Question

Magnetic field varies as $B=2t$ in a square loop of side $20\text{cm}$ as shown in the figure. If the resistance of the loop is $4\Omega$, the total charge that flows in the loop from $t=0$ to $t=2\text{s}$ will be.

• A. $0.16\text{C}$
• B. $0.08\text{C}$
• C. $0.04\text{C}$
• D. $0.01\text{C}$

Answer 1

The correct option is C $0.04\text{C}$

We know that average emf induced in the loop is,

$E_{av}=\left|\frac{\Delta \varphi }{\Delta t}\right|$

$\therefore$ Average current induce in the loop is,

$i_{av}=\frac{E_{av}}{R}=\left|\frac{\Delta \varphi }{R\Delta t}\right|$

$\therefore$ Total charge flown through the loop is,

$q=i_{av}\Delta t=\left|\frac{\Delta \varphi }{R\Delta t}\right|\times \Delta t=\left|\frac{\Delta \varphi }{R}\right|$

Initial flux linked with the loop is,

${\varphi }_1=B_{1}A=2t\times (0.2{)}^2=0[\because t=0]$

Final flux linked with the loop is,

${\varphi }_2=B_{2}A=2t\times (0.2{)}^2$

$=0.08t=0.08\times 2=0.16\text{Wb}[\because t=2\text{s}]$

$\therefore q=\left|\frac{{\varphi }_2-{\varphi }_1}{R}\right|=\left|\frac{0.16-0}{4}\right|$

$\therefore q=0.04\text{C}$

Hence, $\left(C\right)$ is the correct answer.

Why this question : The total charge flown depends upon resistance of the loop, initial flux and final flux only. It does not depend on the time during which the flux changes.