# Two concentric circular coils, $\inline C_{1}$ and $\inline C_{2}$ , are placed in the $\inline XY$ plane. $\inline C_{1}$ has $\inline 500$ turns, and a radius of $\inline 1\; cm$. $\inline C_{2}$ has$\inline 200$ turns and radius of $\inline 20\; cm$. $\inline C_{2}$ carries a time dependent current $\inline I(t)=\left ( 5t^{2}-2t+3 \right )A$ where $\inline t$ is in $\inline s$. The emf indiuced in $\inline C_{1}$  (in $\inline mV$), at the instant $\inline t=1s$ is $\inline \frac{4}{x}$. The value of $\inline x$ is ______. Option: 1 5 Option: 2 4 Option: 3 3 Option: 4 6

$\begin{array}{l} .1=5 t^{2}-2 t+3 \\ \frac{d I}{d t}=10 t-2 \\ \\ \begin{array}{l} \text { At }(t=1 \text { sec })\left(\frac{d I}{d t}\right)=8 \mathrm{~A} / \mathrm{s} \\ \\ \phi=\left(\frac{\mu_{0} \times 200 \times 1 \times 100}{2 \times 20}\right) \times \frac{\pi \times 500}{100 \times 100} \end{array} \end{array}$

$\begin{array}{ll} \therefore \quad & \mathbf{e}=\left|\frac{d \phi}{d t}\right|=\frac{\mu_{0} \times 200 \times 100 \times 500 \times \pi}{2 \times 20 \times 100 \times 100}\left(\frac{d l}{d t}\right) \\ \\ \Rightarrow & e=\left|\frac{d \phi}{d t}\right|=\left(\frac{8}{10}\right) \mathrm{m} \text { volt }=\frac{4}{5} \mathrm{~m} \text { volt } \\ \\ & x=5 \end{array}$

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