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 $....\,mA$.

$A$ 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

$A$ magnetic field $\vec{B} = B_0 \sin(\omega t) \hat{k}$ covers a large region where a wire $AB$ slides smoothly over two parallel conductors separated by a distance $d$ as shown in the figure. The wires are in the $xy$-plane. The wire $AB$ (of length $d$) has resistance $R$ and the parallel wires have negligible resistance. If $AB$ is moving with velocity $v$,what is the current in the circuit? What is the force needed to keep the wire moving at constant velocity?

$A$ uniform but time-varying magnetic field $B(t)$ exists in a circular region of radius $a$ and is directed into the plane of the paper,as shown. The magnitude of the induced electric field at point $P$ at a distance $r$ $(r > a)$ from the centre of the circular region is:

$A$ square loop of side $2.0\,cm$ is placed inside a long solenoid that has $50$ turns per centimetre and carries a sinusoidally varying current of amplitude $2.5\,A$ and angular frequency $700\,rad\,s^{-1}$. The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is $x \times 10^{-4}\,V$. The value of $x$ is $.........$ (Take $\pi = \frac{22}{7}$)

$A$ 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: