Question

A uniform but time varying magnetic field B = $K_1{K}_{2}t$ where $K_1$ & $K_2$ are positive constants and t is time (in seconds), is applied perpendicular to the plane of a circular loop of radius a and resistance R.Find the total charge (in coulomb) that will pass through any point of the loop by the time B becomes zero. [Given a = 2m, R = $\pi \Omega ,K_1$ = 2T]

Answer 1

e = $\frac{d\varphi }{dt}=-\pi a^2\frac{dB}{dt}=\pi a^2{K}_2;$

$I=\frac{e}{R}=\frac{\pi a^2{K}_2}{R}$ ; $

At $t=t_1$ ; $B=0$ $\Rightarrow t_1=\frac{K_1}{K_2}$

So charge flown = $I{t}_1=\left(\frac{\pi a^2{K}_2}{R}\right)\left(\frac{K_1}{K_2}\right)$

$=\frac{\pi a^2{K}_1}{R}=8C$