Question 403: Electrostatics & Capacitance

Question

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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Accepted Answer

We can replace $-$ Q charge at origin by + Q and $-$ 2Q. Now, due to + Q charge at every corner of cube, electric field at centre of cube is zero. So, net electric field at centre is only due to $-$ 2Q charge at origin. Vector form of electric field strength,
$E = {{Kqr} \over {{r^3}}}$
Here, position vector, $r = {a \over 2}(\widehat x + \widehat y + \widehat z)$
$ \Rightarrow \left| r ight| = \sqrt {{{\left( {{a \over 2}} ight)}^2} + {{\left( {{a \over 2}} ight)}^2} + {{\left( {{a \over 2}} ight)}^2}}  = {{\sqrt 3 a} \over 2}$
$ \Rightarrow E = {1 \over {4\pi {\varepsilon _0}}} 	imes {{( - 2Q)a/2} \over {{{\left( {{a \over 2}\sqrt 3 } ight)}^3}}}\,.\,(\widehat x + \widehat y + \widehat z)$
$E = {{-2Q(\widehat x + \widehat y + \widehat z)} \over {3\sqrt 3 \pi {a^2}{\varepsilon _0}}}$

Answer

${{2Q} \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$

Answer

${{ - Q} \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$

Answer

${Q \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$

Answer

${{ - 2Q} \over {3\sqrt 3 \pi {\varepsilon _0}{a^2}}}\left( {\widehat x + \widehat y + \widehat z} \right)$

Electrostatics & Capacitance

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 :

Choose the correct answer: