Electromagnetic Induction and Alternating Current
This chapter joins Faraday's law and Lenz's law with AC circuit theory: reactance, impedance, series LCR resonance, power factor, transformers and the AC generator. Expect one derivation of emf in a rotating coil and one LCR numerical.
Core rules
- Faraday: e = −dΦ/dt; for N turns, e = −N dΦ/dt. Flux Φ = BA cosθ.
- Lenz's law: the induced current opposes the change producing it; this is a statement of energy conservation.
- Motional emf e = Bvl; rotating coil emf e = NBAω sin(ωt), peak e₀ = NBAω.
- Self-induced emf e = −L dI/dt; energy in an inductor U = ½LI². Mutual induction: e₂ = −M dI₁/dt.
- AC: X(L) = ωL, X(C) = 1/ωC, Z = √(R² + (X(L) − X(C))²), tanφ = (X(L) − X(C))/R.
- Series resonance: X(L) = X(C), Z = R minimum, current maximum, f = 1/(2π√(LC)).
- Power P = V(rms)I(rms)cosφ; the power factor is 1 at resonance and zero for a pure inductor or capacitor.
- Transformer: V(s)/V(p) = N(s)/N(p) = I(p)/I(s); ideal efficiency assumed. Step-up raises voltage and lowers current.
Worked example
A series LCR circuit has L = 1 H and C = 1 μF. Resonance frequency f = 1/(2π√(10⁻⁶)) = 1/(2π × 10⁻³) = 159 Hz. At this frequency the impedance equals R alone, so the current peaks and the power factor is 1. For a transformer with 2000 primary and 100 secondary turns on 220 V, the secondary gives 11 V.
Traps
- Dropping the minus sign of Lenz's law and then predicting the wrong current direction.
- Quoting peak values where rms are meant; V(rms) = V₀/√2.
- Adding reactances arithmetically rather than vectorially to get impedance.