Motional EMF (ε = BLv)

Drag a conductor sideways through a magnetic field and every free electron inside it is now a moving charge in a field, so every one of them feels the $qvB$ force. That force points along the wire, so the electrons pile up at one end and leave a deficit at the other, and they keep piling up until the electric field they have built pushes back exactly as hard as the magnetic force pushes. At that balance the wire has a voltage across it: $\varepsilon = BLv$. You have made a battery out of motion. The same result falls out of Faraday's law without mentioning a single electron — a wire of length $L$ moving at speed $v$ sweeps out area at $Lv$ per second, so it sweeps flux at $BLv$ per second — and the fact that the two arguments agree is not a coincidence but one of the observations that led Einstein to special relativity.

An airliner with a 60 m wingspan crossing the vertical component of Earth's field, about 50 µT at mid-latitudes, at 250 m/s develops $50 \times 10^{-6} \times 60 \times 250 = 0.75\ \text{V}$ from wingtip to wingtip. A more workmanlike case: a 0.3 m rod sliding along rails at 4 m/s through a 0.6 T field gives $0.6 \times 0.3 \times 4 = 0.72\ \text{V}$. Close the rails through a 2 Ω load and 0.36 A flows, delivering 0.26 W — and that power has to come from somewhere.

Where it comes from is your arm. The moment current flows in the moving rod, the rod is a current-carrying conductor in a magnetic field, so it feels $F = BIL$ — and Lenz's law guarantees that force opposes the motion. Push the rod at constant speed and the mechanical power you supply, $Fv$, equals the electrical power delivered, to the joule. That is the entire energy accounting of every generator ever built: a turbine does not spin harder when the grid load rises, it spins against more force, and the fuel bill reflects it. This page and the magnetic-force-on-a-wire page describe one machine running in its two directions.

Some cautions, starting with the name. Electromotive force is not a force. It is measured in volts and it is energy per unit charge, and the nineteenth-century name has confused students for a hundred and fifty years — read "EMF" as "the voltage a source generates internally" and it will not mislead you. Nor is that internal voltage the same as the voltage you would measure at the terminals: once current flows, the source's own resistance drops part of it, which is the same reason a car battery reads 12.6 V at rest and 10 V while cranking. Next, $B$ here must be the field component perpendicular to the plane the wire sweeps through; using the total field of a tilted magnet overstates the answer. And the aircraft case is a good lesson in what an EMF is worth without a circuit: the 0.75 V exists, but the whole aeroplane moves together, so there is no closed loop through stationary conductors and nothing useful can be drawn from it.