Electric potential in a particular region of space is $V = 12 x - 3 x^{2} y + 2 y z^{2}$ . The magnitude of electric field at point $P (1 m, 0, - 2 m)$ in $N / C$ is

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Solution

Given, Electric potential as
$V = 12 x - 3 x^{2} y + 2 y z^{2}$
$∴ E_{x} = \frac{δ V}{δ x} = 12 - 6 x y$
and, $E_{y} = \frac{δ V}{δ y} = - 3 x^{2} + 2 z^{2}$
also, $E_{z} = \frac{δ V}{δ z} = 4 y z$
$∴ E at (1, 0, - 2)$ is given by
$E_{x} = 12 - 6 \times 1 \times 0 = 12$
$E_{y} = - 3 \times 1^{2} + 2 \times (- 2)^{2} = 5$
$E_{z} = 4 \times 0 \times (- 2) = 0$
$∴ \to E = 12^i + 5^j + 0^k = 12^i + 5^j$
and, $| \to E | = \sqrt{12^{2} + 5^{2}} = 13 N / C$

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