When an electron travels in circles in a uniform magnetic field, it must lose energy because all accelerated charges radiate, and must therefore spiral down to the center. Is this energy compensated by the magnetic field? Or where does this energy go?
Magnetic Fields – Energy of Electron Spinning in a Magnetic Field
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Best Answer
You are right. An electron in a uniform magnetic field will travel in circles (or in a helix, up to a change in frame of reference), but this means that it is an accelerated charge and it must therefore radiate and lose energy. This radiation is known as synchrotron radiation, and it is a major design issue for particle accelerators. (In fact, it is the reason for a recent trend to go back to linear accelerators, which are less efficient as each accelerating stage only works once per particle, but are not subject to this.) It can also be harnessed to make synchrotron light sources, and with some extra work one can build a free-electron laser using that principle.
In short, then, the electron will spiral down to the centre and lose all its kinetic energy as electromagnetic radiation.
(For the more quantum-mechanical minded, now that Landau eigenstates have joined the fray, this means that all excited Landau states will have to decay through radiative coupling to the ground state with zero angular momentum. Once there, though, the uncertainty principle kicks in and stops the electron getting localized to radii smaller than the characteristic harmonic oscillator length $$x_0=\sqrt\frac{\hbar\omega_c}{m}=\frac{\sqrt{\hbar eB}}{m}$$ corresponding to the cyclotron frequency $\omega_c=eB/m$.)