(a) Definition and SI unit of conductivity
(b) Derivation of the expression for conductivity
Relation between current density and electric field
(a) The conductivity of a material equals to the reciprocal of the resistance of its wire of unit length and unit area of cross section.
Alternatively :
The conductivity ( $\mathit{s}$ ) of a material is the reciprocal of its resistivity $\left(\mathit{\rho }\right)$ ]
(Also accept $\mathit{s}=\frac{1}{\mathit{\rho }}$ )
Its SI unit is
ohm ${}_{}^{-1}{\mathit{m}}_{}^{-1}∕(.$ mho ${\mathit{m}}^{-1})∕$ siemen ${\mathit{m}}^{-1}$
(b) The acceleration, $\overrightarrow{\mathit{a}}=-\frac{\mathit{e}}{\mathit{m}}\overrightarrow{\mathit{E}}$
The average drift velocity, ${\mathit{v}}_{\mathit{d}}$ is given by
${\mathit{v}}_{\mathit{d}}=-\frac{\mathit{e}\mathit{E}}{\mathit{m}}\mathit{\tau }$
( $\mathit{\tau }=$ average time between collisions/ relaxation time)
If $\mathit{n}$ is the number of free electrons per unit volume, the current $\mathit{I}$ is given by
$\mathit{I}=\mathit{n}\mathit{e}\mathit{A}|{\mathit{v}}_{\mathit{d}}|$
$=\frac{{\mathit{e}}^2\mathit{A}}{\mathit{m}}\mathit{\tau }\mathit{n}\left|\mathit{E}\right|$
But $\mathit{I}=|\mathit{j}|\mathit{A}(\mathit{j}=$ $\mathrm{current}\mathrm{density}$)
We, therefore, get
$\left|\mathit{j}\right|=\frac{\mathit{n}{\mathit{e}}^2}{\mathit{m}}\mathit{\tau }\left|\mathit{E}\right|,$
The term $\frac{\mathit{n}{\mathit{e}}^2}{\mathit{m}}\mathit{\tau }$ is conductivity
$\therefore \mathit{\sigma }=\frac{\mathit{n}{\mathit{e}}^2\mathit{\tau }}{\mathit{m}}$
$\Rightarrow \mathit{J}=\mathit{\sigma }\mathit{E}$