Question

The current in a long solenoid of radius R and having n turns per unit length is given by i = i0 sin ωt. 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 ant the coil.

Answer 1

(a) The induced emf in the coil is  $e=\mu_{0} n N \pi R^{2} i_{0} \omega \cos \omega t$

(b) The mutual inductance between the solenoid and the coil is$m=\pi \mu_{0} n N R^{2}$

Explanation:

Given data:

Long Solenoid Radius = R

Number of turns of the long solenoid per unit length = n

In the long current solenoid, then $i=i_{0} \sin \omega t$

Amount of turns in the solenoid low = N

Large Solenoid Scale = R

Inside the long solenoid the magnetic field is given by

$B=\mu_{0} n i$

Flux produced in the tiny solenoid due to the long solenoid,

$\emptyset=\left(\mu_{0} n i\right) \times\left(N \pi R^{2}\right)$

(a) The emf, which is formed in the small solenoid

$e=\frac{d \phi}{d t}=\frac{d}{d t}\left(\mu_{0} n i N \pi R^{2}\right)$

$e=\mu_{0} n i N \pi R^{2} \frac{d i}{d t}$

Substituting $i=i_{0} \sin \omega t$,  in above equation  

$e=\mu_{0} n N \pi R^{2} i_{0} \omega \cos \omega t$

(b) Let the coils have the mutual inductance m.

The flux ∅ connected to the second coil is defined by

$\emptyset=\left(\mu_{0} n i\right) \times\left(N \pi R^{2}\right)$

The flux is also writable as

$\begin{array}{l}{\mathrm{I} \cdot=m i} \\{\therefore\left(\mu_{0} n i\right) \times\left(N \pi R^{2}\right)=m i}\end{array}$

and,  

$m=\pi \mu_{0} n N R^{2}$