Two identical metallic spheres A and B when placed at certain...

Let charge on $A$ and $B$ be ${q}_{A} = {q}_{B} = q$.

Then force acting between them is $F = \dfrac{K{q}^{2}}{{r}^{2}}$.

When $C$ is placed in contact with $A$, charge on $A$ and $C$ will be $= \dfrac{q}{2}$

Now $C$ is placed in contact with $B$, charge on $B$ and $C$ will be $= \dfrac{q + \dfrac{q}{2}}{2} = \dfrac{3q}{4}$

Now, force experienced by sphere $C$ is

${F}^{'} = {F}_{2} - {F}_{1} = \dfrac{K9{q}^{2}}{16 \times \dfrac{{r}^{2}}{4}} - \dfrac{K3{q}^{2}}{8 \times \dfrac{{r}^{2}}{4}} = \dfrac{\left(K\dfrac{3q}{4} - K\dfrac{q}{2}\right)}{\dfrac{{r}^{2}}{4}} \times \dfrac{3q}{4}$

$= \dfrac{3K{q}^{2}}{4{r}^{2}} = \dfrac{3F}{4}$