Beyond Coulomb’s Law: The Interaction Between Overlapping Charged Balls

We give a detailed analysis of the interaction between overlapping charged balls, depending on the distance between them and their sizes. If one of the balls is entirely inside another ball, the interaction force is strictly proportional to the distance between the balls’ centers. This force has a maximum when some partial balls overlap depending on the ratio of their radii. For example, if the balls are the same, then the maximum corresponds to the distance between the centers equal to the balls’ radius. At small relative displacements of oppositely charged balls, they should perform harmonic oscillations around their common center of mass. The paper is addressed to undergraduates who deal with modern electrostatics.