero angular momentum about any point that is not along the line that its center-of-mass is moving on.
Now, if an object is spinning and it has no
net momentum, then in that case the angular momentum that we calculate for different points will all agree. We can generalize this to not just solid objects, but collections of moving objects as well: they can be zipping all over the place, but if the net momentum of all the objects is zero then the
total angular momentum will be constant regardless of what reference point we choose. But the angular momentum of each individual object will NOT be constant, that will still depend on our choice of reference points.
So let's say I've got two balls attached by a string, and they're spinning about each other. They've got angular momentum but no net linear momentum. It's easiest to calculate by using the center of rotation as our reference frame, but any point will do. Now suppose the string breaks, and the balls go flying off in opposite directions. We haven't actually changed the angular momentum. Although the balls are moving in opposite directions, the lines of their motion are offset from each other. So no matter what reference point we pick, the calculated angular momentum of each linearly moving ball won't be the same. They still carry angular momentum.
So all we need is for photons from our disc to do something similar: have their center-of-momentum offset from the center of the disk, and they will carry angular momentum.
ETA: There is another issue associated with this. How does it effect things if the material the charge is placed on is a conductor or an insulator.
If it's an insulator, we can arrange the charges however we want to, and the charges will stay put (at least for a while). If the material is a conductor, then the charges will arrange themselves in order to minimize their electrostatic potential, and they'll do so pretty quickly. This means that if our spinning object is azimuthally symmetric, then for a conductor, the charges are automatically going to arrange themselves in an azimuthally symmetric manner as well. But if you make your conductor non-azimuthally symmetric, then the charges won't be either. As a simple example, if your conductor is a long rod, most of the charge will accumulate at the ends of the rod. So if you rotate that about an axis perpendicular to the axis of the rod, then it WOULD radiate.