Question

A conducting wire of length L, uniform area of cross-section A, and material having n free electrons per unit volume offers a resistance R to the flow of current. m and e are the mass and charge of an electron, respectively. If τ is the mean free time of the electrons in the conductor, the correct formula for resistance R is:

• A. $\quad R = \frac{mL}{e^2 n A \tau} \\\\$
• B. $\quad R = \frac{mA}{e^2 n L \tau} \\\\$
• C. $\quad R = \frac{m \tau}{e^2 n A L} \\\\$
• D. $\quad R = \frac{e^2 n A \tau}{m L}$

Answer 1

Answer: (A)

$\quad R = \frac{mL}{e^2 n A \tau} \\\\$

Answer 2

The resistance R of a conductor can be derived using Drude’s model of electrical conductivity, which relates the electrical properties of materials to their microscopic structure.
According to Drude’s model, the resistivity $\rho$ is given by
$\rho = \frac{m}{n e^2 \tau}$
where:-
m is the mass of an electron,
n is the number density of free electrons,
e is the charge of an electron, and τ is the mean free time between collisions.
The relation between resistivity and resistance is:
$R = \rho \frac{L}{A}$

$R = \frac{m}{n e^2 \tau} \frac{L}{A}$

$\text{Simplifying:} \quad R = \frac{mL}{n e^2 A \tau}$

$\text{Thus, the correct formula for the resistance is } \frac{mL}{n e^2 A \tau}, \text{ which corresponds to option (1).}$