$A$ 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:

$A$ rectangular wire loop of sides $8 \; cm$ and $2 \; cm$ with a small cut is moving out of a region of uniform magnetic field of magnitude $0.3 \; T$ directed normal to the loop. Suppose the loop is stationary but the current feeding the electromagnet that produces the magnetic field is gradually reduced so that the field decreases from its initial value of $0.3 \; T$ at the rate of $0.02 \; T \, s^{-1}$. If the cut is joined and the loop has a resistance of $1.6 \; \Omega$,how much power is dissipated by the loop as heat? What is the source of this power?

The circular wire in the figure below encircles a solenoid in which the magnetic flux is increasing at a constant rate out of the plane of the page. The clockwise emf around the circular loop is $\varepsilon_{0}$. By definition,a voltmeter measures the voltage difference between two points given by $V_{b}-V_{a}=-\int_{a}^{b} E \cdot ds$. We assume that $a$ and $b$ are infinitesimally close to each other. The values of $V_{b}-V_{a}$ along path $1$ and $V_{a}-V_{b}$ along path $2$,respectively,are

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 $....\,mA$.

Two concentric circular coils,$C_{1}$ and $C_{2}$,are placed in the $XY$ plane. $C_{1}$ has $500$ turns and a radius of $1\; cm$. $C_{2}$ has $200$ turns and a radius of $20\; cm$. $C_{2}$ carries a time-dependent current $I(t) = (5t^{2} - 2t + 3)\; A$,where $t$ is in $s$. The $emf$ induced in $C_{1}$ (in $mV$) at the instant $t = 1\; s$ is $\frac{4}{x}$. The value of $x$ is:

$A$ conducting ring of radius $r$ is placed perpendicularly inside a time-varying magnetic field given by $B = B_0 + \alpha t$ as shown in the figure. $B_0$ and $\alpha$ are positive constants. Find the $emf$ produced in the ring.