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This is a question from 2022 AP physics 2 FRQ, I am not here to ask how to solve the question, but I am curious about the assumption made by this question. It says that electrostatic force is the dominant force in this atom, however according to my knowledge, it cannot explain the fact that electron can only exist in certain energy level, so one should introduce quantum mechanics here.

My question is where does (what mass or what length or some other criteria) the classical theory (like Newtonian Mechanics, Electrostatic force etc.) fail to accurately describe phenomenon that are shown by things and we need to introduce quantum mechanics to fix it?

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Accelerated charges radiate energy, and a circular orbit implies constant acceleration and constant energy. Thus, a stable circular orbit for a charged particle is incompatible with classical physics.

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  • $\begingroup$ Was a very shortrange non EM repulsive force that could balance the attraction between the $+$ and $-$ elementary charges ever considered? $\endgroup$
    – hyportnex
    Commented Jul 24 at 2:04
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The electrostatic interaction is the dominant interaction binding the electron to the proton. However, in classical mechanics, there is also an electrodynamic effect that is due to the electron's motion. In particular, an accelerating electron will radiate energy, and that will cause the orbit to decay. Quantum mechanics solves this problem because the solution for the Schrodinger equation with an electrostatic attraction has a minimum energy state, and the electron cannot lower its energy below this by emitting photons.

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  • $\begingroup$ Despite the fact that this was accepted, I don't think it actually answers the question. It gives a particular case where classical mechanics fails, but it does not describe what criteria a system needs to meet for classical theory to work or fail, which much more complicated. $\endgroup$
    – Sturrum
    Commented Jul 24 at 8:03
  • $\begingroup$ @Sturrum That seems like an overly broad interpretation of the OP's intent to me. But regardless, I actually don't think the issue you bring up is that complicated. Classical mechanics succeeds whenever it agrees with experiment. It fails to describe Hydrogen because it predicts the atom would be unstable, which does not agree with experimental evidence about Hydrogen. Broadly speaking, classical mechanics tends to work in processes involving large numbers of particles at high temperatures. But ultimately whether classical mechanics works is an empirical question. $\endgroup$
    – Andrew
    Commented Jul 24 at 12:10

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