$A$ 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.

$A$ magnetic field at a distance $r$ from the $z$-axis is given by $\vec{B} = B_0 r t \hat{k}$,where $B_0$ is a constant and $t$ is time. The magnitude of the induced electric field at a distance $r$ from the $z$-axis is:

$A$ square loop of side $2 \text{ cm}$ is placed in a time-varying magnetic field with magnitude $B = 0.4 \sin(300t) \text{ T}$. The normal to the plane of the loop makes an angle of $60^{\circ}$ with the field. The maximum induced emf produced in the loop is . . . . . . $\text{mV}$.

$A$ magnetic field is changing at the rate of $4 \, T/s$ in a circular region of $5 \, cm$ radius. The value of the induced electric field at a point $P$,which is $10 \, cm$ away from the center $O$ of the region,is ..... $V/m$.

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:

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