$A$ circular wheel with $10$ spokes, with its plane vertical along East-West, is rotating about its natural axis with a uniform speed of $100$ revolutions per minute in the Earth's magnetic field. The radius of the wheel is $0.3 \, m$. If the $EMF$ induced between the centre of the wheel and the rim is $3 \pi \mu V$, what is the angle of dip at that place? (Vertical component of the Earth's magnetic field $B_{V} = 15 \mu T$)

$A$ region in the form of an equilateral triangle (in $x-y$ plane) of height $L$ has a uniform magnetic field $\vec{B}$ pointing in the $+z$-direction. $A$ conducting loop $PQR$,in the form of an equilateral triangle of the same height $L$,is placed in the $x-y$ plane with its vertex $P$ at $x=0$ in the orientation shown in the figure. At $t=0$,the loop starts entering the region of the magnetic field with a uniform velocity $\vec{v}$ along the $+x$-direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graphs best depicts the variation of the induced emf $(E)$ in the loop as a function of the distance $(x)$ starting from $x=0$?

$A$ conducting wire bent in the shape of a semicircle has length $L$ and moves in its plane with constant velocity $v$. $A$ uniform magnetic field $B$ exists in the direction perpendicular to the plane of the wire. The velocity makes an angle $45^{\circ}$ to the diameter joining the free ends,and the emf induced between the ends of the wire is $\Phi = \alpha(B v L)$. The value of the constant $\alpha$ is

$A$ 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 $mV$)

$A$ $1.0 \; m$ long metallic rod is rotated with an angular frequency of $400 \; rad \; s^{-1}$ about an axis normal to the rod passing through its one end. The other end of the rod is in contact with a circular metallic ring. $A$ constant and uniform magnetic field of $0.5 \; T$ parallel to the axis exists everywhere. Calculate the emf developed between the centre and the ring. (in $; V$)

$A$ metallic rod of length $l$ is tied to a string of length $2l$ and made to rotate with angular speed $\omega$ on a horizontal table with one end of the string fixed. If there is a vertical magnetic field $B$ in the region,the $e.m.f.$ induced across the ends of the rod is