Let's say we have a 120 Volt AC source

[shoestring]

That’s a good answer but it still leaves me a bit cheated. We are quick to draw a force arrow to the opposite plate and can explain this arrow but there must exist an arrow with the same (but opposite) magnitude drawing the electrons back. Has anybody ever seen such an arrow with a proper classical explanation? I don’t think we need to go into qm and can stay in electrostatics/classical physics.

It's a good question you're asking. Why indeed don't the electrons go away from a negatively charged metal surface, what's holding them back? I suspect it has to do with quantum mechanics after all, specifically the energy levels for the valence electrons in a metal. Something to do with Fermi surfaces perhaps.

 

[shoestring]

In the free electron model for example, the metal is thought of as a "potential well" for the valence electrons. The permissible energy levels are found by solving the Shroedinger equation. It turns out that the energy levels for the valence electrons, which are free to move all over the metal, are very different from the energy levels in a single atom of the same metal.

I've no idea how many extra electrons, giving a net negative charge of the metal, that can be added without the well "overflowing". If the potential energy for the extra electrons could be described as a smooth function, then I suppose a force could be calculated as minus the gradient of that function. The "extra" electrons are of course interchangeable with all the other valence electrons in the metal, so I don't know if it's possible to have two potential functions, one for the electrons matching the positive charges, and one for the extra electrons responsible for the net negative charge of the metal.

 

[Evil Bunny]

So, I'm still wondering if we're sure about this... Would this separated plate now have a potential just like a thunder cloud? The charge held?

 

[shoestring]

So, I'm still wondering if we're sure about this... Would this separated plate now have a potential just like a thunder cloud? The charge held?

In a conservative field (like in the case of the electric field from a charge distribution where you don't have any time-varying magnetic field to take into consideration) the potential difference between any two points can be found as the line integral of F*ds between the points, i.e. the work it takes to move a charge between the points. (F = qE)

A charged plate suspended in mid air can have a lower or higher potential than earth, or it can have exactly the same potential as the earth. The Earth itself has a negative charge, so a positively charged object will always have a higher potential than the earth. A negatively charged object may have exactly the same potential as the earth, or a higher or lower potential than the earth. (Assuming you don't go too far from the surface of the earth.)

Edit: Relatively near the surface, the Earth is supposed to have an electric field of around 100V/m, pointing downward. http://hypertextbook.com/facts/1998/TreshaEdwards.shtml"

 

[Per Oni]

In the free electron model for example, the metal is thought of as a "potential well" for the valence electrons. The permissible energy levels are found by solving the Shroedinger equation. It turns out that the energy levels for the valence electrons, which are free to move all over the metal, are very different from the energy levels in a single atom of the same metal.

That’s true. The valence electrons become conduction electrons which have continuous energy levels. that’s why I think this problem should be solved by classical rather then with quantum physics. Another argument is that the potential energy is stored in the field between the plates.

 

[shoestring]

Actually, in the free electron model the valence electrons in the metal do have discrete energy levels (but very closely spaced, and more closely spaced the bigger the piece of metal is) and the Pauli exclusion principle is also used to explain the distribution of electrons among the levels. It's of course just a model, but in order to explain bounding energies for electrons, whether they're electrons in atoms, ions, molecules or large metal gitters, I think quantum physics is useful.

Think for example of a negatively charged metal sphere. If the electrons weren't somehow bound to the metal, electrostatic forces would force them away from each other, away from the sphere. It's not that different from an ion with charge -2e. Electrostatically the two extra negative charges repel, but quantum mechanically they're bound to the ion. Something keeps them there. I can't think of any classical way to explain that, but perhaps I'm missing something obvious.

So, I suppose the negative pole must be something of a "potential well" even for the extra electrons on it, because electrostatically those extra electrons repel each other.