Electron cyclotron resonance (ECR) is a phenomenon observed in plasma physics, condensed matter physics, and accelerator physics. It happens when the frequency of incident radiation coincides with the natural frequency of rotation of electrons in magnetic fields. A free electron in a static and uniform magnetic field will move in a circle due to the Lorentz force. The circular motion may be superimposed with a uniform axial motion, resulting in a helix, or with a uniform motion perpendicular to the field (e.g., in the presence of an electrical or gravitational field) resulting in a cycloid. The angular frequency (ω = 2πf ) of this cyclotron motion for a given magnetic field strength B is given (in SI units) by

${\displaystyle \omega _{\text{ce}}={\frac {eB}{m_{\text{e}}}}}$.

where ${\displaystyle e}$ is the elementary charge and ${\displaystyle m}$ is the mass of the electron. For the commonly used microwave frequency 2.45 GHz and the bare electron charge and mass, the resonance condition is met when B = 875 G = 0.0875 T.

For particles of charge q, electron rest mass m_0,e moving at relativistic speeds v, the formula needs to be adjusted according to the special theory of relativity to:

${\displaystyle \omega _{\text{ce}}={\frac {eB}{\gamma m_{0,{\text{e}}}}}}$

where

${\displaystyle \gamma ={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}}$.