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

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answer is 5.

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Detailed Solution

$\begin{array}{l} Flux through 1 due to 2 = \\ φ {=N}_{1} B_{2} A_{1} = (500) (\frac{{200μ}_{0} I}{{2R}_{2}}) {πR}_{1}^{2} \\ Induced emf in coil 1 \\ = E = \frac{dφ}{dt} = \frac{200 \times 500 \times 4 π \times 10^{- 7} \times π \times 10^{- 4}}{2 \times 20 \times 10^{- 2}} \times \frac{dI}{dt} \\ \begin{array}{l} \Rightarrow \frac{4}{x} \times 10^{- 3} = \frac{200 \times 500 \times π^{2} \times 10^{- 7} \times 10^{- 4}}{10^{- 1}} (10 t - 2) \\ Put t = 1 s, \\ \Rightarrow x = 5 .06 ≃ 5 \sec \end{array} \end{array}$

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