Question

A longer solenoid of diameter $0.1m$ has $2\times {10}^4$ turns per meter. At the center 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. $26\pi \mu C$
• 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,
$r_1=\frac{1}{20}m$, $N_1=2\times {10}^4/m$
$r_2=\frac{1}{100}m$, $N_2=100/m$
$R=10{\pi }^2\Omega$
$\frac{dI}{dt}=\frac{4-0}{0.05}=80A/s$
As we know, induced emf $=M\frac{dI}{dt}$
where, $M=\frac{{\mu }_0{N}_1{N}_2{A}_2}{l}$
and $Q=I.dt=\frac{\text{induced emf}}{R}.dt$
So,
$q={\mu }_o{N}_1{N}_2\pi r^2\frac{△I.△t}{l.△t.R}$
$q=[4\pi \times {10}^{-7}\times 2\times {10}^4\times 100\times \pi \times ({10}^{-2}{)}^{-2}\times \frac{4}{0.05}]\frac{1}{10{\pi }^2}\times 0.05$
$q=32\mu C$