Laplace force on a wire

A wire carrying a current in an external magnetic field experiences a force. This is not a new phenomenon: it is simply the sum of the Lorentz forces on all the charges flowing in the wire. Summing the contributions $q\,\vec{v}\times\vec{B}$ of the carriers gives a compact law in terms of the current.

Law — Laplace force

On a segment of wire of length $\Delta\vec{\ell}$ (oriented in the direction of the current $i$) immersed in a field $\vec{B}$, the force acting is: $\ev{\vec{F} = i\,(\Delta\vec{\ell}\times\vec{B})}$

The magnitude is $|F| = i\,\ell\,B\,\sin\theta$, with $\theta$ the angle between the wire and $\vec{B}$: maximum when the wire is perpendicular to the field, zero when it is parallel. The direction, as always with magnetic cross products, is perpendicular to the plane of $\Delta\vec{\ell}$ and $\vec{B}$, with the direction given by the right-hand rule.

This force is the bridge to applications: two parallel wires that attract or repel each other, the current balance, and above all the torque on a loop that makes electric motors turn. When the loop is closed, the forces on opposite sides have zero resultant but non-zero moment arm: they do not translate the loop, they make it rotate.

Collegamenti

Argomenti: Magnetismo Concetti: Forza di Laplace · Forza di Lorentz

Esercizi collegati: Worked exercise — charge in an outward field · Force between two parallel wires with currents in the same direction · Cyclotron frequency of a proton