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Electric Potential Due to Infinite Line of Charge

The electric ...

Question

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

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

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

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

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

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Solution

The correct option is B $\to E =^i (2 x y + z^{3}) + x^{2}^j + 3 x z^{2}^k$
Electric field, $\to E = \frac{- \partial V}{\partial x}^i - \frac{- \partial V}{\partial y}^j - \frac{- \partial V}{\partial z}^k$

Given,
$V = - x^{2} y - x z^{3} + 4$
$\frac{- \partial V}{\partial x} = - (- 2 x y - z^{3}) = 2 x y + z^{3}$
$\frac{- \partial V}{\partial y} = - (- x^{2}) = x^{2}$
$\frac{- \partial V}{\partial z} = - (- 3 x z^{2}) = 3 x z^{2}$
$∴ \to E = (2 x y + z^{3})^i + x^{2}^j + 3 x z^{2}^k$

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