Question

The current in a long solenoid of radius R and having n turns per unit length is given $i=i_0\sin \omega t.$ A coil having N turns is wound around it near the centre. Find (a) the induced emf in the coil and (b) the mutual inductance between the solenoid and the coil.

Answer 1

B = Flux produced due to first coil
$={\mu }_{0}ni$
(a) Emf. Developed $=\frac{d\varphi }{dt}=\frac{d}{dt}\left({\mu }_{0}niN\pi R^2\right)=\left({\mu }_{0}nNa\pi R^2\right)={\mu }_{0}nNa\pi R^2\frac{di}{dt}={\mu }_{0}nN\pi R^2{i}_0\omega \cos \omega t.$
(b) Flux $\varphi$ linked with the second
${\mu }_{0}ni\times NA=m_{0}niN\pi R^2$