Cyclotron Radius Calculator
The radius a charged particle circles at in a magnetic field, with the period, cyclotron frequency and centripetal acceleration — for an electron, proton, alpha particle or any mass and charge you enter.
The magnetic force qvB is always at right angles to the velocity, so it bends the path into a circle without changing the speed: qvB = mv² ÷ r gives r = mv ÷ (qB).
How the cyclotron radius calculator works
The magnetic force qvB is always at right angles to the velocity, so it bends the path into a circle without changing the speed: qvB = mv² ÷ r gives r = mv ÷ (qB). The time round the circle, 2πm ÷ (qB), does not depend on speed — the fact that makes the cyclotron work. Speeds are treated as non-relativistic.
Formula: r = m v ÷ (q B); T = 2π m ÷ (q B); f = 1 ÷ T
Worked examples
| Inputs | Radius of the circle | Note |
|---|---|---|
| Electron at 10⁶ m/s in 0.01 T | 568.563 µm | 0.569 mm |
| Proton at 10⁶ m/s in 1 T | 10.4397 mm | 10.4 mm |
| Alpha particle at 10⁷ m/s in 0.5 T | 414.7269 mm | 0.415 m |
FAQFrequently asked questions
Why does the period not depend on speed?
A faster particle makes a bigger circle in exact proportion, so the time round is the same: T = 2πm ÷ (qB). A cyclotron applies its accelerating voltage at that fixed frequency and every particle stays in step until relativity spoils it.
What happens at relativistic speeds?
The mass in the formula becomes γm, so the radius grows and the frequency falls. Synchrotrons and synchrocyclotrons adjust the field or the frequency to compensate; this calculator stops at 10% of c.
What if the velocity is not perpendicular to the field?
Only the perpendicular component circles; the parallel component is unaffected, so the path is a helix along the field line. Enter the perpendicular component of the speed.
Where is this used?
Mass spectrometers (radius identifies the mass-to-charge ratio), cyclotrons, the Earth’s radiation belts, and the electron gyration that gives the magnetosphere its radio emission.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Physical Laboratory — the UK's national measurement institute
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.