Electric potential existing in space is $V = K (x^{2} y + y^{2} z + x y z)$ . Find the expression of electric field.

zero

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- K [(2 x y + y z) ˆ i + (x^{2} + 2 y z + x z) ˆ j + (y^{2} + x y) ˆ k]

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- K [(\frac{x^{3} y}{3} + \frac{x^{2} y z}{2}) ˆ l + (\frac{y^{2} z^{2}}{2} + \frac{x^{2} y^{2} z}{2}) ˆ k]

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K [(2 x y + y z) ˆ i + (x^{2} + 2 y z + x z) ˆ j + (y^{2} + x y) ˆ k]

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Solution

The correct option is B $- K [(2 x y + y z) ˆ i + (x^{2} + 2 y z + x z) ˆ j + (y^{2} + x y) ˆ k]$
As the electrci field is a conservative field so it is equal to the gradient of a scalar potential V. i.e, $\to E = - \to \nabla V = - (^i \frac{\partial}{d x} +^j \frac{\partial}{d y} +^k \frac{\partial}{d z}) K (x^{2} y + y^{2} z + x y z)$
$\to E = - K [(2 x y + y z)^i + (x^{2} + 2 y z + x z)^j + (y^{2} + x y)^k]$

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