Cyclotron Frequency Calculator
The cyclotron frequency calculator helps you calculate the frequency of circular motion of a charged particle in a magnetic field. Reading the text below, you will learn what a cyclotron is, how to compute the cyclotron frequency, and what cyclotron velocity is. Enter the charge, magnetic field strength, and particle mass to instantly get the frequency, and add a radius to also compute the cyclotron velocity — in both metric (SI) and American (Imperial) units.
What is a cyclotron?
Cyclotrons were one of the first particle accelerators. The particles in a cyclotron move along semicircular paths. An electric field is switched on every time they pass a particular region in order to accelerate them.
To successfully accelerate the particles, we need to know the rate at which they go around so that we switch on the electric field exactly when the particle passes through this region. This rate is the cyclotron frequency. So how do we compute the cyclotron frequency?
How to compute the cyclotron frequency
The cyclotron frequency comes from balancing the Lorentz force and the centrifugal force. (See our Lorentz Force Calculator to learn more about the magnetic force on charged particles.) The equation for the cyclotron frequency is:
f = q × B / (2 × π × m)
Where:
- f — Cyclotron frequency (Hertz, Hz);
- q — Charge of the particle (Coulombs, C);
- B — Strength of the magnetic field (Tesla in SI; Gauss in American units); and
- m — Mass of the particle (kilograms in SI; pounds in American units).
The related angular frequency is ω = q × B / m (in radians per second), and the two are connected by ω = 2 × π × f. The period of one full revolution is T = 1 / f.
This equation does not take into account relativistic effects, and it breaks down if the magnetic field is too strong or the particle mass is too small.
Cyclotron velocity
In the Additional parameters field of the cyclotron frequency calculator, you can compute the cyclotron velocity. It is the velocity of a particle moving in a circle with the cyclotron frequency. To do this, you need to specify the radius of the circular motion. The velocity is then:
v = ω × r = 2 × π × f × r
Beware that if you choose the radius too large, you can easily exceed the speed of light! This is not a sign of a breakdown of special relativity; instead, it signals the need to include relativistic effects.
It turns out that the closer the particle's velocity is to the speed of light, the more energy it takes to accelerate it further. The equation for the cyclotron frequency does not take this effect into account, and that's why it might sometimes give wrong answers.
Unit Systems
Metric (SI) System
- Charge (q): Coulombs (C)
- Magnetic field (B): Tesla (T)
- Mass (m): kilograms (kg)
- Radius (r): meters (m)
- Frequency (f): Hertz (Hz); Velocity (v): meters per second (m/s)
American (Imperial) System
- Charge (q): Coulombs (C) — same as SI, no equivalent imperial unit
- Magnetic field (B): Gauss (G); 1 G = 10⁻⁴ T
- Mass (m): pounds (lb); 1 lb = 0.45359237 kg
- Radius (r): feet (ft); 1 ft = 0.3048 m
- Frequency (f): Hertz (Hz); Velocity (v): feet per second (ft/s)
Frequently Asked Questions
What is the cyclotron frequency?
The cyclotron frequency is the number of revolutions per second that a charged particle makes as it moves in a circle in a uniform magnetic field. It is given by f = qB / (2πm) and, remarkably, does not depend on the particle's speed or the radius of its orbit — only on the charge-to-mass ratio and the magnetic field strength.
Why does the cyclotron frequency not depend on velocity?
As a particle speeds up, it moves in a larger circle. The larger path length exactly compensates for the higher speed, so the time to complete one revolution stays the same. This constant frequency is what makes the classic cyclotron design possible: the accelerating field can be switched at a fixed frequency.
What is the difference between cyclotron frequency and angular frequency?
The angular frequency ω = qB / m is measured in radians per second, while the (ordinary) cyclotron frequency f = qB / (2πm) is measured in Hertz (revolutions per second). They are related by ω = 2πf.
What is the cyclotron frequency of a proton in a 1 T field?
For a proton (q = 1.602×10⁻¹⁹ C, m = 1.6726×10⁻²⁷ kg) in a 1 Tesla field, f = qB/(2πm) ≈ 1.52×10⁷ Hz, or about 15.2 MHz.
When does this formula break down?
The formula is non-relativistic. When the particle's velocity approaches the speed of light, its effective (relativistic) mass increases, so the actual revolution frequency drops below the value this calculator gives. At that point a more advanced treatment (such as a synchrocyclotron or synchrotron) is required.