Question

Given below are two statements, one is labelled as Assertion (A) and other one labelled as Reason (R).
Assertion (A): In the absence of an electric field, the electron gas is in an equilibrium state described by equilibrium distribution functions, viz the fermi-Dirac distribution function for a degenerate electron gas and Maxwell-Boltzmann distribution function for a non-degenerate electron gas.
Reason (R): In a conductor, the number of electrons moving in opposite directions is always the same, their average velocity in any direction is zero and consequently, the distribution functions are symmetric about the axis of ordinates.
In light of the above statements, choose the correct answer from the options given below.

• A. (A) is false but (R) is true.
• B. (A) is true but (R) is false.
• C. Both (A) and (R) are true but (R) is NOT the correct explanation of (A).
• D. Both (A) and (R) are true and (R) is the correct explanation of (A).

Hint:

For Assertion-Reason questions, follow a three-step process: 1. Is A true? 2. Is R true? 3. Does R correctly explain A? In this case, equilibrium (A) is physically manifested by the symmetric random motion and zero net velocity (R).

Answer 1

The Correct Option is D

Step 1: Concept Overview:
The problem requires evaluating an Assertion and a Reason concerning the statistical properties of electrons in a conductor at thermal equilibrium (without external fields).
Step 2: Detailed Analysis:
Assertion (A): The assertion states that, lacking an external field, the electron gas is in equilibrium and can be statistically described. It correctly identifies the Fermi-Dirac distribution for a degenerate electron gas (typical of metals) and the Maxwell-Boltzmann distribution for a non-degenerate gas (relevant for semiconductors under specific conditions or classical gases). This is a fundamental principle in statistical mechanics and solid-state physics. Therefore, Assertion (A) is true.
Reason (R): The reason describes the electron gas's equilibrium state. Without an electric field, electron motion is random. For each electron with velocity \(\vec{v}\), another exists with velocity \(-\vec{v}\). This equal distribution in opposite directions results in a zero average velocity and no net current. This symmetry (equal probability for \(+v_x\) and \(-v_x\)) implies a symmetric velocity distribution function. This accurately depicts the microscopic equilibrium. Therefore, Reason (R) is true.
Connection: Reason (R) explains {why} the system is in equilibrium. The symmetric velocity distribution, leading to zero average velocity, defines the equilibrium state described mathematically by the distribution functions in Assertion (A). The symmetric distribution and zero net flow are precisely what the equilibrium Fermi-Dirac function represents. Hence, (R) correctly explains (A).
Step 3: Conclusion:
Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A).