Relation between drift velocity and current density in a metallic conductor

Introduction
In a metallic conductor, the flow of electric charge is carried by free electrons. When an electric field is applied to the conductor, these free electrons experience a force that causes them to move in a particular direction. This movement of electrons is responsible for the flow of electric current. Two important parameters that describe the behavior of current in a metallic conductor are drift velocity and current density.

Drift Velocity
The drift velocity of electrons in a metallic conductor refers to the average velocity at which they move in response to an applied electric field. It is a measure of the speed at which electrons move through the conductor. The drift velocity is directly proportional to the strength of the electric field and is influenced by factors such as temperature and the density of free electrons in the conductor.

Current Density
Current density, on the other hand, is a measure of the amount of current flowing through a specific area of the conductor. It is defined as the ratio of the current flowing through a cross-sectional area of the conductor to the area itself. Current density is denoted by the symbol 'J' and is expressed in amperes per square meter (A/m^2).

Relation between Drift Velocity and Current Density
The relationship between drift velocity (v_d) and current density (J) in a metallic conductor can be described by the following equation:

J = n * e * v_d

where:
- J is the current density
- n is the number density of free electrons in the conductor
- e is the charge of an electron
- v_d is the drift velocity

Explanation
The equation shows that the current density is directly proportional to both the number density of free electrons and the drift velocity. This means that an increase in either the number of free electrons or the speed at which they move will result in an increase in the current density.

The number density of free electrons mainly depends on the material and its atomic structure. Materials with a higher number density of free electrons will have a larger current density for the same drift velocity.

Similarly, the drift velocity is influenced by the strength of the applied electric field. A stronger electric field will cause the electrons to move faster, resulting in a higher drift velocity and consequently a higher current density.

Conclusion
In conclusion, the drift velocity and current density in a metallic conductor are related through the equation J = n * e * v_d. The current density is directly proportional to both the number density of free electrons and the drift velocity. Understanding this relationship is crucial in analyzing and predicting the behavior of electric current in metallic conductors.