Question

A long solenoid ‘S’ has ‘n’ turns per meter, with diameter ‘a’. At the centre of this coil we place a smaller coil of ‘N’ turns and diameter ‘b’ (where b < a). If the current in the solenoid increases linearly, with time, what is the induced emf appearing in the smaller coil. Plot graph showing nature of variation in emf, if current varies as a function of mt² + C.

Answer 1

Magnetic field due to a solenoid is given by, B = μ₀ni where signs are as usual.

In this problem, the current is varying with time. Due to this an emf is induced in the coil of radius b.

`B_1(t) = mu_0nI(t)`

This varying magnetic field changes flux in the smaller coil.

Magnetic flux in IInd coil

`phi_2 = B(t).A`

= `mu_0nI(t).pib^2`

Induced e.m.f in second coil due to solenoid's varying magnetic field in 1 turn

`ε^' = (-dphi_2)/(dt) = (-d)/(dt) mu_0npib^2I(t)`

= `- mu_0npib^2 d/(dt) (mt^2 + C)`

= `- mu_0npib^2. 2mt`

So net e.m.f. produced in N turns of smaller coil

ε = `- mu_0Nn pi b^2 2mt`