Resistance of a Wire (R = ρL/A)

Resistance behaves like a road: make it longer and the journey costs more; make it wider and traffic moves more easily. Doubling the length of a conductor puts twice as many collisions between the electron and the far end, so resistance doubles. Doubling the cross-sectional area gives the current twice as many parallel paths, so resistance halves. What is left over — the part that depends on the material rather than its shape — is the resistivity $\rho$, and $R = \rho L/A$ is nothing more than those three statements written together. Copper's $1.68 \times 10^{-8}$ Ω·m is the reason it wires the world; aluminium is about 1.6 times higher, silver marginally lower, and a good insulator is some twenty orders of magnitude higher than any of them.

A hundred metres of 2.5 mm² copper comes to $(1.68 \times 10^{-8} \times 100)/(2.5 \times 10^{-6}) = 0.67\ \Omega$. Carrying 16 A, that single conductor drops $16 \times 0.67 = 10.7\ \text{V}$ and dissipates 172 W along its length. Both figures matter, and both are why a long run is specified thicker than the load current alone would suggest — the ampacity keeps the cable from overheating, but it is the voltage drop that usually decides the size.

Resistivity is a bulk property of the material, and the reciprocal quantity, conductivity, is what materials people usually quote. The relation also underpins the strain gauge, whose resistance changes as stretching lengthens it and narrows it; the shunt, a precisely dimensioned low-resistance bar used to measure large currents; and the whole logic of high-voltage transmission, where a fixed $\rho L/A$ in the line means the only way to cut the $I^2R$ loss is to cut $I$.

Four errors, and the first one is nearly universal. When you are computing voltage drop on a circuit, $L$ is the total length of conductor the current traverses, which for an ordinary two-wire supply is twice the run distance — out along one conductor and back along the other. A 30 m run is 60 m of copper, and using 30 halves the answer. Second, resistivity is strongly temperature-dependent: copper rises about 0.39% per degree, so a conductor at 75 °C has roughly 22% more resistance than the 20 °C handbook figure, and electrical codes tabulate drop at operating temperature for exactly this reason. Third, watch the area units. Wire gauges are not areas — 12 AWG is 3.31 mm², 10 AWG is 5.26 mm² — and a diameter must be squared and taken through $\pi d^2/4$ before it can go in the denominator. Fourth, this is a DC resistance. On AC the current crowds toward the surface of the conductor, and above a few hundred hertz, or in large conductors at mains frequency, that skin effect raises the effective resistance measurably; and a long AC run has reactance as well, so the drop you actually measure will exceed what $\rho L/A$ alone predicts.