@RussellMcMahon : By the way, I found out that "evolution" has another meaning: " the action or an instance of forming and giving something off" (merriam-webster.com/dictionary/evolution ). Apparently, this meaning is used in the quote.

"I have found several sources which take this identity as a postulate" What sources? The identity looks suspicious to me: $\psi(x)$ is probably a Dirac spinor, so it is a column matrix, so how can it be multiplied by the unitary matrix from the right?

@JohnDoty : Again, my understanding is nothing in your comments proves my answer wrong. Again, I am saying that there can be no zero charge density areas of a charged conducting surface (at least if the surface is connected), you are saying there can be "almost zero charge density areas". I am not saying your statement is wrong, but I stand by my statement and believe it may be useful for the OP. Unless you prove my statement wrong, I don't quite see the point of our discussion.

@JohnDoty : as worded, your question does not seem to make much sense: of course, the answer is "zero", as, for a fixed total charge of the conducting surface, the area can be arbitrarily large.

@JohnDoty : I don't see any answer of yours, I only see your comments on my answer, and they don't disprove my answer. I am saying that there can be no zero charge density areas of a charged conducting surface (at least if the surface is connected), you are saying there can be "almost zero charge density areas". I don't know if what you say is correct or not, but it does not contradict my answer.

@JohnDoty : The OP considers electrostatics at the level of a certain mathematical model - the Poisson equation. I believe my answer is correct at this level, unless you prove otherwise.

@CompassBearer : Firstly, I would not call midplane a "surface", as it is inside the rod, secondly, I would say "external electric field" is (at least) in tension with the OP's caveat "with proper boundary conditions", and my answer says "at least if the surface is connected", whereas there can be no external electric field without remote (and unconnected) charges.

@JohnDoty : Your comment uses a lot of "almost" and "not much", so it does not disprove my answer.

And what is $U(A)$?

"The Wichmann-Kroll term starts contributing only at..." I would say this term always leads at large distances, where the power law prevails over the exponential law.

@OutisNemo : "a bit low when compared"... I am not sure these two values are comparable at all. In the case of the Sun, it is the global charge, in the case of the Earth, it is the surface charge, and it looks like the global charge of the Earth is very small. Another thing. Electrical conductivity of the Sun is much, much higher than that of the Earth. You see, I cannot be sure 77C is the correct charge of the Sun, but at least it follows from some theory. If you have any solid arguments in favor of the charge of the Sun being much smaller or much greater than 77C, I am all ears.

@JánLalinský : There may be a very fine line between discrete and continuous. I showed how a smooth wave function of an electron can be emulated/approximated by a plasma of discrete electrons and positrons in mdpi.com/2624-960X/4/4/35 (Quantum Rep. 2022, 4(4), 486-508), Section 2.8, and references there.

@JánLalinský : "it has a flaw: it describes electrons as a single continuous fluid" It is not obvious that it is a flaw, as this is how electrons are described in quantum mechanics. And Dirac's results can be extended to quantum mechanics. See, e.g., my article link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4 (Eur. Phys. J. C (2013) 73:2371), Section 2. Electromagnetic field describes a system of interacting electromagnetic field and matter field there.