The Lorentz force

A stationary charge in a magnetic field feels no force at all: magnetism “sees” only moving charges. And even a charge moving along the direction of the field feels nothing. The magnetic force appears only when the velocity has a component perpendicular to $\vec{B}$, and it is given by the famous Lorentz force.

Law — Lorentz force

On a charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ acts the force: $\ev{\vec{F} = q\,\vec{v}\times\vec{B}}$

The cross product packs in three pieces of information. The magnitude is

$|F| = |q|\,v\,B\,\sin\theta$

where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. The direction is perpendicular to both $\vec{v}$ and $\vec{B}$: the force points out of the plane containing the two vectors. The sense is determined with the right-hand rule (and must be reversed if the charge is negative).

Key formula

$|F| = |q|\,v\,B\,\sin\theta$

• if $\vec{v}\perp\vec{B}$: $|F| = |q|\,v\,B$ (maximum force)
• if $\vec{v}\parallel\vec{B}$: $|F| = 0$ (no force)

Since $\vec{F}$ is always perpendicular to $\vec{v}$, the Lorentz force never does any work: it does not change the kinetic energy or the magnitude of the velocity, only the direction of motion. This is the observation from which the whole rich phenomenology of the chapter follows, from circles in the cyclotron to the aurora borealis.

▶ How special relativity makes magnets work — Veritasium

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Topics: Magnetismo Concepts: Forza di Lorentz · Campo magnetico

Related exercises: Vero o falso sulla forza magnetica · Campo magnetico da carica in moto · Aurore boreali