About

A charged particle moving perpendicular to a uniform magnetic field B follows a circular orbit. The frequency of this orbit, the cyclotron frequency f_c, depends only on the particle's charge-to-mass ratio q/m and the field strength. It does not depend on the particle's speed. This property is the operating principle behind cyclotron accelerators and is fundamental to plasma diagnostics, mass spectrometry, and magnetospheric physics. Incorrect charge-to-mass values propagate directly into resonance tuning errors in RF cavities and can cause beam loss or misidentified ion species in spectrometers.

This calculator computes the cyclotron angular frequency ω_c = qB/m, the ordinary frequency f_c, the orbital period T, and the Larmor (gyro) radius r_L given the perpendicular velocity v_⊥. All calculations assume non-relativistic speeds. For particles approaching 0.1c, the relativistic mass increase shifts the actual resonance frequency downward by a factor γ. The tool uses CODATA 2018 recommended values for fundamental constants.

Formulas

The cyclotron angular frequency for a non-relativistic charged particle in a uniform magnetic field:

ω_c = qBm

The cyclotron frequency (ordinary frequency):

f_c = qB2πm

The orbital period:

T = 2πmqB

The Larmor (gyro) radius, which does depend on perpendicular velocity:

r_L = mv_⊥qB

Where q is the absolute charge of the particle in C, B is the magnetic field magnitude in T (tesla), m is the particle rest mass in kg, and v_⊥ is the component of velocity perpendicular to B in m/s. These formulas hold in the non-relativistic regime where v << c. For relativistic particles, replace m with γm where γ = 1/√1 − v²/c².

Reference Data

Particle	Symbol	Charge (q)	Rest Mass (m)	f_c at 1 T
Electron	e⁻	1.602 × 10⁻¹⁹ C	9.109 × 10⁻³¹ kg	27.99 GHz
Proton	p	1.602 × 10⁻¹⁹ C	1.673 × 10⁻²⁷ kg	15.24 MHz
Deuteron	d	1.602 × 10⁻¹⁹ C	3.344 × 10⁻²⁷ kg	7.63 MHz
Triton	t	1.602 × 10⁻¹⁹ C	5.007 × 10⁻²⁷ kg	5.10 MHz
Alpha (α)	α	3.204 × 10⁻¹⁹ C	6.645 × 10⁻²⁷ kg	7.66 MHz
He-3 ion	³He²⁺	3.204 × 10⁻¹⁹ C	5.008 × 10⁻²⁷ kg	10.18 MHz
Muon (μ⁻)	μ⁻	1.602 × 10⁻¹⁹ C	1.884 × 10⁻²⁸ kg	135.5 MHz
O⁺ ion	O⁺	1.602 × 10⁻¹⁹ C	2.657 × 10⁻²⁶ kg	0.960 MHz
Fe²⁺ ion	Fe²⁺	3.204 × 10⁻¹⁹ C	9.274 × 10⁻²⁶ kg	0.550 MHz
Positron	e⁺	1.602 × 10⁻¹⁹ C	9.109 × 10⁻³¹ kg	27.99 GHz
C⁶⁺ ion	C⁶⁺	9.613 × 10⁻¹⁹ C	1.993 × 10⁻²⁶ kg	7.67 MHz
N⁷⁺ ion	N⁷⁺	1.121 × 10⁻¹⁸ C	2.325 × 10⁻²⁶ kg	7.68 MHz

Frequently Asked Questions

As velocity increases, the orbit radius grows proportionally: a faster particle traces a larger circle but completes it in the same time. The radius r_L = mv_⊥/(qB) scales linearly with v_⊥, so the circumference and the velocity cancel in the period expression. This independence breaks down at relativistic speeds where the effective mass γm increases with velocity, causing the frequency to drop. Synchrocyclotrons compensate for this by reducing the RF drive frequency as particles accelerate.

The relativistic correction factor is γ = 1/√(1 − v²/c²). At v = 0.1c (~3 × 10⁷ m/s), γ ≈ 1.005, introducing a 0.5% frequency shift. For electrons this threshold is reached at about 15 keV kinetic energy. For protons it corresponds to about 4.7 MeV. Beyond these energies, use the relativistic corrected frequency f = qB/(2πγm).

In MRI, the Larmor precession frequency of nuclear spins (primarily hydrogen protons) in the scanner's magnetic field is identical in form to the cyclotron frequency: f = γ_gB/(2π), where γ_g is the gyromagnetic ratio. For proton spins, γ_g/(2π) = 42.577 MHz/T. A 1.5 T clinical scanner therefore operates at 63.87 MHz, and a 3 T scanner at 127.73 MHz. Gradient coils produce spatial variation in B, mapping frequency to position.

Effective magnetic confinement requires that the Larmor radius r_L be much smaller than the plasma vessel dimensions. In a tokamak with B = 5 T and ion temperature 10 keV, a deuteron has r_L ≈ 2.9 mm, well within the meter-scale vessel. Alpha particles born at 3.5 MeV have r_L ≈ 5.4 cm. If the minor radius is only tens of centimeters, prompt alpha losses at the edge become significant. Ripple in the toroidal field exacerbates this by creating local regions of weaker confinement.

Yes. Select "Custom Particle" mode and enter the total charge as the charge state times the elementary charge: for Fe³⁺, q = 3 × 1.602 × 10⁻¹⁹ C = 4.806 × 10⁻¹⁹ C. Use the ionic mass (atomic mass minus electron masses, though the difference is negligible for heavy ions). For bare uranium U⁹²⁺ at B = 1 T, the cyclotron frequency is approximately 37.3 MHz.

The Earth's surface field is approximately 25 - 65 μT. At 50 μT, the electron cyclotron frequency is about 1.40 MHz and the proton cyclotron frequency is about 763 Hz. O⁺ ions (dominant in the F-region ionosphere) have a cyclotron frequency near 48 Hz. These frequencies define cutoffs and resonances in radio wave propagation through the ionosphere. VLF waves near the proton gyrofrequency interact strongly with radiation belt particles.