Electromagnetic induction is one of the most productive discoveries in the history of physics — it underlies virtually all electrical power generation and transmission on Earth. At its heart is Faraday's second law: the induced electromotive force (EMF) in a circuit equals the negative rate of change of magnetic flux through it, expressed as EMF = −dΦ/dt. The unit of magnetic flux is the Weber (Wb), and the dimensional formula of flux is [M $L^{2}$ $T^{-2}$ $A^{-1}$]. The negative sign is not merely a mathematical convention — it encodes Lenz's law, which states that the induced current always opposes the change in magnetic flux that produced it. This is a direct consequence of the law of conservation of energy: if the induced current aided the flux change, energy would be created from nothing.

Motional EMF arises when a conducting rod of length l moves with velocity v perpendicular to a uniform magnetic field B. The Lorentz force F = qv × B on free charges in the rod drives positive charges to one end, creating a potential difference ε = Bvl. To determine which end is at higher potential, one applies the right-hand rule to F = qv × B. If the rod forms part of a closed circuit of resistance R, a current I = ε/R flows, and a retarding force F = BIl opposes the motion (Lenz's law). The external agent must do work at rate P = Fv = ε·I = $I^{2}$R to maintain constant velocity — all mechanical work is converted to electrical energy.

For a coil rotating at angular frequency ω in a magnetic field, the induced EMF varies sinusoidally: ε = NBAω sin(ωt) = ε_{0} sin(ωt), where ε_{0} = NBAω is the peak EMF. This is the principle of the AC generator. EMF is maximum when the coil plane is parallel to B (flux is zero but changing fastest), and zero when the coil plane is perpendicular to B (flux is maximum, rate of change is zero).

Self-inductance L of a coil is its tendency to oppose changes in its own current: ε = −L(dI/dt). For a solenoid of N turns, length l, and area A: L = μ_{0}$N^{2}$A/l = μ_{0}$n^{2}$Al. The SI unit is the henry (H), with dimensional formula [M $L^{2}$ $T^{-2}$ $A^{-2}$]. Energy stored in the magnetic field of an inductor carrying current I is U = ½L$I^{2}$, analogous to U = ½$CV^{2}$ for a capacitor. Mutual inductance M describes coupling between two coils; for coaxial solenoids: M = μ_{0}n_{1}n_{2}Al.

Eddy currents are induced in bulk conducting materials by changing magnetic flux. They cause $I^{2}$R heating (exploited in induction furnaces) but are wasteful in transformer cores (reduced by laminating the core into thin, insulated sheets).

In AC circuits, a sinusoidal voltage v = $V_{0}$ sin(ωt) has peak value $V_{0}$, RMS value V_rms = $V_{0}$/√2 ≈ 0.707$V_{0}$, and half-cycle mean V_mean = 2$V_{0}$/π ≈ 0.637$V_{0}$. The RMS value is used for power calculations because P = V_r$ms^{2}$/R matches the DC power formula; it is the "DC equivalent" of AC.

For a pure resistor, V and I are in phase (φ = 0) and average power P = V_rms I_rms. For a pure inductor, the inductive reactance X_L = ωL opposes AC; current lags voltage by 90° — the mnemonic ELI (Voltage E leads current I in an inductance L). Average power is zero (wattless). For a pure capacitor, capacitive reactance X_C = 1/(ωC); current leads voltage by 90° — the mnemonic ICE (Current I leads voltage E in a capacitance C). Average power is again zero.

In a series LCR circuit, the impedance is Z = √($R^{2}$ + (X_L − X_C)^{2}), with phase angle tan φ = (X_L − X_C)/R. When X_L > X_C, the circuit is inductive (I lags V); when X_C > X_L, capacitive (I leads V). Resonance occurs when X_L = X_C, i.e., at frequency f_{0} = 1/(2π√LC). At resonance: impedance Z = R (its minimum value — never zero), current is maximum I = V_rms/R, phase angle φ = 0, and the circuit behaves purely resistively. Individual voltages V_L and V_C at resonance can each exceed the supply voltage by the Q-factor (voltage magnification).

Power in AC circuits: average power P = V_rms I_rms cos φ, where cos φ = R/Z is the power factor. Maximum power factor = 1 at resonance or in purely resistive circuits. The wattless (reactive) current component I_rms sin φ flows but dissipates no power — energy oscillates between the source and the reactive elements.

The transformer exploits mutual inductance to change AC voltage levels: V_s/V_p = N_s/N_p. Crucially, the current ratio is the inverse of the turns ratio: I_s/I_p = N_p/N_s. This follows from power conservation (ideal transformer): V_p I_p = V_s I_s. A step-up transformer (N_s > N_p) increases voltage and decreases current — essential for long-distance power transmission because power loss P_loss = $I^{2}$R decreases dramatically at high voltage (low current). A step-down transformer at the receiving end restores usable voltage levels. Real transformers lose energy through copper loss ($I^{2}$R in windings), eddy current loss in the iron core (reduced by lamination), flux leakage (not all flux links both coils), and hysteresis loss (reduced by using soft iron or silicon steel cores).

The most important exam-ready facts: at resonance Z = R (not zero); transformer current ratio is INVERSE of turns ratio; V in a series LCR circuit is the phasor sum √(V_$R^{2}$ + (V_L − V_C)^{2}), not the arithmetic sum; lamination reduces eddy currents, soft iron reduces hysteresis; ELI the ICE man governs all phase relationships.