A long solenoid of diameter 0 . 1 m has 2 × 10 4 turns per meter. At the centre of the solenoid, a coil of…

In the given case, the solenoid produces a changing magnetic field. This field is linked with the coil inside and hence, there is an induced emf in the coil.

We have, from the laws of EMI:

$\epsilon = -N\frac{d\varphi }{dt}$
$\left|\frac{\epsilon }{R}\right|=\frac{N}{R}\frac{d\varphi }{dt}$

The ratio given above is the induced current in the circuit which can further be written as the rate of flow of charge per unit time.
$dq=\frac{N}{R}d\varphi$
$\mathrm{\Delta }Q=\frac{N\left(\mathrm{\Delta }\varphi \right)}{R}$
$\mathrm{\Delta }Q=\frac{\mathrm{\Delta }{\varphi }_{total}}{R}$
$=\frac{\left(NBA\right)}{R}$
$=\frac{N{\text{μ}}_{0}ni\text{π}{r}^2}{R}$
Putting values
$=\frac{100\times \left(4\pi \times {10}^{-7}\right)\times \left(2\times {10}^4\right)\times 4\times \left[{\pi \times \left(0.01\right)}^2\right]}{10{\text{π}}^2}$
$\mathrm{\Delta }Q=32 \mathrm{\mu C}$