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If we have two like charges separated by some distance $d$, we know there is an force on each due to the other. How can we move from this into the idea of electric field?

In the sense that the electric field means the charge has it's influence throughout space without regard to another charge being present or not? Or is it just a mathematical convenience to define electric field?

I can't imagine an experimental test for this because to detect the field , we must measure, so it is not possible to say if the field causing the force was pre-existing or came into existence due to our measurement via a test charge.

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  • $\begingroup$ The question may not be phrased in the best way, feel free to edit if you think you can explain the idea I have better. $\endgroup$ Jun 23, 2021 at 11:07
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    $\begingroup$ I think that your wording is excellent. Your question is closely related to the nineteenth century debate about action-at-a-distance between charges, where the only role of the intervening space was to provide the separation between the charges, and the field theory of interaction. I suppose that the tide started to turn in favour of the field theory when it was demonstrated that the interaction between distant charges is not instantaneous. But let's hope for some good answers... $\endgroup$ Jun 23, 2021 at 11:27

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There are two main things to say:

  • The concept of electric field is not strictly necessary. Or at least it's not strictly necessary when studying electrostatical or simple electrodynamical phenomena. You could simply think about electrical interactions as a sum of single interactions between point particles. However it's easy to see that conceptualising the electric interaction with a field is practically useful and esthetically pleasing.

  • If you choose to describe electric and magnetic interaction through fields you get the Maxwell's equations. Of course we could in principle express the same physical laws in another form that does not involve fields, but nobody bothered to do it! And that's because theoreticians tend to prefer the mathematically simpler and more elegant way to express laws, and at least for now nobody has come up with a better more elegant way of describing the laws of electromagnetism.

  • Bonus point: The concept of electric and magnetic fields, as well as the correlated concepts of electric potential and magnetic vector potential, become crucial in quantum mechanic. Keep in mind that in QM the concept of "force" lose its meaning and relevance; the dynamics of a QM system is described through the Hamiltonian, constructed with the potentials of our system. So even if the concept of fields is technically avoidable in classical mechanics once you bite the quantum apple it becomes truly essential.

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  • $\begingroup$ While this is a good answer, one can argue that the Hamiltonian became important in classical physics (much before quantum mechanics) precisely because the field description was chosen. So in some sense, if the field description was absent, quantum mechanics may have looked different - assuming we still found it/tested it in the same way and time as was done. $\endgroup$
    – newtothis
    Aug 25, 2021 at 13:48

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