Interconnection between Vacuum Permittivity and Vacuum Permeability
Motivation behind Vacuum Permittivity and Vacuum Permeability
Right from the start we see the equation $c^2 = \dfrac 1 {\varepsilon_0 \mu_0}$ and we understand immediately that these parameters, vacuum permittivity and vacuum permeability, are tightly wound up in the structure of the electromagnetic spectrum.
One of them defines the strength of the electric attraction.
The other defines the strength of the magnetic force.
Hence it is apparent that the units of measurement we use are to a certain extent arbitrary, as the concepts being discussed are more abstract than tangible physical measurements.
But in order to keep the relationships within the grasp of the tangible, we need to make sure the units we use are accessible enough to interface neatly with the existing measurement systems we already have.
Prevalent is the SI system, in which we can set up units of considerable convenience.
We notice that if we define $\mu_0$ as $4 \pi \times 10^{-7}$ in henries per metre, which evaluates to $\dfrac {\mathrm {kg} \times \mathrm m} {\mathrm A^2 \times \mathrm s^4}$, then many of the equations work out with manageable numbers all approximately the same size.
From this convenient number, we define the Ampere, and work out what conditions we require in order to produce it.
We use the Biot-Savart Law to work out what to expect, and then measure the force and work out the rest of the numbers from there.
Hence having chosen $\mu_0$ in that way, we must then define $\varepsilon_0$ as $\dfrac 1 {\mu_0 c^2}$.
Or instead, we could declare what value $\varepsilon_0$ should be based on accurately and reliably reproducible standards, that is $\dfrac {e^2} {2 \alpha h c}$, and then force $\mu_0$ to be $\dfrac 1 {\varepsilon_0 c^2}$.
The latter is the direction it currently lies, after the $2019$ standards redefinition.
We discuss the CGS system in passing, in order to complain about its confusingness and fiddly-smallness.
The author of this page thanks everybody in his educational system that he was never asked to learn it in the first place, coming straight in as they did with amps, coulombs, volts, webers, teslas, farads and henries.