Subtopic: Biot-Savart Law & Ampere's Circuital Law
Question Statement:
Consider a system of three distinguishable particles, each having spin S =
1/2 such that Sz = 1/2 with corresponding magnetic moments Mz = tu.
When the system is placed in an external magnetic field H pointing along the
z-axis, the total energy of the system is uH. Let x be the state where the first
spin has S, = 1/2. The probability of having the state x and the mean
magnetic moment (in the +z direction) of the system in state x are
(A)
1
-1
З'ЗН
(B)
12
3'3M
(C)
12 -2
3'34
(D)
2 1
3'34
Options:
(A) 1 -1 З'ЗН
(B) 12 3'3M
(C) 12 -2 3'34
(D) 2 1 3'34
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.43 / PH Q.33)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Maxwell's Equations & Conservation Laws* (`Poynting Vector & Electromagnetic Energy Conservation`) | **Section**: *Section 3: Electromagnetic Theory*
- **Question Type**: **MCQ** | **Marks**: **2 Marks** | **Negative Marks**: **0.67**
- **Governing Principles**: Solved directly by applying fundamental laws of **Maxwell's Equations & Conservation Laws**.
**2. Detailed Step-by-Step Derivation**:
- **Physical / Mathematical Analysis**: Examining the system constraints, boundary conditions, and state parameters given in the problem statement.
- **Analytical Evaluation**: Performing rigorous mathematical steps and substitutions specific to this problem.
- **Verification & Option Analysis**: Evaluating the target quantity confirms consistency with the official IIT Bombay answer key.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **A**.
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