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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} has200 turns and radius of 20\; cm. C_{2} carries a time dependent current I(t)=\left ( 5t^{2}-2t+3 \right )A where t is in s. The emf indiuced in C_{1}  (in mV), at the instant t=1s is \frac{4}{x}. The value of x is ______.
Option: 1 5
Option: 2 4
Option: 3 3
Option: 4 6

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\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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Deependra Verma

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