Lorentz force

The Lorentz force is the rule that takes a point charge q moving with velocity v through electric and magnetic fields E and B and tells you the force it feels: F = q(E + v×B). The electric piece, qE, points along the field and acts on the charge whether it is moving or not. The magnetic piece, qv×B, is perpendicular to both v and B, vanishes when the charge stands still, and does no work on the charge — it can change the direction of motion but not the speed.

This formula is the single bridge between the field equations of Maxwell (which describe what E and B are at every point of space) and the dynamics of charged particles (which is what every cathode-ray tube, particle accelerator, mass spectrometer, and ionospheric phenomenon ultimately runs on). Plug in a uniform B field and zero E and you get circular motion at the cyclotron frequency ω = qB/m. Add a perpendicular E and you get drift motion — the famous E×B drift that determines plasma dynamics and is the reason aurorae form rings rather than uniform glows. Crank up E and you accelerate the particle along the field; reverse B and you flip the sense of circulation. Every motion of a charged particle in an electromagnetic field is some combination of these ingredients.

The cross-product structure has subtle consequences. Because the magnetic force is always perpendicular to the velocity, magnetic fields cannot speed particles up (that requires E fields, which is why every accelerator is fundamentally an electric-field device) but they can confine them indefinitely in stable orbits (which is the principle of every cyclotron and synchrotron). The Lorentz force is also the right-hand-rule lover's paradise: thumb along v, fingers along B, palm points the way the force pushes a positive charge.

Hendrik Lorentz wrote down the force law in 1895 as part of his electron theory of matter, deliberately separating the electric and magnetic contributions and giving them a unified vector form. Earlier formulations (Maxwell's, Heaviside's) had contained equivalent content but mixed it into the field equations themselves; Lorentz cleanly distinguished the field equations (which describe what fields exist) from the force law (which describes how fields act on matter), an organisational choice that has structured every electromagnetism textbook since.