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I get the grade-school explanation that "the number of electrons equals the number of protons", but the electric field drops off with distance. If the protons are concentrated in the nucleus and the electrons are nebulously around the atom in orbitals, shouldn't the atom have a complicated electrical field that depends on both the position of the electrons and the distance from the nucleus?

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    $\begingroup$ At any point further from the nucleus than all the electrons, the field is $0$ by Gauss's law. Of course, quantum mechanics prevents any location being far enough with certainty, but the field is negligible even a nanometre from the mean centre of mass. $\endgroup$
    – J.G.
    Feb 3, 2023 at 13:56
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    $\begingroup$ Electrons don't have positions. They can be modeled to exist in a shape around the nucleus, where you might find one if you were looking, but that doesn't mean that election is there. They are smeared out. So is their charge. $\endgroup$
    – Stian
    Feb 4, 2023 at 12:23
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    $\begingroup$ @JG Your calculation is only correct if the distribution of charge is spherically symmetric---which say, in the simplest "billiard ball" model of protons and electrons, is impossible. Using a "classical" interpretation of orbitals which is compatible with the question, we have the same problem for non-spherically-symmetric orbitals. $\endgroup$ Feb 5, 2023 at 6:44

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Being neutral does not exclude having an electric field. Any dipole is neutral but still has electric field. Your question is based on a wrong premise. "Neutral" simply means zero net charge. It does not mean no field. If you consider the average field, for a spherical symmetric distribution the field will be zero even though the negative charge is distributed over a larger volume. The instantaneous field may be non-zero due to the fluctuations in the charge distribution. This may produce an instantaneous dipole field. This is the origin of the Van der Waals force between neutral atoms and molecules. If the distribution is non-spehrical then a permanent dipole field will exist even though the net charge is zero so the system is neutral.

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Atoms do have "complicated electric field", and occasionally may lose or gain a few electrons - in which case they are called ions. In fact, this happens all the time in chemical reactions.

However, most of the electron density is concentrated very close to the nucleus, so the nucleus charge is largely screened, and the atom can be seen as approximately neutral (even though the electron cloud of negative charge technically extends infinitely far, and for any finite sphere drawn about the atom, the positive charge inside is a bit bigger than the negative one.)

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Atoms are usually too symmetrical to have notable or complex electric fields. But molecules are often not simple and neither are their electrical fields.

So, even though neutral molecules have the same number of electrons as protons, unless the molecule is very symmetric, there may be significant fields because the distribution of electrons is far from even. This is one of the reasons why chemistry is interesting (certainly far more so then the physics of isolated neutral atoms). Many of the most interesting properties of water (H2O, just three atoms) arise because the electrons are unevenly distributed giving the molecules an uneven field that causes major intermolecular interactions that are responsible for many of the unique properties of the vital liquid.

Even isolated atoms are not perfect in their distribution of electrons and no gas would liquefy if they were. The effects are far smaller than for many molecules, and understanding the detail requires a fair chunk of quantum mechanics, the but van der Waals attractions between atoms that cause gases to liquefy can be thought of as a consequence of fluctuations in the distribution of the electrons in the atoms (this is a little simple, but not grossly wrong).

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As Roger Vadim stated, atoms do have a complicated electric field. But at such a scale, those effects are negligible. (Talking from a classical point of view)

Consider the case of gravity on earth. The acceleration due to gravity(g) on earth varies with height. But the variations(to a certain height) are negligible and don't affect the calculations.

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The total number of protons and the total number of electrons in the nucleus of an atom balance out the total positive and negative charges. This is why an atom is electrically neutral.

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