Question

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

Answer 1

For coil $C_1$,
No. of turns $N_1=500$ and radius, $r=1\text{cm}$
For coil $C_2$,
No. of turns $N_2=200$ and radius, $R=20\text{cm}$
Given current
$\Rightarrow I=(5t^2-2t+3)\Rightarrow \frac{dI}{dt}=(10t-2)$

Flux through small loop is
${\varphi }_{\text{small}}=BA=\left(\frac{{\mu }_{0}I{N}_2}{2R}\right)\left(\pi r^2\right)$
Induced emf in small coil,
$e=\frac{d\varphi }{dt}=\left(\frac{{\mu }_0{N}_2}{2r}\right)\pi r^2{N}_1\frac{di}{dt}=\left(\frac{{\mu }_0{N}_1{N}_2\pi r^2}{2R}\right)(10t-2)$
At $t=1\text{s}$
$e=\left(\frac{{\mu }_0{N}_1{N}_2\pi r^2}{2R}\right)8=4\frac{{\mu }_0{N}_1{N}_2\pi r^2}{R}$
$e=\frac{4\left(4\pi \right){10}^{-7}\times 200}{20}\times 500\times \frac{{10}^{-4}}{{10}^{-2}}\pi$
$e=80\times {\pi }^2\times {10}^{-7}\times 10\times {10}^2\times {10}^{-2}$
$e=8\times {10}^{-4}\text{volt}=0.8\text{mV}=\frac{4}{x}\Rightarrow x=5$