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When discussing about a negatively-charged solid conducting metal sphere, We agree on a statement that

like charges repel so that the negative charges (aka electrons) are evenly distributed on the surface of the sphere.

And the inner of sphere has no electric field (aka negative potential gradient). But the electrons aren't "static" though, they are moving randomly due to thermal motion. So isn't the electrons, at some very very short instance, creating an imbalance of distribution of charge of the sphere? ( like, moving to the inner centre, "clump" together with a closer distance,etc)

When we are talking about the statement, is it because we neglect the size and motion of electrons? Like, analytically, we are treating electrons distribution as a negatively-charged static infinitesimal thin gel on the surface (although the electrons are moving randomly)?

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The thermal motion of the electrons does not lead to macroscopic fluctuations like the one you are considering. It does however lead to Johnson-Nyquist noise.

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  • $\begingroup$ And to add to this, a perfect conductor theoretically makes zero Johnson-Nyquist noise voltage. A good conductor like a metal makes very little. $\endgroup$
    – John Doty
    Commented May 10, 2022 at 15:12
  • $\begingroup$ I should clarify that I assume a limited bandwidth measurement. $\endgroup$
    – John Doty
    Commented May 10, 2022 at 15:20
  • $\begingroup$ @JohnDoty ' I have limited use for textbook idealizations.' John Doty, PSE, not so long ago. $\endgroup$
    – my2cts
    Commented May 10, 2022 at 19:13
  • $\begingroup$ Limited use is not no use. And this may not necessarily count as an idealization since there are such things as superconductors. $\endgroup$
    – John Doty
    Commented May 10, 2022 at 19:33
  • $\begingroup$ @JohnDoty Superconductors and perfect conductors are very different. $\endgroup$
    – my2cts
    Commented May 10, 2022 at 20:11
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An assumption in classical or continuum mechanics is that there are so many electrons or molecules that you can safely neglect the effect of those random fluctuations.

As you add independent random events, the variance decreases. For example if you toss a coin twice, there's a 25% chance each of getting all heads or all tails, if you toss a coin 10 times it's closer to a 0.1% chance. When dealing with coulombs of electrons or moles of molecules, the effect of random fluctuation becomes so small that you can nearly always safely ignore it.

The concept of an electric field only really arises when you've already made that continuum assumption, if you're dealing with small numbers of electrons you get quantum interactions, and things like forces and fields stop being so meaningful.

If you're working with something really small or really low density (like MEMS, cutting-edge transistors or space) it can be worth checking if the typical distance between particles is similar to or bigger than the length of the thing you're working with, in which case you do need to account for random motion and quantum effects.

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