Drift velocity refers to the average speed at which free electrons move within a conductor, influenced by an applied electric field. When an electric field is established across a conductor, conduction electrons are subjected to a force that causes them to accelerate in the direction of the field. However, these electrons do not move uniformly due to frequent collisions with the conductor's atoms. Consequently, while their movement between collisions is random, they possess a net average velocity in the field's direction, which is termed drift velocity.
Mathematically, drift velocity \( v_d \) is expressed as:
\[
v_d = \frac{I}{n e A}
\]
where:
- \( I \) denotes the current.
- \( n \) represents the number density of free electrons in the conductor.
- \( e \) signifies the charge of an electron.
- \( A \) is the cross-sectional area of the conductor.
(b)
Derive the formula for current density in terms of relaxation time \( \tau \) for a conductor of length \( l \) and cross-sectional area \( A \), connected to an ideal battery of emf \( V \).
% Solution
Solution:
Current density \( J \) is defined as the current flowing per unit area of the conductor and is given by:
\[
J = \frac{I}{A}
\]
where:
- \( I \) is the current.
- \( A \) is the cross-sectional area of the conductor.
According to Ohm's law, the current \( I \) in the conductor is related to the applied emf \( V \), resistance \( R \), and length \( l \) by the equation:
\[
I = \frac{V}{R}
\]
The resistance \( R \) of the conductor is determined by:
\[
R = \frac{\rho l}{A}
\]
where \( \rho \) is the resistivity of the material.
Substituting for \( I \) from Ohm's law, we obtain:
\[
J = \frac{I}{A} = \frac{V}{R A} = \frac{V}{\frac{\rho l}{A} \times A} = \frac{V}{\rho l}
\]
The resistivity \( \rho \) of a material is linked to the relaxation time \( \tau \) by the equation:
\[
\rho = \frac{m}{n e^2 \tau}
\]
where:
- \( m \) is the mass of an electron.
- \( n \) is the number density of electrons.
- \( e \) is the charge of an electron.
- \( \tau \) is the relaxation time.
Substituting this expression for \( \rho \) into the formula for current density \( J \):
\[
J = \frac{V}{\left( \frac{m}{n e^2 \tau} \right) l} = \frac{n e^2 \tau V}{m l}
\]
Therefore, the formula for current density in terms of relaxation time \( \tau \) is:
\[
J = \frac{n e^2 \tau V}{m l}
\]