Cyclotron Frequency Calculator

Formula

How this is calculated

When a charged particle moves perpendicularly to a uniform magnetic field B, the Lorentz force qv×B acts as a centripetal force, curving the trajectory into a circle. Balancing Lorentz and centripetal forces gives the radius r = mv/(qB) and, cancelling v, the cyclotron angular frequency ω = qB/m. The ordinary frequency f = ω/(2π) = qB/(2πm) and period T = 2πm/(qB) are both independent of the particle's speed or kinetic energy — a key property exploited in cyclotron particle accelerators.

Because the frequency depends only on the charge-to-mass ratio q/m and the field strength B, all particles of the same species orbit at the same frequency regardless of their speed, allowing a fixed-frequency RF field to accelerate them in resonance. Real cyclotrons are limited by relativistic effects: as a particle approaches relativistic speeds its effective mass increases, raising the period and breaking resonance — synchrocyclotrons or isochronous cyclotrons compensate by sweeping the RF frequency or tapering the magnetic field.

Default values correspond to a proton (q = 1.602×10⁻¹⁹ C, m = 1.6726×10⁻²⁷ kg) in a 1 T field, giving a cyclotron frequency of ≈ 15.24 MHz. An electron at the same field orbits at ≈ 28 GHz because its mass is ≈ 1836 times smaller.