Question

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

• A. $32\pi \mu C$
• B. $64\pi \mu C$
• C. $8\pi \mu C$
• D. $16\pi \mu C$

Answer 1

The correct option is A $32\pi \mu C$

By Lenz's law
$e=-N\frac{d\varphi }{dt}\mathrm{Dividing}\mathrm{by}R\mathrm{on}\mathrm{both}\mathrm{side}\left(\frac{e}{R}\right|=\left(\frac{N}{R}\frac{d\varphi }{dt}\right|where,dq=\frac{N}{R}d\varphi \Delta Q=\frac{N(\Delta \varphi )}{R}\Delta Q=\frac{(\Delta {\varphi }_{total})}{R}=\frac{N\Delta \left(BA\right)}{R}=\frac{NA\Delta B}{R}=\frac{NAR{\mu }_0\Delta I}{R}\Delta Q=\frac{2\times {10}^4\times \pi \times {(0.01)}^2\times 100\times 4\pi \times {10}^{-7}\times 4}{10{\pi }^2}=32\mu C$