Electromagnetic Induction

Faraday's law and Lenz's law

Magnetic flux Φ = B·A = BA cos θ. Faraday's law: induced EMF ε = −dΦ/dt. The minus sign is Lenz's law — the induced current opposes the change that produced it (energy conservation). A coil of N turns gives ε = −N dΦ/dt. Pulling a magnet out of a coil induces a current that tries to keep the flux constant by attracting the magnet back.

Motional EMF

A rod of length L moving at speed v perpendicular to B generates ε = BLv. A 0.5 m rod moving at 4 m/s in a 0.2 T field induces 0.4 V. This is the principle of every generator: rotating a coil at angular speed ω in field B produces ε(t) = NBA·ω·sin(ωt), giving sinusoidal AC with peak ε₀ = NBAω.

Self and mutual inductance

Self-inductance L is defined by ε = −L·dI/dt. Solenoid: L = μ₀N²A/ℓ. A solenoid with 1000 turns, area 2×10⁻⁴ m², length 0.5 m has L ≈ 4π×10⁻⁷·10⁶·2×10⁻⁴/0.5 ≈ 5×10⁻⁴ H. Energy stored U = ½LI². Mutual inductance M between two coils gives ε₂ = −M·dI₁/dt; transformers use this principle.

AC fundamentals

AC voltage V(t) = V₀ sin(ωt). Peak value V₀; RMS value V_rms = V₀/√2 ≈ 0.707 V₀; average value over a cycle is zero. Pakistani mains at 220 V (RMS) have peak voltage 220√2 ≈ 311 V. RMS quantities are used because they give the same heating power as a DC of equal value.

Transformers and AC circuits

An ideal transformer satisfies V_p/V_s = N_p/N_s and I_pN_p = I_sN_s (power in = power out). Step-up has more secondary turns; step-down fewer. Inductive reactance X_L = ωL; capacitive reactance X_C = 1/(ωC). In series LCR circuits, impedance Z = √(R² + (X_L − X_C)²); resonance at ω₀ = 1/√(LC). The MDCAT loves transformer ratio MCQs and resonance-frequency calculations. References: HRW Chapters 30-32, Serway 31-33, FSc Chapter 15.