GATE EC 2014 Electromagnetics Solution Q 2

GATE EC 2014 Electromagnetics Solution Q 2

If \overrightarrow E=-\left(2y^3-3yz^2\right)\widehat x-\left(6xy^2-3xz^2\right)\widehat y+\left(6xyz\right)\widehat z is the electric field in a source free region, a valid expression for the electrostatic potential is

a) xy^3-yz^2

b) 2xy^3-xyz^2

c) y^3+xyz^2

d) 2xy^3-3xyz^2

Ans- d)

Explanation

\overrightarrow E=-\left(2y^3-3yz^2\right)\widehat x-\left(6xy^2-3xz^2\right)\widehat y+\left(6xyz\right)\widehat z

We know that,

\begin{array}{l}E=-\frac{dV}{dx}\\\int dV=-\int Edx\end{array}

But instead of solving such complex integral, we can check each option by the following relation,

\begin{array}{l}E=-\nabla V\\E=-\left(\frac\partial{\partial x}\widehat x+\frac\partial{\partial y}\widehat y+\frac\partial{\partial z}\widehat z\right)V\end{array}

Checking for each; clearly option (d) is the correct option.

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