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We consider an electron "Bob" trying to escape from a metal surface as shown (blue circles represent electrons, red circles represent ions). I know that Bob can't leave the metal surface is because the attraction due to the ions is greater than the repulsion due to the electrons below. My question is: why F(attraction, by ions near the surface) is greater than F(repulse, by electrons)? Thank you!

enter image description here

abcxyzklmn
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    Does this answer your question? What force attracts electrons so that they cannot leave the conductor? – naturallyInconsistent May 14 '23 at 14:03
  • The real physics you are asking for is dominated by quantum behaviour and so a (classical) dynamics answer does not exist. – naturallyInconsistent May 14 '23 at 14:04
  • Thank you, I didn't really know what that had to do with quantum mechanics. Can you explain more about that? – abcxyzklmn May 14 '23 at 14:11
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    Classically, the charge near the surface will induce a localised charge distribution of opposite sign that at small enough distance will result in an attractive force.

    This is show in Jackson, Classical Electrodynamics, (pages 60 to 62 in the third edition), for a charged sphere and a point charge of the same sign. (Posted this by mistake twice in the linked question and now I cannot post it here as an answer because I am asked for an invisible captcha)

     – Peltio May 14 '23 at 14:23
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    That typical answer is by arguing from energy. A not too horrible approximation of a metal is a square well potential, famous from being one of the first few things covered in basic introductory quantum theory. Then you can see that the electrons simply does not have enough energy to escape the potential well, yet is essentially free to move inside. Think of a swimming pool full of freely moving fishes, but they really cannot leave the pool. – naturallyInconsistent May 14 '23 at 14:25

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Why don't you fly apart into your constituent atoms? The same logic applies: the nuclei repel each other, as do the electrons. Why does the attraction between the nuclei and electrons dominate?

One way to rationalize this observation is to understand that the nuclei and electrons can arrange themselves so that, on the average, the nuclei are closer to the electrons and the electrons are closer to the nuclei. A two-dimensional example of that kind of pattern is a chessboard: the white squares are surrounded by black squares and vice-versa.

In such an arrangement, the electrons are more attracted to the nuclei in their neighborhood than they are to the electrons farther away. That keeps them confined to the material.

A more sophisticated story requires quantum mechanics, but this is the essence of it.

John Doty
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  • Can you explain why the distance between electrons is longer than the distance between electrons - ions? I really want to know the reason behind it – newbieatphysics Jun 01 '23 at 14:50
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Classically, the charge near the surface will induce a localised charge distribution of opposite sign that at small enough distance will result in an attractive force. This is show in Jackson, Classical Electrodynamics, (pages 60 to 62 in the third edition), for a charged sphere and a point charge of the same sign.

Jackson goes on to say:

This example exhibits a general property which explains why an excess of charge on the surface does not immediately leave the surface because of mutual repulsion of the individual charges. As soon as an element of charge is removed from the surface, the image force tends to attract it back. If sufficient work is done, of course, charge can be removed from the surface to infinity. The work function of a metal is in large part just the work done against the attractive image force in order to remove an electron from the surface.

Peltio
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  • +1. A different story from mine, but a good one. – John Doty May 14 '23 at 16:51
  • @JohnDoty all credit goes to Jackson, obviously. This also explains why the charge deposited on a conductor won't fly away despite the repulsion. – Peltio May 14 '23 at 19:13
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    The problem with both stories, of course, is that they can't can't quantitatively account for the energy required, since without quantum effects it's infinite. – John Doty May 15 '23 at 21:00