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

In the given circuit below inductance values of $L_1, L_2$ and $L_3$ are same. The magnetic energy stored in the entire circuit is $\left(U_t\right)$ and that stored in the $\mathrm{L}_2$ inductor is $\left(U_l\right)$. $U_t / U_l$ is $\_\_\_\_$ .

(Ignore the mutual inductance if any)

A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T . The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^{\circ}$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is

$\_\_\_\_$ mV . (Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

A conducting circular loop is rotated about its diameter at a constant angular speed of $100 \mathrm{rad} / \mathrm{s}$ in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^{\circ}$ from the horizontal position, the induced EMF is 15.4 mV . The radius of the loop is $\_\_\_\_$ mm.

$$ \left(\text { Take } \pi=\frac{22}{7}\right) $$