Consider the potential U(r) defined as
e-ar
U(r) = -U-
where a and Uo are real constants of appropriate dimensions. According to
the first Born approximation, the elastic scattering amplitude calculated
with U(r) for a (wave-vector) momentum transfer q and a → 0, is
proportional to
(Useful integral:
So sin(qr)e-ar dr = o2
a2+q7)
(A)
19-2
(B) 9-1
(C)
(D) 9?
Options:
(A) 19-2
(B) 9-1
(C)
(D) 9?
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.50 / PH Q.40)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Classical Thermodynamics* (`Entropy & Thermodynamic Potentials (U, H, F, G)`) | **Section**: *Section 5: Thermodynamics and Statistical Physics*
- **Question Type**: **MCQ** | **Marks**: **2 Marks** | **Negative Marks**: **0.67**
- **Governing Principles**: Solved directly by applying fundamental laws of **Classical Thermodynamics**.
**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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