$A$ rod of length $l$ is rotated with angular velocity $\omega$ about one of its ends,in a region perpendicular to a uniform magnetic field of induction $B$. The induced e.m.f. in the rod is:

An aircraft of wing span $40 \ m$ flies horizontally in the Earth's magnetic field of $5 \times 10^{-5} \ T$ at a speed of $500 \ m/s$. The electromotive force (e.m.f.) generated between the tips of the wings of the aircraft is: (in $V$)

$A$ jet plane of wing span $20 \,m$ is travelling towards west at a speed of $400 \,ms^{-1}$. If the earth's total magnetic field is $4 \times 10^{-4} \,T$ and the dip angle is $30^{\circ}$ at that place, the voltage difference developed across the ends of the wing is (in $\,V$)

The figure shows a metal rod $PQ$ resting on the smooth rails $AB$ and positioned between the poles of a permanent magnet. The rails,the rod,and the magnetic field are in three mutually perpendicular directions. $A$ galvanometer $G$ connects the rails through a switch $K$. Length of the rod $= 15 \; cm$,$B = 0.50 \; T$,resistance of the closed loop containing the rod $= 9.0 \; m\Omega$. Assume the field to be uniform.
$(a)$ Suppose $K$ is open and the rod is moved with a speed of $12 \; cm \; s^{-1}$ in the direction shown. Give the polarity and magnitude of the induced $emf$.
$(b)$ Is there an excess charge built up at the ends of the rod when $K$ is open? What if $K$ is closed?
$(c)$ With $K$ open and the rod moving uniformly,there is no net force on the electrons in the rod $PQ$ even though they do experience magnetic force due to the motion of the rod. Explain.
$(d)$ What is the retarding force on the rod when $K$ is closed?
$(e)$ How much power is required (by an external agent) to keep the rod moving at the same speed $(= 12 \; cm \; s^{-1})$ when $K$ is closed? How much power is required when $K$ is open?
$(f)$ How much power is dissipated as heat in the closed circuit? What is the source of this power?
$(g)$ What is the induced $emf$ in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?

$A$ horizontal straight wire $10 \; m$ long extending from east to west is falling with a speed of $5.0 \; m \, s^{-1}$,at right angles to the horizontal component of the earth's magnetic field,$0.30 \times 10^{-4} \; Wb \, m^{-2}$.
$(a)$ What is the instantaneous value of the $emf$ induced in the wire?
$(b)$ What is the direction of the $emf$?
$(c)$ Which end of the wire is at the higher electrical potential?

$A$ $20\,cm$ long metallic rod is rotated with $210\,rpm$ 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 $0.2\,T$ parallel to the axis exists everywhere. The emf developed between the centre and the ring is $.......\,mV$. Take $\pi=\frac{22}{7}$.