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

The Correct Option is A

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:
\(R = \rho \frac{L}{A}\)

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

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

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

Answer 2

The resistance R of a conductor can be expressed in terms of electron properties as:

R = ρ(L/A)

where ρ is the resistivity given by:

ρ = m/(ne²τ)

Substituting ρ into the resistance formula:

R = [m/(ne²τ)] × (L/A)

R = mL/(ne²Aτ)

Rearranging the equation to match the given options:

R = mL/(e²nAτ)

Comparing with the given options, the correct formula is:

R = mL/(e²nAτ)