Two identical conducting spheres \(A\)  and \(B,\)  carry equal charge. They are separated by a distance much larger than their diameter, and the force between them is \(F\) . A third identical conducting sphere, \(C,\)  is uncharged. Sphere \(C\)  is first touched to \(A,\) then to \(B,\)  and then removed. As a result, the force between \(A\)  and \(B\)  would be equal to

\(\therefore \) Force between them, \(\mathrm{F}=\frac{\mathrm{k} \mathrm{qq}}{\mathrm{r}^{2}}\)

When \(A\) and \(C\) are touched, charge on both

\(\mathrm{q}_{\mathrm{A}}=\mathrm{q}_{\mathrm{C}}=\frac{\mathrm{q}}{2}\)

Then when \(\mathrm{B}\) and \(\mathrm{C}\) are touched, charge or \(B\)

\(\mathrm{q}_{\mathrm{B}}=\frac{\frac{\mathrm{q}}{2}+\mathrm{q}}{2}=\frac{3 \mathrm{q}}{4}\)

Now, the force between charge \(q_{A}\) and \(q_{B}\)

\(\mathrm{F}^{\prime}=\frac{\mathrm{k} \mathrm{q}_{\mathrm{A}} \mathrm{q}_{\mathrm{B}}}{\mathrm{r}^{2}}=\frac{\mathrm{k} \times \frac{\mathrm{q}}{2} \times \frac{3 \mathrm{q}}{4}}{\mathrm{r}^{2}}=\frac{3 \mathrm{kq}^{2}}{8} \mathrm{r}^{2}=\frac{3}{8}\, \mathrm{F}\)