I've been trying to convince myself that the assertion that I've read in basic E&M books (Halliday & Resnick, Purcell), and even Griffiths, that the electrostatic potential at a point in space is equal to the sum of the potential contributions from each of the individual charges. My hang-up has been the direction in 3D space that a path integral is brought in from infinity, given that there are multiple charges each creating their own lines with the arbitrary point, and each line extends to infinity along different directions. (I hope my meaning is not lost here).
Would this reasoning starting from a system of one charge, and then adding additional charges, be sufficient to explain it:
1) For a system of one charge, the electrostatic potential at an arbitrary point is the negative path integral of the field dot ds from infinity, in the direction along the line connecting the point to the charge.
2) Adding a second charge, the electrostatic potential at the same arbitrary point is the negative path integral of the sum of the fields dot ds from infinity (in the same direction as 1) to the point.
Can this sum be broken down as such:
-a) the path integral of the sum of the fields is the sum of the path integrals from the field of the original charge E1 and the path integral of the field of the second charge E2
-b) the path integral of the field of the second charge dot ds (along the original direction), because of the nature of conservative fields, is equal to the sum of
---i) the path integral of the field of the second charge along the line aligning the second charge to the arbitrary point, given in the usual form: V = kq/r. This is in a different direction than the path integral for charge 1, but the change in potential is described by the equation here.
---ii) the path integral of field along the connecting arc at infinite distance. This connecting arc path integral has length proportional to r (going to infinity), but field strength inversely proportional to r squared, so that this component becomes negligible.
So the potential is then the sum of the original path integral of one charge and the path integral of the second charge along a different direction to infinity. This logic is repeated for additional charges.
I can draw a picture of my thinking if people suggest that in the comments.