---
id: 2026-05-19T12:23:11-0400
title: 2026-05-19 12:23:11
tags: []
daily: "[[2026-05-19]]"
---
# 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**](https://en.wikipedia.org/wiki/Electrical_resistance_and_conductance)
are properties of electrical "objects" or "elements".
[**Resistivity** and **conductivity**](https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity) are properties of **materials**.[^1]

[^1]: It would be more accurate to describe this relationship
    in terms of [intensive and extensive properties](https://en.wikipedia.org/wiki/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

> 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:

![](https://upload.wikimedia.org/wikipedia/commons/6/68/Resistivity_geometry.png)

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](https://en.wikipedia.org/wiki/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:

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