# What is resistivity

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

For a given material, the resistance of a length of the material is a constant times its length, and divided by its cross-section area.

This constant is the resistivity, $\rho$, of the material.

Resistivity of a material is measured in ohm-metres ($\Omega m$), or volt-metres per amp ($Vm/A$).

The inverse of resistivity is conductivity, $\sigma$, or current density per electric field, and is measured in amps per volt-metre ($A/Vm$) or siemens per metre ($S/m\text{ or }\mho /m$).

Equations

Resistance of a length $l$ cross-section area $A$ and resistivity $\rho$:

$$R\ =\ \frac{l}{A}\,\rho$$

$$\rho\ =\ \frac{A}{l}\,R$$

Conductivity of material of resistivity $\rho$:

$$\sigma\ =\ \frac{1}{\rho}\ =\ \frac{\text{current density}}{\text{electric field}}$$

Ohm's law can be written in terms of resistivity, electric field $E$, and current density $J$:

$E \ = \ J \ \rho$

Extended explanation

Dynamic resistivity:

Resistance (sometimes called static resistance) is voltage per current:

$$R\ =\ V/I$$

Dynamic resistance is the derivative:

$$R_d\ =\ dV/dI$$

Dynamic resistivity is the derivative of resistivity:

$$\rho_d\ =\ d\rho /dI$$

cgs units:

In cgs units (ESU or Gaussian versions), resistivity of a material is measured in seconds.

By comparison, capacitance is measured in cm, and conductance is measured in cm/s … see http://en.wikipedia.org/wiki/Cgs_units#Electromagnetic_units_in_various_CGS_systems for details .

This time is comparable with the time it takes for the the field inside a moderately-sized piece of a conducting material to return to zero when an external electric ﬁeld is applied (ie, for the internal charges to rearrange themselves, creating their own electric field which cancels out the applied field).

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