In substances where the current density **J** is proportional to the applied electric field **E** (i.e. where Ohm's law holds) the *electrical conductivity* σ may be defined,

**J**=σ **E**

σ is expressed in

Siemens per

meter where 1 Siemens is 1

Ampere per

Volt. Electrial conductivity is the inverse of

resistivity ρ.

What follows is the derivation of an expression for σ where a steady electric field is applied to a conductor.

In a conductor there are conduction electrons which experience collisional and electrial forces. They acquire a net drift velocity **v**_{d} in the opposite direction to **J** and **E**. These two forces eventually balance. *(An analogy is terminal velocity in freefall where gravity and viscosity are the forces which balance)*

The collisional force can be expressed (as in kinetic theory) as the momentum m_{e}**v**_{d} divided by the collisional time τ_{c} where m_{e}is the electron mass. The electrical force is -e**E** where e is the electronic charge. Equating these forces one obtains-

-e**E**=m_{e}**v**_{d}/τ_{c}

The

current density **J** of electrons is proportional to the

charge,

velocity of charge carriers (-e) and the

density n

_{e} of charge carriers-

**J**=-n_{e}e** v**_{d}

Making a substitution for the drift velocity it follows-

**E**=(m_{e}/n_{e}e^{2}τ_{c}) **J**

Comparing this expression with the

original one for σ one obtains-

σ=n_{e}e^{2}τ_{c}/m_{e}

From this

equation it is clear that a good conductor (high σ) has a long collisional time.

Some σ values (source *'Electromagnetic Fields and Waves' by P.Lorrain, D.R.Corson, F.Lorrain and published by Freeman*) expressed in Siemens/meter

Aluminium 3.54 X 10^{7}

Copper 5.80 X 10^{7}

Graphite 7.1 X 10^{4}

Seawater 5