Derive an expression for the motional emf in a rod sliding on rails in a uniform magnetic field, first from the change in flux and then from the Lorentz force. Explain why a force is needed to keep the rod moving at constant speed.

1. 1 · State the principle. A moving conductor changes the flux through the loop, so an emf is induced (Faraday's law).
2. 2 · Draw the set-up. [Rails with resistor R; rod PQ of length l; B into the page as crosses; velocity v and distance x marked]
3. 3 · Find the emf from the flux. Φ = Blx; ε = −dΦ/dt = −Bl dx/dt = Blv, since dx/dt = −v.
4. 4 · Find the emf from the Lorentz force. Each free charge feels qvB along the rod; work over length l is qvBl; emf = work per unit charge = Blv.
5. 5 · Explain the opposing force. The induced current I = Blv/R in the rod feels a magnetic force that opposes the motion (Lenz's law), so a steady push is needed; the work done becomes heat.
6. 6 · Real life. Aircraft wings cutting Earth's field carry a small motional emf.
7. 7 · Close in one line. So, both methods give ε = Blv, and the mechanical work done on the rod becomes electrical energy and then heat.

Scoring tip: Show dx/dt = −v; many students lose the sign. Write units in every numerical line.