Question

What is meant by 'relaxation time' of free electrons in a conductor? Show that the resistance of a conductor can be expressed by \( R = \frac{mL}{n e^2 \tau A} \), where symbols have their usual meanings.

Hint:

Understanding the concept of relaxation time helps in comprehending how material impurities and temperature affect the resistance and overall electrical conductivity.

Answer 1

% Explanation of Relaxation Time Relaxation Time: Relaxation time (\( \tau \)) is the average time interval between consecutive collisions of an electron as it moves through a conductor. It is a measure of how long an electron travels freely without interaction, influencing the electrical conductivity of the material.
% Derivation of the Resistance Formula Derivation:
The resistance \( R \) of a conductor can be derived using the Drude model, which relates electrical properties to the behavior of electrons in a material. The resistivity \( \rho \) according to the Drude model is given by:
\[ \rho = \frac{m}{n e^2 \tau} \] where \( m \) is the electron mass, \( n \) is the density of charge carriers, \( e \) is the electron charge, and \( \tau \) is the relaxation time. The formula for resistance \( R \), involving the geometry of the conductor, is:
\[ R = \rho \frac{L}{A} \] Substituting the expression for \( \rho \) gives:
\[ R = \frac{m}{n e^2 \tau} \frac{L}{A} = \frac{mL}{n e^2 \tau A} \] This equation illustrates that resistance is inversely proportional to the density of charge carriers and their relaxation time, and directly proportional to the conductor's length.