Cyclotron Run

A charge q moving at v across a field B feels F = qvB sin θ, always at right angles to the motion. A perpendicular force does no work, so the speed never changes — only the direction does. Setting qvB equal to the centripetal requirement mv²/r cancels one v and leaves r = mv/qB.

The time for a full circle, T = 2πr/v = 2πm/qB, has no v in it at all: a faster ion draws a bigger circle at a higher speed and comes back at the same moment. That cancellation is what makes a cyclotron work.

In the Speed Gate an electric field pushes the ion one way with qE while the magnetic field pushes back with qvB. They balance only at v = E/B. Both q and m cancel, so crossed fields sort on speed alone and cannot tell ²⁰Ne⁺ from ²²Ne⁺. Only the analyser can, because r = mv/qB at one speed becomes r ∝ m — which is exactly the Isotope Split, and exactly how a mass spectrometer measures a mass from a landing distance, m = qBd/2v.

Nothing here is tuned for playability. The ion is stepped forward every 4 ns with an exact rotation for the magnetic part, so its speed is conserved to the last digit, with q = 1.602 × 10⁻¹⁹ C and 1 u = 1.6605 × 10⁻²⁷ kg. The chamber is 1.20 m × 0.75 m and the field runs from 0.020 T to 0.260 T.