$A$ conductor wire $ABCDE$ with each arm $10 \ cm$ in length is placed in a magnetic field of $\frac{1}{\sqrt{2}} \ T$,perpendicular to its plane. When the conductor is pulled towards the right with a constant velocity of $10 \ cm/s$,the induced emf between points $A$ and $E$ is . . . . . . $mV$.

Refer to the figure. The arm $PQ$ of the rectangular conductor is moved from $x=0$ outwards. The uniform magnetic field is perpendicular to the plane and extends from $x=0$ to $x=b$ and is zero for $x>b$. Only the arm $PQ$ possesses substantial resistance $r$. Consider the situation when the arm $PQ$ is pulled outwards from $x=0$ to $x=2b$ and is then moved back to $x=0$ with constant speed $v$. Obtain expressions for the flux,the induced emf,the force necessary to pull the arm,and the power dissipated as Joule heat. Sketch the variation of these quantities with distance.

$A$ wheel with $20$ metallic spokes,each $1 \,m$ long,is rotated with a speed of $120 \,rpm$ in a plane perpendicular to a magnetic field of $0.4 \,G$. The induced emf between the axle and the rim of the wheel will be $\left(1 \;G = 10^{-4} \;T \right)$.

Derive the equation for the induced $emf$ in a rod of length $l$ sliding with velocity $v$ on a $U$-shaped frame placed perpendicular to a uniform magnetic field $B$.

What is motional $emf$?

$A$ train with an axle of length $1.66 \ m$ is moving towards north with a speed of $90 \ km/h$. If the vertical component of the earth's magnetic field is $0.2 \times 10^{-4} \ T$,the emf induced across the ends of the axle of the train is: (in $mV$)