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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).

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    $\begingroup$ The electrostatic repulsion is going to overpower any attractive force (including gravity) in this situation. Having said that, electrons have a magnetic moment - two electrons with their spins aligned anti-parallel might weakly attract each other magnetically, but I'm not sure. $\endgroup$ – Xeren Narcy May 21 '15 at 2:16
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    $\begingroup$ Something something Cooper pair. (Related: physics.stackexchange.com/q/67774/44126) $\endgroup$ – rob May 21 '15 at 3:21
  • $\begingroup$ spinning charged sphere produce magnetic field, electron is zero size particle (rutherford scattering), electron must be a very tiny sphere violating SR due to the electron spin angular momentum (classical physics), then QM introduce their own SR... I'm now too scared to go down this rabbit hole just to find some condensed matter! $\endgroup$ – user6760 May 21 '15 at 3:32
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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.

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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.

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  • $\begingroup$ for this to work I need battery, wires and 2 metal plates, am I right? $\endgroup$ – user6760 May 21 '15 at 5:31
  • $\begingroup$ To accelerate you need metal plates yes, but to see any effect or play with it you might need a cathode ray tube otherwise the ejected electrons will hit the positive plate. $\endgroup$ – Ilaya Raja S May 21 '15 at 7:18
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In certain scenarios there can be a magnetic attraction, but the electrostatic replusion will greatly overpower it.

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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.

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    $\begingroup$ At the center of a sphere, what do you propose to be the direction of that "positive electric field"? Why would the existence of one field cancel the other field (namely, the field of the electrons)? This looks completely wrong to me. $\endgroup$ – Floris Oct 17 '16 at 13:07
  • $\begingroup$ Please explain, supported by formulas, how you suggest that two electrons in a positive electric field lose their mutual repulsion. To me, your statement does not seem very trustworthy without an explicit calculation or argument. $\endgroup$ – Sanya Oct 17 '16 at 13:09

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