Cyclotron Radius & Larmor Radius Calculator
Calculate the Larmor (cyclotron) radius and cyclotron frequency of a charged particle in a magnetic field.
Used in particle accelerators and mass spectrometers.
A charged particle moving perpendicular to a magnetic field experiences a force (the Lorentz force) that continuously redirects it — but never speeds it up. The result is circular motion. The radius of that circle is the Larmor radius, also called the cyclotron radius or gyroradius.
The formula
The magnetic force provides the centripetal force:
qvB = mv^2 / r
Solving for r:
r = mv / (qB)
The cyclotron frequency (how many orbits per second):
f = qB / (2pi x m)
Note: the cyclotron frequency depends only on charge-to-mass ratio and field strength — not on the particle’s speed. This is why cyclotrons work: all particles of the same type orbit at the same frequency regardless of energy, allowing synchronization with an alternating electric field.
Particle constants
Electron: m = 9.109 x 10^-31 kg, q = 1.602 x 10^-19 C. Proton: m = 1.673 x 10^-27 kg (1,836x heavier than electron), same charge. Alpha particle: m = 6.645 x 10^-27 kg, q = 3.204 x 10^-19 C (2 protons + 2 neutrons).
Real-world scales
An electron moving at 10% of c in Earth’s magnetic field (50 microTesla): r ≈ 170 m. In a 1 T laboratory magnet: r ≈ 0.85 mm. In a 10 T MRI magnet: r ≈ 0.085 mm.
Protons in the LHC travel at 0.9999991c in a 8.33 T field. Their relativistic momentum gives a Larmor radius of about 2,800 m — matching the LHC’s 2,814 m radius.
Northern Lights (aurora) are caused by solar wind particles (mostly protons and electrons) spiraling along Earth’s magnetic field lines in tight helices, guided toward the poles.