Where Does the Energy of Cyclotron Radiation Come From? Solving the Paradox

[goran d]

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.

 

[mfb]

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.

The photons carry momentum, everything is fine.

For a perfect mirror, it does not matter if it rotates, by the way.

 

[Vanadium 50]

Since rotating charges produce cyclotron radiation, it emits photons.

A uniformly charged ring rotating about its axis does not radiate. (If you don't believe me, calculate the power radiated)

 

[Jano L.]

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 paradox lies in the question, where does the energy of the photons come from.

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.

 

[sardar]

As a scientist, the first step in addressing this paradox would be to carefully examine the assumptions and conditions of the system described. It is important to note that the concept of a "perfect" toroidal mirror, one that can reflect all photons without any loss, is not physically possible. In reality, there will always be some energy loss due to imperfections in the mirror.

Furthermore, the idea of a charged ring rotating at high speeds is also a simplification of a complex system. In reality, there would be other forces acting on the ring, such as friction, which would also result in energy loss.

With these considerations in mind, it is possible to propose a solution to the paradox. One possibility is that the energy of the photons comes from a combination of the rotational energy of the system and the imperfections in the mirror. This would result in a gradual decrease in the rotation of the ring, as well as a decrease in the intensity of the emitted photons over time.

Another solution could be that the energy of the photons is not coming from the system itself, but from external sources. For example, the charged ring could be placed in a magnetic field, which could provide the necessary energy for the emission of photons. This would not contradict the laws of thermodynamics, as the energy is coming from an external source rather than being created within the system.

In conclusion, the Cyclotron radiation paradox highlights the importance of carefully considering all factors and assumptions in a system before drawing conclusions. It also reminds us that in the complex world of science, seemingly paradoxical situations can often be explained by taking into account a broader range of factors and variables.