$A$ simple pendulum made of a mass of $10 \ g$ and a metallic wire of length $10 \ cm$ is suspended vertically in a uniform magnetic field of $2 \ T$. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^{\circ}$ with the vertical, then the maximum induced $EMF$ between the point of suspension and the point of oscillation is . . . . . . $mV$. (Take $g = 10 \ m/s^2$)

The magnetic field in a region is given by $\vec B = B_0(1 + \frac{x}{a})\hat k$. $A$ square loop of edge length $d$ is placed with its edges along the $x$ and $y$ axes. The loop is moved with a constant velocity $\vec V = V_0\hat i$. The $emf$ induced in the loop is:

$A$ long straight wire carries a current,$I = 2 \text{ A}$. $A$ semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire,the rod,and the rails lie in the same horizontal plane,as shown in the figure. Two ends of the semi-circular rod are at distances $1 \text{ cm}$ and $4 \text{ cm}$ from the wire. At time $t = 0$,the rod starts moving on the rails with a speed $v = 3.0 \text{ m/s}$. $A$ resistor $R = 1.4 \text{ } \Omega$ and a capacitor $C_0 = 5.0 \text{ } \mu\text{F}$ are connected in series between the rails. At time $t = 0$,$C_0$ is uncharged. Which of the following statement$(s)$ is(are) correct? $\left[\mu_0 = 4\pi \times 10^{-7} \text{ SI units}, \ln 2 = 0.7\right]$
$(A)$ Maximum current through $R$ is $1.2 \times 10^{-6} \text{ A}$
$(B)$ Maximum current through $R$ is $3.8 \times 10^{-6} \text{ A}$
$(C)$ Maximum charge on capacitor $C_0$ is $8.4 \times 10^{-12} \text{ C}$
$(D)$ Maximum charge on capacitor $C_0$ is $2.4 \times 10^{-12} \text{ C}$

$A$ rectangular wire loop of sides $8 \text{ cm}$ and $3 \text{ cm}$ with a small cut is moving out of a region of uniform magnetic field of magnitude $0.3 \text{ T}$ directed normal to the plane of the loop. The emf developed across the cut, if the velocity of the loop is $2 \text{ cm s}^{-1}$ in a direction normal to the shorter side of the loop, will be:

$A$ metal rod of length $L$ rotates about one end at origin with a uniform angular velocity $\omega$. The magnetic field radially falls off as $B(r) = B_0 e^{-\lambda r}$; $\lambda$ being a positive constant. The emf induced (neglecting the centripetal force on electrons in the rod) is :

$A$ long rectangular conducting loop of width '$l$',mass '$m$',and resistance '$R$' is placed partly in a perpendicular magnetic field '$B$'. With what velocity should it be pushed downwards so that it may continue to fall without any acceleration? ($g = $ acceleration due to gravity)