19.4 Crossed Electric and Magnetic Fields

In crossed fields, a particle moving through the region simultaneously feels electric and magnetic forces. If geometry is set so they oppose each other, there is one speed where net transverse force is zero and the beam passes undeflected. All other speeds bend up or down depending on which force dominates.

Charge sign still matters for direction. Flipping charge reverses both electric and magnetic force directions. Depending on field orientation, the same speed can therefore deflect opposite ways for positive and negative particles.

Set opposite forces equal in magnitude to obtain the no-deflection condition. Charge magnitude cancels, so selected speed is controlled entirely by field ratio E/B. This is why a selector can pass only particles in a narrow speed band while rejecting faster and slower particles.

If speed is below E/B, electric force dominates and deflection follows electric-force direction. If speed is above E/B, magnetic force dominates and deflection flips. This gives a direct qualitative test of whether your substitution arithmetic is physically plausible.

Beamline designers exploit this as an adjustable gate. Increase E or decrease B and the selected speed rises; decrease E or increase B and selected speed drops. In practical devices, finite slit widths and fringe fields create a finite acceptance window rather than an infinitely sharp cut-off.