Lorentz Force

Lorentz force: F = q·(E + v×B)

The Lorentz force is what a charge feels as it moves through electric and magnetic fields: F = q·(E + v×B). Here q is the charge (C), E the electric field (V/m), v the velocity (m/s) and B the magnetic field (T). The electric part lines up with E and does work. The magnetic part (v×B) sits at a right angle to v, so it does no work at all and just bends the path. With only a B field present, the charge travels in a circle of radius r = mv/(qB) at the cyclotron frequency ω = qB/m. Example: an electron (q = 1.6·10⁻¹⁹ C) moving at v = 10⁶ m/s perpendicular to B = 0.5 T feels F = 8·10⁻¹⁴ N and orbits at r ≈ 11.4 µm.

Applications: accelerators, mass spectrometers, auroras

The Lorentz force is what aimed the electron guns in CRT TVs and oscilloscopes. It bends beams inside particle accelerators (LHC dipoles run around 8 T), sorts ions by mass in mass spectrometers, and pushes cosmic rays along Earth's field to light up the auroras. It's also behind the Hall-effect sensors you'll find in AC motors and smartphones.

FAQ

Why doesn't the magnetic force do work? Because v×B always points at a right angle to v. Work is F·v, and a force perpendicular to the motion adds up to zero.

What if v is parallel to B? Then v×B = 0, the magnetic force disappears, and the charge just keeps going straight along the field.

What's the radius of a circular orbit? r = mv/(qB). Heavier or faster particles trace bigger circles, and a stronger field pulls the radius in tighter. That difference is exactly what lets mass spectrometers tell isotopes apart.

Does the formula depend on the sign of q? It does. Flip the sign and a negative charge curves the other way. That's why, inside a bubble chamber, electrons and positrons spiral off in mirror-image arcs.