Electrostatics of two charged conducting spheres

We prove that two charged conducting spheres will almost always attract each other at close approach, even when they have like charges. The one exception is when the two spheres have the same charge ratio that they would obtain by being brought into contact. In this case, they repel, and we derive an analytical expression for the force at contact, for any size ratio, generalizing a force formula for equal spheres obtained by Kelvin in 1853. We also give the electrostatic energy of two arbitrarily charged spheres, and its analytical forms at large and small separations. Expressions are derived for the surface charge densities of the two spheres. Attraction occurs between two positively charged spheres because of mutual polarization: one of the spheres obtains a negatively charged region (neighbouring the other sphere).