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

A conducting wire has a length \(L\), uniform area of cross-section \(A\), and is made of a material with \(n\) free electrons per unit volume. The resistance \(R\) of the wire is associated with these parameters, along with the electron mass \(m\), electron charge \(e\), and the mean free time \(\tau\) of the electrons. Our objective is to derive the formula for \(R\). 

1. Drift Velocity and Current Density: The current density \(J\) can be expressed by the formula:

\[J=n \cdot e \cdot v_d\]

where \(v_d\) is the drift velocity. Ohm's Law states:

\[J=\sigma \cdot E\]

where \(\sigma\) is the conductivity of the material, and \(E\) is the electric field.

2. Conductivity and Relaxation Time: The relationship between conductivity and relaxation time \(\tau\) is:

\[\sigma=\frac{n e^2 \tau}{m}\]

Therefore, Ohm's Law becomes:

\[J=\frac{n e^2 \tau}{m} \cdot E\]

Equating the two expressions for current density, we get:

\[n e v_d=\frac{n e^2 \tau}{m} \cdot E\]

3. Electric Field and Voltage: For a conductor of length \(L\), the relation between electric field \(E\) and voltage \(V\) is:

\[E=\frac{V}{L}\]

Substituting back, we have:

\[v_d=\frac{e \tau}{m} \cdot \frac{V}{L}\]

4. Resistivity to Resistance Conversion: The resistivity \(\rho\) is the reciprocal of conductivity:

\[\rho=\frac{m}{n e^2 \tau}\]

Resistance \(R\) is given by:

\[R=\rho \cdot \frac{L}{A}\]

Substituting for \(\rho\):

\[R=\frac{m}{n e^2 \tau} \cdot \frac{L}{A}\]

Simplifying gives:

\[R=\frac{mL}{e^2 n A \tau}\]

5. Conclusion: The correct formula for resistance \(R\) is:

\[R=\frac{mL}{e^2 n A \tau}\]

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

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

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

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