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Imagine a negatively charged conductor in a vacuum. The excess electrons will be spread out over the surface such that the net electric field inside the conductor is zero.

What keeps these extra electrons bound to the conductor?

For the case of a neutral conductor, I could intuitively understand some sort of Coulomb force being responsible. But for a negatively charged conductor, I don't understand what (net?) positive entity is exerting the attractive force.

  • Is there an explanation based only on classical electrodynamics (say, at the level of Jackson).
  • If not, is the explanation purely quantum mechanical?

Basically, how does one model this phenomenon?

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What keeps electrons on a negatively-charged conductor from leaving?

It is a quantum mechanical phenomenon. Wherever there exists an electric field potential there exist energy levels , i.e. stable orbital locations which can be occupied by an electron.

How does this happen?

Even the simple Hydrogen atom has a negative ion state, an anion. This is because the potential describing the hydrogen atom, neutral, still has many levels in which the electron can transition, and there exists a probability for an extra electron to be attracted to one of these levels, though one would have to solve the problem " two electrons one proton" energy levels. Even in the ground state the electron orbital has a p state probability, which means that the shape allows regions in space where the positive charge is open to attract other negative charges. These shapes are what allow the molecular bonding into H2.

In neutral atoms/molecules orbitals have shapes given by the complicated spin states necessary for the binding into molecules and solids, and there exist spill over forces that can be attractive to free electrons, i.e. have energy levels that are open to extra electrons. The stability then is guaranteed unless extra interactions occur, as described in the other answers.

One tends when speaking of bulk matter, as in conductors in the question, to forget the underlying quantum mechanical state, but if at the molecular and atomic level there do not exist stable quantum mechanical solutions with energy levels , then there would be no bindings. One speaks of surface tension and work functions etc to describe the many body effect of the underlying quantum mechanical state. The basic stability is due to the energy levels every time.

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  • $\begingroup$ Beautifully explained, @anna v. +1. But in a negatively charged sphere, if all the charges reside on the surface, then there must be a radial field acting on each charge. Then, shouldn't the charges escape, according to a classical Electrostatics POV. Here is the question. physics.stackexchange.com/q/331683 $\endgroup$ – Deechit Poudel Aug 6 '17 at 6:12
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I think that electrons do escape.

For example, Electrostatic Ion Thruster:

Electrons are emitted from a separate cathode placed near the ion beam, called the neutralizer, towards the ion beam to ensure that equal amounts of positive and negative charge are ejected. Neutralizing is needed to prevent the spacecraft from gaining a net negative charge.

I suppose the electrons would be within the conductor attracted by all the positive nuclei but at some point the net nuclear charge would cause it to accelerate out of reach of nuclei and fly away.

I think the specific phenomenon is called Thermionic Emission. I don't know enough about it to give much of an opinion but it seems that at normal (<1000K) temperatures, metal electrons are not energetic enough to actually do the escaping. Which, considering that metals have a very odd bonding structure in which they share a lot of electrons throughout the metal, instead of an electron being stuck with no orbital to get into, isn't that unreasonable.

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Electrons do leave surfaces either due to electrical potentials pulling them as you describe or as a result of a combination of heat plus electrical potentials.

THe electrons are held in because the Fermi level is lower than the vacuum level. Furthermore, the difference in potential between the Fermi level and the vacuum level is 'felt' by the electrons over a short range so that any electric field to pull off the electrons needs to be very strong - because $E = \Delta V / d$ where $\Delta V$ is the potential and $d$ is the short range of probably about $1 nm$ or similar order of magnitude.

One way of making electrons leave the surface in experiments where field emission is used is to use sharp tips to make the $E$ field stronger at the surface.

To return to your question you need a lot of negative charge to generate an $E$ field strong enough to overcome the Fermi level to vacuum level potential barrier.

I think to model you would need to do a quantum mechanical calculation of the surface and given the size/complexity people would often use DFT for this (Density Functional Theory).

Another thing to think about is that this is a bit like STM experiments where electrons pass between a tip and suface by tunneling

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You cannot achieve a stable system putting together a different number of positive and negative charges in a classical context. There will always be some force unbalance and some of them will be expelled from the system.

You need to go quantum. Then particles are forced on energy levels that keeps them confined, so you can a huge variety of (molecular) ions. In a macroscopic conductor you still have bands that allow to collect more electrons giving a net negative charge without having them all flying away immediately.

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I think that the electrostatic dipole is an instructive analogy: Its net charge is zero, as well. But because the positive and negative charge are not at the same position, there is still a resulting potential and even an attractive force to other dipoles.

So the suggested answer would be that for a perfect screening of the attractive force from the positive cores, the electrons would have to be at the same position as the cores. Because they are not, a net attractive force to additional electrons is the result.

(This answer had been deleted by a moderator, because it "doesn't appear to answer the question". Anyway, I think that it addresses the original question, i.e. how can an electrically neutral ensemble be attractive to a charged particle. Therefore I just post it again to have at least the chance to get some comments on it - thanks).

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droplet Extensive modeling, and electrostatic spray data reveal an intimate relationship exists between limiting droplet surface electric field and surface tension for Newtonian liquids. Droplet charging is a quantum-mechanical process associated with the collective behavior of the surface electrons.

ref: Emission Limited Electrostatic Atomization and the Fine Structure Constant, A. J. Kelly, J. Vac Sci. Tech., B 23(40

0 Jul/Aug 2005.

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    $\begingroup$ It might be more useful for you to include the more relevant sections of the paper, than for us to go to the link and read the arguments you present there. $\endgroup$ – Kyle Kanos Feb 16 '15 at 3:20
  • $\begingroup$ Try summarising the relevant arguments. It's been ten years since your paper was published, maybe you have refined your arguments since then. $\endgroup$ – WetSavannaAnimal Feb 16 '15 at 10:11

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