Cyclotron frequency (${\omega }_c=|qB|/m$)

A particle of charge $q$ and mass $m$ in a uniform magnetic field of magnitude $B$ has cyclotron angular frequency ${\omega }_c=|qB|/m$.

In a uniform magnetic field, a nonrelativistic particle of mass $m$ and nonzero charge $q$ completes one circular transverse orbit in time $T_c=2\pi /{\omega }_c$, where ${\omega }_c=|q|B/m$. Parallel velocity changes the helix pitch but not this period. In relativistic motion the constant Lorentz factor multiplies the period: $T_c=2\pi \gamma m/(|q|B)$. The direction of rotation depends on the charge sign; the period is positive.

Cyclotron resonance is resonant absorption by charged particles in a magnetic field when an applied oscillating electric field drives their cyclotron motion. For a parabolic band, the resonance frequency is ${\omega }_c=|q|B/m^{\ast }$, so measuring it determines the cyclotron effective mass. In the quantum description, transverse electric dipole transitions connect adjacent Landau levels, whose separation is $\hslash {\omega }_c$. For an electron and a field along $z$, only the circular polarization rotating in the electron's orbital sense drives the positive-frequency resonance. Scattering broadens the absorption peak; resolving it requires an orbital period short compared with the relaxation time.

An rf field restricted to a metal's skin layer can transfer energy coherently to electrons whose orbits return to that layer. With magnetic field parallel to the surface, resonance can occur at $\omega =n{\omega }_c$, producing harmonics at $B_n=m_c\omega /(ne)$. Surface loss or cavity impedance replaces simple bulk transmission as the measured signal.