Two identical metallic spheres, having unequal opposite charges are placed at a distance of $0.50m$ apart in air. After bringing them in contact with each other, they are again placed at the same distance apart. Now the force of repulsion between them is $0.108N$ . Calculate the final charge on each of them.

Hint: After bringing the metallic spheres in contact, there will be a charge transfer from one sphere to another. The charges on the spheres after transference, will rearrange themselves in such a way that, both the spheres will possess equal magnitude of charge after distribution. That is, the charges distribute in such a manner that it equates the magnitude of electric potential of the two spheres.

Complete answer:
Let the initial charge on the spheres be $({{q}_{1}})and({{q}_{2}})$ .
Now, after bringing the two identical spheres in contact the charges distribute in such a way that the magnitude of charge on each sphere is equal.
Therefore , the final charge on both the spheres will be the same (say$q$ ) .
We know, from Coulomb’s law of electrostatics:
$\Rightarrow F=k\dfrac{{{Q}_{1}}{{Q}_{2}}}{{{r}^{2}}}$
Where,
$F=$ net force of attraction or repulsion between the two charges
$k=$ constant of proportionality also known as the Coulomb’s constant
$r=$ shortest distance between the two charges
And,${{Q}_{1}}$ and ${{Q}_{2}}$ are the respective charges.
It is given in the problem:
$\begin{align}
  & \Rightarrow F=0.108N \\
 & \Rightarrow k=9\times {{10}^{9}}N.{{m}^{2}}/{{C}^{2}} \\
 & \Rightarrow r=0.5m \\
\end{align}$
 And,
$\Rightarrow {{Q}_{1}}={{Q}_{2}}=q$
Putting these values in Coulomb’s law, we get:
 $\begin{align}
  & \Rightarrow 0.108=9\times {{10}^{9}}\dfrac{{{q}^{2}}}{{{(0.50)}^{2}}} \\
 & \Rightarrow {{q}^{2}}=0.25\times 0.108\times \dfrac{1}{9\times {{10}^{9}}} \\
\end{align}$
Taking square root on both the sides and simplifying, we get:
$\begin{align}
  & \Rightarrow q=1.735\times {{10}^{-6}}C \\
 & \Rightarrow q=1.735\mu C \\
\end{align}$

Hence, the final charge on both the identical metallic spheres is $1.735\mu C$ .

Note:
The calculations should be done carefully and basic properties of charge distribution under various circumstances should be known. Also, we should be aware of all the different conditions under which a charge transfer could take place, for example if one of them is earthed or under a certain constant voltage, etc.