Consider a vector field F = (2xz + 3y2)ŷ + 4yz2ẑ. The closed path
(Γ: A → B → C → D → A) in z = 0 plane is shown in figure.
∮ F .
Γ dl⃗⃗⃗ denotes the line integral of F along the closed path Γ. Which of the
following option is/are true?
MSQ
Options:
(A) ∮ F .
Γ
dl⃗⃗⃗ = 0
(B) F is non-conservative
(C) ∇⃗⃗ . F = 0
(D) F can be written as the gradient of a scalar field
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.45)**
**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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