Consider a two -level system with energy states +𝜀 and - 𝜀. The number of
particles at +𝜀 level is N+ and the number of particles at - 𝜀 level is N-. The total
energy of the system is E and the total number of particles is N = N+ + N-. In the
thermodynamic limit, the inverse of the absolute temperature of the system is
(Given: ln N! ≃ N ln N - N)
Options:
(A) kB
2𝜀 ln [
N-E
𝜀
N+ E
𝜀
]
(B) kB
𝜀 lnN
(C) kB
2𝜀 lnN
(D) kB
𝜀 ln [
N-E
𝜀
N+ E
𝜀
]
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.42)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Nuclear & Particle Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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