Motional EMF

When a conductor of length L moves with velocity v through a magnetic field B, the free charges experience a magnetic force F = qv × B that pushes them along the rod.

If v, B, and the rod are mutually perpendicular, charges accumulate at the ends until the resulting E-field cancels the magnetic force.

The end-to-end voltage difference (motional EMF) is ε = BLv.

If the rod is part of a circuit with resistance R: induced current I = ε/R = BLv/R.

The current carrier in the rod (length L, current I) experiences a force F_mag = BIL in the direction OPPOSING its motion — Lenz again.

Power dissipated in the resistor = εI = (BLv)²/R = F_mech · v. All the mechanical work goes to electrical energy.

This is the basic principle behind rail-guns, dynamos, and a generator's armature.