The Lorentz force: definition and components

Definition

The Lorentz force is the total force exerted on a charged particle of charge q, moving at a velocity v, immersed in an electric field E and a magnetic field B:

F = qE + q(v x B)

It can be broken down into two components:

• an electric force F_E = q*E, collinear with E, which accelerates or decelerates the charge depending on the sign of q;
• a magnetic force F_B = q*(v × B), which is always perpendicular to both v and B.

Direction and the right-hand rule

For q > 0, the direction of v × B is determined using the right-hand rule (or corkscrew rule): point your fingers in the direction of v, curl them towards B; your thumb points in the direction of v × B. If q < 0, the direction obtained must be reversed.

Work is always zero

Key property: since F_B is perpendicular to v, it never does any work. dW_B = F_B . v dt = q*(v x B).v dt = 0. Only the electric force can alter the particle’s kinetic energy.

Numerical example

An electron (q = -1.6e-19 C) moves at v = 2e6 m/s along the x-axis in a magnetic field B = 0.5 T along the z-axis. The magnitude of the magnetic force is |q| * v * B = 1.6e-13 N, directed along the -y axis due to the negative sign of q.

Common pitfall

Do not confuse v × B with B × v (opposite directions), and never forget the sign of q, which reverses the final direction of the force.