Motional EMF
ε = BLv. Rod sliding on rails through B field generates current and feels back-force.
Key Notes
When a conductor of length L moves with velocity v through a magnetic field B, the free charges experience a magnetic force F = qv × B that pushes them along the rod.
If v, B, and the rod are mutually perpendicular, charges accumulate at the ends until the resulting E-field cancels the magnetic force.
The end-to-end voltage difference (motional EMF) is ε = BLv.
If the rod is part of a circuit with resistance R: induced current I = ε/R = BLv/R.
The current carrier in the rod (length L, current I) experiences a force F_mag = BIL in the direction OPPOSING its motion — Lenz again.
Power dissipated in the resistor = εI = (BLv)²/R = F_mech · v. All the mechanical work goes to electrical energy.
This is the basic principle behind rail-guns, dynamos, and a generator's armature.
Formulas
Motional EMF (rod ⊥ B ⊥ v)
Standard formula — rod moves perpendicular to a uniform B.
Induced current
R = total loop resistance (rod + rails + external).
Opposing magnetic force on rod
Always opposes v (Lenz).
Power balance
Mechanical power input = electrical power dissipated.
General (v not ⊥ B)
Vector form for arbitrary geometries.
Important Points
BLv is just Φ̇ in disguise: as the rod slides, the loop's enclosed area grows at rate L·v, so dΦ/dt = B·L·v.
If you LET GO of the rod, the magnetic braking force decelerates it. It eventually stops (if friction-free) when KE is fully dissipated as I²R heat.
To keep the rod at constant v, you must apply a force F_ext = BIL = B²L²v/R. The mechanical power F_ext·v = ε²/R appears as resistive heating.
The polarity of the EMF is given by the right-hand rule: fingers along v, curl toward B, thumb points to the + end of the rod.
Even WITHOUT a circuit, motional EMF exists as a voltage between the rod's ends — but no current flows.
Motional EMF is the simplest realisation of Faraday's law — and it's the basic source of EMF in every rotating generator.
Motional EMF notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.