To solve this problem, we will use Faraday's law of electromagnetic induction.$\\$Given:$\\$• Number of turns $N = 1000 \\$• Average radius of the coil $r = 10 \,$m$\\$• Angular velocity $\omega = 2 \,$rad/s$\\$• Vertical component of Earth's magnetic field $B = 2 \times 10^{-5} \,$T$\\$• Electrical resistance of the coil $R = 12.56 \, \Omega \\$First, calculate the area of the coil:$\\ A = \pi r^2 = \pi (10)^2 = 100\pi \,$m$^2 \\$The maximum induced emf (electromotive force) in the coil is given by:$\\ \varepsilon_{max} = N \cdot B \cdot A \cdot \omega \\$Substitute the known values:$\\ \varepsilon_{max} = 1000 \cdot 2 \times 10^{-5} \cdot 100\pi \cdot 2 \\ \varepsilon_{max} = 1000 \cdot 2 \times 10^{-5} \cdot 200\pi \\ \varepsilon_{max} = 4 \times 10^{-2} \cdot 200\pi \\ \varepsilon_{max} = 8\pi \,$V$\\$Now, calculate the maximum induced current using Ohm's law:$\\ I_{max} = \frac{\varepsilon_{max}}{R} \\$Substitute the known values:$\\ I_{max} = \frac{8\pi}{12.56} \\$Simplify:$\\ I_{max} = \frac{8 \cdot 3.14}{12.56} \\ I_{max} = \frac{25.12}{12.56} \\ I_{max} = 2 \,$A$\\$Therefore, the maximum induced current in the coil is $2 \,$A$. \\$This corresponds to Option 4.