# Two spherical conductors $B$ and $C$ having equal radii and carrying equal charges on them repel each other with a force $F$

Two spherical conductors $$B$$ and $$C$$ having equal radii and carrying equal charges on them repel each other with a force $$F$$ when kept apart at some distance. A third spherical conductor having same radius as that of $$B$$ then brought in contact with $$C$$ and finally removed away from both. The new force of repulsion between $$B$$ and $$C$$ is

My question is: Is there any difference between in the answers of conductor and point charge?

The answers will differ, because the charge distribution on each sphere will not be uniform. Effectively, each sphere will have an induced dipole moment due to the presence of the other, which will change the force between them.

This effect of the charge distribution, however, will generally be smaller than the basic effect from treating the spheres as monopoles (point charges). To within an order of magnitude, these corrections to the force will be smaller than the “point charge” force by the ratio of the spheres’ radii $$r$$ to their separation $$D$$. In particular, if the size of the spheres is negligible compared to their separation ($$r \ll D$$), then the approximation off treating the spheres as point charges will be a good one.

Call the new spherical conductor D.

As Michael Seifert implies, the charge on D and the remaining charge on C will depend on where C and D touch.

When they touch, if the distance from B to D equals the distance from B to C, then half of C's charge will flow to D.

If D is between B and C when they touch, then D will have an opposite charge and C will have a stronger charge than before.

If C is between B and D when they touch, then C will have the opposite charge and D will have almost as much charge as C would in the other case.

The point charge approximation does not predict this.