$A$ uniform magnetic field of $0.4 \ \text{T}$ acts perpendicular to a circular copper disc $20 \ \text{cm}$ in radius. The disc is rotating with a uniform angular velocity of $10 \pi \ \text{rad s}^{-1}$ about an axis passing through its centre and perpendicular to the disc. What is the potential difference developed between the axis of the disc and the rim (in $\text{V}$)? $(\pi = 3.14)$

$A$ rectangular conducting loop of length $4 \ cm$ and width $2 \ cm$ is in the $xy$-plane,as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\frac{\sqrt{3}}{2} \hat{x} + \frac{1}{2} \hat{y}$ with a constant speed $v$. The wire is carrying a steady current $I = 10 \ A$ in the positive $x$-direction. $A$ current of $10 \ \mu A$ flows through the loop when it is at a distance $d = 4 \ cm$ from the wire. If the resistance of the loop is $0.1 \ \Omega$,then the value of $v$ is. . . . . . $ms^{-1}$.
[Given: The permeability of free space $\mu_0 = 4 \pi \times 10^{-7} \ NA^{-2}$]

An aeroplane,with its wings spread $10 \, m$,is flying at a speed of $180 \, km/h$ in a horizontal direction. The total intensity of the Earth's magnetic field at that location is $2.5 \times 10^{-4} \, Wb/m^2$ and the angle of dip is $60^{\circ}$. The $EMF$ induced between the tips of the plane's wings will be ...... $mV$.

$A$ $1\,m$ long metal rod $XY$ completes the circuit as shown in the figure. The plane of the circuit is perpendicular to the magnetic field of flux density $0.15\,T$. If the resistance of the circuit is $5\,\Omega$,the force needed to move the rod in the direction indicated with a constant speed of $4\,m/s$ will be $................\,10^{-3}\,N$.

$A$ metal disc of radius $a = 10 \ cm$ rotates with a constant angular speed of $\omega = 200 \ rad \ s^{-1}$ about its axis. The potential difference between the centre and the rim of the disc under a uniform magnetic field $B = 5 \ mT$ directed perpendicular to the disc is: (in $mV$)

$A$ square-shaped conducting wire loop of dimension $a$ moving parallel to the $X$-axis approaches a square region of size $b$ $(a < b)$,where a uniform magnetic field $B$ exists pointing into the plane of the paper (see figure). As the loop passes through this region,the plot correctly depicting its speed $v$ as a function of $x$ is