I'm not fully understanding cyclotron (and therefore synchrotron) radiation, emitted by a charge in a uniform and constant magnetic field. The charge undergoes centripetal acceleration, therefore the term $a^2$ in the Larmor formula is definitely not zero, thus the power emitted is not zero and we have emission of radiation.

Another true fact is that this magnetic field doesn't do work on the charge, so the kinetic energy of the charge stays the same, in fact it moves on a circumference of constant radius, with constant speed, in the plane perpendicular to the direction of the magnetic field.

The energy emitted doesn't come from the magnetic field cause it is unable to do work on the charge, it doesn't come from a loss of kinetic energy of the charge cause that would make it spiral to the center of the circumference, so Where does the energy emitted comes from?

It's a good question. Some of the kinetic energy of the particles is turned into radiation. If you look at the derivation of the Larmor equation (see for example this), it shows that the acceleration of a charge causes a bending of the electric field lines; this lateral motion of the E field is the radiation. But as the field lines bend, they exert a retarding force on the charge - so while the field carries energy with it, the charge is feeling a force from the field.

You may have seen the exercise where people move a heavy rope up and down to generate a wave down the rope. This takes a great deal of effort. The underlying physics is quite similar in form - as the wave travels away, the tension in the rope creates a retarding force; and it takes effort to keep the motion going (sending energy down the rope). (image source)

• Ok, so it's cause of the radiation reaction? In this case I should have used the Abraham–Lorentz equation to calculate the motion of my charge right? In this case I would still obtain circular motion cause of the particle losing energy cause of emission and recovering the same amount cause of self-force? May 24 '17 at 17:35
• I don't understand your comment. If a charged particle is in a uniform magnetic field, it will slowly spiral in unless you accelerate it (which is what we do in cyclotrons / synchrotrons). And of course in a synchrotron there is usually the undulator to create high local acceleration for a intense and focused beam (making the storage loop big reduces the radiation loss around the loop; the undulators create strong radiation at one or more points along the circle). There is no "recovering the same amount cause of self-force" that I can think of. May 24 '17 at 17:39
• The problem I have is not related to the machines, but finding the right theoretical description of the phenomena. In Landau-Lifshitz vol.2 paragraph 21 (motion in a uniform and constant magnetic field) it is shown that the charge follows an helical motion, and its projection onto the plane perpendicular to the magnetic field is a circumference of constant (in time) radius. From the discussion presented there i don't see any spiraling into the center of the circumference, that's why I asked if I maybe had to use different equations to describe the phenomenon. May 24 '17 at 17:51
• It does appear from your comment that your understanding of the operation of the machines may be incorrect. The particle loses energy, and should spiral in to the center. This is prevented by adding an RF cavity - this creates an E field along the path of the particle and accelerates it. The "synchro" part of the synchrotron points to the fact that the RF power has to be synchronized to the synchrotron frequency - this causes spatiotemporal coherence of the particle beam, as only particles that arrive "at the right time" get accelerated. May 24 '17 at 17:55
• I was biased by the derivations i've always seen and i've seen today in L&L, where it says that the velocity stays constant. So i convinced myself that the particle didn't spiral in, but stayed in the same circumference while emitting. That confused me. The fact that it actually loses energy and the circumference shrinks clears almost everything. Thanks. May 24 '17 at 18:20

Another true fact is that this magnetic field doesn't do work on the charge, so the kinetic energy of the charge stays the same [...]

Just because the magnetic field does not perform work on the particles, does not mean their kinetic energy does not change. Due to the radiation they give off, the electrons lose kinetic energy, which is why there are accelerator sections within both these devices.

• Telling more about the accelerator sections will more clearly answer OP's question. Aug 17 '17 at 9:37
• OP asked where the energy for the radiation comes from, and I clearly answered it comes from the kinetic energy of the electrons.
– noah
Aug 17 '17 at 10:45

The energy emitted doesn't come from the magnetic field cause it is unable to do work on the charge,...

You are right about that the magnetic field in sum doesn't contribute any work to the moving and by this deflected electron. The mechanism is more subtle:

1. It is well known that the electron has not only a charge but also a magnetic dipole moment. This moment - in contrast to the electric charge - has an orientation. This north-south orientation is equally distributed in space for the incoming electrons.
2. The cyclotrons magnetic field align the magnetic dipole moments of the moving electrons. At the moment of the alignment a tiny amount of the kinetic energy of the moving electron gets emitted. The electron radiates a photon.
3. The emission of a photons is accompanied with a momentum transfer. This transfer has two consequences for the electron. The electrons magnetic dipole moment gets disaligned again and the electron gets shifted in one direction (which states the Lorentz force).

3.1 A comment. The magnetic dipole moment of subatomic particles is connected with the intrinsic (means every time existing property) spin. Both have a direction and if this directions are defined as parallel for electrons then the directions for the positron is antiparallel. The introduction of spin is due to the fact that if the electron in the cyclotron is deflected to the right then the positron will be deflected to the left.

... it doesn't come from a loss of kinetic energy of the charge cause that would make it spiral to the center of the circumference

As it was explained above the radiation of the charge consumes kinetic energy and the charge indeed spiraling to a center. To be precise it goes in tangerine slices step by step along a spiral path until its kinetic energy is exhausted.

Funny is by the way the imagination that the photons which push the electron into its movement riding biggypack on the electron and get shaked off during the electrons deflection inside a magnetic field.

Another true fact is that this magnetic field doesn't do work on the charge,...

This statement is true from the statistical point of view. But in detail the magnetic field acts like a spring acts. Both are able to accumulate energy. In the case of the cyclotron the alignment and later the disalignment of the electrons magnetic dipole moment makes the external magnetic field sometimes weaker and sometimes stronger but after the electron leave the field it stays unchanged.