Resistivity

In a nutshell

Resistivity is defined as the resistance of a unit length and unit cross sectional area of a material. Conductors have a very low resistivity, insulators have a very high resistivity and semiconductors have a resistivity in between them.

Equations

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Variable definitions

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Resistivity

Resistance is a term that defines the opposition of current flow of a specific component. It can change depending on the size of the component; if the component is longer the resistance will increase since there is more material to get through and thus more collisions for the electrons, if the component has a larger cross-sectional area the resistance will decrease since more electrons can pass through. One can therefore write:

​$$R\propto\dfrac{L}{A}$$​​

The constant of proportionality of this equation is known as the resistivity. It is defined as the resistance of a unit length and unit cross-sectional area of a material or in other words, it is a measure of how much a material opposes current flow. Resistivity is an electrical property specific to every material no matter the size and is symbolised by the greek letter $$\rho$$. Its unit is the ohm-meter $$\Omega\ m$$, and can be calculated with the equation:

​$$\rho=\dfrac{RA}{L}$$​​

Note: The symbol for resistivity $$\rho$$ is also often used for density, be careful not to get confused!

Resistivity in conductors, semiconductors and insulators

Resistivity varies vastly depending on the type of material and its temperature:

• Conductors have a very low resistivity of about $$10^{-8}\ \Omega\ m$$. When the temperature increases so does their resistivity since ions vibrate more increasing the chance of collision with electrons.
• Semiconductors have a moderate resistivity that vary greatly but hover around $$1\ \Omega\ m$$. They have very different properties within each other, for example in some semiconductors a rise in temperature releases more charge carriers decreasing the resistance.
• Insulators have a very high resistivity of about $$10^{16}\ \Omega\ m$$.
  

Example

A metal wire is $$2\ m$$ long, has a cross-sectional of $$5\times10^{-5}\ m^2$$ and a resistance of $$5\times10^{-3}\ \Omega$$. What is the resistivity if this metal?

Firstly, write down all the known values:

$$\begin{aligned}L&=2\ m \\A&=5\times10^{-5}\ m^2\\R&=5\times10^{-3}\ \Omega\end{aligned}$$​​

Next, write down the equation for resistivity:

$$\rho=\dfrac{RA}{L}$$​​

Substitute all the values into the equation:

$$\rho=\dfrac{5\times10^{-3}\times5\times10^{-5}}{2}$$​​

Calculate the resistivity:

$$\rho=1.25\times10^{-7}\ \Omega\ m$$​​

The resistivity of the metal is $$\underline{1.25\times10^{-7}\ \Omega\ m}$$.