cyclotron frequency
The cyclotron frequency is how many times per second a charged particle goes around its circular path in a magnetic field. Remarkably, it does not depend on how fast the particle is moving or how big its circle is: a slow particle on a small circle and a fast one on a large circle complete their loops in the same time. It answers, how quickly does a particle circle in a given magnetic field?
Starting from the circular-motion radius r = m v / (q B) and the fact that the time for one loop is the circumference divided by the speed, the speed v cancels out. The result is an angular cyclotron frequency omega = q B / m (in radians per second), and an ordinary frequency f = q B / (2 pi m) in hertz. The corresponding period is T = 2 pi m / (q B). Notice that only the charge q, the mass m, and the field B appear; the speed is gone. A stronger field makes the particle circle faster; a heavier particle circles slower.
This speed-independence is precisely what makes the classic cyclotron accelerator work: a fixed-frequency electric 'kick' can be applied at the same rhythm to particles of any speed, giving each one an energy boost every half turn as it spirals outward. (At very high speeds the particle's relativistic mass increase makes the frequency drift, which is why modern machines are synchrotrons rather than simple cyclotrons.)
In a 1 T field, an electron circles at f = q B / (2 pi m), about 28 billion times per second (28 GHz). A proton, being about 1836 times heavier, circles that many times more slowly in the same field.
f = q B / (2 pi m): independent of speed, set only by charge, mass, and field.
The independence from speed holds only while the particle is nonrelativistic. Near the speed of light the effective mass grows, the frequency drops, and a simple fixed-frequency cyclotron falls out of step.