Electromagnetic Induction and Alternating Currents
Electromagnetic Induction and Alternating Currents links changing flux to emf, then develops AC circuit analysis: rms and peak values, phasors, series LCR resonance, power factor and transformers.
Core rules
- Faraday's law: ε = −dΦ/dt; for N turns ε = −N dΦ/dt. The minus sign (Lenz's law) guarantees energy conservation.
- Motional emf ε = Blv; a rod rotating about one end gives ½Bωl².
- Self-inductance ε = −L dI/dt with energy U = ½LI²; mutual inductance ε₂ = −M dI₁/dt.
- AC values: I(rms) = I₀/√2, V(rms) = V₀/√2; reactances X_L = ωL and X_C = 1/ωC.
- Series LCR: Z = √(R² + (X_L − X_C)²), tanφ = (X_L − X_C)/R. Resonance at ω₀ = 1/√(LC), where Z = R and current peaks.
- Average power P = V(rms)I(rms)cosφ; cosφ is the power factor. For an ideal transformer V(s)/V(p) = N(s)/N(p).
Worked example
A series LCR circuit has R = 30 Ω, X_L = 80 Ω and X_C = 40 Ω across 200 V rms. Then Z = √(30² + 40²) = 50 Ω, so I = 200/50 = 4 A. The power factor is cosφ = R/Z = 0.6, and the average power is 200 × 4 × 0.6 = 480 W.
Traps
Dropping the Lenz minus sign when direction is asked; adding reactances arithmetically instead of as phasors; using peak values in power formulae; thinking a transformer changes frequency or delivers more power than it receives.