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In electromagnetism, the electric displacement field D represents the distribution of electric charges in a given medium resulting from the presence of an electric field E. Its relation to permittivity in the very simple case of linear, homogeneous, isotropic materials with "instantaneous" response to changes in electric field is $$\epsilon = \frac{D}{E}$$ Where $\epsilon$ is the electric permittivity of the material. In general, permittivity is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity, temperature, and other parameters.

Other hand, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. The concept of permeability arises since in many materials (and in vacuum), there is a simple relationship between the magnetizing field H and the magnetic flux density B at any location or time, in that the two fields are precisely proportional to each other, so $$\mu = \frac{B}{H}$$ where the proportionality factor $\mu$ is the permeability, which depends on the material and its properties.

Question: Why are the electric permittivity $\epsilon_0$ and $\mu_0$ considered as constant? Could not the properties of vacuum be slightly variable in different regions of the universe or at different scales? (for instance, at a Planck scale)?

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    $\begingroup$ How do you define "vacuum" if its properties has spatial variations? Is there any empirical reason to expect such variation? $\endgroup$ Commented Feb 29 at 12:43
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    $\begingroup$ When you study about what kinds of constants have physical meaning and learn that quantities that have units are meaningless, and when you study Maxwell's equations in the context of quantum field theory, you will realise that there is only one appropriate numerical constant associated with "vacuum" that is relevant to electrodynamics, and that is the fine structure constant. By homogeneity and isotropy of spacetime, and from telescope observations, we do not have reason to suspect that it is a function of position. It is, however, a function of energy used to probe. $\endgroup$ Commented Feb 29 at 18:18

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$\epsilon_0$ and $\mu_0$ were constant conversion constants until 2019. At that time, the SI organization changed their definitions to make them dependent on the 'fine structure constant', $\alpha$, whose value is determined by very difficult experiments. Now they could change if alpha varied with time or as you moved through the universe.

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    $\begingroup$ Indeed. The reason why $\alpha$ is preferable over $\mu_0$ and $\epsilon_0$ are multiple: 1) it is easier to measure directly 2) it is a unitless ratio, which are expected to carry more fundamental information (see arxiv.org/abs/hep-th/0208093) 3) it is a gauge coupling and can be directly interpreted as the constant quantifying the strength of the electromagnetic force within the general framework of gauge theories. $\endgroup$ Commented Mar 1 at 8:04
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One of the possible answer to your question is that, from the wave equations you can derive from the Maxwell equations, the speed of light is expressed as

$$c= \frac{1}{\sqrt{\epsilon_0\mu_0}}$$

From Michelson-Morely type experiments, we believe that $c$ is a fundamental constant of nature (on which special relativity is built). Hence if $\epsilon_0$ and/or $\mu_0$ were to change, $c$ would become variable (unless these variations balance each other perfectly to keep constant the product $\epsilon_0\mu_0$.

Note however that "beyond standard model" extensions of electromagnetism propose to study such phenomenon and put tight constraints on it on local or cosmic scales (as e.g. violation of Lorentz invariance, VSL theories, cosmic birefringence ...).

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