Answer:

All points on the rod are moving perpendicular to the magnetic field. Hence, all elementary small elements of the rod induce a small potential difference and the net potential difference in the rod is the integration of the potential differences along the rod.

Motional emf in a conductor moving perpendicular to the field is given by:

ε=Bvl

The potential difference across a small element of rod at a distance l from the center, dε=Bv(dl)

But v=ωl⟹dε=Blωdl

Hence total emf produced across the rod,

ε=∫

0

L

B(lω)dl

=

2

1

BωL

2