Question

In a metal, the charge carrier density is \( 9.1 \times 10^{28} \, \text{m}^{-3} \) and its electrical conductivity is \( 6.4 \times 10^7 \, \text{S m}^{-1} \). When an electric field of \( 10 \, \text{N C}^{-1} \) is applied to the metal, then the average time between two successive collisions of electrons in the metal is:
(Mass of electron \( = 9.1 \times 10^{-31} \, \text{kg} \); charge of electron \( = 1.6 \times 10^{-19} \, \text{C} \))

• A. \( 4.6 \times 10^{-14} \, \text{s} \)
• B. \( 2.5 \times 10^{-13} \, \text{s} \)
• C. \( 4.6 \times 10^{-13} \, \text{s} \)
• D. \( 2.5 \times 10^{-14} \, \text{s} \)

Hint:

The average time between electron collisions can be found using the relation \( \sigma = \frac{n e^2 \tau}{m} \). Rearranging and substituting accurately gives the result.

Answer 1

The Correct Option is D

Step 1: Use the formula for conductivity.
Electrical conductivity \( \sigma \) is given by: \[ \sigma = \frac{n e^2 \tau}{m} \] Where: \( n = 9.1 \times 10^{28} \, \text{m}^{-3} \) 
\( e = 1.6 \times 10^{-19} \, \text{C} \) 
\( m = 9.1 \times 10^{-31} \, \text{kg} \) 
\( \sigma = 6.4 \times 10^7 \, \text{S m}^{-1} \) 
\( \tau = \) average collision time (to be found) 
Step 2: Rearranging to find \( \tau \).
\[ \tau = \frac{m \sigma}{n e^2} \] Step 3: Substitute the known values.
\[ \tau = \frac{(9.1 \times 10^{-31}) (6.4 \times 10^7)}{(9.1 \times 10^{28})(1.6 \times 10^{-19})^2} \] \[ \tau = \frac{5.824 \times 10^{-23}}{(9.1 \times 10^{28}) (2.56 \times 10^{-38})} \] \[ \tau = \frac{5.824 \times 10^{-23}}{2.33 \times 10^{-9}} = 2.5 \times 10^{-14} \, \text{s} \] Step 4: Select the correct option.
Thus, the correct average collision time is \( 2.5 \times 10^{-14} \, \text{s} \), which is option (4).