Motional EMF from a Sliding Bar

Picture two parallel rails connected by a bulb, with a metal bar free to slide across them, all sitting in a magnetic field pointing into the page. Slide the bar and the bulb lights. The bar's free electrons move with it, so each one feels a magnetic force F = qv x B. That force pushes electrons toward one end of the bar, leaving the other end positive. Charge piles up until the electric field inside the bar built by that separation balances the magnetic push on each electron. The result is a steady separation of charge across the bar, which is exactly what an emf is: a source that can push current through a circuit. Its size is emf = B L v, with L the length of bar between the rails.

Close the circuit with a resistor R and current flows, I = BLv/R. That current, now moving through the magnetic field, feels its own force F = BIL, directed to oppose the bar's motion. This is why sliding the bar takes effort: you are fighting a magnetic brake, and the work you do becomes the electrical energy delivered to the resistor. No part of the field is changing in time here. The flux through the circuit grows only because the loop's area grows as the bar slides, so this is motional emf, a different bookkeeping of the same Faraday's law idea you meet with changing fields.

This case sets up the general flux-rule picture that follows: whether flux changes because area changes or because B itself changes, emf = -dPhi/dt. The sliding bar is the cleanest version, where you can trace the force on individual charges by hand instead of just computing a flux derivative.