Two concentric circular coils, $C_1$ and $C_2$, are placed in the $XY$ plane. $C_1$ has $500$ turns, and a radius of $1\ \text{cm}$. $C_2$ has $200$ turns and radius of $20\ \text{cm}$. $C_2$ 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 $t = 1\ \text{s}$ is $\dfrac4x$. The value of $x$ is ______.
Numerical value type. Enter your answer.
Answer: 5
The small coil sits at the centre of the large one, where $B = \dfrac{\mu_0N_2I}{2R}$ is nearly uniform. Mutual inductance:
$$M = \frac{\mu_0N_1N_2\pi r^{2}}{2R} = \frac{4\pi\times10^{-7}\times500\times200\times\pi\times10^{-4}}{0.4} \approx 9.87\times10^{-5}\ \text{H}$$
$\dfrac{dI}{dt} = 10t - 2 = 8$ A/s at $t = 1$ s, so $\varepsilon = M\dfrac{dI}{dt} \approx 7.9\times10^{-4}$ V $= 0.79$ mV.
$\dfrac4x = 0.79 \Rightarrow x \approx 5$.
Solution by Sreeraj P, M.Sc Physics