Magnetic flux (in $Wb$) linked with a closed loop varies with time $t$ (in $s$) as $\phi = 2t^2 + 1$. The magnitude of induced emf at $t = 1 \ s$ is $..... \ V$.

In the following figure, the magnet is moved towards the coil with a speed $v$ and the induced emf is $e$. If the magnet and the coil recede away from one another, each moving with a speed $v$, the induced emf in the coil will be

$A$ square loop of side $10 \; cm$ and resistance $0.5 \; \Omega$ is placed vertically in the east-west plane. $A$ uniform magnetic field of $0.10 \; T$ is set up across the plane in the north-east direction. The magnetic field is decreased to zero in $0.70 \; s$ at a steady rate. Determine the magnitudes of induced emf and current during this time interval.

$A$ long solenoid of diameter $0.1\, m$ has $2 \times 10^4$ turns per meter. At the centre of the solenoid,a coil of $100$ turns and radius $0.01\, m$ is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to $0\, A$ from $4\, A$ in $0.05\, s$. If the resistance of the coil is $10\pi^2\, \Omega$,the total charge flowing through the coil during this time is:

$A$ 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.

If a coil of $40$ turns and area $4.0 \, cm^2$ is suddenly removed from a magnetic field,it is observed that a charge of $2.0 \times 10^{-4} \, C$ flows through the coil. If the resistance of the coil is $80 \, \Omega$,the magnetic flux density in $Wb/m^2$ is: