Conductivity to Resistivity Calculator

Electrical conductivity and resistivity describe the same material property from opposite ends:

$\\sigma = \\frac{1}{\\rho} \\qquad\\Longleftrightarrow\\qquad \\rho = \\frac{1}{\\sigma}$

Symbols and SI units:

• $\\rho$ — resistivity, ohm-metres (Ω·m)
• $\\sigma$ — conductivity, siemens per metre (S/m); one siemens is one reciprocal ohm

There is no constant of proportionality and no unit juggling: $1/(\\Omega\\cdot\\text{m})$ is S/m. Copper at 20 °C sits at $\\rho = 1.68 \\times 10^{-8}$ Ω·m, so $\\sigma = 5.96 \\times 10^{7}$ S/m.

Conductor metals are often quoted as a percentage of the International Annealed Copper Standard, $\\%\\text{IACS} = \\sigma / (5.80 \\times 10^{7}\\ \\text{S/m}) \\times 100$.

The operating assumption: a homogeneous, isotropic material at a stated temperature. Both quantities drift with temperature, so a value without a temperature is incomplete.

Resistivity becomes a resistance once you add the geometry:

$R = \\frac{\\rho L}{A} = \\frac{L}{\\sigma A} \\qquad\\Longrightarrow\\qquad \\rho = \\frac{RA}{L}$

with $R$ in ohms, $L$ in metres and $A$ in square metres. The last form is how resistivity is measured in practice: take a sample of known length and cross-section, measure its resistance, and divide.

Temperature is handled with a linear coefficient about a reference temperature $T_0$:

$\\rho_T = \\rho_{0}\\left[1 + \\alpha (T - T_0)\\right]$

For copper, $\\alpha \\approx 0.00393$ per °C referenced to 20 °C.

Watch the units people trip on. Water and process conductivity is quoted in µS/cm, not S/m; $1$ S/m $= 10{,}000$ µS/cm. Cable tables often use Ω·mm²/m, where copper is $0.0168$ — numerically the same as $1.68 \\times 10^{-8}$ Ω·m, because $1$ Ω·mm²/m $= 10^{-6}$ Ω·m.

The mistake people make: measuring a two-wire loop and dividing by the one-way length. Use the full conductor length the current actually travels.

• Inverting the wrong quantity — a small resistivity means a large conductivity. If $\\rho = 10^{-8}$ gives $\\sigma = 10^{-8}$, a reciprocal was skipped.
• Mixing µS/cm with S/m — divide µS/cm by $10{,}000$ to get S/m before inverting.
• Using mm² for $A$ in an SI formula — $10$ mm² is $1 \\times 10^{-5}$ m², a factor of a million.
• Ignoring temperature — a conductor running at $75$ °C has about $22\\%$ more resistance than the same conductor at $20$ °C, which matters for voltage-drop work.
• Treating resistivity as a fixed constant for an alloy — it varies with composition, temper and impurity level, so use the supplier's figure for the actual material.
• Confusing resistivity with sheet resistance — sheet resistance is $\\rho/t$ in ohms per square and applies to a thin film of thickness $t$.