Motional emf

A conductor moving through a magnetic field also has an emf across it, even if the field is steady. This is motional emf.

For a straight rod of length l, moving at speed v at right angles to B (and to itself):

ε = Blv

Why: the rod's free electrons move with it, so each feels a magnetic (Lorentz) force q(v × B) along the rod. Electrons pile up at one end, leaving the other end positive. That separation is the emf.

Example: a rod 0.5 m long moving at 4 m/s through 0.2 T: ε = 0.2 × 0.5 × 4 = 0.4 V.

The same result comes from Faraday's law: the rod sweeps out area lv every second, so the flux through the circuit changes at the rate Blv.

A rod rotating about one end, with angular speed ω, in a field along the axis of rotation: points further out move faster (speed ωr), so we take the average. The emf is:

ε = ½ B ω R²

Example: R = 0.5 m, ω = 100 rad/s, B = 0.2 T: ε = ½ × 0.2 × 100 × 0.25 = 2.5 V.

A wheel with many spokes gives the same emf as one spoke: the spokes are in parallel between the axle and the rim.

If the rod is part of a closed circuit, a current flows, and the field pushes back on that current (Lenz). So a force is needed to keep the rod moving, and the work done becomes the electrical energy.