Sketch of a two-dimensional vector field V⃗ is shown below. Here, length and arrow head
of the arrows denote magnitude and direction of the vector field, respectively. Which of
the following statements is correct for ∇⃗⃗ × V⃗ ?
Options:
(A) It is zero everywhere in the two-dimensional space.
(B) Its magnitude is non-zero and its direction is out of the two-dimensional plane.
(C) Its magnitude is non-zero and its direction is into the two-dimensional plane.
(D) It points in opposite directions above and below the x-axis.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.22)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (1 Mark, 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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