The relaxation time $\tau$ is nearly independent of the applied $E$ field,whereas it changes significantly with temperature $T$. The first fact is (in part) responsible for Ohm's law,whereas the second fact leads to the variation of resistivity $\rho$ with temperature. Elaborate why?

(N/A) Relaxation time $\tau$ is defined as the average time interval between two successive collisions of an electron with the lattice ions.
$1$. Dependence on $E$ field: When an external electric field $E$ is applied,the drift velocity $v_d$ acquired by electrons is very small (order of $10^{-3} \ m/s$) compared to their random thermal velocity (order of $10^5 \ m/s$). Since the collision frequency is primarily determined by the random thermal motion,the relaxation time $\tau$ remains nearly independent of the applied $E$ field. This constancy of $\tau$ ensures that the current density $J$ is directly proportional to $E$ $(J = \sigma E)$,which is the microscopic form of Ohm's law.
$2$. Dependence on $T$: As temperature $T$ increases,the random thermal velocity of electrons increases significantly. This leads to more frequent collisions between electrons and lattice ions,thereby decreasing the relaxation time $\tau$. According to the relation $\rho = \frac{m}{n e^2 \tau}$,since $\tau$ decreases with an increase in temperature,the resistivity $\rho$ of the conductor increases.