2026-05-19 12:23:11

Resistivity and Conductivity

In 2026-04-14_15-50-06 I described the relationship between the resistance and conductance...

Resistance and conductance are properties of electrical "objects" or "elements". Resistivity and conductivity are properties of materials.¹

Resistivity is commonly represented by the Greek letter ρ (rho). The SI unit of electrical resistivity is the ohm-meter (Ω⋅m).

Electrical conductivity (or specific conductance) is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letter σ (sigma), but κ (kappa) (especially in electrical engineering) and γ (gamma) are sometimes used. The SI unit of electrical conductivity is siemens per meter (S/m).

The meaning of these units are not intuitive, but are better understood from the ideal case diagrammed below:

The resistance of the conductor is directly proportional to its length \ell, and inversely proportional to its cross-sectional area A.


R \propto {\frac{\ell}{A}}

Let electrical resistivity \rho be the constant of proportionality.


R = \rho \frac{\ell}{A}

(This equation is known as Pouillet's law, after Claude Pouillet)


\Leftrightarrow \rho = R \frac{A}{\ell},

where

R is the electrical resistance of a uniform specimen of the material
\ell is the length of the specimen
A is the cross-sectional area of the specimen

The meaning of the ohm-meter (Ω⋅m) in this context is difficult to grok. Wikipedia describes it thus:

...ohms multiplied by square meters (for the cross-sectional area) then divided by meters (for the length).

The conductance of the conductor is inversely proportional to its length \ell, and directly proportional to its cross-sectional area A.


G \propto {\frac{A}{\ell}}

Let electrical conductivity \sigma be the constant of proportionality.


\begin{aligned}
R &= \sigma \frac{A}{\ell} \\
\Leftrightarrow \sigma &= G \frac{\ell}{A},
\end{aligned}

where

G is the electrical resistance of a uniform specimen of the material
\ell is the length of the specimen
A is the cross-sectional area of the specimen

Conductivity, \sigma, is the inverse of resistivity:


\sigma = \frac{1}{\rho}

It would be more accurate to describe this relationship in terms of intensive and extensive properties. The resistance and conductance of a copper bar would not change if the cube doubled in size, but its resistivity and conductivity ↩︎

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2026-05-19 12:23:11

Resistivity and Conductivity

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