Two concentric circular coils, $C_1$ and $C_2$, are placed in the $XY$ plane. $C_1$ has $500$turns, and a radius of $1 \mathrm{cm}$. $C_2$ has 200 turns and radius of $20 \mathrm{cm}\text{.} C_2$ carries a time dependent current $I\left(\mathit{t}\right)=\left(5t^2-2t+3\right)\mathrm{A}$ where $t$ is in s. The emf induced in $C_1\left(\mathrm{in} \mathrm{mV}\right)$ at the instant $t=1$s is $\frac{4}{x}.$ The value of $x$ is ..........

An infinitely long straight wire carrying current $I$, one side opened rectangular loop and a conductor $C$ with a sliding connector are located in the same plane, as shown in the figure. The connector has length $l$ and resistance $R$. It slides to the right with a velocity $v$. The resistance of the conductor and the self-inductance of the loop are negligible. The induced current in the loop, as a function of separation $r$, between the connector and the straight wire is:

A part of a complete circuit is shown in the figure. At some instant, the value of current I is $1\mathrm{A}$ and it is decreasing at a rate of ${10}^2{\mathrm{As}}^{-1}$. The value of the potential difference ${\mathrm{V}}_{\mathrm{p}}-{\mathrm{V}}_{\mathrm{Q}},$ (in volts) at that instant is-