Cyclotron orbits

If an electron travels with velocity v across a magnetic field B, the Lorentz force −ev × B causes an acceleration perpendicular to the velocity, which produces circular motion. Newton’s second law for the circularly accelerated motion gives f = m_ev²_⊥/r = ev_⊥B, so the cyclotron frequency f_c = v_⊥/2πr is proportional to the field:

.........(1)

The angular frequency ω_c equals 2πf_c. Any component of the electron velocity parallel to the magnetic field is uninfluenced by the Lorentz force, so the trajectory is a helix around the field direction. Electrons which follow cyclotron orbits in vacuum radiate energy of frequency f_c. The cyclotron frequency is 28 GHz T⁻¹.
The cyclotron radius r_c = m_ev_⊥/eB is of order a few micrometres for electrons in metals when B ≈ 1 T. It is less for semiconductors and semimetals, where a smaller electron density leads to a lower Fermi velocity and the effective electron mass m may be less than m_e.
One example of the use of cyclotron radiation is the domestic microwave oven. Another is the synchrotron, where electrons accelerated to a large multiple γ_e of their rest energy and constrained to move in curved paths by bending magnets, emit linearly polarized white radiation in a narrow beam of width 1/γ_e radians. Synchrotron radiation is a valuable tool for probing the electronic structure of solids.