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 **v**_{d} = a τ

_{d}= 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 τ_{1} be its relaxation time. Say, the **thermal velocity** of the electron = u_{1}.

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 :

v_{1} = u_{1} + a τ_{1}

**=> v _{1} = u_{1} +**

**a τ**

_{1}…………….. (2)We know, that the average velocity of the electrons is called drift velocity v_{d}. Now, considering N number of electrons we get the drift velocity as:

v_{d} = ( v_{1} + v_{2} + …. + v_{n}) / N

v_{d} = [( **u _{1} +**

**a τ**) + (

_{1}**u**

_{2}+**a τ**) + …. + (

_{2}**u**

_{n}+**a τ**)] /N

_{n}v_{d} = [(**u _{1}** +

**u**

_{2}+… +

**u**

_{n}) /N] + ( aτ

_{1}+ aτ

_{2}+ ……… + aτ

_{n}) /N

As, the **average thermal velocity** of all the free electrons is zero, hence [(**u _{1}** +

**u**

_{2}+… +

**u**

_{n}) /N] = 0

So, v_{d} = ( aτ_{1} + aτ_{2} + ……… + aτ_{n}) /N

v_{d} = a( τ_{1} + τ_{2} + ……… + τ_{n}) /N

Here, ( τ_{1} + τ_{2} + ……… + τ_{n}) /N = average relaxation time = τ

Hence, drift velocity v_{d} = a τ

Drift velocity formula derivation (in terms of relaxation time)

=> v_{d}= a τ ………. (3)

τ = average relaxation time

v_{d} = average velocity of electrons = drift velocity of electrons

v_{d} = a τ

Drift velocity formula derivation (in terms of electric field)

=> v_{d}= ( Ee/m) τ ……………… (4)