Velocity Selectors and Mass Spectrometers

Put a charged particle between a parallel-plate capacitor and a magnetic field pointing perpendicular to it, with the electric force pushing one way and the magnetic force pushing the opposite way. The electric force qE does not care about speed, but the magnetic force qvB grows with speed. Only one speed makes the two forces cancel exactly, so only particles at that speed travel straight through undeflected. Every other speed gets deflected into a plate or wall and never makes it out. That shared exit slit is the whole trick: it filters a messy beam down to one velocity.

Setting the forces equal gives qE = qvB, and the charge cancels, leaving v = E/B. That is the selector's one formula: speed depends only on the field strengths, not on the mass or charge of whatever is passing through. Feed in any charged particle and if it is not moving at E/B, it does not survive the trip.

Once particles emerge at that single speed, a mass spectrometer sends them into a region with only a magnetic field, where they bend into a circular arc. Setting the magnetic force equal to the centripetal force, qvB = mv²/r, gives r = mv/(qB). Since v, B, and q are now fixed, the radius depends only on mass: heavier isotopes swing out in wider arcs, lighter ones curl tighter. Detectors placed around the arc record where each particle lands, which sorts a mixed sample by mass. This links directly to the next idea in the unit, the simple circular motion of a charge in a uniform magnetic field alone, since the spectrometer is really that motion put to work as a measuring tool.