Gce Model Questions Vol1 2020 Physics Question 6

An insulated square loop of side \(a = 0.15\ \text{m}\) and mass \(m = 0.050\ \text{kg}\) carries a current \(I\). The loop is hinged along one edge and can rotate in a vertical plane about this hinge. The loop is placed in a uniform magnetic field of magnitude \(B = 0.40\ \text{T}\) whose direction is perpendicular to the plane of the loop when the loop is in its equilibrium position (i.e., when the plane of the loop is parallel to the field). The loop is displaced by an angle \(\theta = 30^{\circ}\) from this position and released. A torsion spring attached to the hinge provides a restoring torque \(\tau_s = \kappa \theta\) with spring constant \(\kappa = 1.5\times10^{-4}\ \text{N·m·rad}^{-1}\). Determine the magnitude of the current \(I\) (in amperes, to three significant figures) required so that the magnetic torque on the loop exactly balances the combined restoring torque from the spring and the gravitational torque due to the weight of the loop acting at its centre of mass.