|MadSci Network: Physics|
The amount of kinetic energy the electron has is not the determining factor for photon emission; electrons emit photons when they are accelerated in a magnetic field. An electron moving at a constant velocity will not emit photons no matter how much kinetic energy it has. Your electron in the cyclotron emits photons because it is accelerating; even at constant speed, the electron is accelerating because the direction of its velocity is changing. The frequency of the photon released (or absorbed, in the case of the electron gaining energy) is E/h, where E is the change in energy of the electron and h is Planck's constant. The energy change depends on why the energy is changing. In the case of a cyclotron, the acceleration (and thus the change in energy) is due to the application of a magnetic field. You can find the equation that will give you the change in energy over time of an electron in a cyclotron at this wikipedia page. When an electron emits a photon, its energy is decreased by E=f/h, where f is the frequency of the photon. This can cause it to slow down, and/or to change the size and shape of its orbit. In an atom, there are only certain allowed energy levels, which control the possible values of E, and thus the possible frequencies of photons that can be emitted. In an atom, this emission (or absorption) of photons can be either stimulated or spontaneous, but in either case it generally is not continuous; that is, an electron emits a photon, changes energy level, and that's it (until the next time it randomly emits if it's not in the ground state). In the cyclotron of your example, a magnetic field is being constantly applied, so photons will be continuously emitted. Note that the emission causes the electrons to lose energy, so the machine that is producing the magnetic field must do work to bring the electron's energy back up, thus energy is conserved in this machine.
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