# Relation Between Current and Drift Velocity - Derivation

The concept of the relation between electric current and drift velocity is essential in the study of electrical conductivity, the behavior of electrons in electrical devices and circuits, and current flow in conductors. In this article, we will learn about the relation between electric current and drift velocity. Before that let us first learn about definitions of current and drift velocity.

## Electrical Current :

The displacement or flow of charged particles usually electrons, through a conductive material is called an electric current. It is defined as the rate at which electric charge flows. The unit of electric charge is Coulomb, hence Coulomb per second is the unit of electric current generally known as Ampere.

Current density is the measure of electric current per unit cross-section area of conductor denoted by the symbol J. The formula of current density (J) is given as J = I/A, where I is the amount of current and A is the area of the cross-section of the conductor. The unit of current density is amperes per square meter (A/m2).

## Drift Velocity :

Drift velocity is defined as the average velocity that a charged particle such as electrons attains in a conductor due to an electric field. In other words, the drift velocity of electrons refers to the slow movement of free electrons in the conductor towards the positive terminal when a voltage or emf is applied.

When an electric field is applied across a conductor, the free electrons moving in random directions exert a force in the direction of the applied field. Due to this force, the free electrons with their random movement shift towards the higher potential of the conductor. As a result, each drifting electron achieves a net velocity towards the higher potential of the conductor. This net velocity acquired by each electron is said to be known as the Drift Velocity of electrons.

However, when electrons drift they collide with other particles, and the imperfections in the material cause obstruction to the motion of electrons. Due to this, the electrons do not achieve a constant or uniform velocity, but exhibit a random motion with an overall average velocity towards the higher potential of the conductor.

The drift velocity depends upon factors such as the number of free electrons per unit volume in the material, the strength of the electric field, characteristics of the material, conductivity, and mobility of charge carriers.

## Relation Between Current and Drift Velocity :

Let us consider a conductor of length L with cross-sectional area A connected across the battery of voltage V. The free electrons in the conductor start drifting towards the positive terminal due to the electric field set up by the battery as shown in the below figure.

#### In the above figure,A = Area of the cross-section of the wire,L = length of a portion of conductor,n = Electron density (number of free electrons per unit volume),e = Charge on each electron,vd = Drift velocity of free electrons.

Current flowing through the conductor is given by,

I = Total Charge / Time ...(1)

In the above figure, the total number of free electrons within the length L flowing through the cross-section at P will be nAL. Therefore, the total free charge flowing through the cross-section at P is given as,

Q = nAL × e ...(2)

The time taken by the above free charge to pass through the cross-section at P is given as,

T = L/vd ...(3)

Substituting equation 2 and 3 in equation 1 we get,

I = Q/t = nALe/(L/vd)
∴ I = nAevd

### Example :

A copper wire with an area of cross-section of 4mm2 and a length of 4m long carries a current of 10A. The density of free electrons in the conductor is 8 × 1028 m-3. How much time is required by an electron to travel the length of the wire?

#### Given,Current I = 10A,Area of cross-section A = 4mm2 = 4 × 10-6 m2,Electron density n = 8 × 1028 m-3,Charge on each electron e = 1.6 × 10-19 C.

Thus, drift velocity is given by,

vd = I/nAe = 1.95 × 10-4 ms-1

Therefore, the time taken by an electron to travel the length of the wire is,

t = L/vd = 4/1.95×10-4 = 2.05 × 104s = 5.7hours

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