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. $16\mu C$
• B. $32\mu C$
• C. $16\pi \mu C$
• D. $32\pi \mu C$

Answer 1

The correct option is B $32\mu C$

Given, $s=2x{10}^4$, $i=4A$
Initially, $i=0A$ or $\varphi =0$
Finally, the magnetic field at the centre of the solenoid is given as
$B_f={\mu }_{0}ni$$B_f=4\pi \times {10}^{-7}\times 2\times {10}^4\times 4T=32{\pi }^2\times {10}^{-3}T$
Final magnetic flux through the coil is given as
${\varphi }_f=NBA=100\times 32\pi \times {10}^{-3}\times \pi \times {\left(0.01\right)}^2$
${\varphi }_f=32{\pi }^2\times {10}^{-5}T{m}^2$
Induced charge,
$q=\frac{\left(}{\Delta \varphi }=\frac{\left(}{{\varphi }_f-{\varphi }_i}=\frac{32{\pi }^2\times {10}^{-5}}{10{\pi }^2}=32\times {10}^{-6}C=32\mu C$