Lorentz force

Remarks

The Lorentz force law is a law of physics, not a theorem; together with Maxwell's equations it is the basis of classical electrodynamics. The magnetic part does no work, since $(v \times B) \cdot v = 0$; so a magnetic field changes the direction of motion but not the speed. Example: in a uniform field $B$ with $E = 0$, a particle of mass $m$ moving perpendicular to $B$ with speed $u$ follows a circle. Its radius is $m u / (\abs{q} \abs{B})$ and its angular frequency is $\abs{q} \abs{B} / m$, the cyclotron frequency. For a current density $J$, the force per unit volume is $\rho E + J \times B$, with charge density $\rho$.

Going deeper

For a student

With Newton's second law, $m \dot v = q (E + v \times B)$ gives the equations of motion of charged particles, used for cyclotrons, mass spectrometers and plasmas. For constant crossed fields with $\abs{E} < c \abs{B}$, the particle drifts with the velocity $E \times B / \abs{B}^2$. In special relativity the law becomes covariant, with the electromagnetic tensor acting on the four-velocity, and $m \dot v$ replaced by the rate of change of relativistic momentum.