A long solenoid of diameter 0.1 m has $2 imes 10^{4}$ turns per meter. At the centre of the soleno

Question

A long solenoid of diameter 0.1 m has $2 	imes 10^{4}$ turns per meter. At the centre of the solenoid, a coil of 100 turns and radius 0.01 m is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to 0 A from 4 A in 0.05 s . If the resistance of the coil is $10 \pi^{2} \Omega$, the total charge flowing through the coil during this time is\

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(b) : Given $n=2 	imes 10^{4}, I=4 \mathrm{~A}$ Initially $I=0 \mathrm{~A} \quad 	herefore B_{i}=0$ or $\phi_{i}=0$\ Finally, the magnetic field at the centre of the solenoid is given as $$ B_{f}=\mu_{0} n I=4 \pi 	imes 10^{-7} 	imes 2 	imes 10^{4} 	imes 4=32 \pi 	imes 10^{-3} \mathrm{~T} $$ Final magnetic flux through the coil is given as $$ \phi_{f}=N B A=100 	imes 32 \pi 	imes 10^{-3} 	imes \pi 	imes(0.01)^{2} $$ $$ \phi_{f}=32 \pi^{2} 	imes 10^{-5} \mathrm{~T} \mathrm{~m}^{2} $$ Induced charge, $q=\frac{|\Delta \phi|}{R}=\frac{\left|\phi_{f}-\phi_{i}ight|}{R}=\frac{32 \pi^{2} 	imes 10^{-5}}{10 \pi^{2}}$ $$ =32 	imes 10^{-6} \mathrm{C}=32 \mu \mathrm{C} $$