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My friend and I were studying for our EM test when we started to think about what happens to the electric field near an infinite line of charge.

$$E = \frac{\lambda}{2\pi\rho\epsilon_{0}}$$

As you get close to the line of charge, it seems like the electric field strength goes to infinity.

We were wondering if this means that, even in the classical picture of physics, two electrons could never touch because it would require an infinite amount of energy. Is this correct?

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    $\begingroup$ It is the inverse of the conundrum that generated the need for quantum mechanics, i.e. two opposite charges and their attraction. In this case the infinite energy is enough to save classical mechanics. In the opposite charges there would have been no atoms if not for QM $\endgroup$ – anna v Sep 22 '14 at 3:26
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    $\begingroup$ This question borders on the philosophical: does an electron have a physical diameter? $\endgroup$ – Carl Witthoft Sep 22 '14 at 11:49
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    $\begingroup$ Note that in "real life" electrons have a non-zero probability of being anywhere, including "inside" other particles, including protons. Classical electrodynamics is a lot simpler. $\endgroup$ – OrangeDog Sep 22 '14 at 16:55
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You are correct when you concluded that two classical point electrons could never touch each other. It would take infinite energy.

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  • $\begingroup$ True, if electron radius is 0, which it's not true. $\endgroup$ – Roee Gavirel Sep 22 '14 at 14:36
  • $\begingroup$ As far as we can tell electrons are true points $\endgroup$ – AlanZ2223 Oct 5 '14 at 2:55
  • $\begingroup$ In truth, classical E&M has nothing at all to say about the nature of charge. Electrons have to be added, and for that we need a model for the electron. Evidence provides an upper limit on the electron radius of $10^{-22}$ m. We can't say it's a point particle, and we can't say that it's not. But we can say that for any calculation or experiment that we can do with today's equipment, it is valid to use a point particle as a model. For all practical purposes, with equipment available today, electrons can't touch. And they may never. $\endgroup$ – garyp Oct 5 '14 at 14:20
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There is another 'infinity' (among others) lurking in classical electrodynamics which is evident when one calculates the electrostatic energy $W$ of a uniform spherical charge distribution of radius $a$ and total charge $Q$

$$W = \frac{3}{5}\frac{Q^2}{4\pi \epsilon_0 a}$$

Thus, by this result, a point (zero radius) particle of charge Q has 'infinite' self-energy. Even in a quantum formulation of electrodynamics, formally infinite self-energy must be addressed by a mathematical procedure that essentially 'subtracts' this infinity in order to produce finite, meaningful predictions.

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    $\begingroup$ One of my favorite results in physics is a paper that Arnowitt, Deser, and Misner did in the '60s -- if you calculate the potential energy to assemble a spherical point mass in general relativity, the electrostatic self-force cancels against the gravitational self-force. $\endgroup$ – Jerry Schirmer Sep 22 '14 at 3:44
  • $\begingroup$ @JerrySchirmer: Interesting. I'd like to look at the paper. Do you have a reference? Is this for zero spin? This doesn't seem relevant to actual subatomic particles, since their masses and spins would require them to be naked singularities (not black holes), and they don't seem have the properties of naked singularities. $\endgroup$ – user4552 Sep 22 '14 at 4:49
  • $\begingroup$ I can go look the thing up. The proof required using isotropic coordinates, so it was certainly the zero spin case (also, to my knowledge, there's no existing analytic extension of the Kerr metric to the interior of a star) $\endgroup$ – Jerry Schirmer Sep 22 '14 at 13:01
  • $\begingroup$ @JerrySchirmer Did you find the ADM reference? $\endgroup$ – Selene Routley Oct 5 '14 at 3:01
  • $\begingroup$ @WetSavannaAnimalakaRodVance : it appears to actually be done in THE ADM paper. Section seven: arxiv.org/pdf/gr-qc/0405109.pdf $\endgroup$ – Jerry Schirmer Oct 5 '14 at 4:23
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That is a unusual case but it can be achieved in a particle accelerator. when they give electrons enough kinetic energy to overcome the electric force they have on each other.
keep in mind that electrons, small as they are, do have size. so for them to "touch" you will need A LOT of energy, but not infinite.

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The answer to the main question is YES. Two electrons will "touch" each other when their centers are at a separation equal to one electron diameter. Since the diameter of an electron is not zero, an infinite amount of energy, is not required to make them "touch." With a (calculated) electron diameter = to $2.82 \times 10^{-15}$ m, the required energy can be calculated. Note: although a more modern calculation sets the upper limit at $10^{-22}$ m, the idea is the same.

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