Question

The electrical conductivity of a metal is

• A. directly proportional to the mean free path
• B. directly proportional to the mass of electron
• C. inversely proportional to the relaxation time
• D. inversely proportional to the mean free path
• E. directly proportional to the average speed of free electrons

Hint:

Relaxation time ($\tau$) and mean free path ($\lambda$) are both measures of how "easy" it is for electrons to flow. Higher values for either will increase conductivity.

Answer 1

The Correct Option is A

Concept:
Electrical conductivity ($\sigma$) is defined as $\sigma = \frac{ne^2\tau}{m}$. The relaxation time ($\tau$) is the average time between collisions, related to the mean free path ($\lambda$) by $\tau = \frac{\lambda}{v_d}$.

Step 1: Analyzing the proportionality.
Substituting $\tau$ into the conductivity formula: \[ \sigma = \frac{ne^2}{m} \left( \frac{\lambda}{v_d} \right) \] From this, we observe that $\sigma \propto \lambda$.

Step 2: Evaluating options.
A longer mean free path means electrons travel further without scattering, which directly increases the conductivity of the material. This makes option (A) correct.