Cyclotron Frequency Calculator

Background.

A charged particle moving across a magnetic field is pushed sideways, at right angles to both its motion and the field, and that push bends its path into a circle. The surprising part, which Ernest Lawrence built the first cyclotrons around in the early 1930s, is how long each lap takes. A faster particle swings round a proportionally bigger circle, so the time per orbit does not depend on speed at all. It depends only on the charge, the field strength and the mass: f = |q|B/(2πm).

That independence from speed is what makes the frequency so useful. In a cyclotron, an alternating voltage timed to it gives particles a kick every half-turn as they spiral outwards. In a Penning trap or a Fourier-transform ion cyclotron resonance mass spectrometer, measuring how fast a trapped ion circles is a way of weighing it with great precision. In a plasma, microwaves tuned to the electron cyclotron frequency are absorbed strongly wherever the field has just the right strength, which is how electron cyclotron resonance heating delivers energy to a chosen spot inside a fusion device.

The numbers are worth knowing. An electron in a 1-tesla field circles about 28 billion times a second, squarely in the microwave band; a proton in the same field manages about 15.2 million, a radio frequency, because it is roughly 1,836 times heavier.

The formula used here is the nonrelativistic one. As a particle approaches the speed of light its inertia grows by the Lorentz factor γ and the true frequency falls by the same factor, already about 2% for a 10 keV electron, which is why fixed-frequency cyclotrons cannot take electrons far. It also assumes a uniform field at right angles to the motion; gradients, collisions and the direction of rotation, which depends on the sign of the charge, are outside the calculation.

What is cyclotron frequency calculator?

The cyclotron frequency is the number of complete circles per second a charged particle makes when it moves perpendicular to a uniform magnetic field. It is fixed by the particle's charge-to-mass ratio and the field strength, f_c = |q|B/(2πm), and does not depend on the particle's speed or the size of its orbit, provided the speed is well below that of light. Multiplying by 2π gives the angular frequency, ω_c = |q|B/m, in radians per second.

How to use this calculator.

1. 01Enter the charge magnitude in coulombs. An electron or a proton carries 1.602176634 × 10⁻¹⁹ C; for an ion in charge state z, multiply that by z. The sign is not needed, since it only decides which way the particle turns.
2. 02Enter the magnetic flux density in tesla. Fields quoted in gauss convert at 10,000 G per tesla, so 875 G is 0.0875 T.
3. 03Enter the rest mass in kilograms: 9.1093837139 × 10⁻³¹ kg for the electron, about 1.6726 × 10⁻²⁷ kg for the proton, or an ion's mass in atomic mass units multiplied by 1.6605 × 10⁻²⁷ kg.
4. 04Read the frequency in hertz. For a particle moving at a sizeable fraction of light speed, divide by its Lorentz factor γ to get the real orbital frequency.

The formula.

f_c = |q|B / (2πm). A charge q moving at speed v perpendicular to a field B feels a magnetic force |q|vB, and that force supplies the centripetal force mv²/r needed for circular motion. Setting the two equal gives the orbit radius, r = mv/(|q|B). One lap is 2πr long and takes 2πr/v = 2πm/(|q|B) seconds — the speed has cancelled — and the frequency is the reciprocal of that period. The result is proportional to charge and field and inversely proportional to mass, so doubling B doubles f_c, and a particle twice as heavy with the same charge circles half as fast. The units work out cleanly: coulombs times tesla divided by kilograms reduce to reciprocal seconds, or hertz.

A worked example.

Frequently asked questions.

References& sources.

1. [1]U.S. Naval Research Laboratory, 2019 NRL Plasma Formulary.
2. [2]OpenStax, University Physics Volume 2, section 11.4, Motion of a Charged Particle in a Magnetic Field.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.