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Mutual inductance $M$ couples two circuits: $\varepsilon_2 = -MdI_1/dt$. Reciprocity: $M_{12} = M_{21}$. For coaxial solenoids: $M = \mu_0 n_1 n_2 A_{\text{inner}} l$ (only the inner coil's area matters). General relation: $M = k\sqrt{L_1 L_2}$ where $k$ is the coupling coefficient ($0 \leq k \leq 1$).

The transformer exploits mutual induction with tight coupling ($k \approx 1$) on a common iron core. Voltage ratio: $V_s/V_p = N_s/N_p$. Power conservation (ideal): $V_p I_p = V_s I_s$. Step-up ($N_s > N_p$): increases voltage, decreases current. Step-down: the reverse.

Four loss mechanisms: (1) Copper losses ($I^2R$ in windings — use thick, low-resistance wire). (2) Eddy current losses (circulating currents in core — use laminated sheets). (3) Hysteresis losses (repeated magnetization — use soft iron with narrow hysteresis loop). (4) Flux leakage (imperfect coupling — wind tightly on common core). Practical transformers achieve 90-99% efficiency.

Transformers work only with AC (constant DC produces no flux change). This is why AC won the "War of Currents" — transformers enable efficient long-distance power transmission by stepping voltage up (reducing current and $I^2R$ losses).