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I learned in this video and in this answer that electrons repel from each other by exchanging photons. This makes sense when it comes to electrons, but what happens when two protons or two positrons repel each other? Do positively charged particles also exchange photons?

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Yes. Photons are the carriers for the electromagnetic force, regardless of the charges involved.

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  • $\begingroup$ Unless of course the charge is zero ;) $\endgroup$
    – gj255
    Commented Oct 24, 2016 at 19:01
  • $\begingroup$ @gj255 reading your comment I'm getting curious how neutrons get scattered of each other? $\endgroup$ Commented Oct 24, 2016 at 20:57
  • $\begingroup$ I don't know the answer to this, but I can suggest two possibilities. Neutrons are composed of quarks, which are electrically charged. Hence, if two neutrons get sufficiently close together that their internal composition becomes important, the component quarks can exchange a photon. It's also possible, however, that the scattering involves the exchange of some particle other than the photon. At the level of 'effective theory', nucleon interactions can in fact be described by exchange of pions. $\endgroup$
    – gj255
    Commented Oct 24, 2016 at 21:02
  • $\begingroup$ As I understand it, neutron-neutron (and neutron-proton) interactions involve the strong nuclear force, not electromagnetism at all. $\endgroup$ Commented Oct 24, 2016 at 21:47
  • $\begingroup$ @HolgerFiedler In addition to the strong nuclear force, neutrons experience a (feeble) dipole-dipole magnetic interaction mediated via photons. Would make an interesting follow-up question. $\endgroup$
    – rob
    Commented Oct 24, 2016 at 22:33

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