(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 ( $s$ ) of a material is the reciprocal of its resistivity $(\rho)$ ] (Also accept $s=\frac{1}{\rho}$ ) Its SI unit is $\left(\frac{1}{\text{ohm-metre}}\right) /$ ohm $^{-1} m^{-1} / (.$ mho $m^{-1}) /$ siemen $m^{-1}$ (b) The acceleration, $\vec{a}=-\frac{e}{m}\vec{E}$ The average drift velocity, $v_d$ is given by $v_d=-\frac{eE}{m}\tau$ ( $\tau=$ average time between collisions/ relaxation time) If $n$ is the number of free electrons per unit volume, the current $I$ is given by $I = n e A |v_d|$ $= \frac{e^2 A}{m} \tau n |E|$ But $I = |j| A (j =$ $\text{current density}$) We, therefore, get $|j| = \frac{n e^2}{m} \tau |E|,$The term $\frac{n e^2}{m} \tau$ is conductivity $\therefore \sigma = \frac{n e^2 \tau}{m}$ $\Rightarrow J = \sigma E$