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.