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Cyclotron Frequency Calculator

Find the cyclotron frequency of a particle.

How it is calculated

The tool evaluates f = q × B ÷ (2π × m) in SI units: charge q in coulombs, magnetic field B in tesla and mass m in kilograms. The result is a frequency in hertz (revolutions per second). The angular frequency would be ω = q × B ÷ m, which is 2π times larger; this tool reports the ordinary frequency f.

Charge and mass must be positive, and the field may be zero or more (a zero field gives 0.00 Hz). Very small numbers can be typed in scientific notation, for example 1.602e-19. The formula is the non-relativistic one, so it is accurate only while the particle's speed is well below the speed of light. The result does not depend on speed or orbit radius.

Worked example

Take a proton in a 1.5 T field: q = 1.602e-19 C, m = 1.6726e-27 kg. Numerator: q × B = 1.602e-19 × 1.5 = 2.403e-19. Denominator: 2π × 1.6726e-27 ≈ 1.0509e-26. Dividing gives 22,865,558.31 Hz, about 22.9 MHz. Every proton in that field circles at this rate whether it is slow or fast, which is why a cyclotron can use one fixed drive frequency.

Try this example →

Common mistakes

  • Entering mass in grams or atomic mass units instead of kilograms. A proton is about 1.67e-27 kg, not 1.67.
  • Using gauss rather than tesla; 1 T equals 10,000 gauss.
  • Entering the charge in elementary-charge units (1) rather than coulombs (1.602e-19), which makes the answer about 6 quintillion times too large.
  • Reporting the result as angular frequency. This tool gives f in Hz, not rad/s.
  • Using it for particles near light speed, where relativistic mass increase lowers the real frequency.

About this calculator

The cyclotron frequency is the frequency at which a charged particle circles around magnetic field lines, given by f = q·B / (2π·m), where q is the particle's charge, B is the magnetic field strength, and m is its mass. Remarkably, at non-relativistic speeds this frequency depends only on the charge-to-mass ratio and field strength — not on the particle's speed or the radius of its circular path, which is what makes it possible to accelerate particles to ever-higher energies using a fixed driving frequency.

This relationship is the physical basis of the cyclotron particle accelerator, used in nuclear physics research and in medical facilities that produce radioisotopes for PET scans and proton therapy, and it underlies ion cyclotron resonance mass spectrometry (FT-ICR MS), an ultra-high-precision technique used by analytical chemists to identify molecules by mass. Physics students and plasma physicists also use the cyclotron frequency formula when studying charged-particle motion in magnetic fields, including in fusion reactor and space plasma research.

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