electrostatics - Superposition Principle for Electric Fields

If there is a collection of charges $q_1,q_2,q_3....q_n$, and we want to calculate the total Electric Field due to all these charges at a point $P$ ,then the we sum them all up by the principle of superposition.

$$E_{tot}=E_{1P}+E_{2P}....E_{nP} $$ Where $E_{nP}$ is the field due to the $n^{th}$ particle at point $P$.

I know that this superposition principle has been proved experimentally, but is there a more deeper reason as to why this happens? Why is the field at a given point independent of the fields due to other charges. I mean it could have also been something like: $E_{tot}=E_{1P}$ x $E_{2P}$ x .... x $E_{nP} $. Is there a particular reason for nature to act this way? I am sorry if this is a stupid-silly question.

Answer

The effect of each charge appears to be completely independent of the effects of other charges.

So those effects get summed.

The effects of different charges sort of cross right over each other without affecting each other. Like ripples on a pond cross each other and continue, each unaffected by the others.

It could have been that they affected each other, and that would be more complicated -- if you had to take account of the way they affect each other that would require more work than when they don't affect each other at all.

But it didn't turn out that way.

It sounds like you are asking for a complete theory about electric fields that would have this result fall out of it. You could imagine "OK, THIS is what electric fields are like! Now I know.".

You could get a theory like that, but it would have to be compatible with relativity theory, since the numbers come out right from that. People have a hard time envisioning relativity, and if you had an alternative that got the same results, it might likely be hard to envision that too.

It would only be a way to think about it. The math gets the right experimental results within experimental error. Concepts about what's happening with the math are useful for your intuition, and they are likely to suggest interesting applications of the math.

But probably there will be multiple concepts that fit, with no way to show which of them is right. They're just ways to think about it that are compatible with the math.

The particular thing we're seeing here is that the forces that each charge puts on all other charges after a delay, do not change each other. They all act independently. That's what the math describes.