105 lines
3.0 KiB
Markdown
105 lines
3.0 KiB
Markdown
---
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id: 2026-05-19T12:23:11-0400
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title: 2026-05-19 12:23:11
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tags: []
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daily: "[[2026-05-19]]"
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---
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# 2026-05-19 12:23:11
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## Resistivity and Conductivity
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In [[2026-04-14_15-50-06]] I described the relationship
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between the resistance and conductance...
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[**Resistance** and **conductance**](https://en.wikipedia.org/wiki/Electrical_resistance_and_conductance)
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are properties of electrical "objects" or "elements".
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[**Resistivity** and **conductivity**](https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity) are properties of **materials**.[^1]
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[^1]: It would be more accurate to describe this relationship
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in terms of [intensive and extensive properties](https://en.wikipedia.org/wiki/Intensive_and_extensive_properties).
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The resistance and conductance of a copper bar
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would not change if the cube doubled in size,
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but its resistivity and conductivity
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> Resistivity is commonly represented by the Greek letter ρ (rho).
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> The SI unit of electrical resistivity is the ohm-meter (Ω⋅m).
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> Electrical conductivity (or specific conductance)
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> is the reciprocal of electrical resistivity.
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> It represents a material's ability to conduct electric current.
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> It is commonly signified by the Greek letter σ (sigma),
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> but κ (kappa) (especially in electrical engineering) and γ (gamma)
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> are sometimes used.
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> The SI unit of electrical conductivity is siemens per meter (S/m).
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The meaning of these units are not intuitive,
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but are better understood from the ideal case
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diagrammed below:
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The **resistance** of the conductor
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is directly proportional to its length $\ell$,
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and inversely proportional to its cross-sectional area $A$.
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$$
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R \propto {\frac{\ell}{A}}
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$$
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Let electrical resistivity $\rho$ be the constant of proportionality.
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$$
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R = \rho \frac{\ell}{A}
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$$
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(This equation is known as **Pouillet's law**,
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after [Claude Pouillet](https://en.wikipedia.org/wiki/Claude_Pouillet))
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$$
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\Leftrightarrow \rho = R \frac{A}{\ell},
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$$
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where
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* $R$ is the electrical resistance of a uniform specimen of the material
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* $\ell$ is the length of the specimen
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* $A$ is the cross-sectional area of the specimen
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The meaning of the ohm-meter (Ω⋅m) in this context is difficult to grok.
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Wikipedia describes it thus:
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> ...ohms multiplied by square meters (for the cross-sectional area)
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> then divided by meters (for the length).
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%%
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The **conductance** of the conductor
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is _inversely_ proportional to its length $\ell$,
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and _directly_ proportional to its cross-sectional area $A$.
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$$
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G \propto {\frac{A}{\ell}}
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$$
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Let electrical conductivity $\sigma$ be the constant of proportionality.
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$$
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\begin{aligned}
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R &= \sigma \frac{A}{\ell} \\
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\Leftrightarrow \sigma &= G \frac{\ell}{A},
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\end{aligned}
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$$
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where
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* $G$ is the electrical resistance of a uniform specimen of the material
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* $\ell$ is the length of the specimen
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* $A$ is the cross-sectional area of the specimen
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%%
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Conductivity, $\sigma$, is the inverse of resistivity:
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$$
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\sigma = \frac{1}{\rho}
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$$ |