Using Faraday's law of electromagnetic induction, derive an equation for motional emf.

Explanation

Let us consider a rectangular conducting loop of width l in a uniform magnetic field `vecB` which is perpendicular to the plane of the loop and is directed inwards. A part of the loop is in the magnetic field while the remaining parts is outside the loop.
When the loop is pulled with a constant velocity `vecv` to the right the area of the portion of the loop within the magnetic field will decrease. Thus, the flux linked with the loop will also decrease. According to Faraday.s law, an electric current is induced in the loop which flow in a direction so as to oppose the pull of the loop.

Let x be the lenght of the loop which is still within the magnetic field, then its area is lx. The magnetic flux linked with the loop is
`phi_(B)=intvecB.dvecA=BAcostheta`
Here `theta=0^(0)`and `cos0^(0)=1`
`=BA`
`phi_(B)=Blx" "...(1)`
As this msgnetic flux decrease due to the movement of the loop, the magnitude of the induced emf is given by
`e=(dphi_(B))/dt=d/dt(Blx)` Here, both B and l are constants. Therefore,
`e=Bl(dx/dt)`
`e=Blv" "...(2)`
where `v=dx/dt` is the velocity of the loop. This emf is known as motional emf since it is produced due to the loop in the magnetic field.