Description of the paradox: We have a charged ring, which rotates at high speed. Since rotating charges produce cyclotron radiation, it emits photons. There is a toroidal mirror around the ring, spinning together with it. The mirror reflects the photons back inside, preventing them from escaping. The paradox lies in the question, where does the energy of the photons come from. There are two possible answers: 1. The energy comes from the rotational energy of the system. This way, the energy is conserved, but then, the system has to slow down its rotation, which contradicts the law of conservation of angular momentum. 2. The system does not reduce it's rotation. Angular momentum is conserved, but then, the energy of the photons comes from nothing, which contradicts thermodynamics.
The photons carry momentum, everything is fine. For a perfect mirror, it does not matter if it rotates, by the way.
A uniformly charged ring rotating about its axis does not radiate. (If you don't believe me, calculate the power radiated)
Vanadium is right, if the electric current was constant, there would be no radiation according to Maxwell's equations. In fact it is said that in the first years of cyclotrons, nobody expected radiation from them. It was discovered accidentally. For more realistic description of the cyclotron radiation, we can replace the continuous current by a series of bunches of point-like electrons. One bunch contains many (>##10^9## ?) electrons. These move more-less like one big charged body and because of the separations between different bunches, there are points in space where the current density is no longer constant in time. Then we get radiation of energy from the electrons. The energy of radiation comes from the energy of the circling electrons, which consists of their kinetic energy and the electromagnetic energy of the field near them. There is no paradox with conservation of angular momentum, since the radiation emitted carries the lost angular momentum. I think that perfect mirror does not exist, so sooner or later the radiation will leak out, but if it was there as a sort of spatial restriction (toroidal universe), the angular momentum of particles + field would be constant too.