Problem 7: A long solenoid with 15 turns / cm has a small loop of area 2.0 cm² placed inside the solenoid normal to its axis. If the current created by the solenoid changes steady from 2.0 A to 4.0 Amp in 0.1 s. What is the induced emf in the loop when the current is changing.

Solution

Given data                     Number of turns of the solenoid = n = 15/cm = 1500/m

                                         Area of the loop = A = 2.0 cm² = 2.0 × (10^-2)² = 2.0 × 10^-4 m

                                         Change in current = ∆I = 4.0 – 2.0 = 2.0 A

                                         Time required for the change in the current = 0.1 s

Required data              Induced emf = ε =?

Formulae             (i) B = μ₀nI           (ii) Δϕ = Δ(BA)                    (iii) ϵ = (Δϕ)/Δt

Now from the given data, use formula (i),

B₁ = μ₀nI₁             and        B₂ = μ₀nI₂,            Therefore, ΔB = B₂ – B₁ = μ₀n(I₂ – I₁)

Now use formula (ii), Δϕ = AΔB …  (Since area A is constant and can be taken out). Put the values from the given data,

Now apply formula (iii) to find the induced emf.