Electrostatics Part 1

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

r3

Here, position vector, r=‌

a

2

(

∧

x

+

∧

y

+

∧

z

)
⇒‌‌|r|=√(‌

a

2

)2+(‌

a

2

)2+(‌

a

2

)2=‌

√3a

2

⇒‌‌E=‌

1

4πε0

×‌

(−2Q)a∕2

(‌ a 2 √3)3

⋅(

∧

x

+

∧

y

+

∧

z

)
E=‌−2Q(

∧

x

+

∧

y

+

∧ z )

3√3πa2ε0