Motional EMF
A moving conductor becomes a battery.
Lesson at a glanceShow the key points
- ℰ = vBl (38.3) from qv⃗ × B⃗ on free electrons.
- The moving rod acts as a battery: i = vBl/(R + r).
- Retarding force ilB; mechanical power Fv = i²R.
Where does an induced emf actually come from? When a conductor moves through a field, the answer is simply the magnetic force on its moving electrons.
Two ways to change the flux
Flux can change because (a) the field is steady but the loop (or part of it) moves, (b) the loop is still but the field changes, or both. The mechanism is different in the two cases. This lesson covers (a), motional emf.
A rod moving through a field
A rod PQ of length l moves at velocity v⃗ through a field B⃗, with rod, velocity and field all mutually perpendicular. Its free electrons share the velocity v⃗, so each feels qv⃗ × B⃗ along the rod. Electrons pile up at one end (P) and leave positive charge at the other (Q). The resulting electric field E inside the rod pushes back. Charge stops building up when the two balance: qE = qvB, i.e. E = vB.
Join P and Q by a resistor and current flows. The magnetic force keeps driving electrons back up the rod, maintaining the potential difference. With external resistance R and rod resistance r: i = vBl/(R + r).
Faraday’s law agrees: if x is the length of the circuit in the field, Φ = Blx and |dΦ/dt| = Bl dx/dt = vBl.
Forces and energy
Once current i flows in the rod, the field pushes on it with ilB, against its motion (Lenz again). To keep it moving at constant v something must push with force F = ilB = B²l²v/R. That force delivers power Fv = B²l²v²/R, and the circuit produces heat i²R = (vBl/R)²R = B²l²v²/R. Exactly equal: your mechanical work becomes the heat in the circuit.
A 25 cm rod slides at 4.0 m/s across rails in a 0.60 T field; the total resistance of the circuit is 2.0 Ω. Find the emf, current, the force needed to keep it moving and the power supplied.
A rectangular loop of total resistance R, with its side MN (length l) still inside a field B and the rest outside, is pulled out at constant speed v. Find the current, the magnetic force, the pulling force and the powers.
- 1Motional emf comes from the magnetic force on the conductor's moving free electrons: ℰ = vBl (38.3).
- 2The rod acts as a battery; i = vBl/(R + r).
- 3Induced current makes a retarding force ilB; mechanical power Fv equals the heat i²R.
Check yourself
Keep exploring
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