Question

The magnetic field $B=2t^2+4t^2$ (where $t=$ time) is applied perpendicular to the plane of a circular wire of radius $r$ and resistance $R$. If all the units are in SI the electric charge that flows through the circular wire during $t=0s$ to $t=2s$ is

• A. $\frac{6\pi r^2}{R}$
• B. $\frac{24\pi r^2}{R}$
• C. $\frac{32\pi r^2}{R}$
• D. $\frac{48\pi r^2}{R}$

Answer 1

The correct option is B $\frac{24\pi r^2}{R}$

Magnetic field is given as $B=2t^2=4t^2=6t^2$

Differentiating w.r.t t we get, $\frac{dB}{dt}=12t$

Area of the circular coil $A=\pi r^2$

Change in rate of flux from $t=0$ to $t=2$ s, $\frac{d\varphi }{dt}=A\frac{dB}{dt}{|}_o^2$

$\therefore$ $\frac{d\varphi }{dt}=\left(\pi r^2\right)\left(12t\right){|}_o^2=\left(\pi r^2\right)\times 12(2-0)=24\pi r^2$

Current flowing $I=\frac{E}{R}=\frac{\frac{d\varphi }{dt}}{R}=\frac{24\pi r^2}{R}$