A paramagnetic material containing paramagnetic ions with total angular
momentum J =
1
2 is kept at absolute temperature T. The ratio of the magnetic field
required for 80% of the ions to be in the lowest energy state to that required for
having 60% of the ions to be in the lowest energy state at the same temperature is
Options:
(A) 2ln2
ln (3
2)
(B) ln2
ln (3
2)
(C) 3ln2
ln (3
2)
(D) ln3
ln (3
2)
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.23)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Electromagnetic Theory** within *Section 3: Electromagnetic Theory* (1 Mark, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Electromagnetic Theory:
- 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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