$A$ coil of resistance $400 \Omega$ is placed in a magnetic field. If the magnetic flux $\phi$ (Wb) linked with the coil varies with time $t$ $(s)$ as $\phi = 50t^2 + 4$,the current in the coil at $t = 2 \ s$ will be: (in $A$)

$A$ coil of resistance $8 \Omega$,number of turns $250$,and area $120 \ cm^2$ is placed in a uniform magnetic field of $2 \ T$ such that the plane of the coil makes an angle of $\frac{\pi}{6}$ with the direction of the magnetic field. In a time of $100 \ ms$,the coil is rotated until its plane becomes parallel to the direction of the magnetic field. The current induced in the coil is (in $A$)

$A$ magnetic field $B$ is confined to a region $r \le a$ and points out of the paper (the $z$-axis),$r = 0$ being the centre of the circular region. $A$ charged ring (charge $= Q$) of radius $b$,$b > a$ and mass $m$ lies in the $xy$-plane with its centre at the origin. The ring is free to rotate and is at rest. The magnetic field is brought to zero in time $\Delta t$. Find the angular velocity $\omega$ of the ring after the field vanishes.

$A$ uniform magnetic field $\vec{B}$ is perpendicular to the plane of a circular loop of diameter $10 \text{ cm}$ formed from a wire of diameter $2 \text{ mm}$ and resistivity $2 \times 10^{-8} \Omega \text{ m}$. If a current of $11 \text{ A}$ is to be induced in the loop, then the rate at which $\vec{B}$ is to be changed is: (in $\text{ T s}^{-1}$)

If magnetic flux $\phi = (3t^2 - 2t + 5) \ Wb$,then what is the induced emf at $t = 2 \ s$?

Derive the formula for induced charge and prove that it is independent of the rate of change of magnetic flux.