Right-hand rule

The cross product $\vec{c} = \vec{a}\times\vec{b}$ has a sense that no scalar formula can give: a geometric convention is needed. The right-hand rule provides it reliably.

Principle — Right-hand rule for the cross product

For $\vec{c} = \vec{a}\times\vec{b}$:

• Index finger in the direction of $\vec{a}$ (first factor)
• Middle finger in the direction of $\vec{b}$ (second factor)
• Thumb gives the direction of $\vec{c}$ (result)

The angle between index and middle finger must be less than $180^\circ$.

Applied to the Lorentz force $\vec{F} = q\,\vec{v}\times\vec{B}$: index finger along $\vec{v}$, middle finger along $\vec{B}$, the thumb gives the direction of $\vec{F}$ for a positive charge. For a negative charge the direction is reversed: simply flip the final result.

Hitchhiker's rule

To find the sense of a rotation: curl the fingers of the right hand in the sense of the rotation; the thumb, extended like a hitchhiker’s, gives the direction of the vector (angular velocity, angular momentum, field inside a solenoid, etc.). It is the same right hand, used for “axial” quantities rather than for cross products between two given vectors.

The most common mistake is using the left hand, or swapping the order of the factors: both flip the direction. When in doubt, it is best to redo the gesture slowly, checking that the index and middle fingers really do point along $\vec{v}$ and $\vec{B}$.

Links

Topics: Magnetismo Concepts: Forza di Lorentz Skills: Regola della mano destra

Related exercises: Esercizio svolto — carica in campo uscente · Frequenza di ciclotrone di un protone · Ranking della forza di Lorentz (quattro angoli)