To solve this problem, we need to calculate the total charge flowing through the coil due to the changing magnetic field in the solenoid.$\\$Given:$\\$• Diameter of the solenoid $= 0.1 \,$m$\Rightarrow$Radius $r_s = 0.05 \,$m$\\$• Turns per meter of the solenoid $= 2 \times 10^4 \\$• Initial current in the solenoid $I_i = 4 \,$A$\\$• Final current in the solenoid $I_f = 0 \,$A$\\$• Time interval $\Delta t = 0.05 \,$s$\\$• Number of turns in the coil $N_c = 100 \\$• Radius of the coil $r_c = 0.01 \,$m$\\$• Resistance of the coil $R = 10 \pi^2 \, \Omega \\$The magnetic field inside the solenoid is given by:$\\ B = \mu_0 n I \\$where $\mu_0 = 4\pi \times 10^{-7} \,$T m/A$$ is the permeability of free space,$\\ n = 2 \times 10^4 \,$turns/m$$ is the number of turns per meter, and $I$ is the current.$\\$The change in magnetic field $\Delta B$ is:$\\ \Delta B = \mu_0 n (I_f - I_i) = \mu_0 n (0 - 4) = -4 \mu_0 n \\$The magnetic flux through the coil is given by:$\\ \Phi = B \cdot A \\$where $A = \pi r_c^2$ is the area of the coil.$\\$The change in magnetic flux $\Delta \Phi$ is:$\\ \Delta \Phi = \Delta B \cdot A = (-4 \mu_0 n) \cdot (\pi r_c^2) \\$The induced emf $\mathcal{E}$ in the coil is given by Faraday's law:$\\ \mathcal{E} = -N_c \frac{\Delta \Phi}{\Delta t} \\$Substitute the values:$\\ \mathcal{E} = -100 \cdot \frac{(-4 \cdot 4\pi \times 10^{-7} \cdot 2 \times 10^4 \cdot \pi \cdot (0.01)^2)}{0.05} \\$Simplify:$\\ \mathcal{E} = 100 \cdot \frac{4 \cdot 4\pi \times 10^{-7} \cdot 2 \times 10^4 \cdot \pi \cdot 10^{-4}}{0.05} \\ \mathcal{E} = 100 \cdot \frac{32 \pi^2 \times 10^{-7} \cdot 2 \times 10^4 \cdot 10^{-4}}{0.05} \\ \mathcal{E} = 100 \cdot \frac{64 \pi^2 \times 10^{-7} \cdot 10^{-4}}{0.05} \\ \mathcal{E} = 100 \cdot \frac{64 \pi^2 \times 10^{-11}}{0.05} \\ \mathcal{E} = 100 \cdot 128 \pi^2 \times 10^{-11} \\ \mathcal{E} = 128 \pi^2 \times 10^{-9} \,$V$\\$The total charge $Q$ flowing through the coil is given by:$\\ Q = \frac{\mathcal{E} \cdot \Delta t}{R} \\$Substitute the values:$\\ Q = \frac{128 \pi^2 \times 10^{-9} \cdot 0.05}{10 \pi^2} \\ Q = \frac{128 \times 0.05 \times 10^{-9}}{10} \\ Q = \frac{6.4 \times 10^{-9}}{10} \\ Q = 6.4 \times 10^{-10} \,$C$\\ Q = 64 \times 10^{-10} \,$C$\\ Q = 64 \times 10^{-6} \,$C$\\ Q = 64 \, \mu$C$\\$Therefore, the total charge flowing through the coil is $64 \, \mu$C$. \\$This corresponds to Option 2.