Flux $\phi$ (in weber) in a closed circuit of resistance $10 \, \Omega$ varies with time $t$ (in $s$) according to the equation $\phi = 6t^2 - 5t + 1$. What is the magnitude of the induced current at $t = 0.25 \, s$ (in $, A$)?

Is relative motion an absolute condition for inducing $emf$?

$A$ uniform magnetic field $B$ exists in a cylindrical region of radius $10\, cm$ as shown in the figure. $A$ uniform wire of length $80\, cm$ and resistance $4.0\,\Omega$ is bent into a square frame and is placed with one side along a diameter of the cylindrical region. If the magnetic field increases at a constant rate of $0.010\, T/s$,find the current induced in the frame.

$A$ square loop of side $2.0\,cm$ is placed inside a long solenoid that has $50$ turns per centimetre and carries a sinusoidally varying current of amplitude $2.5\,A$ and angular frequency $700\,rad\,s^{-1}$. The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is $x \times 10^{-4}\,V$. The value of $x$ is $.........$ (Take $\pi = \frac{22}{7}$)

$A$ circular loop of radius $20 \text{ cm}$ and resistance $2 \text{ } \Omega$ is placed in a time-varying magnetic field $\vec{B} = (2t^2 + 2t + 3) \text{ T}$. The plane of the loop is perpendicular to the magnetic field. The induced current in the loop at $t = 3 \text{ s}$ is $\frac{\alpha}{50} \text{ A}$. The value of $\alpha$ is . . . . . . .

The radius of the circular conducting loop shown in the figure is $R.$ The magnetic field is decreasing at a constant rate $\alpha.$ The resistance per unit length of the loop is $r.$ Find the current in the wire $AB,$ where $AB$ is one of the diameters.