Due to electrostatic repulsion the two electrons will repel each other as they both possess similar charges (lets leave gravitational attractive force out of the picture).
My question is: can there be any attractive force(s) between the electrons? I know the reason that the particles stated above cannot share the same quantum state is due to the Pauli Exclusion Principle (PEP).
you can check the discussion here. There is a certain case in which a phonon mediates attraction between two electrons. Indeed, acoustic phonons correspond to a slowly varying in-space displacement of atoms which produces a charge. This charge, in turn, results in an electric potential for the electrons. This means that the electron distorts the crystal lattice (builds up a positive charge around itself) which, at the end, attracts other electrons.
Two electrons when they move experience these forces $$ F_{electrostatic repulsion } = \frac{ke^2}{r^2}$$ And, $$ F_{magnetic attraction} = \frac{μ_0 . e^2 v^2}{4 \pi . r^2}$$
As you can see from the formulae for attraction there must be a velocity.
For the two forces to be the same the speed of the electrons must as fast as light, practically these two electrons will move in a double helix with increasing radius.
In certain scenarios there can be a magnetic attraction, but the electrostatic replusion will greatly overpower it.
We are currently exploring a hypothesis that suggests two electrons can exhibit attractive forces under certain conditions, specifically when considering their magnetic dipole moments and specific spin configurations. Our team is actively seeking a laboratory equipped with the necessary technology and instrumentation to empirically validate this theory.
A positive electric field exists at the center of a positively charged sphere. Electrons immersed within this field lose their mutual repulsion for each other and could touch each other due to gravitation.
