A

To determine the amplitude of the induced current in the loop, we start by calculating the magnetic flux through the loop and then use Faraday's law of electromagnetic induction.

**Step 1: Calculate the area of the loop.**
The loop consists of two circular arcs with inner radius (b) = 10 cm = 0.1 m and outer radius (a) = 20 cm = 0.2 m. The cross-sectional area of the loop can be approximated as the area of the outer circle minus the area of the inner circle:
\[ \text{Area} (A) = \pi a^2 - \pi b^2 = \pi (0.2^2) - \pi (0.1^2) = \pi (0.04 - 0.01) = \pi (0.03) = 0.09424778 \, m^2 \]

**Step 2: Find the magnetic flux (Φ) through the loop.**
The magnetic field is given as \( B(t) = B_0 \sin(\omega t) \), where \( B_0 = 10 \, mT = 0.01 \, T \) and \( \omega = 100 \, rad/s \). The magnetic flux is given by:
\[ \Phi = B(t) \cdot A = 0.01 \sin(100t) \cdot 0.09424778 \, m^2 \]
\[ \Phi = 0.0009424778 \sin(100t) \, Wb \]

**Step 3: Calculate the induced EMF (ε).**
According to Faraday's law:
\[ \varepsilon = -\frac{d\Phi}{dt} \]
Taking the derivative:
\[ \varepsilon = -0.0009424778 \cdot 100 \cos(100t) = -0.09424778 \cos(100t) \, V \]

**Step 4: Calculate the resistance of the loop.**
The total length of the loop (L) can be approximated as the average circumference of the inner and outer circles:
\[ L = \pi (a + b) = \pi (0.2 + 0.1) = \pi (0.3) = 0.9424778 \, m \]
The resistance (R) is given by the resistance per unit length times the length:
\[ R = (50 \times 10^{-3} \, \Omega/m) \cdot 0.9424778 \, m = 0.04712389 \, \Omega \]

**Step 5: Calculate the amplitude of the current (I).**
Using Ohm's law \( I = \frac{\varepsilon}{R} \), where \( \varepsilon \) is the peak EMF:
The peak EMF is \( 0.09424778 \, V \). So:
\[ I = \frac{0.09424778}{0.04712389} = 2 \, A \]

Therefore, the amplitude of the induced current in the loop is 2 A.
Thus, the final answer is option A.