For an arbitrary charged object, it seems to be the case that we express it as a continuous sum (sum on the reals/integral) of point charges $dq$ that have a canonical Coulomb's law force.
That is to say, for an arbitrary charged object, we split it up into tiny $dq$'s (located at $\vec r'$, with the force exerted on reference point $P$ at $\vec r$ by them equal to..
$\text{let} \ \vec r - \vec r' = \vec \zeta$
$$F_{dq} = k \ dq \ \frac{\vec{\zeta}}{\zeta^2}$$
Implying..
$$\vec E = k \int \frac{1}{\zeta^2} \vec \zeta dq$$
But why do we assume that $dq$ exhibits the form $F_{dq}$? It's almost like there's a fundamentally point-like charged particle composing all charged objects.. aha! Electrons. But wasn't this theory established independent of electrons? How could we justify them without electrons? Do we need to? Is that even the justification for it? Why are we allowed to assume all charged objects are made of infinitesimal point charges and do electrons have anything to do with it?
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