Given a cube with point charges q on each of its vertices. Calculate the force exerted on any of the charges due to rest of the 7 charges.

Explanation

The net force particle a can be given by vector sum of force experienced by this particle due to all the other charges on vertices of the cube. For this we use vector from of coulomb's law `vec(F)=(kq_(1)q_(2))/(|vec(r)_(1)-vec(r)_(2)|^(3))(vec(r)_(1)-vec(r)_(2))`
From the figure the different force acting on A are given as `vec(F)_(A_(1))=(kq^(2)(-ahat(k)))/a^(3)`
`vec(F)_(A_(2))=(kq^(2)(-ahat(j)-ahat(k)))/((sqrt(2)a)^(3)), vec(F)_(A_(3))=(kq^(2)(-ahat(i)-ahat(j)-ahat(k)))/((sqrt(3)a)^(3)), vec(F)_(A_(4))=(kq^(2)(-ahat(i)-ahat(k)))/((sqrt(2)a)^(3))`
`vec(F)_(A_(5))=(kq^(2)(-ahat(i)))/a^(3), vec(F)_(A_(6))=(kq^(2)(-ahat(i)-ahat(j)))/((sqrt(2)a)^(3)), vec(F)_(A_(7))=(kq^(2)(-ahat(j)))/a^(3)`
The net force experienced by A can be given as
`vec(F)_("net")=vec(F)_(A_(1))+vec(F)_(A_(2))+vec(F)_(A_(3))+vec(F)_(A_(4))+vec(F)_(A_(5))+vec(F)_(A_(6))+vec(F)_(A_(7))=(-kq^(2))/a^(2)[(1/(3sqrt(3))+1/sqrt(2)+1)(hat(i)+hat(j)+hat(k))]`