$A$ conductor loop of radius $R$ is present in a uniform magnetic field $B$ perpendicular to the plane of the ring. If the radius $R$ varies as a function of time $t$ as $R = R_0 + t$,the induced $e.m.f.$ in the loop is:

$A$ $10 \, m$ wire kept in the east-west direction is falling with a velocity of $5 \, m/s$ perpendicular to the magnetic field of $0.3 \times 10^{-4} \, Wb/m^2$. The induced e.m.f. across the terminals of the wire will be:

$A$ cycle wheel of radius $0.5 \; m$ is rotated with a constant angular velocity of $10 \; rad/s$ in a region of magnetic field of $0.1 \; T$ which is perpendicular to the plane of the wheel. The $EMF$ generated between its centre and the rim is.....$V$

$A$ horizontal loop $abcd$ is moved across the pole pieces of a magnet as shown in the figure with a constant speed $v$. When the edge $ab$ of the loop enters the pole pieces at time $t = 0 \text{ s}$,which one of the following graphs correctly represents the induced emf in the coil?

$A$ conducting wire $XY$ of mass $m$ and negligible resistance slides smoothly on two parallel conducting wires as shown in the figure. The closed circuit has a resistance $R$ due to $AC$. $AB$ and $CD$ are perfect conductors. There is a magnetic field $\vec{B} = B(t) \hat{k}$.
$(i)$ Write down the equation for the acceleration of the wire $XY$.
$(ii)$ If $\vec{B}$ is independent of time,obtain $v(t)$,assuming $v(0) = u_0$.
$(iii)$ For $(ii)$,show that the decrease in kinetic energy of $XY$ equals the heat lost in $R$.

$A$ square loop $ABCD$ is moving with constant velocity $\vec{v}$ in a uniform magnetic field $\vec{B}$ which is perpendicular to the plane of paper and directed outward. The resistance of the coil is $R$. What is the rate of production of heat energy in the loop? [$L$ = length of side of the loop]