A

Step 1: Identify the area of the loop. The loop is composed of two concentric circles with radii a = 0.2 m and b = 0.1 m. The area (A) of each is given by:
A_{outer} = \pi a^2 = \pi (0.2)^2 = 0.04\pi \, m^2
A_{inner} = \pi b^2 = \pi (0.1)^2 = 0.01\pi \, m^2
Effective Area = A_{outer} - A_{inner} = 0.04\pi - 0.01\pi = 0.03\pi \, m^2

Step 2: Calculate the rate of change of magnetic flux (\Phi) through the loop. The magnetic field (B) is given by B = B_{0} \sin(\omega t). Therefore, the magnetic flux is:
\Phi = B \cdot A = B_{0} A \sin(\omega t) = (10 \times 10^{-3})(0.03\pi) \sin(100 t).

Step 3: Differentiate the flux to find the induced emf (\varepsilon):
\varepsilon = -\frac{d\Phi}{dt} = -\frac{d}{dt}\left[(10 \times 10^{-3})(0.03\pi) \sin(100 t)\right] = -\left(10 \times 10^{-3})(0.03\pi) (100 \cos(100 t)) = -0.03 \pi\, \cos(100 t).

Step 4: The amplitude of the induced emf is:
\varepsilon_{max} = 0.03\pi \, V.

Step 5: Using Ohm's law (V = IR), where R is the resistance and can be calculated by R = \rho \frac{L}{A}, where \rho is the resistance per unit length and L is the perimeter of the loop.
The perimeter (L) of the loop is:
L = 2\pi a + 2\pi b = 2\pi(0.2) + 2\pi(0.1) = 0.6\pi\, m.
Therefore, the total resistance:
R = (50 \times 10^{-3})(0.6\pi) = 0.03\pi \, \Omega.

Step 6: Finally, we can find the amplitude of the current (I) using Ohm's law:
I = \frac{\varepsilon_{max}}{R} = \frac{0.03\pi}{0.03\pi} = 1 \, A.

Therefore, the amplitude of the current induced in the loop is 1 A.