In two cases, two identical conducting spheres are given equal charges, in one case of the same type whereas in another case of opposite type. The distance between the spheres is not large comparing with the diameter. Let $F_1$ and $F_2$  be the magnitude of the force of interaction between the spheres, as shown, then

 

The medium is affects force. We know this from the constant, k, in Coulomb's Law, that depends on the medium. If you place a charge in an insulator or dielectric medium, like water, the force between them will decrease. This decrease can be taken into account by the medium's permittivity. 

The polarity of the charges that are interacting help us understand the direction. Like charges, whether it's both positive or negative, will always repel. Unlike charges, when one is positive and the other is negative, will always attract each other. The force acts along the line connecting the two

The main condition would be that the point charges have to be at rest or be in a stationary position, for Coulomb's Law to work. The thing is, if the charge particles move, we have to consider the impact of magnetic forces then. 

According to question, we can write

^{$F_{CA}=F_{CB}=\frac{KqQ}{x^2+{\left(\frac{d}{2}\right)}^2}\Rightarrow F=2F_{CA}cos\theta =\frac{2KqQx}{{\left[x^2+{\left(\frac{d}{2}\right)}^2\right]}^{\frac{3}{2}}}$}                         

For maxima of force  $\frac{dF}{dx}=0,so$  

$x=\frac{d}{2\sqrt[]{2}}$