A long solenoid of radius r consists of n turns per unit length. A current I = I_0 sin ω t flows in the solenoid. A coil of N turns is wound tightly…

Finding

magnetic field produced in the solenoid near the center

\[\mathrm{B}=\mu_{\mathrm{o}} n \mathrm{I} \]

Flux linked with the coil wound over solenoid

\[\begin{aligned} \phi & =\mathrm{NBA}=\mathrm{N} \pi \mathrm{r}^{2} \mathrm{~B} \\ & =\mathrm{N} \pi \mathrm{r}^{2} \mu_{\mathrm{o}} n \mathrm{I} \end{aligned} \]

\[\begin{aligned} \text { Induced emf } \mathrm{e}=\frac{-d \phi}{d \mathrm{t}} & =-\pi \mathrm{r}^{2} \mathrm{Nn} \mu_{\mathrm{o}} \frac{d \mathrm{I}}{d \mathrm{t}} \\ & =-\mu_{\mathrm{o}} \pi \mathrm{r}^{2} \mathrm{nN} \mathrm{I}_{\mathrm{o}} \omega \cos \omega \mathrm{t} \end{aligned} \]