Ohm's law

Ohm's law is the linear relation between current and voltage in metallic conductors: V = IR, where V is the potential difference across the conductor in volts, I is the current through it in amperes, and R is a geometry-and-material constant called the resistance in ohms. Ohm measured it in 1827 with torsion-balance galvanometers of his own design, and it remains the first equation every electrical engineer ever writes down. The law is not a fundamental principle — it is an empirical statement that happens to hold for most metals at ordinary temperatures and modest currents, with violations appearing in semiconductors, electrolytes, vacuum tubes, and any conductor pushed hard enough to matter.

The microscopic explanation came only in 1900 with Drude's free-electron model: electrons in a metal are accelerated by the applied electric field, scatter off lattice vibrations and defects with mean free time τ, and settle into a drift velocity v_d = eEτ/m. The current density J = nev_d = (ne²τ/m)E is then proportional to E with conductivity σ = ne²τ/m, which is Ohm's law at the microscopic level (J = σE). The linearity is a direct consequence of the fact that v_d depends linearly on E, which in turn depends on the field being small enough that the scattering time τ is essentially field-independent. Push the field high enough (say, by driving very large currents or approaching breakdown voltages) and τ itself becomes field-dependent, and the material becomes nonlinear.

Ohm's law defines the concept of resistivity, a material property ρ such that R = ρℓ/A for a wire of length ℓ and cross-section A. Resistivity is what a materials scientist measures; resistance is what an engineer reads off a meter. At room temperature, ρ ≈ 1.7 × 10⁻⁸ Ω·m for copper, 2.7 × 10⁻⁸ for aluminium, 10⁻⁵ for nichrome (heating elements), and as small as you can detect in a superconductor. The law's practical consequence — that networks of conductors obey linear algebra — is what makes Kirchhoff's two sentences from 1845 enough to solve any DC circuit by hand.

Georg Ohm published the linear V–I relation in his 1827 book Die galvanische Kette, mathematisch bearbeitet. The book was received with hostility in Germany — "unphysical speculation" was the standard verdict — and he lost his teaching post. Fifteen years later the Royal Society awarded him the Copley Medal, and the German academic establishment reversed course. The SI unit of resistance was named the ohm in 1881. The Drude model gave the microscopic picture in 1900; it is still the standard undergraduate explanation, though modern treatments use the Boltzmann transport equation instead.