Electricity and magnetism

The Lorentz force

The Lorentz force is the electromagnetic force on a moving charge, $\vec F=q(\vec E+\vec v\times\vec B)$.

The Lorentz force is the electromagnetic force on a moving charge, $\vec F=q(\vec E+\vec v\times\vec B)$.

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The electric part $q\vec E$ can change kinetic energy. The magnetic part $q\vec v\times\vec B$ is perpendicular to velocity and does no work. For $\vec v\perp\vec B$ in a uniform magnetic field alone, the path is circular, with $r=mv/(|q|B)$ and $T=2\pi m/(|q|B)$. The charge sign determines the rotation sense.

The magnetic term $q\vec v\times\vec B$ is always perpendicular to velocity, so magnetic force does no work even though it can bend a trajectory. For a particle entering a uniform field perpendicularly, the force supplies centripetal acceleration and the orbit radius is $r=mv/(|q|B)$. If velocity also has a component parallel to $\vec B$, that component remains unchanged and the particle follows a helix around the field line.

Changing the charge sign reverses Lorentz-force direction without changing its magnitude. Thus an electron moving through the same $\vec E$ and $\vec B$ feels force opposite to a proton with the same velocity.

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