The instantaneous magnetic flux associated with a closed loop of resistance $10 \ \Omega$ is given by $\phi = 2t^2 - 5t + 1$. The magnitude of the induced current at $t = 0.25 \ s$ will be . . . . . . . (in $A$)

$A$ coil having $500$ square loops each of side $10\, cm$ is placed normal to a magnetic flux which increases at the rate of $1.0\, T/s$. The induced $e.m.f.$ in volts is:

Lenz's law applies to

$A$ wire loop of area $0.2 \, m^2$ has a resistance of $20 \, \Omega$. $A$ magnetic field pointing normal to the loop has a magnitude of $0.25 \, T$ and is reduced to zero at a uniform rate in $10^{-4} \, s$. What is the induced emf and the resulting current?

$A$ highly conducting ring of radius $R$ is perpendicular to and concentric with the axis of a long solenoid as shown in the figure. The ring has a narrow gap of width $d$ in its circumference. The solenoid has a cross-sectional area $A$ and a uniform internal magnetic field of magnitude $B_0$. Starting at $t = 0$,the solenoid current is steadily increased such that the field magnitude at any time $t$ is given by $B(t) = B_0 + \alpha t$,where $\alpha > 0$. Assuming that no charge can flow across the gap,the end of the ring which has an excess of positive charge and the magnitude of the induced e.m.f. in the ring are respectively:

Derive the relation between induced charge and change in magnetic flux.