Article — Conductivity to Resistivity Calculator
Conductivity to Resistivity Calculator
Conductivity (σ) and resistivity (ρ) are reciprocals: ρ = 1/σ. Copper has σ = 5.96 × 10⁷ S/m and ρ = 1.68 × 10⁻⁸ Ω·m. The two values describe the same material — one says how easily current passes through, the other says how strongly it is opposed.
What conductivity and resistivity mean
Resistivity is a bulk property of a material that quantifies how strongly it opposes current. A 1 m cube of copper with current entering one face and leaving the opposite face has 1.68 × 10⁻⁸ Ω of resistance — that number is the resistivity. Geometry has been stripped away; only the substance matters.
Conductivity flips the perspective. A material with high σ moves charge easily. Both quantities measure the same physics, just inverted, so converting between them is a single arithmetic step. Engineers use whichever value is more convenient: σ when discussing solutions and electrolytes, ρ when sizing solid conductors.
The reason both terms persist is historical. Nineteenth-century work on metals naturally produced resistivity (high values, easy to measure with bridges). Twentieth-century work on solutions produced conductivity (a more intuitive direct reading on a conductance meter). Today both are equally legitimate, and you choose by context.
The span from silver (best ordinary conductor, σ = 6.3 × 10⁷ S/m) to fused quartz (σ ≈ 10⁻¹⁸ S/m) covers 26 orders of magnitude. No other common physical property varies so wildly across materials.
The conductivity-resistivity formula
The relationship is ρ = 1/σ, with σ in siemens per meter and ρ in ohm-meters. If you have one, dividing 1 by it gives the other exactly. There is no approximation, no fitted constant, no temperature term. Both values change with temperature, but their reciprocal relationship holds at any single temperature.
A more physical form comes from the Drude model: σ = n × e × μ, where n is the density of mobile charge carriers, e is the elementary charge (1.602 × 10⁻¹⁹ C), and μ is mobility (m²/V·s). Copper has roughly 8.5 × 10²⁸ free electrons per m³ — an enormous reservoir of mobile charge carriers.
Units used for conductivity and resistivity
The SI units are siemens per meter (S/m) for conductivity and ohm-meter (Ω·m) for resistivity. One siemens equals one ampere per volt, so S/m has units of A/(V·m).
- S/m = base SI conductivity unit
- mS/cm = milli-siemens per cm, common in water analysis: 1 S/m = 10 mS/cm
- μS/cm = micro-siemens per cm, used for ultrapure water: 1 S/m = 10,000 μS/cm
- Ω·m = base SI resistivity unit
- Ω·cm = ohm-centimeter, common for semiconductor wafers: 1 Ω·m = 100 Ω·cm
- μΩ·m = micro-ohm-meter, used for excellent conductors: copper is 16.8 nΩ·m (0.0168 μΩ·m)
Materials by conductivity, from silver to rubber
Materials fall into four bands. Conductors (σ > 10⁶ S/m) include all bulk metals. Semiconductors (10⁻⁴ to 10² S/m) include silicon, germanium, and gallium arsenide — the bedrock of modern electronics. Electrolytes and ionic solutions sit between 10⁻³ and 10² S/m. Insulators (σ < 10⁻⁸ S/m) include rubber, glass, dry wood, and most ceramics.
Aluminum (3.77 × 10⁷ S/m) is only 63% as conductive as copper but a third of the weight. For overhead power lines where weight matters more than wire diameter, aluminum wins.
How temperature changes resistivity
Metals get more resistive when heated. Thermal vibrations of the crystal lattice scatter the conduction electrons, lengthening their mean free path. The relationship is nearly linear: ρ(T) = ρ₀ × [1 + α × (T − T₀)] with α around 0.004 per °C for most metals.
Semiconductors do the opposite. Heat liberates more charge carriers from their bound states, so conductivity rises with temperature. This is why a tungsten light-bulb filament has roughly 10× higher resistance when hot than cold, but a silicon thermistor decreases in resistance as it warms.
Copper +0.00393Aluminum +0.00429Tungsten +0.0045Nichrome +0.0004Constantan +0.00002Carbon −0.0005Calculating wire resistance from resistivity
Resistivity becomes resistance when you specify a geometry. The formula R = ρ × L / A turns a material property into a circuit component. A 100 m run of 14 AWG copper (2.08 mm² cross-section) carries R = 1.68 × 10⁻⁸ × 100 / 2.08 × 10⁻⁶ = 0.81 Ω. At 15 A that yields a voltage drop of 12 V, which is why long runs use heavier gauge.
The same formula applied to electrolytic cells lets engineers size conductivity sensors. Cell constant K (in cm⁻¹) is geometry-dependent; measured conductance G converts to conductivity via σ = K × G. A standard cell with K = 1 cm⁻¹ reads conductance directly as conductivity in S/cm.
Cable sizing tables in electrical codes are nothing more than systematic applications of R = ρL/A combined with a voltage-drop limit and a temperature derating. The NEC, IEC 60364, and equivalent national standards all start from the resistivity of copper or aluminum at conductor operating temperature.
The IACS — International Annealed Copper Standard — defines 100% conductivity as 5.8 × 10⁷ S/m at 20°C. Modern high-purity copper exceeds 101% IACS. Aluminum is rated about 61% IACS, the AAC overhead-line alloys around 56%.
Common conductivity and resistivity mistakes
Three errors dominate practical work. First, mixing units: ρ in Ω·cm is 100× ρ in Ω·m, and σ in mS/cm is 10× σ in S/m. Reading a datasheet without checking the unit produces wrong answers by orders of magnitude.
Semiconductor wafers are spec'd in Ω·cm, while metallurgists use μΩ·m and electrical engineers use Ω·m. A 10 Ω·cm wafer is 0.1 Ω·m, not 10. Always confirm the unit before quoting a resistivity value.
Second, ignoring temperature. Quoted resistivity is almost always at 20°C. A 100 m copper cable in a 60°C attic has 16% more resistance than the same cable at 20°C — enough to trip a tight voltage-drop budget. Third, confusing resistivity (a material property, Ω·m) with resistance (a component property, Ω). Resistance depends on length and cross-section; resistivity does not.
A fourth pitfall trips up water-quality work: forgetting that conductivity is referenced to 25°C in most standards, while pure water itself dissociates more at warmer temperatures and reads higher. Lab instruments apply automatic temperature compensation, but reading raw values without correction can suggest contamination that is not there.