Motional EMF

Motional EMF is the induction mechanism that operates when a conductor physically moves through a static magnetic field. Every free charge q in the conductor feels the Lorentz force F = qv×B, where v is the velocity of the conductor relative to the field source. In a straight rod moving perpendicular to both B and its own length ℓ, this force drives positive charges toward one end of the rod and negative toward the other, setting up an electrostatic field inside the rod that, in steady state, exactly balances the magnetic force on each free charge. The potential difference between the ends is ε = (v × B) · ℓ — and if the rod is part of a closed circuit, this potential difference is an EMF that drives a current around the loop.

The canonical example is the "rails-and-rod" circuit: a U-shaped conducting rail sits in a uniform B-field, and a straight conducting rod slides along the rails, closing the loop. As the rod moves with velocity v, the flux through the loop changes at rate dΦ/dt = B · ℓ · v (the rate at which new area is swept out), and the induced EMF is Bℓv. Plug it into the loop resistance R and the current is I = Bℓv/R. The force on the rod due to this current in the magnetic field is F_mag = IℓB, pointing opposite the velocity — exactly what you'd expect from Lenz's law, and exactly the retarding force you need to apply to keep the rod moving at constant v. The mechanical power you supply, F_mag · v = B²ℓ²v²/R, equals the electrical power dissipated in the circuit, I²R. Energy conservation closes the book.

Motional EMF is the operating principle of every electric generator ever built. Rotate a multi-turn coil in a magnetic field (or, equivalently, rotate a magnet past a stationary coil), and the v × B term varies sinusoidally in time, producing the AC voltage that reaches your wall socket. It is also why a loop of wire walked through the Earth's magnetic field picks up a tiny EMF (microvolt scale, but real and measurable). Faraday's law in its rate-of-flux form includes motional EMF automatically when the boundary of the loop is moving; the two cases — stationary circuit with changing B, moving circuit with constant B — are unified by the covariant formulation in special relativity, where the distinction between v × B and ∂A/∂t is frame-dependent.