A cube of side 'a' has point charges +Q located at each of its… | JEE Main 2021 (24 Feb, Shift 1)

Q30JEE Main 2021 · 24 FEB · SHIFT 1Medium

Electric Field & Field Lines

A cube of side 'a' has point charges +Q located at each of its vertices except at the origin where the charge is $-$ Q. The electric field at the centre of cube is:

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Answer: (d)

Write $-Q$ at the origin as $+Q$ plus $-2Q$ . Eight equal $+Q$ charges give zero field at the centre.
So only $-2Q$ at the origin acts; distance to centre $= \dfrac{\sqrt{3}a}{2}$ .
$E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2Q}{3a^2/4} = \dfrac{2Q}{3\pi\varepsilon_0 a^2}$ , directed towards the origin, i.e. along $-\dfrac{(\hat{x}+\hat{y}+\hat{z})}{\sqrt{3}}$
$\vec{E} = \dfrac{-2Q}{3\sqrt{3}\pi\varepsilon_0 a^2}(\hat{x} + \hat{y} + \hat{z})$
So the answer is (d).

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