Gce Model Questions Vol1 2021 Physics Question 41

A long solenoid of length 0.50 m and radius 0.020 m has an unknown number of turns \(N\). When a current of 2.0 A flows, the magnetic field at the centre is measured to be \(1.6\times10^{-3}\,\text{T}\). The current is then increased to 3.0 A and a short iron rod (length 0.10 m, cross‑sectional area \(1.0\times10^{-4}\,\text{m}^2\)) is inserted completely into the solenoid. The inductance of the solenoid is found to increase by 15 %. Assuming the rod is magnetically linear with susceptibility \(\chi\), determine (i) the number of turns \(N\) (to the nearest whole turn), (ii) the magnetic susceptibility \(\chi\) of the iron, and (iii) the average force required to insert the rod (take the insertion distance as the rod length).