A metallic square loop PQRS moving in its own plane with velocity v in a uniform magnetic field perpendicular to its plane .If v_p,v_q,v_r,v_s the potential of points P,Q,R and S then what is the relation between v_p,v_q,v_r and v_s?

Since the loop is neutral initially, the vertical portion of the wire say PS and QR will have induced EMF due to the accumulation of equal and opposite charges on the opposite ends of the wire (according to the lorentz force equation). Since the magnetic field is uniform, both the vertical parts will have equal EMF induction.
hence,

also,

also, and
therefore,

Thanks and Regards
Apoorva Arora
IIT Roorkee
askIITians Faculty