Question

For a vector $\overline{A}$, which of the following statement is NOT TRUE?

• A. If $\nabla \cdot \overline{A} = 0$ then $\overline{A}$ is called solenoidal
• B. If $\nabla \times \overline{A} \neq 0$ then $\overline{A}$ is called rotational
• C. If $\nabla \times \overline{A} = 0$ then $\overline{A}$ is called irrotational
• D. If $\nabla \cdot \overline{A} = 0$ then $\overline{A}$ is called irrotational

Hint:

Solenoidal = Zero Divergence. Irrotational = Zero Curl.

Answer 1

The Correct Option is D

Step 1: Concept
This evaluates the definitions of solenoidal and irrotational vector fields.

Step 2: Meaning
A field is solenoidal if its divergence is zero ($\nabla \cdot \overline{A} = 0$). A field is irrotational if its curl is zero ($\nabla \times \overline{A} = 0$).

Step 3: Analysis
Option (D) incorrectly claims that zero divergence implies an irrotational field.

Step 4: Conclusion
Since zero divergence defines a solenoidal field, not an irrotational one, Statement (D) is false. Final Answer: (D)