Cyclotron Frequency Calculator

A cyclotron is one of the earliest types of particle accelerators. It uses a combination of a static magnetic field and an alternating electric field to accelerate charged particles along an outward spiral path. Cyclotrons are widely used in nuclear physics research and medical isotope production.

The cyclotron frequency (also called the gyrofrequency) is the frequency at which a charged particle orbits in a uniform magnetic field. It depends only on the particle's charge-to-mass ratio and the magnetic field strength — not on the particle's speed or the radius of its orbit.

The cyclotron frequency is calculated using the formula 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 result is in Hertz (Hz).

For a proton (charge ≈ 1.602×10⁻¹⁹ C, mass ≈ 1.673×10⁻²⁷ kg) in a 1 T magnetic field, the cyclotron frequency is approximately 15.24 MHz. This is a commonly referenced benchmark in particle physics.

When a charged particle moves faster in a magnetic field, its orbital radius increases proportionally, so the time to complete one full orbit remains constant. This isochronous property is the key principle that allows cyclotrons to synchronize the accelerating electric field with the particle's orbit without adjusting for speed.

The cyclotron frequency f is measured in Hertz (cycles per second), while the angular frequency ω = 2πf is measured in radians per second. Both describe the same circular motion — angular frequency is simply the radian-based equivalent and is often more convenient in physics equations.

A cyclotron uses a fixed magnetic field and a fixed-frequency electric field, relying on the isochronous property of cyclotron motion. A synchrotron, on the other hand, increases both the magnetic field and the RF frequency as particles accelerate to account for relativistic effects, allowing it to reach much higher energies.

Enter the particle charge in Coulombs (C), mass in kilograms (kg), and magnetic field strength in Tesla (T). For reference, the elementary charge is 1.602×10⁻¹⁹ C, a proton's mass is 1.673×10⁻²⁷ kg, and typical cyclotron magnets range from 1 to 3 Tesla.