The figures represent vector fields X, Y, Z, respectively. Which statement is true?

$\nabla\cdot\mathbf X=0,\ \nabla\times\mathbf Y\ne0,\ \nabla\times\mathbf Z=0$

$\nabla\cdot\mathbf X\ne0,\ \nabla\times\mathbf Y=0,\ \nabla\times\mathbf Z\ne0$

$\nabla\cdot\mathbf X=0,\ \nabla\times\mathbf Y=0,\ \nabla\times\mathbf Z=0$

GATE 2017 EE (EE2) – Question 13

Electromagnetic Fields · Biot-Savart law, Ampere law, Curl, Faraday law, Lorentz force · 1 mark · Multiple choice

The figures represent vector fields $\mathbf X,\mathbf Y,\mathbf Z$, respectively. Which statement is true?

Diagram for GATE 2017 EE (EE2) question 13

$\nabla\cdot\mathbf X=0,\ \nabla\times\mathbf Y\ne0,\ \nabla\times\mathbf Z=0$
$\nabla\cdot\mathbf X\ne0,\ \nabla\times\mathbf Y=0,\ \nabla\times\mathbf Z\ne0$
$\nabla\cdot\mathbf X\ne0,\ \nabla\times\mathbf Y\ne0,\ \nabla\times\mathbf Z\ne0$
$\nabla\cdot\mathbf X=0,\ \nabla\times\mathbf Y=0,\ \nabla\times\mathbf Z=0$

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Show answer and explanation

Correct answer: (C) $\nabla\cdot\mathbf X\ne0,\ \nabla\times\mathbf Y\ne0,\ \nabla\times\mathbf Z\ne0$

Explanation

X has outward radial arrows, indicating a source and nonzero divergence. Y circulates around the centre, indicating nonzero curl.

Z combines outward spreading with rotation: its curved spiral arrows retain a circulating component, so its curl is also nonzero. The three nonzero conditions are precisely those in **C**. Divergence measures source strength, while curl measures local circulation.

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