Let us consider a small element at a distance from the centre of the rod rotating with angular velocity about its axis perpendicular bisector.

Formula used: The emf induced is given by

The emf is induced in the rod because of the small element is

Where is the emf induced in that small element

B is the magnetic field

is the angular velocity of the small element

is the length of the small element

The emf induced across the centre and the end of the rod is found by the integration of the above relation with respect to x from limit o to

The potential difference between the centre of the rod and an end is .

Thus, option B is correct.