Question

Define the term 'drift velocity' of conduction electrons in a conductor.

Hint:

The drift velocity represents the average speed of electrons in a conductor when subjected to an electric field. The current density depends on the relaxation time, charge density, and the electric field.

Answer 1

Drift velocity is the average velocity with which free electrons move in the direction of the applied electric field in a conductor. When an electric field is applied to a conductor, the conduction electrons experience a force that accelerates them in the direction of the field. However, due to collisions with the atoms of the conductor, their motion is not uniform. The electrons move randomly between collisions but have a net average velocity in the direction of the field, called the drift velocity. Mathematically, the drift velocity \( v_d \) can be given by the relation: \[ v_d = \frac{I}{n e A} \] where: - \( I \) is the current, - \( n \) is the number density of free electrons in the conductor, - \( e \) is the charge of an electron, - \( A \) is the cross-sectional area of the conductor. (b) A conductor of length \( l \) and area of cross-section \( A \) is connected across an ideal battery of emf \( V \). Derive the formula for the current density in terms of relaxation time \( \tau \).
% Solution Solution:
The current density \( J \) is the amount of current flowing per unit area of the conductor, and it is given by: \[ J = \frac{I}{A} \] where: - \( I \) is the current, - \( A \) is the cross-sectional area of the conductor. Now, using Ohm's law, the current \( I \) in the conductor is related to the applied emf \( V \), the resistance \( R \), and the length \( l \) of the conductor by the equation: \[ I = \frac{V}{R} \] The resistance \( R \) of the conductor is given by: \[ R = \frac{\rho l}{A} \] where \( \rho \) is the resistivity of the material. From Ohm's law, we substitute for \( I \) and get: \[ J = \frac{I}{A} = \frac{V}{R A} = \frac{V}{\frac{\rho l}{A} \times A} = \frac{V}{\rho l} \] Now, the resistivity \( \rho \) of a material is related to the relaxation time \( \tau \) by the equation: \[ \rho = \frac{m}{n e^2 \tau} \] where: - \( m \) is the mass of the electron, - \( n \) is the number density of electrons, - \( e \) is the charge of the electron, - \( \tau \) is the relaxation time. Substituting this into the expression for current density \( J \), we get: \[ J = \frac{V}{\left( \frac{m}{n e^2 \tau} \right) l} = \frac{n e^2 \tau V}{m l} \] Thus, the formula for the current density in terms of relaxation time \( \tau \) is: \[ J = \frac{n e^2 \tau V}{m l} \]