$A$ wire $cd$ of length $l$ and mass $m$ is sliding without friction on conducting rails $ax$ and $by$ as shown. The vertical rails are connected to each other with a resistance $R$ between $a$ and $b$. $A$ uniform magnetic field $B$ is applied perpendicular to the plane $abcd$ such that $cd$ moves with a constant velocity $v$. Find the value of $v$.

$A$ $10\,m$ long horizontal wire extends from North-East to South-West. It is falling with a speed of $5.0\,m/s$,at right angles to the horizontal component of the earth's magnetic field,which is $0.3 \times 10^{-4}\,Wb/m^2$. The value of the induced $emf$ in the wire is:

$A$ metal rod of length $l$ rotates about one of its ends in a plane perpendicular to a magnetic field of induction $B$. If the e.m.f. induced between the ends of the rod is $e$,then the number of revolutions made by the rod per second is:

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?

$A$ long horizontal metallic rod with length along the east-west direction is falling under gravity. The potential difference between its two ends will

$A$ frame $CDEF$ is placed in a region where a magnetic field $\vec{B}$ is present. $A$ rod $PQ$ of length $l = 1 \, m$ moves with a constant velocity $v = 20 \, m/s$ and the strength of the magnetic field is $B = 1 \, T$. The power spent in the process is .............. $kW$ (take $R = 0.2 \, \Omega$ and assume all other wires and the rod have zero resistance).