Coulomb's law fails when the distance $r$ between to point charges vanishes. As $r\to 0$, the electric field between two point changes increases without limit. I should be bothered about the limit $r\to 0$ only if that limit is a physical limit. For example, does anyone ever directly have to use Coulomb's law to determine the force between two overlapping electrons? Even in quantum mechanics, the singularity of the Coulomb potential at $r=0$ do not pose any problem. Hydrogen atom exists and is solvable. So my question is are there situations in physics where $r\to 0$ can be a physical limit and one needs to worry (particularly, classical physics)?
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