In the application of an external **electric field**, free electrons move with some velocity in a conductor. The velocities of all the electrons are not equal at every point in the conductor at every time. So, we need to consider the average value of the velocity of electrons. This is nothing but the **drift velocity** of the electrons inside the conductor. In this article, we are going to discuss the definition and expression for drift velocity of electrons in a conductor. This formula for drift velocity will give the relation between the drift velocity and electric **current density**.

**Contents in this article:**

**What is drift velocity?****Derive the expression for drift velocity of electrons in a conductor****Relation between drift velocity and current****Relation between drift velocity and electric field****Relaxation time and Drift velocity**

**Drift velocity of electron**

Drift velocity of a free electron is defined as the average velocity of the electron with which it moves through a conductor to carry an electric current. We need to consider the average velocity of electrons because the velocity of an electron is not the same at every point in a conductor. Again, the velocities of all the electrons are not equal at any time or any point in the conductor.

**Expression for drift velocity**

Let electrons are moving through a current-carrying wire of current **I**. This wire can be considered as a cylinder of cross-section area **A.** We consider that **n** number of electrons of charge **e** is moving per unit volume of the wire with the drift velocity **V _{d}. **Since

**V**is the drift velocity, it implies that the electrons move

_{d}**V**length of the wire per unit time. So, the volume of occupation of the electrons per unit time is

_{d}**A**.

**V**_{d}So, the total number of electrons crossing the volume **AV _{d}** per unit time is

**N=nA**

**V**_{d}Now, each electron has the charge of **e**. So, the amount of electric charge crossing the volume per unit time is **eN = enA V_{d}**

Since the amount of charge flowing per unit time is the electric current then the electric current through the wire is, **I=enA V_{d}**

or, the drift velocity, ** V_{d}=I/enA** ………………… (1)

This is the relation between drift velocity and current.

Again, the current per unit area is the current density (**J**). So, **J=I/A**.

So, another expression for drift velocity of electrons in a wire is, ** V_{d}=J/en **……………… (2)

**en **is the charge density. So, one can use \small {\color{Blue} \rho =en} in equation (1) and (2).

**Unit of drift velocity** **of electrons**

Drift velocity is one kind of velocity. So, its unit is the same as the unit of normal velocity. SI unit of drift velocity is m/s and CGS unit of drift velocity is cm/s.

**Relation between drift velocity and current**

The relation between drift velocity and electric current is, ** V_{d}=I/enA** or,

**I=enAV**

_{d}Again, charge density, \small \rho=en . Then, the relation between drift velocity and electric current is, I=\small \rhoAV_{d}

**Relation between drift velocity of electrons and electric current density**

current density, **J=I/A**

Then the relation between drift velocity and current density is, ** V_{d}=J/en** or,

**J=enV**

_{d}Again, en is the charge density. So, **J=\small\rhoV _{d}. **This is the relation between the current density and charge density.

**Relation between drift velocity and electric field**

From the differential form of Ohm’s law we get the relation between the current density and **electric field** is **J=\small\sigmaE**

Then, from above relation, **\small\sigmaE=enV _{d} **or, drift velocity,

**V**

_{d}=\small\frac{\sigma}{en}EAgain, **Mobility**, \small\mu=\frac{\sigma}{en}

Then, relation between the drift velocity, electric field and the **mobility of electron** is, **V _{d}=**\small\mu

**E**

**Relation between drift velocity and relaxation time**

**Relaxation time** is the average time interval between the two successive collisions of an electron with other electrons during their motion in the conductor.

If a voltage **V **is applied across a conducting wire of length **L**, then electric field **E=V/L **will appear inside the conductor. Let an electron of charge **e** (magnitude) is moving under the electric field **E**. Then the electric force on the charge is **F=eE**.

If **m** be the mass of the electron then the acceleration of the electron is, **a=F/m =eE/m**

Now, if **t **be the relaxation time then the average velocity of electrons between two successive collisions or the drift velocity of electrons is,

**V _{d}=at**

or, drift velocity, **V _{d}=eEt/m** ……………… (3)

This is the relation between the drift velocity of electron and the relaxation time.

This is all from this article on the expression for drift velocity of electrons in a conductor. If you have any doubt on this topic you can ask me in the comment section.

Thank you!

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