Earnshaw's theorem says that:

A collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges.

I also know about this theorem being proved visually by Gauss' law, where there is this idea about a negative charge being there. But I want that case where there is a positive charge surrounded by other positive charges.

I know I am getting something wrong, but could you please explain what and why this is true or false?


The answer is double:

  1. Atoms and molecules are not in stationary equilibrium; electrons are moving all around. An electron moving on circular orbit is in stable equilibriun, altough not stationary, so can circumvent Earnshaw. I think that with orbiting electrons, it is possible to devise a somehow stable chemical bond between to positive charges. The new issue here is that the orbiting electrons should radiate electromagnetic waves, thus lose energy and decay, so the susytem is again not stable! This was gigantic issue in the atomic theory at the beginning of 20th century.

  2. By classical electromagnetism, Earnshaw theorem holds, so as you said, atoms and chemical bonds are terribly puzzling! To solve this and other issues, quantum mechanics has been invented; electrons must respect a new "exclusion rule" that put each electron in quite fixed and stable orbitals around atoms. In chemical bonds, some external electrons orbit around both involved atoms, making the bond stable; and the quantization of the orbits keep electrons form decaying.

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