A long solenoid S has n turns per metre, with radius a. At the centre of this coil, we place a smaller coil of N turns and radius b (where $b). If the current in the solenoid increases linearly with time, what is the induced emf appearing in the smaller coil. Plot a graph showing nature of variation in emf, if current varies as a function of $mt^{2}+C.$

Magnetic field caused by a solenoid is given by

$B=\mu _{0}ni$.

Magnetic flux in the smaller coil is

$\phi _{m}=NBA$

where, $A=\pi b^{2}$

Applying Faraday's law of EMI, we have

So, $e=\frac{-d\phi }{dt}=\frac{-d}{dt}(NBA)$

$=-N\pi b^{2}\frac{d(B)}{dt}$

Where, $B=\mu _{0}Ni$

$\Rightarrow e=-N\pi b^{2}\; \mu _{0}n\frac{di}{dt}$

Since, current varies as a function of time, so

$i(t)=mt^{2}+C$

$\Rightarrow \; e=-Nn\pi \mu _{0}b^{2}\frac{d}{dt}(mt^{2}+C)$

By solving$e=-\mu _{0}Nn\pi b^{2}2mt$

The negative sign signifies opposite nature of induced emf.

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