Lorentz force

The Lorentz force is the force on a charged particle moving through electric and magnetic fields:

$$\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B}).$$

The electric part $q\mathbf{E}$ pushes along or against the electric field depending on the sign of the Charge. The magnetic part $q\mathbf{v}\times \mathbf{B}$ is perpendicular to both the velocity and the magnetic field, with direction set by the right-hand rule for positive charge.

Work and energy

The rate of doing work is

$$P=\mathbf{F}\cdot \mathbf{v}=q\mathbf{E}\cdot \mathbf{v},$$

because $\mathbf{v}\cdot (\mathbf{v}\times \mathbf{B})=0$. Magnetic fields bend trajectories but do no work on a point charge by themselves.

Mechanics bridge

This is where Newton’s Laws of Motion meets Electromagnetism MOC. Given $\mathbf{E}$, $\mathbf{B}$, and initial conditions, use $m\mathbf{a}=\mathbf{F}$ to find the motion. In advanced mechanics, the same physics can be written using Lagrangian mechanics and Hamiltonian mechanics, but the momentum can differ from simple $m\mathbf{v}$.

Common pitfalls

• Reverse the magnetic-force direction for a negative charge.
• A magnetic field can change velocity direction without changing speed.
• Do not confuse electric potential energy with magnetic bending; the magnetic force is perpendicular to instantaneous motion.

Source trail: Susskind The Theoretical Minimum index; reference book: Classical Mechanics: The Theoretical Minimum, Lecture on electric and magnetic forces.