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Coulomb's Law states that : $$F_e = K_e \dfrac{q_1q_2}{d^2}$$ where $F_e$ means the electrostatic force between two charges of magnitude $q_1$ and $q_2$ and $d$ is the distance between them.
Scientist Charles Augustine de Coulomb derived it in the following manner:

He observed that $F_e$ was directly proportional to the product of the charge of the particles. So, $$F_e \propto q_1q_2$$ He also observed that $F_e$ is inversely proportional to the square of the distance between the particles (inverse-square law). So, $$F_e \propto \dfrac{1}{d^2}$$ Using these two proportions hence obtained, he concluded that :$$F_e \propto \dfrac{q_1q_2}{d^2}$$ which can be expressed in the form of an equation like this : $$F_e = K_e \dfrac{q_1q_2}{d^2}$$ where $K_e$ is the constant of proportionality and is commonly known as Coulomb's Constant or the Electric Constant.

Now, Coulomb observed two factors that affect the force of attraction between any two charged particles. But how did he conclude that there were only $2$ factors that affected the electrostatic force between particles? Could there not be a third factor which affects it too?
Or are we still not sure about the existence of any other factor?

Thanks!

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Controlled experimentation/observation is generally the best way to go when setting up a mathematical equation for the phenomenon.

So in case of the electric force, while experimenting they considered various parameters that the force could potentially depend on and tested for it. All of that ultimately led to the form Coulomb got.

Like in all of science, the laws that we have are actually hypothesis that are continuously tested. And so far, Coulomb’s law holds in all those cases.

See this answer for regime where it is violated.

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  • $\begingroup$ Thanks a lot! I'd be glad if you could give some examples of the parameters that you referred to in your answer. :) $\endgroup$ – Rajdeep Sindhu May 30 at 14:13
  • $\begingroup$ @RajdeepSindhu well things like temperature, mass and such. $\endgroup$ – Superfast Jellyfish May 30 at 19:27
  • $\begingroup$ Thanks...${}{}$ $\endgroup$ – Rajdeep Sindhu Jun 1 at 2:24
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Check Casimir force. It is a force due to quantized electromagnetic fields, and it is a mutual force between particles, static or not. So, Coulomb did understandably miss this.

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  • $\begingroup$ Thanks! I didn't get it much from that Wikipedia page though. It would be really nice if you could elaborate. $\endgroup$ – Rajdeep Sindhu May 30 at 8:06
  • $\begingroup$ @RajdeepSindhu It arises from quantized electromagnetic fields, which suggests our vacuum is not totally nothing. The vacuum stores energy, and when we change the shape of it, we also change how much energy can be stored. In your example, when we move away two objects, the overall energy usually gets smaller, which means we need contributing some "force" when moving them apart, otherwise these two objects would attract each other. This Casimir force is independent of electric charge though. $\endgroup$ – naijing May 30 at 8:13
  • $\begingroup$ I'm so sorry, but I don't think I still get it (I'm just in 10th grade), but is that in some manner signaling that there are factors other than the magnitude of charges and the distance between them that affects the electrostatic force? $\endgroup$ – Rajdeep Sindhu May 30 at 8:42
  • $\begingroup$ @RajdeepSindhu Electrostatic force is basically a synonym for the force described by Coulomb's law. So, are there other factors affecting electrostatic force? No. Are there other electromagnetic forces that affect static objects? Yes. $\endgroup$ – naijing May 30 at 8:48
  • $\begingroup$ Thanks! Just verifying this : Do you mean that electrostatic force has no other factors affecting it because we defined it with respect to Coulomb's Law? $\endgroup$ – Rajdeep Sindhu May 30 at 10:01

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