Two identical metallic spheres A and B when placed at certain distance in air repel each other with a force of F. Another identical uncharged sphere C is first placed in contact with A and then in contact with B and finally placed at midpoint between spheres A and B. The force experienced by sphere C will be

Consider two metallic spheres A and B both have charge q.
The force between two sphere will be $F=\frac{k{q}^2}{r^2}$  
When sphere C is placed in contact with sphere A. charge on sphere A and sphre C will be $=\frac{q}{2}$ 
Now sphere C is placed in contact with sphere B, charge on the sphere B and sphre C will be  $=\frac{q=\frac{q}{2}}{2}=\frac{3q}{4}$  Now,
Then  $F_1=\frac{\frac{kq}{2}.\frac{3q}{4}}{\frac{r^2}{4}}$   $F_2=\frac{\frac{k3q}{4}\frac{3q}{4}}{\frac{r^2}{4}}$  
 
The force experienced by sphere C,  $F\text{'}=F_2-F_1=\frac{(k\frac{3q}{4}-k\frac{q}{2})}{\frac{r^2}{4}}.\frac{3k{q}^2}{4r^2}=\frac{3F}{4}$