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The text says that each charge carrier, q, in the rod has the...
velocity v and since q moves in a magnetic field it experiences a
lorentz force F_M= qv cross B. The text then states that the work
done by this force pushing the charges along the rod is F_M * L=
qvBL. Since emf is energy per charge, the motional emf between the
ends of the rod is E= vBL.
I prefer the explanation of calculating the changing flux, Phi_M=
BLx, through the circuit where x is the position of the rod measured
along the rails from the resistance. This gives the emf E=
-dPhi_M/dt= BL(dx/dt)= BLv.
After all this yacking, my reason for posting is to get a range of
opinions of this text's derivation.