GATE EC 2015 Electromagnetics Solution Q 18 - Grad Plus

# GATE EC 2015 Electromagnetics Solution Q 18

A vector \overrightarrow P is given by \overrightarrow P=x^3y{\overrightarrow a}_x-x^2y^2{\overrightarrow a}_y-x^2yz{\overrightarrow a}_z Which of the following statements is True?

a) \overrightarrow P is solenoidal,but not irrotational

b) \overrightarrow P is irrotational,but not solensoidal

c) \overrightarrow P is neither solenoidal nor irrotational

d) \overrightarrow P is both solenoidal and irrotational

Ans-a

Explanation

\overrightarrow P=x^3y{\overrightarrow a}_x-x^2y^2{\overrightarrow a}_yx^2yz{\overrightarrow a}_z

For solenoidal, \nabla\cdot\overrightarrow P=0

\begin{array}{l}\Rightarrow\nabla\cdot\overrightarrow P=\frac{\partial P_x}{\partial x}+\frac{\partial P_y}{\partial y}+\frac{\partial P_z}\partial\\\;\;\;\;\;\;\;\;\;\;\;\;\;=3x^2y-2x^2y-x^2y\\\;\;\;\;\;\;\;\;\;\;\;\;\;=0\end{array}

\Rightarrow\overrightarrow P is solenoidal

For irrotational, \nabla\times\overrightarrow P=0

\Rightarrow\nabla\times\overrightarrow P=\begin{vmatrix}{\overrightarrow a}_x&{\overrightarrow a}_y&{\overrightarrow a}_z\\frac\partial{\partial x}&\frac\partial{\partial y}&\frac\partial{\partial z}\x^3y&-x^2y^2&-x^2yz\end{vmatrix} \begin{array}{l}={\overrightarrow a}_x\left(-x^2z\right)+{\overrightarrow a}_y\left(2xyz\right)+{\overrightarrow a}_z\left(-2x^2-x^3\right)\\\neq0\\\Rightarrow\overrightarrow P\;is\;not\;irrotational\end{array}
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