Cyclotron Radius & Larmor Radius Calculator

A charged particle moving perpendicular to a magnetic field experiences a force (the Lorentz force) that continuously redirects it without ever speeding 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: at everyday speeds the cyclotron frequency depends only on the charge-to-mass ratio and the field strength, not on how fast the particle is going. This is why cyclotrons work at all: particles of the same type orbit at the same rate regardless of energy, so one fixed drive frequency keeps the whole beam in step. That stops being true once the speed climbs. The radius stretches by γ and the frequency drops by the same factor, so the beam falls out of step with a fixed drive. The calculator below reports the corrected figure.

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).

Relativity, and when it starts to matter

Above roughly 10% of the speed of light the momentum is no longer mv but γmv, where the Lorentz factor γ = 1 / √(1 − v²/c²). The radius formula becomes:

r = γmv / (qB)

At 10% of c, γ is 1.005 and the correction is half a percent, small enough to ignore for most purposes. At 90% of c, γ is 2.29 and the classical answer is less than half the truth. This calculator applies the correction automatically, so its answers stay right at any speed you can enter.

Real-world scales

Take an electron at 10% of c (about 30 million m/s):

Notice the span. The same electron traces a circle you could walk around in Earth’s field and one narrower than a human hair in a strong magnet. Radius scales as 1/B, so a field 200,000 times stronger gives a circle 200,000 times smaller.

Protons in the Large Hadron Collider carry 7 TeV, which puts them at β = 0.999999991 with γ ≈ 7,454. In the LHC’s 8.33 T dipoles that gives a radius of about 2,800 m, matching the machine’s 2,804 m bending radius. Without the γ factor the same calculation returns 0.38 m, which is the difference between a particle accelerator and a desk ornament.

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.