# Cyclotron Frequency Calculator

Created by Miłosz Panfil, PhD
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

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 cyclotron frequency and what cyclotron velocity is.

## What is a cyclotron?

Cyclotrons were one of the first particle accelerators. The particles in a cyclotron are moving in semicircular paths. An electric field is switched on every time they pass a particular region to accelerate them. To successfully accelerate the particles, we need to know the rate at which the particles 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 to compute cyclotron frequency? Diagram showing how a cyclotron works. The size of the magnets have been reduced to show the particle's path.

## How to compute the cyclotron frequency

The cyclotron frequency comes from balancing the Lorentz force and centrifugal force. See our Lorentz Force Calculator and this centrifugal force calculator to learn more. The equation for the cyclotron frequency is:

$\small f = \frac{qB}{2m}$

where:

• $f$ – Cyclotron frequency;
• $q$ – Charge of the particle;
• $B$ – Strength of the magnetic field; and
• $m$ – Mass of the particle.

This equation does not take into account the relativistic effects and breaks down if the magnetic field is too strong or the particle mass is too small.

## Cyclotron velocity

In the advanced mode of the cyclotron frequency calculator, you can compute the cyclotron velocity. It is a velocity of a particle moving in a circle with cyclotron frequency. To do this, you need to specify the radius of the circular motion.

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 the relativistic effects.

It turns out that the closer the particle's velocity 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.

## FAQ

### What is the cyclotron frequency?

The cyclotron frequency is the frequency at which a charged particle revolves in a uniform magnetic field perpendicular to the direction of the movement.

The cyclotron frequency is also the swapping frequency of the electric field in a cyclotron that allows the acceleration of the particle on a spiraling path. By tuning the field at such frequency, the linear segments in the machine would always increase the particle's speed.

### How do I calculate the cyclotron frequency?

To calculate the cyclotron frequency, follow these easy steps:

1. Calculate the centripetal force: Fc = m × v²/r, where:
• m — The mass of the particle;
• v — Its speed; and
• r — The radius of the revolution.
2. Calculate the Lorentz magnetic force: Fm = q × B × v; where:
• q — The charge of the particle; and
• B — The intensity of the magnetic field.
3. Equate the two quantities and calculate the angular frequency with the following formula:
ω = v/r = (q × B)/m
4. To compute the frequency, divide ω by 2π.

### What is the cyclotron frequency for a proton moving in a 1 T field?

15.4 MHz. To calculate the result, consider the following quantities:

• The proton charge: q = e = 1.602e-19 C;
• The proton mass: m = 1 u = 1.672e-27 kg;
• The field strength: B = 1 T.

Apply the formula for the cyclotron frequency to find the result:
f = (q × B)/(2π × m) = (1.602e-19 C × 1 T)/(2π × 1.627e-27 kg) = 15.4e6 Hz = 15.4 MHz

### What is the difference between cyclotron and synchrotron?

A cyclotron consists of a constant magnetic field and a variable rf-electric field. This configuration imprints a continuous acceleration to the particle till relativistic effects break the synchrony between the motion and the field switching. Particles in a cyclotron move in spiraling trajectories.

In a synchrotron, the magnetic field varies in strength to maintain synchrony between the particle angular frequency and the field switching. Particles in a synchrotron move in closed orbits.

Miłosz Panfil, PhD
Charge
e
Magnetic field strength
T
Mass
u
Cyclotron frequency
MHz
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