Cyclotron Frequency Calculator

Enter a charged particle's mass, charge, and the magnetic field strength to find its cyclotron frequency, angular frequency, orbital period, and orbit radius using f = qB / (2πm).

Quick Facts

Cyclotron frequency
f = qB / (2πm)
Depends only on the charge-to-mass ratio and field strength — not on speed or orbit radius.
Proton in a 1 T field
≈ 15.2 MHz
A textbook reference value used to calibrate cyclotrons and mass spectrometers.
Electron in a 1 T field
≈ 28.0 GHz
The basis for electron cyclotron resonance (ECR) heating and ion sources.

Your Results

Calculated
Cyclotron Frequency
-
f = qB / (2πm)
Angular Frequency
-
ω = qB / m, in rad/s
Orbital Period
-
T = 2πm / (qB)
Orbit Radius
-
r = mv / (qB), from your speed

Ready

Enter the particle's mass, charge, and the magnetic field strength, then press Calculate.

How to Calculate Cyclotron Frequency

A charged particle moving through a uniform magnetic field feels a Lorentz force that always points perpendicular to its velocity. That force never speeds the particle up or slows it down — it only bends its path, curling it into a circle. The rate of that circular motion, the cyclotron frequency, is one of the most useful numbers in plasma physics, mass spectrometry, and particle accelerator design, because it depends only on the particle's charge, its mass, and the magnetic field strength — not on how fast the particle happens to be moving. This calculator uses the classical (non-relativistic) formula f = qB / (2πm) to compute the frequency, angular frequency, orbital period, and orbit radius for any charged particle.

Deriving f = qB / (2πm) from the Lorentz force

For a charge q moving at speed v perpendicular to a field B, the magnetic force has magnitude F = qvB. This force supplies exactly the centripetal force needed to hold the particle on a circular path: qvB = mv² / r. Solving for the radius gives the Larmor radius, r = mv / (qB). Since the angular speed of the orbit is ω = v / r, substituting the radius back in cancels the velocity term completely: ω = qB / m. Converting to ordinary frequency (f = ω / 2π) gives f = qB / (2πm), and its reciprocal gives the orbital period, T = 2πm / (qB). Notice that v drops out of f, ω, and T entirely — only the orbit radius depends on how fast the particle is moving.

Practical notes and limits

  • Non-relativistic assumption: f = qB/(2πm) is accurate when the particle's speed is well below the speed of light — the usual case for lab electromagnets and mass spectrometers.
  • Relativistic correction: at high energy, replace m with γm, where γ = 1/√(1 − v²/c²). The true frequency f = qB/(2πγm) drops as the particle speeds up, which is why real accelerators (synchrocyclotrons, synchrotrons) must ramp the magnetic field or RF drive frequency to stay in step with the particle.
  • Applications: cyclotron and synchrocyclotron particle accelerators, mass spectrometers (inferring mass-to-charge ratio from a measured frequency), electron cyclotron resonance (ECR) heating and ion sources in fusion research, magnetrons, and ion/electron gyrofrequencies in space and ionospheric physics.

Frequently Asked Questions

What is cyclotron frequency and why is it independent of speed?
Cyclotron frequency is the rate at which a charged particle circles around magnetic field lines under the Lorentz force. Although a faster particle traces a larger circle (radius r = mv/(qB)), it also travels farther on each lap, and the two effects cancel exactly — so the time per orbit, and therefore the frequency f = qB/(2πm), depends only on the charge-to-mass ratio and the field strength, never on the particle's speed.
What is the formula for cyclotron frequency?
The non-relativistic cyclotron frequency is f = qB / (2πm), where q is the particle's charge in coulombs, B is the magnetic field strength in tesla, and m is the particle's mass in kilograms. The angular form is ω = qB/m (rad/s), and the period is its reciprocal, T = 2πm/(qB).
Why is the electron cyclotron frequency so much higher than the proton's?
Frequency scales inversely with mass (f ∝ 1/m), and a proton is about 1,836 times heavier than an electron. In a 1 tesla field, an electron orbits at roughly 28 GHz while a proton orbits at roughly 15.2 MHz — almost exactly that same 1,836:1 ratio, since both particles carry the same magnitude of charge.
Does cyclotron frequency change at relativistic speeds?
Yes. The formula above assumes the particle's speed is much less than the speed of light. At relativistic speeds the particle's effective mass grows by the Lorentz factor γ = 1/√(1 − v²/c²), so the true frequency is f = qB / (2πγm), lower than the classical estimate. This is why accelerators like synchrocyclotrons and synchrotrons must adjust their magnetic field or driving RF frequency as particles approach the speed of light.

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