What is the difference between 1 4 pi epsilon and 1 k?
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Everything I typed was lost when this was supposed to be a long question.Here goes.
Defining the symbol $k$ in Coulomb's law is perfectly allowed when one understands it as a definition of $The motivation for this definition is that when you work out the forces between two plates, they come out as F.
When you look into a theory of capacitance, you find that the voltage between the plates is the same as the one in A/d$.If you want to put a dielectric in between the plates, you need to know the electric permittivity.The permittivity of free space is simply what we mean by it.
Why is the "derived" unit, $epsilon_0$, treated as more "fundamental" than the original $k$?The permittivity of free space was easier to measure during the 19th and early 20th century when electrical research was more focused on circuit-based technologies.
The time duration of a certain number of periods of radiation emitted from a particular type of electron transition between energy levels in an isotype of Cesium is defined by the unit of the second.
It is an assumption that light travels at a constant speed $c$ independent of one's reference frame, so now we can define a unit of length: the meter is the distance light travel in $1/299792548.
The SI unit of current is defined so that the permeability of free space takes on a desired value.
We can then define varepsilon _0 and k as well.Is it worth it?
You don't have to fix a system of units to do this.Any system of units will be held in the above definitions.To see that these definitions don't end up being circular, it helps to know that we can define them in terms of purely physical phenomena.For the above definitions to make sense, we had to know that we could define both of them.This can be done with the above definition of SI units.
An equally valid question could be why the "$4pi$" in the Coulomb constant is magnetic.
Maybe a clue can be found in the equation for the speed of light in a vacuum.
The electric permitivity is related to magnetic permeability in the vacuum.
The reason for the "$4pi$" appearing here and in Coulomb's constant is so that the equations can be written without any factors.
Consider how Gauss' law describes the "flux through a closed surface enclosing the charge", compared to Coulombs law, which describes field intensity at a distance squared.