Circular Motion in a Magnetic Field

The magnetic force on a moving charge is F = qv x B. That cross product guarantees F is perpendicular to v at every instant. A force perpendicular to velocity can never speed up or slow down the particle, it only changes direction. Speed stays constant, and the only motion consistent with constant speed plus constant-magnitude sideways force is uniform circular motion. The field does no work, ever, no matter how the particle moves through it.

Once you know the motion is circular, the magnetic force is just doing the job of centripetal force. Set them equal: qvB = mv²/r. Solve for r and the v cancels once, leaving r = mv/(qB). A faster particle still needs more centripetal force, but it also sweeps a bigger circle, and those two effects balance in a specific way. Push that further into the period: T = 2 π r / v = 2 π m/(qB). The v's cancel completely. Speed drops out entirely, so a slow proton and a fast proton in the same field finish one loop in exactly the same time, the slow one just on a tighter circle. This speed-independent period is called the cyclotron frequency idea, and it is the reason cyclotrons work: you can accelerate a particle on each pass and its orbital timing never drifts out of sync with a fixed-frequency driving voltage.

This sits right after the basic magnetic force law and right before motion where E and B fields combine, like velocity selectors and mass spectrometers, which reuse r = mv/(qB) directly to measure mass or momentum.