# Drift velocity Derivation

Last updated on May 29th, 2023 at 05:37 am

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

**Drift velocity formula**

The drift velocity formula in terms of Relaxation Time is as follows: **v _{d} = a τ** [we will derive this here]

The drift velocity formula in terms of Electric Field is as follows: ** v_{d} = ( Ee/m) τ** [we will derive this here]

The drift velocity formula in terms of electric current: **v _{d}**

**= I/[neA]**. We have a separate post on this here: Derive drift velocity equation in terms of electric current

## Drift velocity derivation | derive drift velocity in terms of relaxation time | 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 the 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 derivation in terms of electric field**

From equation (1) we get **a = F/m = E e/m**. Putting this in equation (3) we get another equation of drift velocity as follows:

Drift velocity formula derivation (in terms of electric field)

=> v_{d}= ( Ee/m) τ ……………… (4)

**Drift velocity derivation in terms of electric current**

We have a separate post on this. Read it here: Derive drift velocity equation in terms of electric current