Cyclotron Radius Calculator

When a charged particle enters a uniform magnetic field with velocity perpendicular to B, Lorentz force qvB provides centripetal acceleration: qvB = mv²/r. Solving gives orbit radius r = mv/(|q|B). Heavier particles or higher speeds curve more gently; stronger B tightens the circle.

The motion is uniform circular with angular frequency ω = |q|B/m and period T = 2πm/(|q|B), independent of speed v in classical mechanics — the basis of the cyclotron accelerator (particles gain speed each crossing of the gap while orbital period stays fixed).

Electron example: m = 9.11×10⁻³¹ kg, v = 10⁶ m/s, B = 1 T gives r ≈ 5.7×10⁻³ m = 5.7 mm. Proton at same v and B has r ≈ 10⁴ times larger because of greater mass.

If velocity has a component v∥ along B, the path is a helix: radius r from the perpendicular part v⊥, pitch determined by v∥. Velocity selectors use crossed E and B so only particles with v = E/B enter the deflection region undeflected.

Mass spectrometers measure r (or time of flight) to identify ions by mass-to-charge ratio. Cloud chambers and bubble chambers show curved tracks whose curvature reveals momentum.

At relativistic energies use r = p/(|q|B) with p = γmv. Synchrotrons vary B and RF frequency as particle mass increases with energy.