Two concentric circular coils,
Question:

Two concentric circular coils, $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$, 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 current $20 \mathrm{~cm} . \mathrm{C}_{2}$ carries a time dependent current $\mathrm{I}(t)=\left(5 t^{2}-2 t+3\right)$ A Where $t$ is in $\mathrm{s}$.

The emf induced in $\mathrm{C}_{1}$ (in $\mathrm{mV}$ ), at the instant $t=1 \mathrm{~s}$ is $\frac{4}{x}$. The value of $x$ is______

Solution:

(5)

For coil $C_{1}$, No. of turns $N_{1}=500$ and radius, $r=1 \mathrm{~cm}$.

For coil $C_{2}$, No. of turns $N_{2}=200$ and radius, $R=20 \mathrm{~cm}$

$I=\left(5 t^{2}-2 t+3\right) \Rightarrow \frac{d I}{d t}=(10 t-2)$

$\phi_{\text {small }}=B A=\left(\frac{\mu_{0} I N_{2}}{2 R}\right)\left(\pi r^{2}\right)$

Induced emf in small coil,

$e=\frac{d \phi}{d t}=\left(\frac{\mu_{0} N_{2}}{2 r}\right) \pi r^{2} N_{1} \frac{d i}{d t}=\left(\frac{\mu_{0} N_{1} N_{2} \pi r^{2}}{2 R}\right)(10 t-2)$

At $t=1 \mathrm{~s}$

$e=\left(\frac{\mu_{0} N_{1} N_{2} \pi r^{2}}{2 R}\right) 8=4 \frac{\mu_{0} N_{1} N_{2} \pi r^{2}}{R}$

$=\frac{4(4 \pi) 10^{-7} \times 200}{20} \times 500 \times \frac{10^{-4}}{10^{-2}} \pi$

$=80 \times \pi^{2} \times 10^{-7} \times 10 \times 10^{2} \times 10^{-2}$

$=8 \times 10^{-4}$ volt $=0.8 \mathrm{mV}=\frac{4}{\mathrm{x}} \Rightarrow x=5 .$