A solenoid of length $l$, cross-sectional radius $r$, and $n$ turns per unit length carries current $I$. Its self-inductance is:

$\varepsilon = -L\frac{dI}{dt}$ (back-EMF opposes change in current; Lenz\'s law). $\Lambda = LI$ (flux linkage). $L$ depends only on geometry and medium, not on current. Solenoid: $L = \mu_0 n^2 V$ (volume $V = Al$). Toroid: $L = \mu_0 N^2 A/2\pi r$. Inductors in series: $L_{eq} = \sum L_i$. In parallel: $1/L_{eq} = \sum 1/L_i$ (no mutual inductance).

Resistor: $V = IR$, in phase. Inductor: $V = IX_L$ where $X_L = \omega L$; $V$ leads $I$ by 90°. Capacitor: $V = IX_C$ where $X_C = 1/\omega C$; $V$ lags $I$ by 90°. Series LCR: $Z = \sqrt{R^2 + (X_L-X_C)^2}$; $\tan\phi = (X_L-X_C)/R$. Resonance: $X_L = X_C$ → $\omega_0 = 1/\sqrt{LC}$, $Z = R$ (minimum impedance). Power factor $\cos\phi = R/Z$; average power $P = V_{rms}I_{rms}\cos\phi$.

Transformer: $V_s/V_p = N_s/N_p = I_p/I_s$ (for ideal, lossless transformer). Step-up: $N_s > N_p$ → $V_s > V_p$, $I_s < I_p$. Step-down: $N_s < N_p$ → $V_s < V_p$, $I_s > I_p$. Power: $P_{in} = P_{out}$ (ideal). Losses: eddy currents (reduced by lamination), hysteresis (heat from repeated magnetisation), resistance heating in windings. Efficiency typically 95-99% for large power transformers.

Electric generator: mechanical energy → electrical energy (Faraday\'s law). $\varepsilon = NBA\omega\sin(\omega t) = \varepsilon_0\sin(\omega t)$ for rotating coil in magnetic field. Electric motor: electrical → mechanical (current-carrying conductor in field experiences force). Induction cooktop: alternating current in coil induces eddy currents in the ferromagnetic cooking vessel, which heats up. MRI: strong, uniform magnetic field + radio-frequency pulses excite nuclear spins → signals detected.

Self-inductance is defined by $\phi = LI$, and the induced emf is $e = -L\frac{dI}{dt}$. For a long solenoid $L = \mu_0 n^2 A l$, so inductance depends on the geometry and the core material, never on the current flowing. Note the $n^2$: doubling the turns per unit length quadruples the inductance. Inserting an iron core multiplies L by the relative permeability, which is why practical inductors are wound on cores.

Building up a current against the back-emf requires work, and that energy is stored in the magnetic field as $U = \frac{1}{2}LI^2$. The form mirrors $\frac{1}{2}CV^2$ for a capacitor and $\frac{1}{2}mv^2$ for a moving mass — inductance is the electrical analogue of inertia. This is also why current through an inductor cannot change instantly, and why breaking an inductive circuit produces a spark as the stored energy is forced out.

Mutual inductance $M$ relates the emf in one coil to the rate of change of current in another, and it is the principle behind the transformer, where $\frac{V_s}{V_p} = \frac{N_s}{N_p}$. Transformers work only on AC, because a steady current produces no changing flux. Real losses come from copper heating, eddy currents (reduced by laminating the core), hysteresis and flux leakage — a list worth having ready, since it is a standard three-mark question.