Explain the drift of electrons and drift velocity. Derive the equation for electric current in terms of the cross-sectional area of a conductor.

(N/A) $1$. Drift of Electrons: In a conductor,free electrons move randomly due to thermal energy and collide with positive ions. Their average velocity is zero. When an external electric field $E$ is applied,these electrons experience a force $F = -eE$,causing them to drift slowly in the direction opposite to the electric field. This net slow motion is called drift.
$2$. Drift Velocity $(v_d)$: It is defined as the average velocity acquired by free electrons in a conductor under the influence of an external electric field. If $\tau$ is the average relaxation time,then $v_d = -\frac{eE\tau}{m}$.
$3$. Derivation of Current $(I)$: Consider a conductor of cross-sectional area $A$ and length $L$. Let $n$ be the number density of free electrons. The total number of electrons in a volume $A \Delta x$ is $n A \Delta x$. Since $\Delta x = v_d \Delta t$,the total charge $\Delta Q$ passing through the cross-section in time $\Delta t$ is $\Delta Q = (n A v_d \Delta t) e$. The current $I$ is given by $I = \frac{\Delta Q}{\Delta t} = n e A v_d$.