Drift Velocity:

Drift velocity refers to the average velocity at which charges (such as electrons or ions) move through a conductor in response to an electric field. It is an important concept in understanding the flow of current in a conductor. The expression for drift velocity can be derived using the concept of current density.

Current Density:

Current density is defined as the amount of current flowing per unit cross-sectional area of a conductor. It is denoted by the symbol J and is given by the equation:

J = I / A

where J is the current density, I is the current flowing through the conductor, and A is the cross-sectional area of the conductor.

Charge Carriers and Drift Velocity:

In a conductor, such as a metal wire, there are free charge carriers (electrons or ions) that can move. When an electric field is applied across the conductor, these charge carriers experience a force due to the field and start moving. However, due to the presence of collisions with the lattice structure of the conductor, their motion is not uniform.

Deriving the Expression:

To derive the expression for drift velocity, we consider the average velocity of the charge carriers. The average velocity can be defined as the total distance traveled divided by the total time taken. In the case of charge carriers, the total distance traveled is the length of the conductor, L, and the total time taken is the relaxation time, τ.

Therefore, the average velocity is given by:

v_avg = L / τ

The relaxation time, τ, represents the average time between successive collisions of charge carriers with the lattice structure. It depends on the material properties and temperature.

Relation between Current and Charge Carriers:

The current flowing through a conductor can be expressed as the product of charge carriers and their average velocity. Mathematically, it can be represented as:

I = n * q * A * v_avg

where n is the number density of charge carriers, q is the charge of a single carrier, A is the cross-sectional area of the conductor, and v_avg is the average velocity.

Expressing Drift Velocity:

From the above equations, we can express the drift velocity, v_d, as:

v_d = v_avg = I / (n * q * A)

This expression shows that the drift velocity is inversely proportional to the number density of charge carriers, the charge of a single carrier, and the cross-sectional area of the conductor. It is directly proportional to the current flowing through the conductor.

Conclusion:

In conclusion, the expression for drift velocity is derived by considering the average velocity of charge carriers in a conductor. It is an important parameter that describes the motion of charge carriers in response to an electric field. The expression for drift velocity provides insights into the relationship between current, charge carriers, and the properties of the conductor.