Why does $epsilon_0=1$ in CGS units?

Question

This Wikipedia article gives a table of certain constants in different systems of units. I noticed that in Gaussian and electrostatic CGS the value of $epsilon_0$ equals the dimensionless $1$.

I wondered whether anyone could provide justification for this? I was under the impression that $epsilon_0$ is not even necessary in these units since it's essentially just a dimensional conversion factor in the SI, however I suppose it would still be possible to convert it into CGS units. How is this done?

jim · Accepted Answer

Possibly the best way to see this is to consider Coulomb's Law that gives the force F (in Newtons) between two charges $Q_1, Q_2$ (in Coulomb) separated by a distance $R$ (in m): $$F = frac{Q_1 Q_2}{4 pi epsilon_0 R^2}.$$
In cgs units the force is measured in dyne, $1 N = 10^5 dyne$, the unit of length is measured in cm, $1 m = 100 cm$. So if you measure force, f, in dyne and separation, r, in cm then
$$10^{-5} f = frac{Q_1 Q_2}{4 pi epsilon_0 (10^{-2} r)^2}.$$
You can now define a charge $q$ by $q = Q/sqrt{10^{
+5} times 4 pi epsilon_0 times 10^{-4}}$ then Coulombs Law becomes $$f = frac{q_1 q_2}{r^2}.$$ It follows that the relationship between charge in SI units and cgs units is $$q (Fr) equiv 2997919999.934 ,,, Q (C)$$

Pangloss · Answer

The value of $epsilon_o$ alone is meaningless,
but the question becomes meaningful if examined in a wider perspective.
For historical reasons electromagnetism was developed following
various formal schemes (characterized by different definitions of some quantities).
Consequently many mathematical formulae, dimensions of quantities and
systems of units don't match.
This may be a bit annoying when looking at different textbooks
and when we need to compare units with not equivalent dimensions.
This well known formal problem can be solved by building an "ancestor" system
having three arbitrary constants, as shown by Wikipedia and Jackson.
Each formal scheme is uniquely determined by a suitable set of constants,
that enable us to write all electromagnetic relations
in every form, as well as to find all dimensional relationships (allowing
an automatic conversion of units belonging to totally different systems).
A good choise of the parameters is tabulated below.
Here the three constants $k_o,epsilon_o,mu_o $ are indipendent,
instead  $chi = csqrt{(epsilon_o mu_o)}$ and $xi = frac{1}{(epsilon_o k_o)} $
are two useful dependent parameters.
begin{array}{|l||c|c|c||c|c|c|}
hline
text{Formal system}  rule{0pt}{14pt} rule[-8pt]{0pt}{12pt}  
& quad k_oquad & quad k_oepsilon_o quad & quad epsilon_omu_o quad &  chi  &  xi  
hline
text{BIPM - SI}  rule{0pt}{18pt}  & k_o & frac{1}{4pi} & frac{1}{c^2}   & 1 & 4pi 
text{ES abs}   rule{0pt}{14pt}  & 1   & 1   & frac{1}{c^2} &   1 & 1 
text{EM abs}   rule{0pt}{14pt}  & c^2 & 1 & frac{1}{c^2} &  1 & 1 
text{Gauss}         rule{0pt}{14pt}  & 1   & 1 & 1 &  c & 1 
text{Heaviside}     rule{0pt}{14pt}
             rule[-10pt]{0pt}{10pt}   & frac{1}{4pi} & frac{1}{4pi} & 1   & c & 4pi 
hline
end{array}
So e.g. Maxwell's equations can be written in a general form,
which easily includes all schemes as special cases.
begin{cases}
begin{aligned}
& nablacdotboldsymbol{D}   = frac{4pi}{xi} rho  
	&& boldsymbol{D}     = epsilon_oboldsymbol{E} + frac{4pi}{xi}boldsymbol{P} 
& nablacdotboldsymbol{B}   = 0 
& nablatimesboldsymbol{E}  = -frac{1}{chi}frac{partialboldsymbol{B}}{partial t} 
& nablatimesboldsymbol{H}  = frac{4pi}{xichi}boldsymbol{J} + 
frac{1}{chi},frac{partialboldsymbol{D}}{partial t} qquad 
	&& boldsymbol{H} = frac{boldsymbol{B}}{mu_o} - frac{4pi}{xi}boldsymbol{M}  
end{aligned}
end{cases}

Why does $epsilon_0=1$ in CGS units?

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