Question

Derive an expression for the torque acting on a rectangular current loop suspended in a uniform magnetic field.

Hint:

The torque on a current loop in a magnetic field is maximum when the magnetic moment is perpendicular to the magnetic field. It tends to align the loop with the magnetic field.

Answer 1

The torque \( \tau \) on a current loop in a magnetic field is expressed as:\[\tau = \vec{m} \times \vec{B}\]Here, \( \vec{m} \) represents the magnetic moment of the loop, and \( \vec{B} \) denotes the magnetic field. The magnetic moment \( \vec{m} \) is defined as:\[\vec{m} = I A \hat{n}\]where \( I \) is the current, \( A \) is the loop's area, and \( \hat{n} \) is the unit vector perpendicular to the loop's plane.The magnitude of the torque is given by:\[\tau = m B \sin \theta\]with \( \theta \) being the angle between the magnetic moment and the magnetic field. Substituting the expression for \( m \), we obtain:\[\tau = I A B \sin \theta\]Therefore, the torque acting on the rectangular current loop is given by:\[\tau = I A B \sin \theta\]