Cyclotron Frequency Calculator
A charged particle moving across a magnetic field is bent into a circle, and it completes each loop at a fixed rate called the cyclotron frequency. The striking feature is that this rate does not depend on the particle's speed or orbit size, only on its charge-to-mass ratio and the field strength. This calculator takes the charge, the magnetic field and the particle mass, then returns the cyclotron frequency in hertz and the angular frequency in radians per second. It applies to accelerators, mass spectrometers, magnetrons and plasma physics.
Cyclotron frequency formula
f = q * B / (2 * pi * m)
omega = q * B / m = 2 * pi * f
q = charge (C), B = field (T)
m = particle mass (kg)
The frequency depends only on the charge-to-mass ratio and the field, not on speed or radius. The angular frequency is two pi times the ordinary frequency in hertz.
Cyclotron motion context
- The cyclotron frequency is independent of the particle's speed at non-relativistic energies.
- The elementary charge is 1.602176634 times ten to the minus nineteenth coulombs.
- The electron mass is about 9.109 times ten to the minus thirty-first kilograms.
- The proton is roughly 1,836 times heavier than the electron, lowering its cyclotron frequency.
- Measuring an ion's cyclotron frequency in a known field gives a precise mass measurement.
Cyclotron frequency: frequently asked questions
What is the cyclotron frequency?
The cyclotron frequency is the rate at which a charged particle circles in a uniform magnetic field. It is f = q * B / (2 * pi * m), where q is the charge, B the field strength and m the particle mass. Remarkably it does not depend on the particle's speed or orbit radius, only on its charge-to-mass ratio and the field.
Why is cyclotron frequency independent of speed?
As a particle speeds up, its orbit radius grows in exact proportion, so the time to complete one loop stays the same. This is why the early cyclotron worked with a fixed-frequency accelerating field: every orbit took equal time regardless of energy, at least until relativistic speeds break the simple relation.
What is the angular cyclotron frequency?
The angular cyclotron frequency, omega, equals q * B / m, or two pi times the ordinary frequency. It is the form that appears in the equation of motion for the circular orbit, expressed in radians per second, while the ordinary frequency in hertz counts complete loops per second.
What charge and mass should I enter?
Enter the particle's charge in coulombs and mass in kilograms. The default values are for an electron: charge 1.602 times ten to the minus nineteenth coulombs and mass 9.109 times ten to the minus thirty-first kilograms. For protons or ions, substitute their charge and mass.
Where is the cyclotron frequency used?
It underpins particle accelerators, mass spectrometers, magnetrons in microwave ovens and electron cyclotron resonance plasma sources. Measuring the cyclotron frequency of an ion in a known field is also one of the most precise ways to determine its mass.
Official sources
- U.S. National Institute of Standards and Technology: Elementary charge.
- U.S. National Institute of Standards and Technology: Electron mass.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.