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The interaction (of an electron with an external electromagnetic field) is because of the electron's charge, but charge causes an electric field, so, by Syllogism one can say the interaction is because of the electron's electric field.

Now we can say that, by linearity, the external field and the field of the electron sums together. So how does a force arise from this summing process?

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    $\begingroup$ If you wish to use this approach to modeling the interaction, note that the field is the vector sum of the fields, but the energy in the field is proportional to the square of the magnitude of the field. What is the consequence? $\endgroup$
    – John Doty
    Commented Jun 3, 2023 at 12:11
  • $\begingroup$ There is a conceptual error in the problem statement: there is an (1) external EM field and a (2) seperate EM field due to charge of electron. (1) is not produced by charge of electron. However, both (1) and (2) can affect motion of electron, see Wald's Advanced classical electromagnetism $\endgroup$
    – paul230_x
    Commented Jun 3, 2023 at 12:30
  • $\begingroup$ I don't know. What is the consequence? $\endgroup$
    – talanum1
    Commented Jun 17, 2023 at 9:17

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Nature likes to minimize a lot of things. One thing in particular that it likes to minimize is the magnitude of electric potential at any given point in spacetime.

Moving like charges apart allows for the magnitude of electric potential to be minimized. Ergo, electric repulsion.

Similarly, moving opposite charges closer allows for the magnitude of electric potential to be minimized. Ergo, electric attraction.

Since $\mathbf{E}=-\mathbf{\nabla}V$ so we can just as well make these same predictions in terms of electric fields, but it's just easier to make these statements in terms of electric potential.

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    $\begingroup$ Nature likes to minimize the total potential energy of a system but that runs into the problem of the infinite self-energy of an ideal test charge, I believe. I suspect that that's the reason why we don't like to teach it this way. I don't know if that problem still exists if we use the proper action principle, but I suspect that nothing on the classical level removes this problem for fields with point charges. I might be wrong. $\endgroup$
    – FlatterMann
    Commented Jun 3, 2023 at 20:13
  • $\begingroup$ Knowing the external electromagnetic field and the electromagnetic field of an electron, how do you derive the force law? $\endgroup$
    – talanum1
    Commented Jun 17, 2023 at 9:42

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