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How does classical physics justify the existence of an electric field at $r=0$?

Is this an edge case, an ambiguity, a "does not exist"?

Is this a trivial case or indicative of an actual fault in classical electrodynamics?

Obviuosly the math breaks down because the denominator is $r^2$...What I want to know, is this significant or a trivial case?

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    $\begingroup$ The force goes like $1/r^2$ for two point-like charges very far away from each other: once the charges approach, their sizes and the charge distributions come into play and the field is more than regular at $r=0$. $\endgroup$ – gented Nov 14 '16 at 21:21
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To make it simple, it does not exist, there are no real point-like classical charged particles. That's why we learn, for example, the electric field of a homogeneous charged sphere right after the one for a point charge.

To put it in another way. A point charge $e$ could be thought of as made up by many $de$ tiny charges at one spot, but you'll require an infinite amount of energy to bring two (or more) of these charges from infinity to a single and the same point.

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  • $\begingroup$ Point particles has problems with causality too. The self-interaction during a constant force into the body leads to action before the cause, advanced acceleration (see Griffiths, pg. 467). Conclusion: The awareness of very tiny point charge concentration leads to quantum mechanics? $\endgroup$ – Nogueira Oct 20 '15 at 18:28
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    $\begingroup$ @Nogueira see also physics.stackexchange.com/questions/208799/… $\endgroup$ – Rol Oct 21 '15 at 7:53
  • $\begingroup$ Ow. GR came first! But who know what is happening inside a block hole? $\endgroup$ – Nogueira Oct 21 '15 at 19:15
  • $\begingroup$ Note that exactly the same thing occurs with all the 1/r² laws, from gravity to vorticity. $\endgroup$ – Fabrice NEYRET Oct 29 '15 at 17:52

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