What is cyclotron frequency?

Cyclotron frequency (gyrofrequency) is the rate at which a charged particle orbits around a magnetic field line. It depends only on the particle's charge-to-mass ratio and the field strength: f_c = |q|B / (2πm). It is independent of the particle's speed in the non-relativistic limit.

What is the difference between cyclotron frequency and gyrofrequency?

They are the same quantity. "Cyclotron frequency" is common in accelerator physics; "gyrofrequency" is preferred in plasma and space physics. Both refer to f_c = |q|B/(2πm), and the associated angular rate ω_c = 2πf_c = |q|B/m.

How does relativity affect the cyclotron frequency?

At relativistic speeds the effective mass increases by the Lorentz factor γ = 1/√(1−v²/c²), so the relativistic cyclotron frequency becomes f_c = |q|B/(2πγm). In a synchrotron the RF frequency must track γ as the particle accelerates, unlike a classical cyclotron where f_c is constant.

What is the Larmor (gyro) radius?

The Larmor radius r = mv_⊥/(|q|B) is the radius of the circular orbit a charged particle traces when moving perpendicular to a magnetic field. A faster or heavier particle has a larger orbit; a stronger field tightens it. In the relativistic case, replace m with γm.

What is a typical electron cyclotron frequency in a fusion device?

In a tokamak with B ≈ 3–5 T the electron cyclotron frequency is roughly 84–140 GHz (millimeter-wave band). ECR heating uses microwave sources tuned to this frequency to transfer energy directly to electrons.

Cyclotron Frequency Calculator

Enter a particle type and magnetic field strength to compute cyclotron (gyro) frequency and gyroradius instantly.

Particle & Field

Particle

Magnetic field strength

Particle speed (optional — for gyroradius)

Quick examples

Cyclotron Frequency

Angular frequency ω_c

—

rad/s

Linear frequency f_c

—

Period T

—

Relativistic γ

—

Lorentz factor

Set a particle and field strength, then click Calculate.

Summary

The cyclotron frequency (also called gyrofrequency) is the angular rate at which a charged particle orbits in a uniform magnetic field: ω_c = |q|B/m, giving a linear frequency f_c = |q|B/(2πm). Simultaneously, the Larmor (gyro) radius r = mv_⊥/(|q|B) sets the physical size of that orbit. These two quantities govern wave–particle interactions in space plasmas, determine the design of cyclotrons and synchrotrons, and appear in electron cyclotron resonance (ECR) heating of fusion plasmas.

How it works

Select a particle preset (electron, proton, alpha, or enter custom charge and mass).
Enter the magnetic field strength B in Tesla (or nanotesla for space physics scenarios).
Optionally enter the particle speed (as a fraction of c or in m/s) to compute the Larmor radius.
The tool computes the angular frequency ω_c = |q|B/m and linear frequency f_c = ω_c / (2π).
If a speed is provided, the gyroradius r = mv / (|q|B) is shown; the relativistic correction γ is applied when v > 0.01c.
Results are shown in Hz, MHz, GHz, and the period T = 1/f_c for easy cross-reference.

Use cases

Determine the ECR heating frequency for a given magnetic mirror field in fusion research.
Check cyclotron resonance conditions for RF wave injection into a plasma.
Estimate Larmor radii of solar wind particles at 1 AU (~5 nT field).
Verify cyclotron frequency ranges when designing RF cavities for particle accelerators.
Study how magnetic field strength and particle species shift the gyrofrequency in astrophysical jets.
Compute proton gyroradii in Earth's magnetosphere for space weather modeling.
Convert between angular and linear gyrofrequency in plasma physics homework.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu