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There are two neutral atoms set separated at a long distance $R$ and let's consider them phenomenologically through Bohr model. Let's also assume that the nuclei (charged $+q$) of the atoms are fixed and punctate and that their surrounding electrons (totally charged $-q$, mass $m_-$) have perfect circular orbitals at a short distance $r_0$ ($\ll R$). The net electromagnetic (EM) force between the two is approximately zero due to their electrical neutrality. The setup is shown below:

problem setup

Remark: notice that the term "atoms" may describe electrically neutral entities (i.e. whatever it may be) with a central static charge $+q$ and an orbiting charge $-q$ of opposite sign $-$ consider them as microscopic (i.e. quantum-mechanically) or macroscopic (large, classical) objects as well.

Let's see some different characteristic mutual orientations of the orbitals of electrons which are otherwise statistically randomly set:

characteristic orientations of the orbitals

From the picture above, one may notice that the force associated with a configuration of orbitals has its symmetrical counterpart which usually cancels it out when treated classically, e.g. like constant electron currents around a fixed nucleus or two electromagnets made of circular wires with identical currents (which repel/attract each other in order to minimize the EM energy in the field). Also, the electrons may have different mutual phases regarding their orbital "motion", especially if their hosting atoms have different charges (thus those electrons have different frequencies orbiting the atoms).

If we involved Special relativity (SR) to these cases, the symmetry would be broken (here shown for the parallel orbitals only):

relativistic approach

For example, one electron with velocity $\vec v$ would see the other one which is in phase with the same velocity $\vec v$ as static, because they share the same frame of reference. In the case that their velocities have opposite directions ($\vec v$ vs. $-\vec v$), one has to account for relativistic effects, i.e. to relativistically add $\vec v$ and $-\vec v$.

  • Is there a calculation of the net force done which treats the system described as above using either SR or QFT ?

    or

  • Does this make sense at all ?

Of course, the net force is assumed to be averaged over time, over all orbital phases, over small differences of the distance $R$ and over all possible mutual orientations of the orbitals. Further, one might calculate it for all possible atoms (on both sides) with their different charges (i.e. atomic numbers) and therefore different orbital speeds of their electrons.


Motivation: to estimate order of strength of the possible EM force asymmetry compared to gravity, therefore it shall be calculated precisely since the strength of the EM force for the electron is about $10^{40}$ times stronger than gravity.

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The Bohr model is not an accurate model, electrons do not move in circles with a statistical distribution of circles.

Even if they did, just because two charges have the same velocity does not mean the other one sees the other as static.

The force one charge $q_1$ feels do to another charge $q_2$ depends on the position and velocity right now of the charge $q_1$ that feels it and also on depends on the position, velocity, and acceleration of the charge $q_2$ (back in the past) causing the force.

And for a fixed velocity and acceleration it is the force due to the acceleration that is the strongest when you get super far away.

If you want to learn how people keep track of long range forces between neutral atoms, study van der Waals forces.

Also keep in mind that the proton and the electron of the neutral atom both matter, and they have spins with intrinsic magnetic dipoles that also interact magnetically.

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  • $\begingroup$ Thanks for your answer $-$ your arguments hold, all these things are complicating the picture, while van der Waals forces are more surface like (behaving like capacitance, e.g.) due to polarization. Using the Bohr picture here is just for phenomenological pointing out the major mechanism and that is the asymmetry risen from relativistically treated moving of two charged particles (the orbiting electrons, $\vec v_1$, $\vec v_2$ vs. $\vec v_1$, $-\vec v_2$ etc). Does it have an adequate image in QFT, at least approximately ? $\endgroup$ – gox Aug 5 '15 at 9:36

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