The figure shows a circular region of radius $R$ in which a uniform magnetic field $B$ exists. The magnetic field is increasing at a rate $\frac{d B}{d t}$. The magnitude of the induced electric field at a distance $r$ from the centre for $r < R$ 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 $, A$)?

In the given figure,the magnetic flux through the loop increases according to the relation $\phi_{B}(t) = 10t^{2} + 20t$,where $\phi_{B}$ is in milliwebers $(mWb)$ and $t$ is in seconds $(s)$. The magnitude of the current through the $R = 2\,\Omega$ resistor at $t = 5\,s$ is $....\,mA$.

$A$ 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:

$A$ coil is placed in a time-varying magnetic field. The power dissipated due to current induced in the coil is $P_1$. If the number of turns is doubled and the radius of the wire is halved,the power dissipated is $P_2$. Then $P_1: P_2$ is