In absence of Electric field, electrons in a
conductor will be moving due to thermal motion during which they collide with
the fixed ions. An electron colliding with an ion emerges with the same speed
as before the collision. However, the direction of its velocity after the
collision is completely random.

If we consider all the electrons, their average
velocity will be zero since their directions are random. Thus, if there are N
electrons and the velocity of the ith electron (i = 1, 2, 3, ... N) at a given
time is v

_{i,}then
Where –e is the charge and m is the mass of an
electron.

Consider again the ith electron at a given time t.
This electron would have had its last collision some time before t, and let

**be the time elapsed after its last collision. If***t*_{i}**was its velocity immediately after the last collision, then its velocity***v*_{i}**V**at time t is_{i}
The average of

**’s is zero since immediately after any collision, the direction of the velocity of an electron is completely random.***v*_{i}
Let τ is the time between to collisions. The average value of t

_{i}then is τ (known as**relaxation time**).
Thus, averaging over the N-electrons at any given time t gives
us for the average velocity

*v*_{d}
This is the phenomenon of drift and the velocity

**v**_{d}_{ }is called the**drift velocity.**
Because of the drift, there will be net transport
of charges across any area perpendicular to E. Consider a planar area A,
located inside the conductor such that the normal to the area is parallel to E
amount of time ∆t, all electrons to the left of the area at distances upto

**would have crossed the area.***|v*_{d}|∆t
If n is the number of free electrons per unit
volume in the metal, then there are

**A such electrons.***n ∆t |v*_{d}|
Since each electron carries a charge –e, the total
charge transported across this area A to the right in time ∆t is

**. E is directed towards the left and hence the total charge transported along E across the area is negative of this.***–ne A|v*_{d}|∆t
It's a nice post about Drift Velocity. It's really helpful. Thanks for sharing it.

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