How this instrument works
Resistance is not a property of copper. It is a property of one particular piece of copper — this length, this thickness, drawn on this day. Resistivity is what belongs to the metal itself: ohms measured edge to edge across one cubic metre of it, current entering by one face and leaving by its opposite. Multiply by how far charge must travel, divide by how much room it has for travelling, and geometry converts material into component. Stretch a wire to twice its length and it thins to half its area, so resistance quadruples — which is exactly how every strain gauge bonded to an aircraft spar earns its keep.
Few engineering quantities span this much ground. Silver sits at 1.59 × 10⁻⁸ Ω·m, copper at 1.68, gold at 2.44, aluminium near 2.65; nichrome heating wire runs around 1.1 × 10⁻⁶, sea water near 0.2, fused quartz somewhere past 10¹⁸ — twenty-six orders of magnitude between the bench's best conductor and whatever insulates it. Augustus Matthiessen spent much of the 1860s pinning down that metallic end, and found alloying spoils conductivity far more brutally than anyone had guessed: his rule splits resistivity into a temperature-dependent lattice term plus a fixed impurity term, which is why traces of arsenic in smelted copper matter commercially, and why industry eventually settled on an International Annealed Copper Standard in 1913 — 1.7241 × 10⁻⁸ Ω·m at 20 °C, still today's 100% mark against which conductor grades get quoted.
Four assumptions hide inside that single division. Cross-section must be uniform with current density even across it — alternating current disobeys, crowding toward the surface until only its skin conducts; at 50 Hz that skin reaches roughly 9 mm into copper, so the middle of any fat busbar is largely decoration. Material must be homogeneous and isotropic, ruling out graphite measured along its wrong axis or any conductor gone corroded at heart. Length must mean whatever path charge takes, not endpoint separation — stranded cable winds its strands into helices a percent or two longer than that cable measures. And every published resistivity carries a temperature with it, conventionally 20 °C; quote one without that qualifier and you have quoted half a number.
- Enter Resistivity (Ω·m) for your material — 1.68 × 10⁻⁸ for copper, 2.65 × 10⁻⁸ for aluminium, higher again for anything alloyed.
- Set Conductor length to whatever path current actually takes. A supply-and-return circuit uses both legs, so a 30 m run means 60 m of conductor.
- Give Cross-sectional area, switching its unit to mm² if you are reading off cable markings — entering 2.5 where 2.5 × 10⁻⁶ m² is meant costs six orders of magnitude.
- Read Resistance, flipping its unit to milliohms for busbar links or kilohms for resistance wire and long instrument leads.
- Sanity-check per metre: 2.5 mm² copper sits close to 6.7 mΩ/m, so divide by your length and see whether that order of magnitude survives.
Worked example — 100 metres of 2.5 mm² copper
Consider wiring reaching out to a garden workshop: 100 metres of 2.5 mm² copper, the size used for socket circuits across much of Europe. Put 1.68 × 10⁻⁸ into Resistivity (Ω·m), 100 into Conductor length, and 2.5 × 10⁻⁶ m² — that is 2.5 mm² — into Cross-sectional area. R = (1.68 × 10⁻⁸ × 100) ⁄ (2.5 × 10⁻⁶) = (1.68 × 10⁻⁶) ⁄ (2.5 × 10⁻⁶) = 0.672 Ω.
Two-thirds of an ohm sounds like nothing until something loads it. That figure covers one conductor; a real circuit needs its return leg too, making 200 m and 1.344 Ω. Draw 16 A through that pair and 21 volts never reach your workshop — nine percent of a 230 V supply, well outside any wiring regulation, which is precisely why such a run gets specified in 4 mm² or 6 mm² instead.
Your remedy is visible in the algebra. Resistance falls in proportion to area, so 6 mm² brings that loop down to 0.56 Ω and its drop to 9 V, under four percent. Length is an unforgiving term by comparison: distance cannot be negotiated, only paid for in copper.
Questions
What is the difference between resistivity and resistance?
Resistivity belongs to the material, resistance to the object. Resistivity is quoted in ohm-metres and does not change when you cut a wire shorter; resistance is quoted in ohms and does. Two conductors of identical resistance can be made from wildly different metals, and two pieces of one metal will differ in resistance if either is longer or thinner. This formula is a bridge between them: feed it a material figure plus the two dimensions that matter, and it returns an object figure.
Why is my answer a million times too large or too small?
Almost certainly area. Cable is labelled in square millimetres while this formula wants square metres, and one m² holds 10⁶ mm², not 10³. A 2.5 mm² conductor is 2.5 × 10⁻⁶ m². Switch the Cross-sectional area unit to mm² and let this instrument do that conversion. Second most common slip: entering a diameter. For round wire, area is πd²⁄4, so a 1.78 mm strand gives 2.49 mm², nowhere near 1.78.
Is aluminium worth using instead of copper?
Aluminium resistivity runs near 2.65 × 10⁻⁸ Ω·m against copper at 1.68 × 10⁻⁸, so it needs roughly 58% more area for equal resistance — usually two size steps up. What that buys is weight: aluminium has about a third of copper's density, so an equal-resistance aluminium conductor still weighs around half as much. That trade explains why nearly every overhead transmission line is aluminium stranded over a steel core. Building wiring is fussier, because aluminium creeps under clamping pressure and its oxide insulates, which is how terminations quietly go high-resistance and hot.
Does temperature change my answer?
Yes, by more than most people expect. Metallic resistivity is published against a reference temperature, conventionally 20 °C, and climbs with a coefficient near 0.004 per kelvin for copper and aluminium alike — windings at 100 °C are therefore roughly one third more resistive than any table admits. Platinum turns that nuisance into an instrument: its resistance-temperature curve repeats so reliably that a Pt100 sensor, 100 Ω at 0 °C, serves as a defined interpolation standard for the international temperature scale from the triple point of hydrogen up to silver's freezing point.
Why does a four-wire measurement read differently?
Because the two-wire meter measures its own test leads and probe contacts alongside your sample. Lead resistance easily reaches 0.1 to 0.5 Ω, meaningless against a 10 kΩ part yet comparable to this whole worked answer of 0.672 Ω. A four-wire, or Kelvin, connection sends current down one pair and senses voltage on a separate pair carrying almost none, so lead drop never enters that reading. Below about 10 Ω, four-wire is not refinement — it is the only honest method.
Does this hold for alternating current?
At mains frequency and ordinary conductor sizes, closely enough. Accuracy drifts once skin depth becomes comparable to conductor radius: at 50 Hz copper conducts mainly within about 9 mm of its surface, so a 40 mm bar has an effective area well below its geometric one. Proximity effect worsens matters where conductors run bundled. At radio frequencies that discrepancy is total, which is why tubing performs as well as solid rod and why litz wire exists. Reactance is a separate question again — this sheet returns resistance, never impedance.