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A vector \(\vec{P}\) is given by \(\vec{P}\) = \(x^3 \vec{a_x}-x^2 y^2 \vec{a_y}-x^2 y z \vec{a_z}\). Which one of the following statements is correct?

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UPSC ESE (Prelims) Electronics and Telecommunication Engineering 19 Feb 2023 Official Paper
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  1. \(​\vec{P}\) is solenoidal, but not irrotational.
  2. \(​\vec{P}\) is irrotational, but not solenoidal.
  3. \(​\vec{P}\) is neither solenoidal nor irrotational.
  4. \(​\vec{P}\) is both solenoidal and irrotational.

Answer (Detailed Solution Below)

Option 3 : \(​\vec{P}\) is neither solenoidal nor irrotational.
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Detailed Solution

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Given that,

\(\vec P = X^3 a_x - X^2y^2\vec A_y - X^2 YZ\vec a_z\)

Div\(\vec P = \frac{\partial} {\partial X} (x^3) - \frac{\partial}{\partial y}(x^2y^2) - \frac{\partial }{\partial z}(x^2yz)\)

= \(3x^2 -2X^2 - x^2y\)

=\(a_x (-X^2Z -0)- a_y (-2XYZ-0)+ a_z(-2xy^2-0)\)

\(\ne 0\)

\(\vec P\)  is neither solenoidal nor irrotational.

Here, option 3 is correct.

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