# Drift Velocity of Electrons with Derivation

Every material is made up of atoms which are in turn composed of negatively charged electrons. These negatively charged electrons move in random directions within the atom. This movement of electrons generates electricity. But due to their random motion, the average velocity of electrons in a material becomes zero. It was observed that when a potential difference is applied to the ends of a material, electrons present in the material acquire a certain amount of velocity which causes a small net flow in one direction. This velocity that causes the electrons to move in a certain direction is known as Drift Velocity.

## What is A Drift Velocity?

The average velocity attained by random moving electrons when the external electric field is applied, which causes the electrons to move towards one direction is called the Drift Velocity.

Every conductor material contains free, randomly moving electrons at a temperature above the absolute zero. When the external electric field is applied around the material the electrons attain velocity and tend to move towards the positive direction, and the net velocity of the electrons will be in one direction. The electron will move in the direction of the applied electric field. Here electron does not give up its randomness of motion but shifts towards higher potential with their random motion.

The current produced by this movement of electrons towards the higher potential is called the Drift Current. Thus, one can say that every current produced in a conductor material is a Drift Current.

### Derivation

To derive the expression for drift-velocity, its relation with the mobility of electrons and the effect of applied external electric field have to be known. Mobility of an electron is defined as it’s Drift Velocity for a unit electric field. The electric field is proportional to the current. Thus the Ohm’s law can be written as

**F = -μE.——(1)**

where μ is the mobility of the electron measured as m^{2}/ V.sec

E is the electric field measured as V/m

we know that F =ma, substitute in (1)

**a = F/m = -μE/m———-(2)**

final velocity u = v+at

Here v = 0, t = T, which is the relaxation time of electron

Therefore u =aT, substitute in (2)

**∴ u =-(μE/m)T**

Here, u is the Drift velocity, measured as m/s.

This gives the final expression for the Drift Velocity.

**Formula**

This formula is used to find the drift velocity of electrons in a current-carrying conductor. When electrons with density n and charge Q causes a current ‘I’ to flow through a conductor of cross-sectional area A, Drift velocity v can be calculated through the formula I = nAvQ.

An increase in the applied external electric field intensity causes the electrons to accelerate more rapidly towards a positive direction, opposite to the direction of the applied electric field.

### Relationship with Electric Current

Every conductor contains randomly moving free electrons in it. Movement of electrons in one direction caused by the Drift velocity generates current. The drift velocity of an electron is very small usually in terms of 10^{-1}m/s. Thus, with this amount of velocity, it will take an electron usually 17 mins to pass through a conductor of length one meter.

That means if we switch on an electric bulb it should turn on after 17 mins. But we can turn on the electric bulb in our home at a lightning speed with a flick of a switch. This is because the speed of the electric current does not depend on the drift velocity of the electron.

Electric current moves with a speed of light. It is not established with the drift velocity of the electrons in the material. Thus, the drift velocity may vary in material but the speed of electric current always established on the speed of light.

### Relationship with Current Density

Current density is defined as the total amount of current passing per unit time through per unit cross-sectional area of the conductor. From the formula of drift velocity, the current is given as

**I = nAvQ**

so, the current density J when cross-sectional area and drift velocity are given can be calculated as

**J= I/A =nvQ**

where v is the drift velocity of the electrons. Current density is measured as Amperes per square meter. Thus, from the formula, it can be said that the Drift velocity of the electrons of a conductor and its current density is directly proportional to each other. As the Drift-velocity increases with the increase in the electric field intensity, the current flowing through per cross-sectional area also increases.

### Relationship with Relaxation Time

In a conductor, electrons move randomly as gas molecules. During this motion, they collide with each other. The relaxation time of the electron is the time required by the electron to return to its initial equilibrium value after the collision. This relaxation time is directly proportional to the applied external electric field strength. Larger the electric field time, more the time needed by electrons to come to initial equilibrium after the field is removed.

Relaxation time is also defined as the time for which the electron can move freely between successive collisions with other ions.

When the force due to the applied electric field is eE, then V can be given as

**V = (eE/m)T**

where T is the relaxation time of the electrons.

**Expression**

When the mobility μ of the charge carriers and the strength of the applied electric field E are given, then Ohm’s law in terms of drift velocity can be expressed as

**V = μE**

The S.I units for the mobility of electron are m^{2}/V-s.

S.I units of electric field E are V/m.

Thus the S.I unit for v is m/s. This S.I unit is also known as the Axial Drift Velocity.

Electrons present in the conductor move randomly even when no external electric field is applied. But the net velocity produced by them get canceled due to random collisions so, the net current will be zero. Thus, the relationship between electric current, current density and drift-velocity helps in the proper flow of electric current through the conductor. What is a Drift current?