(i). The mobile free electrons in arm RS are driven from S to R, according to Fleming’s left hand rule. 

Each charge ‘q’ within the arm RS moves with velocity \(\vec{V}\) and experiences a magnetic force, 

F = |q| vB.

Work done in moving the charge +q from R to S, 

W = F × RS

= (q v B)× l

∴ emf induced = work done per unit charge

i.e.,

e = \(\frac{W}{q}\)

= \(\frac{qvBl}{q}\)

= Blv.

(ii). If R is the resistance of the loop PQRS at a given instant, the induced current is :

I = \(\frac{e}{R}\)

= \(\frac{Blv}{R}\)

∴The magnitude of force on the conductor RS moving in the magnetic field is :

F = BIl 

= B(\(\frac{Blv}{R}\)).l 

= \(\frac{B^2l^2v}{R}\)

The direction of this force is opposite to the velocity of the conductor.

∴ |F_ext|= |F| 

(According to Newton’s 3rd law)

F_ext = \(\frac{B^2l^2v}{R}\) 

(iii) Power required to push the conductor with velocity 'v':

P =F_ext × v 

= \(\frac{B^2l^2v}{R}\) \(\times\) v

= \(\frac{B^2l^2v^2}{R}\)

As the conductor is pushed mechanically,

Power dissipated as heat,

P = I²R

= \((\frac{Blv}{R})^2.R\) 

= \(\frac{B^2l^2v^2}{R}\)