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Do you know how the values were measured and if they where determined by particular conditions/ restraints? They can be put to 1 (or to any values, I suppose); what escapes me is why 1/ε × μ must be equal to c2. Was this an arbitrary choice just to prove the speed of propagation?

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The vacuum permittivity $\epsilon_0$ is at first glance just a constant related to electric forces. It can be measured through Coulomb's law that provides an expression for the electric force between two charged point objects (with charges $q_1,q_2$: $$F=K_{\rm C} \frac{q_1q_2}{r^2},$$ where $r$ is the distance between the charges and $K_{\rm C}$ is the proportionality Coulomb constant and you can write it as you wish (depending on your unit system). In modern times, we write $K_{\rm C}=1/4\pi \epsilon_0$ (SI units).

The vacuum permeability $\mu_0$ comes from magnetic effects. The easiest way to measure it is to use Ampere's force law, which describes the magnetic force between two wires carrying DC currents $I_1, I_2$, given by $$f_{\rm A}=2 K_{\rm A}\frac{I_1I_2 }{r}$$

where $f$ has units of force per unit length. In modern times, we write $K_{\rm A}=\mu_0/{4 \pi}$.

If electric forces and magnetic forces were unrelated, we would be able to choose whatever we want for $K_{\rm C}$ and $K_{\rm A}$, but that is not the case. The discovery of electromagnetic induction, imposed equations between electric and magnetic phenomena.

James Clerk Maxwell compiled all the known equations for electricity , magnetism and electromagnetism (today known as Maxwell's equations), and in 1865 he discovered that in the case of empty space you could put them together to write a wave equation. This wave equation indicates the speed of the waves as a function of the constants of electromagnetism, as is given by the ratio

$$c=\sqrt{\frac{K_C}{K_{\rm A}}}$$.

When Maxwell calculated $c$ from that formula, he found that the speed of these electromagnetic waves coincides with the speed of light measured by the experiments of Fizeau and Foucault some years before.

Thus in modern times, when we choose to write everything with $\epsilon_0$ and $\mu_0$, $c=(\mu_0\epsilon_0)^{-1/2}$, this is not a choice. The laws of electromagnetism impose that this constants have to be related by speed of light (only one of the two can be chosen freely).

Additional notes

  • Weber and Kohlrausch seem to have empirically noticed the relation between c, $K_{\rm C}$ and $K_{A}$ before Maxwell in 1835. Maxwell himself cites them, but finds a theoretical reasoning.

  • In "A Dynamical Theory of the Electromagnetic Field", Phil. Trans. R. Soc. Lond 1865, Maxwell uses the relation of Weber and uses the units $K_{\rm A}=4\pi$ and $K_{\rm C}=4\pi c^2$.

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  • $\begingroup$ Useful nuggets of information in your answer but it does not address the key question about the empirical measurement of the two constants. On original reference which talks the actual experimental set up and measurments would be truly useful. $\endgroup$
    – AChem
    May 20, 2022 at 20:25
  • $\begingroup$ @AChem I tried to provide an answer to the second question. OP's first questions is unclear and would need more focus, "measured" by whom?. $\endgroup$
    – Mauricio
    May 20, 2022 at 22:37
  • $\begingroup$ Mauricio, He is trying toask how did physicists measure permeability and permittivity? $\endgroup$
    – AChem
    May 20, 2022 at 23:31
  • $\begingroup$ @AChem the answer for vacuum is up there, just used Coulomb's law or Ampère law. $\endgroup$
    – Mauricio
    May 21, 2022 at 11:12
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Maxwell's measurements of permeability and permittivity were used in his estimate of the speed of electromagnetic waves and to support his theory that electromagnetism and light were the same phenomena. Here is how Brain Clegg described the process in his biography of Maxwell titled Professor Maxwell's Duplicitous Demon. Unfortunately, it is a bit light on technical details.

[...] The equivalent properties for electromagnetism were known as the magnetic permeability and the electric permittivity of space, and the values of these properties were relatively poorly known, so the result had a significant uncertainty.

With Charles Hockin, an electrical engineer based in Cambridge, Maxwell devised and experiment that would pin down these values to far greater accuracy than ever before. The experiment balanced out the attraction from the electrical charge on two metal plates with the repulsion between two electromagnets with like poles facing each other. The bigger the effect, the more accurate the measurement could be - which meant hunting down an extremely high-voltage source.

Rather surprisingly, the owner of the most powerful batteries in the country turned out not to be a physics laboratory or an electrical power company, but a wine merchant based in Clapham, London. John Gassiot had spent his fortune on constructing an extravagant private laboratory. It was Gassiot who provided Maxwell and Hockin with a vast battery of cells, 2,600 in all, which between them put out around 3,000 volts.

The experiment proved highly successful, though the batteries went flat with unnerving rapidity, meaning Maxwell and Hockin had to take measurements furiously quickly before the charge ran out. These more accurate values for magnetic permeability and electric permittivity resulted in a calculated speed for electromagnetic waves of 288,000 kilometres per second.

When Maxwell later learned that Foucault had obtained a value of 298,000 kilometres per second for the speed of light, he was more certain than ever that his waves were light.

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  • $\begingroup$ I well recall decades ago sitting by a physicist on a long flight and his mentioning teaching undergraduate physics students how the speed of light could be derived in this way, without actually empirically measuring it, and them being truly awed. As big a deal as special relativity. If Maxwell had lived long enough to speak with Einstein just as Faraday met Maxwell. Who knows what JCM would have found in another few decades? $\endgroup$
    – releseabe
    May 29, 2023 at 14:43

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