Helical Paths, Velocity Selectors, and Mass Spectrometers

Split a charge's velocity into a piece parallel to B and a piece perpendicular to it. The perpendicular piece feels the magnetic force qv x B and gets bent into a circle. The parallel piece feels nothing from B, since v x B is zero along that direction, so it just cruises along unchanged. Add the two together and the path is a circle that also drifts forward: a helix, like a spring. No matter what angle the charge enters at, this split is what happens.

A velocity selector traps a charge between a uniform electric field and a uniform magnetic field pointed crossways to it, so the electric force qE and magnetic force qvB point opposite ways. For most speeds these forces don't cancel and the charge curves off course into a wall or plate. Only the speed where qE = qvB, giving v = E/B, feels zero net force and sails through the slit in a straight line. Change v and the balance breaks.

A mass spectrometer bolts a velocity selector onto a second magnetic field region with nothing else acting on the charge, so it's forced into a clean circular arc of radius r = mv/(qB), landing a distance d = 2r from where it entered. Measure that landing spot, and since q, v, and B are all known, the radius hands you the mass directly. Some designs replace the selector with an accelerating voltage V, using energy conservation qV = (1/2)mv² to fix v instead, then solve the same circular-motion equation for m. Either way, the whole point of this unit's loop tricks is the same: turn a measurable geometric quantity, a radius, into a physical one, a mass.