Resistivity and R = ρL/A

By the end of this lesson

By the end you can explain why resistance depends on the dimensions of a specimen, define resistivity as a property of the material, and use $R = \rho L / A$ in both directions: to calculate a resistance from a length and cross-sectional area, and to calculate the resistivity of a material from measurements on a wire.

Why length and area matter

Take a wire and think of it as a stack of identical slices, end to end.

Make it longer. Every extra slice is more material for the charge carriers to push through, and the slices are in series, so their resistances add. Double the length and you double the resistance:

$R \propto L$

Make it thicker. A thicker wire is like several thinner wires joined side by side: extra parallel paths for the current, so the resistance falls. Doubling the cross-sectional area halves the resistance:

$R \propto \frac{1}{A}$

Putting the two together introduces a constant that belongs to the material itself:

$R = \frac{\rho L}{A}$

where $\rho$ (Greek "rho") is the resistivity of the material.

Definition

Resistivity (ρ)

The resistivity of a material is the resistance of a specimen of that material of length $1\ \text{m}$ and cross-sectional area $1\ \text{m}^2$ — equivalently, the resistance of a unit cube of the material measured between opposite faces.

Its unit is the ohm metre ( $\Omega\,\text{m}$ ), from $\rho = RA/L$ : $(\Omega \times \text{m}^2)/\text{m} = \Omega\,\text{m}$ .

Resistivity is a property of the material (and its temperature), not of the particular wire. Copper has a very low resistivity (about $1.7 \times 10^{-8}\ \Omega\,\text{m}$ ); nichrome, used in heating elements, has about $1.1 \times 10^{-6}\ \Omega\,\text{m}$ .

Why "ohm metre" and not just "ohm"?

Resistance alone cannot describe a material, because the same copper gives you a different resistance depending on how long and how thick you cut it. Two wires can have the same resistance while being made of completely different materials — one a short, thin piece of nichrome, the other a long, thick piece of copper.

Resistivity removes the geometry. $\rho = RA/L$ combines the measured resistance with the dimensions, so what is left depends only on the substance. That is why the unit carries a length: $\Omega\,\text{m}$ is resistance multiplied by an area and divided by a length.

Because resistivity depends on temperature, it is quoted "at 20 °C" in data tables. For a metal, heating makes the lattice vibrate more and $\rho$ rises; for a thermistor, $\rho$ falls sharply as temperature rises.

Worked example: Resistance of a copper wire

A copper wire is $2.00\ \text{m}$ long and has a diameter of $0.50\ \text{mm}$ . The resistivity of copper is $1.7 \times 10^{-8}\ \Omega\,\text{m}$ . Calculate its resistance.

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