$$F=k\frac{qQ}{r^2},$$ if $r\rightarrow 0$ then $F\rightarrow \infty$.
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5If the electromagnetic force were the only force acting, and the points have zero size, then yes. But that's not the universe we live in. – lemon May 02 '16 at 09:12
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The electron is not a point particle. It has a wave nature. See answers like this. – John Duffield May 02 '16 at 11:41
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As others have hinted, classical E&M has this embarrassing flaw: divergences at every point particle! To study interactions as r->0 requires a more sophisticated picture than classical E&M – anon01 May 02 '16 at 11:52
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1"Hey, can you get that Pauli guy on the phone? My hydrogen's gone inert again." – Asher May 02 '16 at 13:30