Two concentric circular coils,

Question:

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

Solution:

$B=\frac{\mu_{0} N I}{2 R}$

$\phi=\frac{\mu_{0} \mathrm{NN}^{\prime} \mathrm{I}}{2 \mathrm{R}} \pi \mathrm{r}^{2}$

$\varepsilon=\frac{\mathrm{d} \phi}{\mathrm{dt}}=\frac{2 \pi \times 10^{-7} \times 10^{5} \times \pi \times 10^{-4}}{0.2}$

$=8 \times 10^{-4}=0.8 \mathrm{mV}$

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