A long solenoid with 25 turns per cm has a small loop of area $2.0\, cm^2$ placed inside the solenoid normal to its axis. If the current carried by the solenoid changes steadily from 3.0 A to 5.0 A in 0.1 s. The induced emf in the loop due to change in the current is

$4π × 10^{-6} V$

The correct answer is Option (3) → $4π × 10^{-6} V$

Number of turns per unit length: $n = 25 \; \text{turns/cm} = 2500 \; \text{turns/m}$

Area of loop: $A = 2.0 \; \text{cm}^2 = 2 \times 10^{-4} \; \text{m}^2$

Change in current: $\Delta I = (5 - 3) = 2 \; \text{A}$

Time interval: $\Delta t = 0.1 \; \text{s}$

Magnetic field in solenoid: $B = \mu_0 n I$

Induced emf in loop: $\mathcal{E} = A \frac{\Delta B}{\Delta t} = A \mu_0 n \frac{\Delta I}{\Delta t}$

Substitute values:

$\mathcal{E} = (2 \times 10^{-4})(4\pi \times 10^{-7})(2500)\left(\frac{2}{0.1}\right)$

$\mathcal{E} = (2 \times 10^{-4})(4\pi \times 10^{-7})(2500)(20)$

$\mathcal{E} = (4\pi \times 10^{-7})(10)$

$\mathcal{E} = 4\pi \times 10^{-6} \; \text{V}$