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+---
+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:
+
+$$
+\sigma = \frac{1}{\rho}
+$$