GATE PH 2013 • Q15
MCQ • 1 Mark
Section 1: Mathematical Physics
Linear Algebra & Vector Calculus
Subtopic: Vector Differential Operators (Grad, Div, Curl)
Question Statement:
For a scalar function ߮satisfying the Laplace equation, ߮has
Options:
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(A) zero curl and non-zero divergence
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(B) non-zero curl and zero divergence
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(C) zero curl and zero divergence
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(D) non-zero curl and non-zero divergence
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2013 Q.15)**
**1. Vector Calculus & Laplace's Equation**:
- Given scalar potential $\phi$ satisfying Laplace's equation:
$\nabla^2 \phi = 0$
- Define vector field $\vec{F} = \vec{\nabla}\phi$.
**2. Divergence and Curl Evaluation**:
- **Divergence of $\vec{F}$**:
$\vec{\nabla} \cdot \vec{F} = \vec{\nabla} \cdot (\vec{\nabla}\phi) = \nabla^2 \phi = 0$
Therefore, $\vec{\nabla}\phi$ has **zero divergence** (solenoidal field).
- **Curl of $\vec{F}$**:
$\vec{\nabla} \times \vec{F} = \vec{\nabla} \times (\vec{\nabla}\phi) = 0$
By vector calculus identity, the curl of any gradient field is identically zero (irrotational field).
- Consequently, $\vec{\nabla}\phi$ is both **divergenceless and curl-free**.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **Option (C)** (`both zero divergence and zero curl`).
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