# Cyclotron resonance

Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillating electric field tuned to this resonance to add kinetic energy to charged particles.

## Cyclotron frequency

The cyclotron frequency or gyrofrequency is the frequency of a charged particle moving perpendicular to the direction of a uniform magnetic field B (constant magnitude and direction). Since that motion is always circular,[1] the cyclotron frequency is given by equality of centripetal force and magnetic Lorentz force

${\displaystyle {\frac {mv^{2}}{r}}=qBv}$

with the particle mass m, its charge q, velocity v, and the circular path radius r, also called gyroradius.

By substitution for the circulation frequency ${\displaystyle f={\frac {v}{2\pi r}}}$ which defines the cyclotron frequency, this leads to

${\displaystyle f={\frac {qB}{2\pi m}}}$,

or the angular frequency

${\displaystyle \omega =2\pi f={\frac {qB}{m}}}$.

It is notable that the cyclotron frequency is independent of the radius and velocity and therefore independent of the particle's kinetic energy - all particles with the same charge-to-mass ratio rotate around magnetic field lines with the same frequency.

The cyclotron frequency is also useful in non-uniform magnetic fields, in which (assuming slow variation of magnitude of the magnetic field) the movement is approximately helical - in the direction parallel to the magnetic field, the motion is uniform, whereas in the plane perpendicular to the magnetic field the movement is, as previously, circular. The sum of these two motions gives a trajectory in the shape helix.