$A$ circular loop of radius $20 \text{ cm}$ and resistance $2 \text{ } \Omega$ is placed in a time-varying magnetic field $\vec{B} = (2t^2 + 2t + 3) \text{ T}$. The plane of the loop is perpendicular to the magnetic field. The induced current in the loop at $t = 3 \text{ s}$ is $\frac{\alpha}{50} \text{ A}$. The value of $\alpha$ is . . . . . . .

$A$ conducting ring of radius $r$ is placed in a varying magnetic field perpendicular to the plane of the ring. If the rate at which the magnetic field varies is $x$,the electric field intensity at any point on the ring is:

$A$ line charge $\lambda$ per unit length is lodged uniformly onto the rim of a wheel of mass $M$ and radius $R$. The wheel has light non-conducting spokes and is free to rotate without friction about its axis. $A$ uniform magnetic field extends over a circular region within the rim. It is given by,
$B = -B_{0} \hat{k}$ for $r \leq a$ (where $a < R$)
$B = 0$ otherwise.
What is the angular velocity of the wheel after the field is suddenly switched off?

Flux $\phi$ (in weber) in a closed circuit of resistance $10 \, \Omega$ varies with time $t$ (in $s$) according to the equation $\phi = 6t^2 - 5t + 1$. What is the magnitude of the induced current at $t = 0.25 \, s$ (in $, A$)?

$A$ conducting ring of radius $1\,m$ is placed in a uniform magnetic field $B$ of $0.01\,T$ oscillating with a frequency of $100\,Hz$,with its plane at right angles to $B$. What will be the induced electric field in $V/m$?

$A$ non-conducting ring of radius $R$ and mass $m$ having charge $q$ uniformly distributed over its circumference is placed on a rough horizontal surface. $A$ vertical time-varying uniform magnetic field $B = 4t^2$ is switched on at time $t=0$. The coefficient of friction between the ring and the table,if the ring starts rotating at $t = 2 \, s$,is: