Gaussian Unit of Charge and Force

Question

I just read that in the Gaussian Units of charge The Final equation in Coulomb's law is as simple as $$\boldsymbol{F}=\frac{q_1q_2}{r^2}$$

No $\epsilon_0$ no $4\pi$ like you have in the $\mbox{SI}$ units of measurement .

The permittivity constant was the factor in the $\mbox{SI}$ system of Coulombic Force that determined the intensity of force in a medium.

In the Gaussian system i see no such constant . So that would mean that some other factor would govern the Quantity of charge on a body.

What is that factor ?

(I vaguely remember it as being related to the speed of light $c_0$)

Answer 1

I believe you're thinking of the statcoulomb, which is the CGS/Gaussian system unit of charge.

The article linked above details some methods of conversion, which you can read for yourself, but probably the most important ones to know that:

1) $1\ Statcoulomb = \frac{{(gram)^{1/2}}(cm)^{3/2}}{sec}$

2) $1\ Coulomb \leftrightarrow \sqrt{4\pi \epsilon_0}\frac{c}{(1m/s)} Statcoulomb$

Answer 2

You said it clearly enough.

Let $q$ refer to the Gaussian charge and $e$ to the charge in SI units. Then $$\frac{qq'}{r^2} = \frac{ee'}{4πε_0r^2}.$$ Therefore, $$qq' = \frac{ee'}{4πε_0}.$$ The same proportion is used uniformly for all the charges, so, $$q = \frac{e}{\sqrt{4πε_0}}, \hspace 1em q' = \frac{e'}{\sqrt{4πε_0}}.$$

The permittivity $ε_0$ is mixed up with the charges in Gaussian units.

Now, we get to the sources of one of the biggest misunderstandings and miscommunications in contemporary physics. This will put your other comment "some other factor would govern the Quantity of charge on a body" in perspective.

Quoting Weinberg (as a typical example reflecting the view of this issue commonly stated in the theoretical literature):

"[⋯] in order for the charge $q_ℓ$ to characterize the response of the charged particles to a given renormalized electromagnetic field, we should define the renormalized charges by $$q_ℓ = \sqrt{Z_3} q_{Bℓ}\tag{10.4.18}.$$

We see that the physical charge $q$" [the "dressed" charge] "of any particle is just proportional to a parameter $q_B$" [the "bare" charge] "related to those appearing in the Lagrangian, with a proportionality constant ${Z_3}^{-1}$ that is the same for all particles."

That's Gaussian units! And, on account of the mixing up of the permittivity in with the charge is totally confused and wrong. The actual statement is seen clearly in SI. So, here it is, rewritten:

"[⋯] in order for the charge $e/\sqrt{4πε_ℓ}$ to characterize the response of the charged particles to a given renormalized electromagnetic field, we should define the renormalized permittivity in it by $$\frac{e}{\sqrt{4πε}} = \frac{e}{\sqrt{4πε_0{Z_3}^{-1}}} \tag{10.4.18}.$$ We see that the physical permittivity $ε$" [the "dressed" vacuum permittivity] "is just proportional to a parameter $ε_0$" [the "bare" vacuum permittivity] "related to that appearing in the Lagrangian, with a proportionality constant $Z_3$."

There's no need to explain how or why it "is the same for all particles" - because it's not the particles' properties that are being renormalized at all, but the permittivity! There is no distinction between bare and dressed charge - except the Gaussian ones, which are mixed up. The distinction is between the bare versus dressed vacuum!!

Definitions have consequences; and the wrong definitions - as is the case with Gaussian units - lead to the wrong conclusions and wrong physics.

You see it more clearly here. Let $a_μ$ refer to the Gaussian version of the electromagnetic potential and $A_μ$ to the SI form. They are related by $$a_μ = \sqrt{\frac{4π}{μ_0}}A_μ = \sqrt{4πε_0}cA_μ,$$ since $μ_0ε_0 = 1/c^2$. Now, let's look a little further up in Weinberg (with the change in notation from $A$ to $a$)

"The renormalized electromagnetic field (defined to have a complete propagator whose pole at $p^2 = 0$ has unit residue) is conventionally written in terms of $a^μ_B$ as $$a^μ = {Z_3}^{-1/2} a^μ_B, \tag{10.4.17}$$ so in order for the charge $q_ℓ$ [⋯]"

Again, things are mixed up and confused. In SI, where that confusion is removed, this reads:

"The electromagnetic field with the renormalized permittivity (defined to have a complete propagator whose pole at $p^2 = 0$ has unit residue) is conventionally written in terms of $\sqrt{4πε_0}cA^μ_B$ as $$\sqrt{4πε}cA^μ = \sqrt{4πε_0{Z_3}^{-1/2}}c{A^μ}, \tag{10.4.17}$$ so in order for the charge $e/\sqrt{4πε_ℓ}$ [⋯]"

Again: there is no distinction between "bare" versus "dressed" fields, charges or anything else like that. The distinction is between the bare versus dressed vacuum! A similar comment can be made for the gauge fields, and even for the fermion fields, where there is another coefficient of that type that's hidden from view (the spinor metric) by mixing it up into the fields.

Now, you can begin to see what I don't do anything with Gaussian units - and never did. They've likewise mostly disappeared from other contexts, except the one lone hold-out: the theoretical literature. It has contributed nothing, there, but confusion and false attributions, the cited excerpts being a major case in point.

The quotes are from section 10.4 "Renormalized Charge And Ward Identities" of chapter 10 "Non-Perturbative Methods" of Volume I (Foundations) of "The Quantum Theory Of Fields", Weinberg, Cambridge University Press, 1995.