A circular loop of radius $$r = 20$$ cm and resistance $$R = 2\,\Omega$$ is placed in a time varying  magnetic field $$B = (2t^2 + 2t + 3)$$ T. At $$t = 0$$, for the plane of the loop being perpendicular to the magnetic field and, The induced current in the loop  at $$t = 3$$ s is $$\frac{\alpha}{50}$$ A. The value of $$\alpha$$ is__________.
(Take $$\pi = 22/7$$)

$$\Phi_B = B \cdot A = (2t^2 + 2t + 3) \cdot (0.04\pi)$$

$$\varepsilon = \frac{d\Phi_B}{dt} = 0.04\pi \cdot \frac{d}{dt}(2t^2 + 2t + 3)$$

$$\varepsilon = 0.04\pi \cdot (4t + 2)$$

$$\varepsilon_{t=3} = 0.04\pi \cdot (4(3) + 2) = 0.04\pi \cdot (12 + 2) = 0.04\pi \cdot 14$$

$$\varepsilon_{t=3} = 0.56\pi\text{ V}$$

$$I = \frac{\varepsilon}{R} = \frac{0.56\pi}{2} = 0.28\pi\text{ A}$$

$$I = 0.28 \times \frac{22}{7} = 0.04 \times 22 = 0.88\text{ A}$$

$$\frac{\alpha}{50} = 0.88$$

$$\alpha = 0.88 \times 50 = 44$$