 question and Solutions for gate paper 2017 Electromagnetics

GATE EC 2017 Electromagnetics Solution If the vector function $\overrightarrow F={\widehat a}_x\left(3y-K_1z\right)+{\widehat a}_y\left(k_2x-2z\right)-{\widehat a}_z\left(k_3y+z\right)$ is irrotational,then the values of the constants $K_1,K_2\;and\;K_3$ respectively, are

a) 0.3,-2.5,0.5

b) 0.0,3.0,2.0

c) 0.3,0.33,0.5

d) 4.0,3.0,2.0

Ans-b

Explanation $\overrightarrow F={\widehat a}_x\left(3y-k_1z\right)+{\widehat a}_y\left(k_2x-2z\right)-{\widehat a}_z\left(k_3y+z\right)$ $\nabla\times\overrightarrow F=0\;\left(irrotational\right)$ $\nabla\times\overrightarrow F=\begin{vmatrix}{\widehat a}_x&{\widehat a}_y&{\widehat a}_z\\\frac\partial{\partial x}&\frac\partial{\partial y}&\frac\partial{\partial z}\\3y-k_1z&k_2x-2z&-\left(k_3y+z\right)\end{vmatrix}$ $={\widehat a}_x\left[\frac\partial{\partial y}\left[-\left(k_3y+z\right)\right]-\frac\partial{\partial z}\left(k_2x-2z\right)\right]$ $-{\widehat a}_y\left[\frac\partial{\partial x}\left[-\left(k_3y+z\right)\right]-\frac\partial{\partial z}\left(3y-k_1z\right)\right]$ $+{\widehat a}_z\left[\frac\partial{\partial x}\left(k_2x-2z\right)-\frac\partial{\partial y}\left(3y-k_1z\right)\right]$ ${\widehat a}_x\left[-k_3+2\right]-{\widehat a}_y\left[k_1\right]+{\widehat a}_z\left[k_2-3\right]=0$ $\begin{array}{l}\Rightarrow k_3=2,\;k_1=0,\;k_2=3\\\\or\;\;k_1=0,\;k_2=3,k_3=2\end{array}$

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