$A$ conducting circular loop is placed in the $X-Y$ plane in the presence of a magnetic field $\overrightarrow{B} = (3t^3 \hat{j} + 3t^2 \hat{k})$ in $SI$ units. If the radius of the loop is $1 \ m$,the induced emf in the loop at time $t = 2 \ s$ is $n\pi \ V$. The value of $n$ is:

$A$ conducting rod $AB$ of length $l = 1\,m$ is moving with a velocity $v = 4\,m/s$. The velocity vector makes an angle of $30^o$ with the length of the rod. $A$ uniform magnetic field $B = 2\,T$ exists in a direction perpendicular to the plane of motion. Then:

$A$ metallic cube of side $15\,cm$ is moving along the $y$-axis at a uniform velocity of $2\,m/s$ in a region of uniform magnetic field of magnitude $0.5\,T$ directed along the $z$-axis. In equilibrium,the potential difference between the faces of higher and lower potential developed because of the motion through the field will be $..........mV$.

$A$ frame $CDEF$ is placed in a region where a magnetic field $\vec{B}$ is present. $A$ rod $PQ$ of length $l = 1 \, m$ moves with a constant velocity $v = 20 \, m/s$ and the strength of the magnetic field is $B = 1 \, T$. The power spent in the process is .............. $kW$ (take $R = 0.2 \, \Omega$ and assume all other wires and the rod have zero resistance).

$A$ conducting wire is moving towards the right in a magnetic field $B$. The direction of the induced current $i$ in the wire is shown in the figure. The direction of the magnetic field will be

Two metallic rings of radius $R$ are rolling on a metallic rod. $A$ magnetic field of magnitude $B$ is applied in the region. The magnitude of the potential difference between point $A$ and point $C$ on the two rings (as shown) will be: