Motional EMF and Generators

Push a metal bar along two parallel rails sitting in a magnetic field, and the free charges inside the bar move along with it. A moving charge in a magnetic field feels a force, F = qv x B, so the electrons in the bar get shoved toward one end. Charge piles up until the electric field it creates balances the magnetic push, and that separation of charge is a voltage: the motional emf, emf = BLv, where L is the length of bar crossing the field. Connect the rails into a loop and that emf drives current around it, no battery needed.

A generator is this same trick done in a circle instead of a straight line. Spin a coil of wire inside a steady magnetic field, and each side of the coil is a little moving bar sweeping through the field. As the coil turns, the angle between its velocity and the field keeps changing, so the emf rises, falls, crosses zero, reverses sign, and repeats twice every revolution. Plotted against angle or time, the output is a sine wave, which is exactly why wall power is alternating current rather than a steady voltage.

Flip the picture and you get a motor. A motor takes current flowing through a coil sitting in a field and gets a torque out of it, turning electrical energy into motion. A generator takes motion of a coil in a field and gets current out of it, turning motion into electrical energy. Same wires, same magnets, same physics, just read in opposite directions, which is why a spinning motor also generates a back emf that fights the current driving it.