Why is the constant of proportionality in Gauss's law exactly $1/\epsilon_0$

Question

$\epsilon_0$ is epsilon naught, or permittivity of free space.

Let me preface this by saying that I've just started to learn about electromagnetism. When I first saw Coulomb's law, I was incredibly confused to why the proportionality constant was exactly $1/4\epsilon_0$. Then I saw that this was derived from Gauss's law. In Gauss's law though, the constant of proportionality is $1/\epsilon_0$. Then I wondered how that is true? I researched more and find that most people derive Gauss's law from Coulomb's law. So it seem's to be more like circular reasoning?

This might be similar to this question: What was discovered first - The Coulomb constant or Gauss law?

But I find the answer here not satisfactory in that it never explains why the constant of proportionality is $1/\epsilon_0$. I kind of get that Gauss's law is a mathematical theorem and that it comes from something called the Divergence theorem that is then applied to electrostatics. I don't understand the math because it is not at my level but I can understand why the general form of the equation is like that. What I don't get is why that formula has $\epsilon_0$ as the proportionality constant. Where did it come from?

I hope my question makes sense. To summarize: What I want to know is, why is $\epsilon_0$ in these equations and how?

Answer 1

Gauss's law is not a mathematical theorem. It is an empirical law. It is an observed fact that the total flux out of a surface is equal to a constant times the contained charge. You cannot derive this constant ($\epsilon_0$) from pure math alone. At least, not with any current theory - some physicists hope that we might one day find a theory of the universe that explains all the constants we see in nature. But so far, Gauss's law is an observed fact, not an essential consequence of any mathematical theorem.

Coulomb's law is just the same. It's a mathematical equation that we observe works for describing reality.

If we assume Coulomb's law, then we can derive Gauss's law (in the way you allude to, using the divergence theorem). If we assume Gauss's law, we can derive Coulomb's. In some sense, they encode the same information, and so it is not surprising that they both have the same constant $\epsilon_0$ in them. But ultimately, you must start from one or the other. Neither they, nor the constant $\epsilon_0$, can be derived from pure math.

At some level, you can't ask "why" they're true. They're true because that's how we've discovered the universe happens to function.

(Also, to clear up some potential confusion: there is a related mathematical theorem that is sometimes called "Gauss's theorem" - and that's the divergence theorem that you mention. This theorem has many applications in electromagnetism, but it does not contain $\epsilon_0$, and it is not the same as the "Gauss's law" that physicists refer to.)