A circular coil of 200 turns and radius 0.1 m is rotated at a...

The area of the circular coil is $A = \pi r^2 = \pi (0.1)^2 = 0.01\pi \text{ m}^2$.

The maximum induced EMF is $\varepsilon_{\text{max}} = NAB\omega$.
$\varepsilon_{\text{max}} = 200 \times 0.01\pi \times 0.5 \times 10 = 10\pi \text{ V}$.

The maximum induced current in the coil is $I_{\text{max}} = \dfrac{\varepsilon_{\text{max}}}{R} = \dfrac{10\pi}{10} = \pi \text{ A}$.

To maintain constant rotation, the external torque must balance the opposing magnetic torque. The magnetic torque on the coil is $\tau = MB\sin\theta = (NIA)B\sin\theta$.

The maximum magnetic torque occurs when the current is maximum and $\sin\theta = 1$:
$\tau_{\text{max}} = N I_{\text{max}} A B$
$\tau_{\text{max}} = 200 \times \pi \times 0.01\pi \times 0.5 = \pi^2 \text{ N m}$.

Answer: $\pi^2 \text{ N m}$