$A$ long solenoid with $2000$ turns per meter has a small loop of radius $3 \,cm$ placed inside the solenoid normal to its axis. If the current through the solenoid increases steadily from $1.5 \,A$ to $5.5 \,A$ in $\frac{\pi^2}{100} \,s$, the induced emf in the loop is (in $\,mV$)

$A$ metal wire of length $2500 \ m$ is kept in east-west direction,at a certain height from the ground. If it falls freely on the ground,then the current induced in the wire when its speed is $10 \ m/s$ is (Resistance of wire $= 25 \ \Omega$,$g = 10 \ m/s^2$ and Earth's horizontal component of magnetic field $B_{H} = 2 \times 10^{-5} \ T$). (in $A$)

$A$ square frame of metallic wire is moving in a uniform magnetic field $(\vec{B})$ acting perpendicular to the paper inward as shown. $LP$ and $QN$ are also metallic wires. Find the potential difference between $L$ and $N$.

$A$ 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 $R$.

$A$ conducting rod of length $L$ rotates with angular speed $\omega$ in a uniform magnetic field of induction $B$ which is perpendicular to its motion. The induced emf developed between the two ends of the rod is

An aeroplane having a wing span of $30 \ m$ flies due north with a speed of $170 \ m/s$. If the vertical component of the Earth's magnetic field is $B = 3.6 \times 10^{-5} \ T$, the potential difference between the tips of the wings will be: (in $mV$)