In a certain region of space, potential is given by : V = k[2x^2 - y^2 + z^2]. electric field at poi

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2. Electric Potential and Capacitance

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In a certain region of space, the potential is given by : $V = k[2x^2 - y^2 + z^2].$ The electric field at the point $(1, 1, 1) $ has magnitude =

A

$k\sqrt 6 $

B

$2k\sqrt 6 $

C

$2k$$\sqrt 3 $

D

$4k\sqrt 3 $

Solution

Given, $V=k\left[2 x^{2}-y^{2}+z^{2}\right]$

Electric field $, \vec{E}=-\left(\frac{d V}{d x} \hat{i}+\frac{d V}{d y} \hat{j}+\frac{d V}{d z} \hat{k}\right)$

or $, \vec{E}=-k(4 x \hat{i}-2 y \hat{j}+2 z \hat{k})$

$\operatorname{or}, \vec{E}_{(1,1,1)}=-k(4 \hat{i}-2 \hat{j}+2 \hat{k})$

Magnitude of electric field $=\left|\vec{E}_{(1,1,1)}\right|=\sqrt{k^{2}(16+4+4)}=k \sqrt{24}=2 k \sqrt{6}$

Standard 12

Physics

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