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Coulomb's law says that as the distance between two electrons approaches zero, the force becomes infinite. However this is just an approximation, and electrons are better described as fuzzy clouds of charge distributed through space. But bringing two electrons close together would increase the replusive force between them. If the force is not infinite, what is it?

My guess is that as you force two electrons closer and closer together, at some point there is so much energy localised in one spot you spontaneously create a black hole. And presumably there would be some way to calculate this value?

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Your guess is within classical mechanics, but electrons are elementary quantum mechanical point (not fuzzy) particles and when the dimensions are commensurate with Plancks constant , classical estimates of its behavior no longer work, and quantum mechanics takes over.

In quantum mechanics two electrons interact with specific rules and once the energy of the interaction gets too large, more elementary particles are created, again with specific rules given in the standard model of particle physics. See the second page here to see how the repulsion is calculated at the quantum level, when the energy is not enough to create more particles.

So no, it cannot become a black hole, within the validated theories we have up to now.

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  • $\begingroup$ Just to be more specific, I would also add that the correct description is quantum electrodynamics (Dirac equation), rather than quantum mechanics alone (Schrödinger equation). $\endgroup$
    – sintetico
    Commented Nov 2, 2020 at 11:16
  • $\begingroup$ I have heard that there is a unimaginably small but non-zero chance the LHC could have created a microscopic black hole by colliding protons together. Would that same logic apply to smashing two electrons hard enough? $\endgroup$
    – F Ham
    Commented Nov 3, 2020 at 23:06
  • $\begingroup$ @FHam cms.cern/news/… . In principle one could make a model. $\endgroup$
    – anna v
    Commented Nov 4, 2020 at 5:00
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The repulsion of 2 electrons is described in the form of exchange of photons(same spin). When the electrons exchange a photon they scatter. A more detailed explanation is given by a Feinmann diagram. The energy of each photon and electron are proportional to the Planck constant so their energy can never be infinite. Also if the 2 electrons get really close the non linearity of the very strong electric field makes sure Coulombs law isn't an approximation.

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