Resistivity (Conductor Resistance)

Multiply the resistivity of the material by the length and divide by the cross-section: R = ρ·l/A. Using the practical unit ρ in Ω·mm²/m you insert l in metres and A in mm² and obtain R directly in ohms. Example: 50 m of copper wire (ρ = 0.0178 Ω·mm²/m) with a 1.5 mm² cross-section has R = 0.0178 × 50/1.5 ≈ 0.59 Ω. The formula shows the two levers: a longer wire has more resistance, a thicker one less. Always check whether your table value is given in Ω·mm²/m or in Ω·m; A must then be in mm² or m² accordingly.

A thick conductor acts like many thin conductors side by side, i.e. like a parallel circuit. In a parallel circuit the reciprocals of the resistances add, so the total resistance drops. Doubling the cross-section exactly halves R, which is why A sits in the denominator. Microscopically, more parallel paths are available to the electrons; the current spreads over more cross-sectional area. In practice this is why car starter cables are thick: at currents of 100 A and more a thin cable would develop too much heat according to P = I²·R and its voltage drop would cripple the starter. Length works the other way round: double length, double resistance (series connection).

Rearrange the formula for the cross-section: A = ρ·l/R_max, where R_max is the highest permissible line resistance. Example: a 20 m copper feed line is to have at most 0.2 Ω: A = 0.0178 × 20/0.2 = 1.78 mm², so you choose the next standard size 2.5 mm². Two subtleties matter in practice: first, the current flows out and back, so for the voltage drop the round-trip length counts. Second, the conductor heats up under load, and the resistivity of copper rises by about 0.4 % per kelvin, so a hot line has noticeably more resistance than the 20 °C table promises.

ρ is the material parameter of electrical conduction, independent of the geometry of the specific wire. Small values mean good conductors: silver leads at 0.016 Ω·mm²/m, just ahead of copper (0.0178) and aluminium (0.027); that is why wiring is copper and overhead lines are the lighter aluminium. Iron sits noticeably higher at about 0.10. Alloys like constantan (0.5) are deliberately poor conductors with an almost temperature-independent ρ, ideal for measuring resistors. Heating elements like Kanthal use high ρ for targeted heat generation. Insulators like glass reach values beyond 10¹⁶ Ω·mm²/m. A span of more than 20 orders of magnitude makes ρ one of the most variable material properties of all.

In metals the number of free electrons is practically independent of temperature, but the lattice ions vibrate more strongly when hot. The drifting electrons therefore collide more often with the vibrating ions, their mean free path shrinks, and the resistance rises, for copper by about 0.4 % per kelvin. An incandescent lamp shows the effect drastically: cold, its filament has only about one tenth of its operating resistance, so the switch-on current is correspondingly high. Semiconductors behave the other way round because heat releases additional charge carriers there. Metrology exploits this contrast: platinum resistance thermometers (Pt100) measure temperature via the metallic rise, NTC thermistors via the semiconductor fall.