The average velocity of all the free electrons in the conductor is called the drift velocity of free electrons of the conductor. When a conductor is connected to a source of emf an electric field is established in the conductor, such that E=V/L
When V= potential difference across the conductor and L=length of the conductor
The electric field exerts an electrostatics force ‘-Ee’ on each free electron in the conductor
The acceleration of each electron is given by
$$
\bar{a}=-\frac{e E}{m}
$$
Where, e=electric charge on the electron and $\mathrm{m}=$ mass of electron
Acceleration and electric field are in opposite directions, so the electrons attain a velocity in addition to thermal velocity in the direction opposite to that of electric field.
$$
\begin{array}{l}
\overrightarrow{\mathrm{v}}_{\mathrm{d}}=\frac{\mathrm{cE}}{\mathrm{m}} \tau_{\ldots \ldots \ldots \ldots \ldots(\mathrm{i})} \\
\mathrm{E}=\frac{-\mathrm{V}}{\mathrm{L}} \ldots \ldots \ldots \ldots(\mathrm{ii})
\end{array}
$$
Where $\tau=$ relaxation time between two successive collision
Let $\mathrm{n}=$ number density of electrons in the conductor
No. of free electrons in the conductor =nAL
Total charge on the conductor, $\mathrm{q}=$ nALe
Time taken by this charge to cover the length $\mathrm{L}$ of the conductor, $\mathrm{t}=\frac{\mathrm{L}}{\mathrm{v}_{\mathrm{d}}}$
Current $\mathrm{I}=\frac{\mathrm{q}}{\mathrm{t}}$
$$
\begin{array}{l}
=\frac{\mathrm{n} \mathrm{A} \mathrm{Le}}{\mathrm{L}} \times \mathrm{v}_{\mathrm{d}} \\
=\mathrm{n} \mathrm{Aev}_{\mathrm{d}}
\end{array}
$$
Using equation (i) and (ii), we get that
$$
\begin{array}{l}
\mathrm{I}=\mathrm{nAe} \times\left(-\frac{\mathrm{e}(-\mathrm{V})}{\mathrm{mL}} \tau\right) \\
=\left(\frac{\mathrm{ne}^{2} \mathrm{~A}}{\mathrm{~mL}} \tau\right) \mathrm{V}
\end{array}
$$
​