Two identical conducting spheres, A and B, carry equal charge. They are stationary and are separated by a distance much larger than their diameters. A third identical conducting sphere, C, is uncharged. Sphere C is first touched to A, then to B, and finally removed (to a far away distance). If before the sphere C is introduced the electrostatic force between A and B is Fi , how does Ff , the electrostatic force between A and B after C is removed, compare to Fi?

When two identical spheres are touched to each other, they equally share the total charge. Therefore, When neutral C is first touch to A, they share the initial charge of A equally.

Let us denote that the initial charge of A and B are Q. Then after C is touched to A, their respective charges are Q/2.

Then, C is touched to B, and they share the total charge of Q + Q/2 = 3Q/2. Their respective charges afterwards is 3Q/4 each.

The electrostatic force, Fi, in the initial configuration can be calculated as follows.

[tex]F_i = \frac{1}{4\pi\epsilon_0}\frac{q_Aq_B}{r^2} = \frac{1}{4\pi\epsilon_0}\frac{Q^2}{r^2}[/tex}

The electrostatic force, Ff, in the final configuration is

[tex]F_f = \frac{1}{4\pi\epsilon_0}\frac{q_Aq_B}{r^2} = \frac{1}{4\pi\epsilon_0}\frac{3Q^2/8}{r^2}[/tex}

Therefore, the relation between Fi and Ff is as follows

[tex]F_i = F_f\frac{3}{8}\\8F_i = 3F_f[/tex]