Derive equation of induced emf in the rod which is sliding on two sides of U shaped frame placed perpendicular to magnetic field.

Explanation

Due to some kind of motion may flux linked with coil changes and emf is induced in coil then such emf is called motional emf.
Let us consider a straight conductor moving in a uniform and time-independent magnetic field. Figure shows a rectangular conductor PQRS in which the conductor PQ is free to move. The rod PQ is moved towards the left with a constant velocity v as shown in the figure. Assume that there is no loss of energy due to friction.

PQRS forms a closed circuit enclosing an area that changes as PQ moves. It is placed in a uniform magnetic field B which is perpendicular to the plane of this system. If the length RQ = x and RS = l, the magnetic flux `Phi_B` enclosed by the loop PQRS will be
`Phi_B=Blx` [`because Phi_B=ABcostheta` and `A=lx , theta=0^@`]
Since x is changing with time, the rate of change of flux `Phi_B` will induce an emf given by,
`epsilon=(-dPhi_B)/(dt)=-d/(dt)(Blx)`
`=-Bl(dx)/(dt)`
`epsilon=Blv " " (because (dx)/(dt)=-v)`
`(dx)/(dt)=-V` Here with increase in time x decrease hence V is negative.
Where we have used dx/dt = - v which is the speed of the conductor PQ. The induced emf Blv is called motional emf.