Current density: Current density is the current per unit area of cross-section of a conductor. $J = \frac{I}{A}$ Relaxation time: Relaxation time is the time interval between two successive collisions of electrons in a conductor. Expression for resistivity: $v_d = \frac{e E}{m} \tau$ or, $v_d = \frac{e V}{m l} \tau \; \text{(since, } E = \frac{V}{l} \text{)}$ or, $V = \frac{v_d m l}{e \tau}$ or, $I R = \frac{v_d m l}{e \tau}$ (since from Ohm's law, $V = I R$ ) or, $R = \frac{v_d m l}{e \tau I}$ $\frac{\rho l}{A} = \frac{v_d m l}{e \tau I} \quad \text{(since } R = \frac{\rho l}{A}\text{)}$$\rho = \frac{v_d m A}{e \tau I}$$\;$ $\text{ or, } \rho = \frac{v_d m A}{e \tau (n e A v_d)}$ $\quad \text{ (since, } I = n e A v_d \text{ ) }$ $\therefore \rho = \frac{m}{n e^2 \tau}$