Relation between Electric Current and Drift Velocity

In this article, we discuss the electric current, the drift velocity of electrons, current density and the relation between electric current and drift velocity, and the relation between current density and drift velocity.

Electric Current

The rate of change flow of electric charge is called electric current. The electric current is denoted by symbol ‘I (constant current)’ and ‘i (time-varying current)’. Mathematically, the electric current is given by the electric charge divided by time, i.e.

`\I=Q/t`

Where, Q is the electric charge in coulomb and t is the time in seconds. Therefore, the current is measured in coulomb per second (C/s), where,

1 C/s = 1 Ampere

Drift Velocity

The drift velocity of free electrons is defined as the average velocity with which the free electrons get drifted, i.e. moved in a metallic conductor in a specific direction under the influence of an electric field. It is denoted by v_d and is measured in meters per second.

Generally, the drift velocity of the free electrons is of the order of 10^-5 m/s. However, the drift velocity of electrons is very small, but it is entirely responsible for the flow of electric current in the metallic conductors.

Relation between Current and Drift Velocity

Consider a small portion of a metallic conductor (say copper wire) through which a current of I amperes is flowing as shown in the following figure.

Let,

l = length of the wire

A = area of cross-section of wire

Therefore, the volume (V) of the conductor wire is

`\V=A×l`

If n is the electron density, i.e. number of free electrons per unit volume of the conductor, then,

Total number of electrons in the conductor = n × A × l

Therefore,

Total charge in the conductor, Q = n × A × l × e

Where, e is the charge on one electron and is equal to –1.6×10^-19 C.

If the electron takes t seconds to cross the conductor, then we have,

`\t=l/v_d`

Where, v_d is the drift velocity of free electrons.

Now, by the definition of electric current, we have,

`\I=Q/t`

`\I={n eAl}/((l⁄v_d ) )`

Therefore, the relation between current and drift velocity will be,

`\∴I=n eAv_d`

Since, for a given conductor n, e, A are constant. Therefore,

`\I∝v_d`

i.e. the electric current flowing through a conductor is directly proportional to the drift velocity of free electrons.

Current Density

The current density is defined as the current per unit area. It is denoted by the symbol J and is measured in Ampere per square meter (A/m²).

`\J=I/A`

Hence, the relation between current density and drift velocity can be established as follows,

`\J=(n eAv_d)/A`

`\∴J=n ev_d`

Numerical Example – The density of conduction electrons in a copper wire is `\8.5×10^28" m"^(-3)`. If the radius of the wire is 0.8 mm and it is carrying a current of 3 A, what will be the average drift velocity of electrons?

Solution – Given data,

`\n=8.5×10^28" m"^(-3)`

`\I=3" A"`

`\r=0.8" mm"`

Therefore, the area of cross-section (A) of the wire is

`\A=πr^2`

`\⟹A=π×(0.8×10^(-3) )^2`

`\⟹A=2×10^(-6)" m"^2`

Now, since we know that the relation between current and drift velocity is given by,

`\I=n eAv_d`

Therefore, the drift velocity of electrons is given by,

`\v_d=I/{n eA}`

`\⟹v_d=3/((8.5×10^28 )×(1.6×10^(-19) )×(2×10^(-6) ) )`

`\∴v_d=1.1×10^(-4)" ms"^(-1)`

Hence, in this article, we discussed about the electric current, drift velocity, and the relationship between current and drift velocity.