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. \(C_2\) has 200 turns and radius of 20 cm. \(C_2\) carries a time dependent current \(I(t) = (5t^2 - 2t + 3)\) A where \(t\) is in s. The emf induced in \(C_1\) (in mV), at the instant \(t = 1\) s is \(\dfrac{4}{x}\). The value of \(x\) is ___ .
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Correct Answer: 5
Solution and Explanation
Flux linked with \(C_1\): \(\Phi = N_1 B\,\pi r_1^2 = \dfrac{\mu_0 N_1 N_2 \pi r_1^2}{2R_2}\,I\).
emf \(= \dfrac{\mu_0 N_1 N_2 \pi r_1^2}{2R_2}\dfrac{dI}{dt}\).
\(\dfrac{dI}{dt} = 10t - 2 = 8\) A/s at \(t = 1\) s.
emf \(= \dfrac{4\pi\times10^{-7}\times 500\times 200\times \pi\times (10^{-2})^2}{2\times 0.2}\times 8\)
\(= \dfrac{4\pi^2\times10^{-7}\times 10^5\times 10^{-4}}{0.4}\times 8 = 80\pi^2\times10^{-6}\) V \(\approx 7.9\times10^{-4}\) V \(= 0.79\) mV.
\(\dfrac4x = 0.79 \Rightarrow x \approx 5.07 \approx 5\).
Answer: 5
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