A cylindrical metal rod of radius R carries a current I. The velocity of free electrons drifting at a distance r(< R) from the axis of the rod is given by  $\mathrm{v}={\mathrm{v}}_0\left(1+\frac{\mathrm{r}}{\mathrm{R}}\right)$ where v₀ is a constant. The average drift velocity of the electrons drifting in the rod is

Consider a small element of thickness dr at a distance r from the axis of the rod as shown in Fig.

The current flowing through the element is

$\begin{array}{c}\mathrm{dI}=\mathrm{en} \mathrm{dA} v\\ =\mathrm{en}\times 2\mathrm{\pi rdr}\times {\mathrm{v}}_0\left(1+\frac{\mathrm{r}}{\mathrm{R}}\right)\\ =\frac{2{\mathrm{\pi env}}_0}{\mathrm{R}}\left(\mathrm{Rrdr}+{\mathrm{r}}^2\mathrm{dr}\right)\end{array}$

Integrating

$\begin{array}{c}\mathrm{I}=\left(\frac{2{\mathrm{\pi env}}_0}{\mathrm{R}}\right)\left[\mathrm{R}{\int }_0^{\mathrm{R}} \mathrm{rdr}+{\int }_0^{\mathrm{R}} {\mathrm{r}}^2\mathrm{dr}\right]\\ =\frac{2{\mathrm{\pi env}}_0}{\mathrm{R}}\left(\frac{{\mathrm{R}}^3}{2}+\frac{{\mathrm{R}}^3}{3}\right)\\ =\mathrm{en}{\mathrm{\pi R}}^2{\mathrm{v}}_0\left(\frac{5}{3}\right)\\ \mathrm{I}=\text{ en }\mathrm{A}\times \frac{5{\mathrm{v}}_0}{3}\end{array}$

Comparing with $\mathrm{I}=\mathrm{enA} {\mathrm{v}}_{\mathrm{d}}$, we get

                      ${\mathrm{v}}_{\mathrm{d}}=\frac{5{\mathrm{v}}_0}{3}$