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I understand that if a conductor was charged with a charge of say 1 coulomb, the charge would spread uniformly over it's entire surface to neutralize any difference in potential. However, and though this would obviously not be something done practically, what would happen if you added just one electron to the metal sphere.

Obviously the electron is incapable of splitting to distribute charge uniformly. So would the electron just keep moving around the surface of the conductor? This would imply that the electric field just outside the conducting sphere wouldn't need to be perpendicular to it at all points.

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    $\begingroup$ When you add one electron, all the other electrons will also move a little. There is also thermal movement and, since conductors have to be described with quantum mechanics, a slightly more complicated picture of the charge state in such systems than electromagnetism alone suggests. $\endgroup$
    – CuriousOne
    Jul 22, 2016 at 3:46
  • $\begingroup$ Well it probably breaks down at that point. But it helps to remember that the situation is different than a single free electron; you have a sea of ions and electrons in most cases, which means the 'one additional charge' can be distributed, and needn't be localized. $\endgroup$
    – anon01
    Jul 22, 2016 at 3:49
  • $\begingroup$ So the charge would be uniformly distributed nonetheless? The surface charge density of the sphere surface would increase slightly but still remain constant for all area elements on the sphere? $\endgroup$ Jul 22, 2016 at 3:50
  • $\begingroup$ What @CuriousOne is saying is basically what I've tried to say in different words. $\endgroup$
    – anon01
    Jul 22, 2016 at 3:50
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    $\begingroup$ Certainly more uniform than a single electron on a sphere, but electrostatics likely will not hold, in such simple form, in this low charge carrier limit. the macroscopic treatment you're referring to treats charge like a continuous 'fluid'; if a single electron becomes important, this model may no longer hold. $\endgroup$
    – anon01
    Jul 22, 2016 at 3:54

2 Answers 2

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If electrons obeyed classical mechanics, they would rearrange in a new configuration in order to maximize the distance between them. They would not stay still because of thermal motion, as pointed out by CuriousOne, but on average they will still maximize this distance.

However, electrons don't obey classical mechanics, but quantum mechanics. The behavior of electrons in a conductor is more like that of waves (a plane wave in the free electron model, which is the crudest approximation). So the electron are delocalized on the whole surface of the conductor and it is probably more meaningful to talk about a single charge density function $\rho(\mathbf{r})$ rather than talking about individual electrons.

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Electrons in metals have states which people call Bloch states. If we want to describe these states we should bear in mind that electrons have wave like behavior. Bloch states are a subset of the many infinite wave states which electrons can have in general. They're such that when an electron is in a Bloch state, it somehow fills everywhere in the metal. If you are familiar with the problem of a quantum mechanical particle in a box, you understand what I mean. So when we add a new electron to a metal, it spreads all over the metal, and hence it produces a uniform charge density(if we put away the effect of boundaries).

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