Resistance vs Resistivity and Conductance vs Conductivity

As we saw on the previous page, the resistance, $R$ of a piece of material depends on its length, $\ell$, and cross-sectional area, $A$, with the relationship $R \propto \frac{\ell}{A}$.

We would like to be able to talk about the properties of a material independently of the size and geometry of the sample. For this we define an intrinsic material property, the resistivity, denoted $\rho$ with the following relationship to resistance:

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

Sometimes we talk about the conductivity, $\sigma$, of a material instead which is just the inverse of the resistivity: $\sigma=\frac{1}{\rho}$. Likewise, we can talk about the conductance, G, of a specific sample of material which will depend on the sample geometry.

The table below summarizes these terms:

Value	Symbol	Unit	Material Property?	Relationship
Resistance	$R$	$\Omega$	No	$R=1/G$
Resistivity	$\rho$	$\Omega –\text{m}$	Yes	$\rho=RA/\ell$
Conductance	$G$	$\Omega^{-1}$ or $\text{S}$	No	$G=1/R$
Conductivity	$\sigma$	$\Omega^{-1}–\text{m}^{-1}$ or $\text{S}–\text{m}^{-1}$	Yes	$\sigma=G\ell/A$

Value

Symbol

Unit

Material Property?

Relationship

Resistance

$R$

$\Omega$

$R=1/G$

Resistivity

$\rho$

$\Omega –\text{m}$

Yes

$\rho=RA/\ell$

Conductance

$G$

$\Omega^{-1}$ or $\text{S}$

$G=1/R$

Conductivity

$\sigma$

$\Omega^{-1}–\text{m}^{-1}$ or $\text{S}–\text{m}^{-1}$

Yes

$\sigma=G\ell/A$

What determines the resistivity of a material?

Now that we have distinguished between resistance/conductance which depend on the geometry of a material sample and resistivity/conductivity which are intrinsic material properties, we are ready to get into the materials science of what determines the resistivity of a material. There are many factors we will discuss, but we'll start by introducing models to explain why some materials conduct electricity at all and others don't. That is, why some materials are conductors and some are insulators.