Neco 2025 Physics Question 4

A long solenoid 0.5 m in length has 800 turns and carries a current \(I\).\n(a) Derive an expression for the magnetic field \(B\) at the centre of the solenoid in terms of \(I\).\n(b) If \(I = 5\,\text{A}\) and a small magnetic dipole with moment \(\mu = 2.0\times10^{-3}\,\text{A\,m}^2\) is placed at the centre, find the magnitude of the torque on the dipole when its axis makes an angle of \(30^{\circ}\) with the field direction.\n(c) The current is increased until the field at the centre becomes \(1.2\,\text{T}\). Assuming the field varies linearly along the length, estimate the magnitude of the force on the dipole (use \(F \approx \mu\,\frac{dB}{dx}\) with \(\frac{dB}{dx}\approx \frac{B}{L}\)).