Question

A uniform but time varying magnetic field $B=Kt-C$; $(0\le t\le C/K)$, where K and C are constants and t is time, is applied perpendicular to the plane of the circular loop of radius 'r' and resistance R. Find the total charge that will pass around the loop in coulombs?

Answer 1

emf = rate of change of flux = $\frac{d\left(\pi r^{2}B\right)}{dt}=\pi r^2\frac{dB}{dt}=\pi r^{2}K$
current induced = $\frac{emf}{R}=\frac{\pi r^{2}K}{R}$
total charge that will flow = ${\int }_0^{C/K}idt={\int }_0^{C/K}\frac{\pi r^{2}K}{R}dt=\frac{\pi r^{2}K}{R}\frac{C}{K}$coulombs