electrical conductivity

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_ev_d divided by the collisional time τ_c where m_eis the electron mass. The electrical force is -eE where e is the electronic charge. Equating these forces one obtains-

-eE=m_ev_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_ee v_d

Making a substitution for the drift velocity it follows-

E=(m_e/n_ee²τ_c) J

Comparing this expression with the original one for σ one obtains-

σ=n_ee²τ_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 10⁷
Copper 5.80 X 10⁷
Graphite 7.1 X 10⁴
Seawater 5