# How can moving electrons participate in electrostatic interaction?

People say that there is an electrostatic force between electrons and atomic nuclei. However, electrostatic force applies to static charges, i.e. charges at rest.

Question: How can electrostatic force exist given that electrons aren't stationary?

• In an atom nucleus which is positive charge is at rest. Not even nucleus is at rest. It is considered to be so - it is an approximation. Further you can ask why the electrons do not fall into the nucleus? And then you have to consider the quantum nature of particles in order to answer to that question.
– Nemo
Jun 13, 2018 at 16:31
• Your logic assumes that the electron and nucleus are classical charges. Applying classical electromagnetism to the atom would tell you that the accelerating electron radiates and eventually falls into the nucleus. This doesn't happen, so the picture in your head (with electrostatic "forces") is incorrect. In quantum mechanics, particles are not localized, and so the concept of a "force" is not particularly useful. As @annav said, the quantum-mechanical approximate description of the atom has the electron wavefunction bound with an electrostatic potential. Jun 13, 2018 at 16:38
• @probably_someone, the question assumes only that electrons are not stationary, not that they are "classical charges" (I am not sure what exactly you mean by that). Electrons not being stationary and having some non-zero expected average momentum or orbital angular momentum is a valid assumption in quantum theory too, for example, if atom is in field of external EM wave, expected average momentum of electron at some time $t$ is not generally zero. Jun 18, 2018 at 15:33
• @JánLalinský What I meant by "classical charge" here was that the concept of the Coulomb force acting on a wavefunction doesn't really make sense; hence, the only way to sensibly interpret the question was to say that the force was acting on classical particles or distributions. Jun 18, 2018 at 22:29
• SHOBH asks about a statement containing electrostatic forces and electrons which he does not understand. He does not require that this statement is to be understood/explained assuming electrons are classical particles - that is your interpretation. Another way to explain the statement is that the "electrostatic force" is a common figure of speech for using Coulomb potential. If you think that statement is wrong for some reason, I think that would be best communicated in an answer. The question itself is fine, jumping to conclusions in a comment is not warranted. Jun 18, 2018 at 23:34

The Coulomb interaction remains valid when velocities are much smaller than the speed of light. This is called the "quasistatic" regime, and it's the one that applies in the analysis of the electron-nucleus interaction.

When talking of electron and nucleus, one has to use quantum mechanics to model the interaction. In quantum mechanics the coulomb potential enters the differential equations which will give the eigenfunctions of the system .

The Bohr model was developed on the classical thoughts, as described by Zero , but the correct model is the quantum mechanical which gives the spectra of the atoms and the probabilities of interactions.

• Hi Anna, classical mechanics strikes back! By replacing the electrostatic force with a simpler version of the electromagnetic force, quantum mechanics can be solved with simplified orbital mechanics. You can read my answer to this question or read about it in my blog: unlimitedphysicsblog.wordpress.com/2018/03/19/… Jun 18, 2018 at 9:44

To a good approximation the nucleus can be considered at rest and the nuclear potential to be static.

You are absolutely correct, it does not make any sense to use the electrostatic force on an electron racing around the nucleus.

The electrostatic force was formulated by Coulumb and is therefore also called Coulumb's force. Coulumb measured the attractive and repulsive forces between charged iron spheres. In such a charged object, electrons will repel each other, and the force work in all directions. In a stationary charged iron sphere, force, current and magnetic field, goes in all directions, as the individual electrons move in all directions.

For a current in a wire we can't use the electrostatic force, as the charges are moving in the same direction. As the current move in one direction, force and magnetism act in perpendicular directions as described by the right hand rule. For charges moving in the same direction we therefore use a simpler version of the electromagnetic force.

It was Niels Bohr which used the electrostatic force as the force between the electron and the nucleus. And with it he constructed a solar system like Newtonian model of the atom, where electrons are confined to Bohr orbitals. But now we photograph the atom, and there are no Bohr orbitals, we observe an electron haze, so Bohr's assumption was wrong.

So we might update Bohr's postulates with our new observations. If we simply multiply Bohr's electrostatic force with the Bohr radius, we set the electron free from the Bohr orbitals and allow electrons to orbit the nucleus in an electron cloud:

We can then also see how the electromagnetic force works different in an atom, in a current and in a charged object, all due to the elementary charges, adding up. The relationship on how elementary charges add up can be described like this:

By using this simpler electromagnetic force Quantum Mechanics can be solved with simplified orbital mechanics. It even gives a new and simple term of the fine structure constant. Here is a link to the publication which solves QM in the simplest way: http://vixra.org/abs/1804.0050