A long circular tube of length 10m and radius 0.3m carries a current I along its curved surface as shown. A wire-loop of resistance 0.005Ω and of radius 0.1m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as I = I₀cos(300t) where I₀ is constant. If the magnetic moment of the loop is Nμ₀I₀sin(300t), then 'N' is

According to Ampere's circuital law the magnetic field inside the tube is B = μ₀I/L where L is the length of the tube.
Flux linked the wire loop is Φ = Bπr² where r is the radius of the loop
Φ = μ₀I L πr² = μ₀πr²I₀cos300t/L
Induced emf in the loop is ε = dΦ/dt = -d/dt(μ₀Lπr²I₀cos300t) = μ₀πr²I₀300sin300t/L
Induced current in the loop is i = ε/R = 300μ₀πr²I₀sin300t/LR where R is the resistance of the loop
Magnetic moment of the loop M = iπr² = 300π²r⁴μ₀I₀sin300t/LR
Substituting the given value, we get
m = 300 × 10 × (0.1)⁴/10 × 0.005 μ₀I₀sin300t (Take π² = 10)
= 6μ₀I₀sin300t
M = Nμ₀I₀sin300t
∴N = 6