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 \( \tau \) acting on a current loop in a magnetic field is given by: \[ \tau = \vec{m} \times \vec{B} \] where \( \vec{m} \) is the magnetic moment of the loop, and \( \vec{B} \) is the magnetic field. The magnetic moment \( \vec{m} \) is given by: \[ \vec{m} = I A \hat{n} \] where \( I \) is the current, \( A \) is the area of the loop, and \( \hat{n} \) is the unit vector normal to the plane of the loop. The magnitude of the torque is: \[ \tau = m B \sin \theta \] where \( \theta \) is the angle between the magnetic moment and the magnetic field. Substituting for \( m \), we get: \[ \tau = I A B \sin \theta \] Thus, the expression for the torque acting on the rectangular current loop is: \[ \tau = I A B \sin \theta \]