My model of Conductors in "static" condition

Question

My textbook presents an idealization of a conductor as made up of infinitesimal units of charge and derives results. I was not convinced, so I started thinking of how electric fields are in real metals. Here is what I think now:

i) My description of a "static" situation - There are electric fields inside a conductor, but no net electric field that does anything meaningful. Electrons are flying around left, right, up and down, but no net current is present. If you take a reasonably macroscopic chunk of metal, no net current will flow.

ii) The entire metal is at a lower potential energy than let's say, air, that surrounds it. Therefore, there exists a work function to pull out electrons. It's like a well. You need to pull hard enough to yank an electron out. Precisely, you have to do more work than the force that holds the electron in.

iii) The interior of a metal cannot have excess charge in a static condition - If the interior does have charge, the charges will repel until they reach a point of equilibrium, which means the metal isn't static. This is almost like a definition.

iv) Excess charge lies at the surface of a conductor, in static situation - Since the metal is static, all the inside of the metal should behave like a regular neutral metal. For a sphere for example, this means that excess charges must form a ring of equal charge density. This is similar to saying that electric field is zero inside a ring. The excess charges will not contribute any field

v) The excess charges exert a field on each other. For a sphere for example, every charge exerts charge on every other charge, so that net electric field points outside. The existence of the work function means that any attempts to pull out charges will be countered. This countering force (essentially due to protons in atoms) will counter outside field. An equilibrium is reached and therefore, those charges are more or less still.

vi) The electric field inside is more or less zero, and the excess charge on the outside experiences zero field too, so the metal is an equipotential surface.

vii) Any external electric field is countered using by an arrangement of charge that yields zero field inside, and these charges do not fly out because they are held in.

I know there are many approximations in this model. But I think it beats the unrealistic idealization given by my textbook isn't really connected to real life. Are there any flaws in it? I will learn better models when I do solid state physics, right?

Answer 1

You are exactly correct! However, the electrostatic n-body problem when $n\approx 10^{23}$ is not exactly tractable, and we get a lot of useful solutions by assuming that the electrons are so small and numerous that we can treat them as a continuum.

This approach also allows us to use differential calculus: if we treated the charges as point sources, we couldn't take a derivative like $\nabla\cdot E$ and get a nice, smooth result.

In fact, your textbook's model is connected to the real world. Just take the limit of your model as the size of the charge carriers goes to 0 and their number goes to infinity.

In solid-state physics the interactions between individual electrons and atoms come into play, and quantum effects are accounted for. The models you use there will be different: better for those problems, but worse (less useful) for electrostatics.