A long solenoid $S$ has $n$ turns per meter, with diameter $a$. At the centre of this coil, we place a smaller coil of $N$ turns and diameter $b$ (where $b< a$ ). If the current in the solenoid increases linearly, with time, what is the induced emf appearing in the smaller coil. Plot graph showing nature of variation in emf, if current varies as a function of $m t^2+C$.
Magnetic field due to a solenoid $S, B=\alpha_0 n l$ where signs are as usual.
Magnetic flux in smaller coil $\phi=$ NBA where
$$A=\pi b^2$$
Applying Faraday's law of EMI, we have
$$\begin{aligned} &\text { So, }\\ &\begin{aligned} e & =\frac{-d \phi}{d t}=\frac{-d}{d t}(N B A) \\ & =-N \pi b^2 \frac{d(B)}{d t} \end{aligned} \end{aligned}$$
where,
$$\begin{aligned} B & =\propto_0 N i \\ & =-N \pi b^2 \propto_0 n \frac{d l}{d t} \\ & =-N n \pi \propto_0 b^2 \frac{d}{d t}\left(m t^2+C\right)=-\propto_0 N n \pi b^2 2 m t \end{aligned}$$
$$\begin{aligned} &\text { Since, current varies as a function of } m t^2+C \text {. }\\ &e=-\propto_0 N n \pi b^2 2 m t \end{aligned}$$