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: Understanding the Concept:
This question asks to evaluate an Assertion and a Reason related to the statistical description of an electron gas in a conductor in thermal equilibrium (i.e., no external fields).
Step 2: Detailed Explanation:
Assertion (A): This statement claims that in the absence of an external field, the electron gas is in equilibrium and can be described by statistical distribution functions. It correctly identifies the Fermi-Dirac distribution for a degenerate electron gas (which is the case for metals) and the Maxwell-Boltzmann distribution for a non-degenerate gas (which applies to electrons in semiconductors under certain conditions or classical gases). This statement is a fundamental concept in statistical mechanics and solid-state physics. Thus, Assertion (A) is true.
Reason (R): This statement describes the physical situation of the electron gas in equilibrium. In the absence of an electric field, the motion of electrons is random. For every electron moving with a certain velocity \(\vec{v}\), there is another electron moving with velocity \(-\vec{v}\). This means the number of electrons moving in opposite directions is the same. Consequently, the average velocity of the electron gas is zero, resulting in no net current flow. This symmetry in velocity distribution (equal probability for \(+v_x\) and \(-v_x\)) means that the distribution function of velocities is symmetric. This statement accurately describes the microscopic picture of equilibrium. Thus, Reason (R) is true.
Connection: The reason (R) explains \textit{why} the system is in equilibrium. The symmetric velocity distribution with zero average velocity is the very definition of the equilibrium state that is described mathematically by the equilibrium distribution functions mentioned in the assertion (A). The fact that the distribution is symmetric and the net flow is zero is precisely what the equilibrium Fermi-Dirac function describes. Therefore, (R) provides the correct physical explanation for the situation described in (A).
Step 3: Final Answer:
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).