A conducting Rod Moving through a Uniform Magnetic Field

Consider a conducting rod ab of length l moving with velocity v parallel to the direction to Y-axis in a uniform field of magnetic induction B. The length of the rod is along z-axis. Since the rod is a conductor, it contains charged particles. Then, a particle of charge q will experience a force,

            F = q (v X B)                                … (1)

The force F pushes the negatively charged electrons towards one end of the rod. Consequently, the other end of the rod becomes positively charged. This goes on until these separated charges themselves create and electric field inside the rod. In the state when the conductor moves with a constant velocity v, the force F is equal to the force due to electric field E created inside the rod in the opposite direction. Then, the resultant force on every charge carrier is zero.

i.e.,                 qE + q (v x B) = 0
                qE = - q (v x B)
                 E = - v x B                            … (2)

The e.m.f. induced between the two ends of the rod is   
    e = ∫ E.dl =| - ∫ (v x B) .dl | = v/B                                … (3)

This e.m.f. is called a ‘motional’ e.m.f.