Last updated: 2026-07-01

What the cyclotron frequency describes

A charged particle moving through a uniform magnetic field feels a force perpendicular to its velocity. That sideways push never speeds the particle up or slows it down — it only bends the path, curling it into a circle. The cyclotron frequency (also called the gyrofrequency) is the number of complete loops the particle makes each second. It sets the rhythm of everything from particle accelerators to the plasma in a fusion reactor and the electrons trapped in Earth's magnetosphere.

The formula

Setting the magnetic force equal to the centripetal force required for circular motion, qvB = mv²/r, and using f = v / (2πr), the speed cancels and leaves:

The angular frequency and the orbital period follow directly:

Why the frequency ignores the particle's speed

The most surprising feature of the formula is what is missing from it: neither the speed nor the orbit radius appears. A faster particle sweeps out a wider circle, since r = mv / (qB) grows with speed, but it also has proportionally farther to travel each lap. The two effects cancel exactly, so every particle of the same charge-to-mass ratio circles at the same frequency regardless of how energetic it is. This is precisely what made Ernest Lawrence's classical cyclotron possible: a single fixed-frequency voltage could keep kicking particles as they spiralled outward to ever-larger radii.

At very high energies the particle's effective mass grows with relativity, the frequency drifts downward, and the trick breaks — which is why modern machines are synchrocyclotrons or synchrotrons that adjust their frequency to match.

Worked example

Find the cyclotron frequency of an electron in a 0.5 T magnetic field.

1. Identify the values: q = 1.602 × 10⁻¹⁹ C, B = 0.5 T, m = 9.109 × 10⁻³¹ kg.
2. Apply the formula: f = qB / (2πm) = (1.602 × 10⁻¹⁹ × 0.5) / (2π × 9.109 × 10⁻³¹).
3. Compute the numerator and denominator: 8.011 × 10⁻²⁰ / 5.724 × 10⁻³⁰.
4. Result: f ≈ 1.400 × 10¹⁰ Hz, or about 14 GHz.

The angular frequency is ω = 2πf ≈ 8.79 × 10¹⁰ rad/s, and the period is T = 1/f ≈ 71.4 picoseconds per orbit.

Electrons versus protons

Because electrons and protons carry equal-magnitude charges, the ratio of their cyclotron frequencies is set entirely by their masses. A proton is roughly 1836 times heavier than an electron, so in the same 0.5 T field its cyclotron frequency drops from about 14 GHz to only about 7.6 MHz. Heavier ions circle even more slowly. This mass sensitivity is the basis of ion cyclotron resonance mass spectrometry, where measuring an ion's cyclotron frequency reveals its mass-to-charge ratio.

Where it shows up

• Particle accelerators. Cyclotrons and their descendants time their accelerating voltage to the cyclotron frequency so each pass adds energy in step.
• Plasma physics and fusion. The gyrofrequency governs how charged particles spiral along magnetic field lines in tokamaks and stellarators, and it appears in the resonances used to heat the plasma.
• Space physics. Electrons trapped in planetary magnetic fields gyrate at the local cyclotron frequency, producing radio emissions that spacecraft detect.

Related calculations

The cyclotron frequency builds on the magnetic force that bends the particle's path, so computing that force is a natural companion calculation. The field itself often comes from a current-carrying wire or coil, and the changing flux of a rotating charge connects to Faraday's law of induction.