# Drift velocity formula derivation

In this post, we will derive the drift velocity formula and understand the concepts of the drift velocity of an electron.

## Drift velocity formula derivation | derive drift velocity equation | derive vd = a τ

The free electrons in metals move at random due to thermal agitation. During this motion, the free electrons collide with stationary positive ions, and the direction of motion of the free electrons changes after each collision. Hence, the average thermal velocity of all the free electrons is zero. It means that there is no net motion of the free electrons in any particular direction.

When a battery is connected across the metal wire, a Potential Difference (PD) is established between the ends of the wire, and an electric field is produced at every point of the wire. Each free-electron experiences an electric force. Due to this electric force, the electrons get accelerated in the direction opposite to the direction of the electric field. The acceleration of the electron of mass m can be expressed as a = F/m = eE/m.

a = F/m = E e/m …………. (1)

Let’s consider an electron under the effect of the applied electric field E.

Let’s τ1 be its relaxation time. Say, the thermal velocity of the electron = u1.

Its acceleration due to the electric force = a

So, the velocity it acquires after time interval = τ1 under the influence of the electric force can be expressed as :

v1 = u1 + a τ1

=> v1 = u1 + a τ1 …………….. (2)

We know, that the average velocity of the electrons is called drift velocity vd. Now, considering N number of electrons we get the drift velocity as:

vd = ( v1 + v2 + …. + vn) / N

vd = [( u1 + a τ1) + (u2 + a τ2) + …. + (un + a τn)] /N

vd = [(u1 + u2 +… + un) /N] + ( aτ1 + aτ2 + ……… + aτn) /N

As, the average thermal velocity of all the free electrons is zero, hence [(u1 + u2 +… + un) /N] = 0

So, vd = ( aτ1 + aτ2 + ……… + aτn) /N

vd = a( τ1 + τ2 + ……… + τn) /N

Here, ( τ1 + τ2 + ……… + τn) /N = average relaxation time = τ

Hence, drift velocity vd = a τ

=> vd = a τ ………. (3)

Drift velocity formula derivation (in terms of relaxation time)

τ = average relaxation time

vd = average velocity of electrons = drift velocity of electrons

vd = a τ

=> vd = ( Ee/m) τ ……………… (4)

Drift velocity formula derivation (in terms of electric field)
Drift velocity formula derivation
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