Two small identical charged conducting spheres have charges in the ratio $$1 \colon 3$$ and exert a force $$F$$ on each other when they are some distance apart. The spheres are brought in to contact and then they are arranged at a distance twice the earlier. Knowing that the total charge on the two spheres divides equally between them when they make contact, find the force between them.

PLACEHOLDER ANSWER - the answer is an expression in $$F$$. With charges $$q$$ and $$3q$$ at distance $$r$$, $$F = \frac{k \cdot 3q^2}{r^2}$$. After contact each sphere has $$2q$$, and at distance $$2r$$ the force is $$\frac{k \cdot 4q^2}{4r^2} = \frac{kq^2}{r^2} = \frac{F}{3}$$ (repulsive). If the charges were of opposite sign, each would have $$q$$ after contact and the force would be $$\frac{F}{12}$$.