The electric potential at a point $P (x, y, z)$ is given by $V = - x^{2} y - x z^{3} + 4$ . The electric field $\to E$ at that point is

(2 x y + z^{3})^i + x^{2}^j + 3 x z^{2}^k

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2 x y^i + x^{2}^j + 3 x z^{2}^k

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(2 x y + z^{3})^i + (x^{2} + y^{2})^j + (3 x y - y^{2})^k

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2 x y - z^{3}^i + x y^{2}^j + 3 z^{2} x^k

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Solution

The correct option is A $(2 x y + z^{3})^i + x^{2}^j + 3 x z^{2}^k$
$V = - x^{2} y - x z^{3} + 4$

$\to E = - \frac{\partial V}{\partial x}^i - \frac{\partial V}{\partial y}^j - \frac{\partial V}{\partial z}^k$

Now, $\frac{\partial V}{\partial x} = - 2 x y - z^{3}$

$\frac{\partial V}{\partial y} = - x^{2}$

$\frac{\partial V}{\partial z} = - 3 x z^{2}$

Hence,

$\to E = (2 x y + z^{3})^i + x^{2}^j + 3 x z^{2}^k$

Answer-(A)

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