A solenoid of length ℓ has N closely wound turns carrying a steady current I. The solenoid is sufficiently long so that edge effects can be ignored.
a. (i) Derive an expression for the magnetic field B at an interior point on the axis of the solenoid in terms of μ0,N,ℓ and I. State clearly any approximations used.
b. (ii) If the interior of the same solenoid is completely filled with a magnetic material of relative permeability μr, obtain the new value of the magnetic field magnitude.
c. (iii) Hence find the ratio of magnetic energy density inside the solenoid with the core to that without the core, keeping the current unchanged.