Cyclotron Frequency Calculator

Created by Miłosz Panfil, PhD
Reviewed by Steven Wooding
Last updated: Aug 22, 2022

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 of a cyclotron particle accelerator
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. The equation for the cyclotron frequency is:

f=qB2m\small f = \frac{qB}{2m}

where:

  • ff – Cyclotron frequency;
  • qq – Charge of the particle;
  • BB – Strength of the magnetic field; and
  • mm – 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.

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