GATE EC 1993 Electromagnetics Solution Q 1

GATE EC 1993 Electromagnetics Solution Q 1

Given, \mathbf{\overrightarrow v=x\cos^2y\widehat i+x^2e^z\widehat j+z\sin^2y\widehat k} and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axis, the value of the integral is \iint_C\overrightarrow v\cdot\widehat ndS\;

Ans- 1

Explanation

According to Divergence Theorem for any close surface, Now the given surface is surface of a cube which is the close surface. So we have,

\oint\overrightarrow V\cdot\overrightarrow{ds}=\int\left(\nabla\cdot\overrightarrow V\right)\;dv \nabla\cdot\overrightarrow V=\cos^2y+\sin^2y=1 \int_v\left(1\right)dv=\int_vdv=Volume\;of\;Cube=1

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