# Who established that $K_e = \frac{1}{4 \pi \varepsilon _m}$?

Coulomb's Law states that : $$F_e = K_e\dfrac{q_1q_2}{r^2}$$ where $$q_1$$, $$q_2$$ are magnitudes of the two point charges, $$r$$ is the distance between them and $$K_e$$ is Coulomb's Constant (aka Dielectric Constant). In my Physics textbook, it's mentioned that $$K_e$$ can also be written as $$\dfrac{1}{4\pi\varepsilon_m}$$, where $$\varepsilon_m$$ is ther permittivity of a medium $$m$$ but no details about its establishing or derivation are given.

So, I'd like to know who established this (I think it was Gauss, though), thanks!

PS : If it was Gauss, was it established by using Gauss's Law?

• $4\pi r^2$ is the surface area of a sphere of radius $r$. – Spencer Jun 4 '20 at 13:11

It is not that a particular person established this relation. This relation is a result of the historical development of the idea of assigning units or rather dimensionality to charge $$q$$. According to the Wikipedia article on Vacuum permittivity:

The experiments of Coulomb and others showed that the force $$F$$ between two equal point-like "amounts" of electricity, situated a distance $$r$$ apart in free space, should be given by a formula that has the form

$$F=k_{\text{e}}{\frac {Q^{2}}{r^{2}}}$$

where $$Q$$ is a quantity that represents the amount of electricity present at each of the two points, and $$k_e$$ is the Coulomb constant. If one is starting with no constraints, then the value of $$k_{\text{e}}$$ may be chosen arbitrarily. For each different choice of $$k_e$$ there is a different "interpretation" of $$Q$$: to avoid confusion, each different "interpretation" has to be allocated a distinctive name and symbol.

In one of the systems of equations and units agreed in the late 19th century, called the "centimetre–gram–second electrostatic system of units" (the cgs esu system), the constant $$k_e$$ was taken equal to 1, and a quantity now called "gaussian electric charge" $$q_s$$ was defined by the resulting equation

$$F={\frac {{q_{\text{s}}}^{2}}{r^{2}}}$$

Thus, charge was expressed in esu, a derived CGS unit with dimensions as $$[M]^{1/2}[L]^{3/2}[T]^{-1}$$.

The idea subsequently developed that it would be better, in situations of spherical geometry, to include a factor $$4π$$ in equations like Coulomb's law, and write it in the form:

$$F=k'_{\text{e}}{\frac {{q'_{\text{s}}}^{2}}{4\pi r^{2}}}$$

The quantities $$q'_s$$ and $$k'_e$$ are not the same as those in the older convention. Putting $$k_e = 1$$ generates a unit of electricity of different size, but it still has the same dimensions as the cgs esu system.

This is the Lorentz–Heaviside system of units. Lorentz–Heaviside units, unlike Gaussian units, are rationalised, meaning that there are no factors of $$4π$$ appearing explicitly in Maxwell's equations.

The next step was to treat the quantity representing "amount of electricity" as a fundamental quantity in its own right, denoted by the symbol $$q$$, and to write Coulomb's Law in its modern form:

$$F={\frac {1}{4\pi \varepsilon _{0}}}{\frac{q^{2}}{r^{2}}}$$

This is the SI system with Ampere as a fundamental unit for electric current. Coulomb is the unit of charge with one Coulomb being equal to one Ampere-second. This makes the dimensionality of charge as $$[I]^{1}[T]^{1}$$.