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. $22\mu C$
• C. $16\pi \mu C$
• D. $32\pi \mu C$

Answer 1

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

$ds=0.1m$, $n_5=2\times {10}^4/m$
$n_c=100,r_e=0.01m$ $0A$ from $4A$ in $0.5s$
$R=10{\pi }^2$
Field at centre of solenoid
$B={\mu }_o{n}_{5}i$
Hence through loop $=\theta =BA$
$={\mu }_0{n}_{5}i\times \pi (0.01{)}^2\times n_c$
$e=\frac{d}{dt}={\mu }_0{n}_{5}n\theta \pi (0.01{)}^2\times \frac{△i}{△t}$
$i=\frac{e}{R}$, $\theta (-i)\times △t$ $Q=\frac{c△t}{R}$
$Q={\mu }_0{n}_5{n}_0\pi (0.01{)}^2\times △i\frac{1}{R}$
$Q=4\pi \times 106-7\times 2\times {10}^4\times 100\times 100\pi \times {10}^{-4}\times 4\times \frac{1}{10\pi }$
$Q=32\mu C$