Magnetic field and electromagnetic induction
Magnetic field and induction questions combine the force on moving charges and currents with Faraday's and Lenz's laws. The exam typically uses a rod sliding on rails, a rotating coil or a solenoid, and asks for the induced emf, the direction of the current and the energy balance.
Core rules
- Lorentz force: F = qv × B, magnitude qvB sinθ; it is always perpendicular to v, so it does no work on the charge.
- A charge moving perpendicular to a uniform B follows a circle of radius r = mv/(qB) and period T = 2πm/(qB), independent of speed in the non-relativistic case.
- Force on a straight wire: F = BIL sinθ; parallel currents attract, antiparallel currents repel.
- Long solenoid: B = µ₀nI with n the turns per metre; long straight wire: B = µ₀I/2πr.
- Flux: Φ = B·S = BS cosθ. Faraday's law: ε = −dΦ/dt. Lenz's law gives the direction: the induced current opposes the change of flux.
- Motional emf for a rod of length l moving at v perpendicular to B: ε = Blv.
Worked example
A 0.50 m rod slides at 4.0 m/s along horizontal rails in a vertical field B = 0.80 T, closing a circuit of resistance 0.40 Ω. Then ε = Blv = 0.80 × 0.50 × 4.0 = 1.6 V and I = ε/R = 4.0 A. The magnetic force on the rod is F = BIl = 0.80 × 4.0 × 0.50 = 1.6 N, opposing the motion, so an external agent supplies P = Fv = 6.4 W, exactly matching the Joule heating I²R = 16 × 0.40 = 6.4 W. This equality is the quickest check that the direction chosen by Lenz's law is right.
Marks lost here
Using the full B instead of its perpendicular component in the flux. Reversing the induced current because Lenz's law was applied as 'along the change' rather than 'against' it. Believing the magnetic force speeds a charge up, when it only curves the path. Forgetting that the induced emf is zero when the flux is constant, however large B is. Ignoring that the period of a cyclotron orbit does not depend on speed.