Which field gives maximum torque

Problem

A rectangular loop carrying a constant current is placed in a uniform field $\vec B$. The magnetic torque acting on the loop is maximum. Which of the three configurations in the figure corresponds to this case?

Solution

The torque on the magnetic dipole is

$M = \mu B \sin\beta$

where $\beta$ is the angle between the magnetic moment $\vec\mu$ (perpendicular to the plane of the loop) and the field $\vec B$. The torque is maximum when $\sin\beta = 1$, i.e. $\beta = 90^\circ$: $\vec\mu \perp \vec B$.

Since $\vec\mu$ is perpendicular to the plane of the loop, $\vec\mu \perp \vec B$ means that the PLANE of the loop is PARALLEL to $\vec B$, i.e. the field lines lie in the plane of the loop. This is the tilted configuration B, in which the field lines cross the loop while remaining in its plane.

$\ev{\text{configuration B}}$

Links

Topics: Magnetism Concepts: Magnetic dipole · Torque