A conducting rod of length l is hinged at point O. It is free to rotate in a vertical plane. There exists a uniform magnetic field B in horizontal direction. The rod is released from the position. What is the potential difference between the two ends of the rod is proportional to?

 The gravitational potential decreases to increase the kinetic energy of the rod. The equation can be written as,

mg 1/2sinθ = 1/2Iω2 = 1/2(ml2/3)ω2

⟹ ω=√3g/lsinθ

Hence the magnetic field and the direction of velocity of rods is perpendicular. 

Total potential difference developed across the ends =

B(xω)dx 

=Bl2/2√ 3g/lsinθ ∝ l3/2√sinθ.