Question

Derive the expression for motional EMF induced in a conductor moving in a uniform magnetic field.

Answer 1

Let us consider a straight conductor moving in a uniform and time-independent magnetic field.
The figure shows a rectangular wire PQRS in which the conductor PQ is free to move.
Rod PQ is moved towards the left with a constant velocity v.
PQRS forms closed circuit enclosing an area that changes as PQ moves.
It is placed in a uniform magnetic field B perpendicular to the plane of our system.
Let the length RQ =x and RS =l,
the magnetic flux ${\varphi }_B$ enclosed by the loop PQRS
${\varphi }_B=Blx$
As x is changing with time, the rate of change of flux ${\varphi }_B$ will induce an emf given according to Faraday's law of electromagnetic induction :
$ϵ=-\frac{d{\varphi }_B}{dt}=-\frac{d}{dt}\left(Blx\right)=-Blv$
This induced emf Blv is motional emf.