The relaxation time in conductors

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?

What is current density? Derive Ohm's law in the form of current density.

Derive the equation of mobility in terms of electric current.

Two cylindrical rods of uniform cross-sectional area $A$ and $2A$,having free electrons per unit volume $2n$ and $n$ respectively,are joined in series. $A$ current $I$ flows through them in a steady state. Then the ratio of the drift velocity of free electrons in the left rod to the drift velocity of electrons in the right rod is $\left( \frac{v_L}{v_R} \right)$.

The current density in a cylindrical wire of radius $r = 4.0 \, mm$ is $J = 1.0 \times 10^{6} \, A/m^{2}$. The current through the outer portion of the wire between radial distances $r/2$ and $r$ is $x \pi \, A$,where $x$ is ..........