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I wonder whether there is any explanation for e.g. magnetic permeability of the vacuum. Is it just as it is, so that we take its properties as a given constant, or will we ever be able to find a reason for its properties?

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The value of the magnetic permeability can be seen as being fixed by special relativity. Maxwell's equations, describing electromagnetic phenomena, fall naturally from SR. And in these equations, the speed of light $c=(\varepsilon_0 \mu_0)^{-1/2}$ is one empirical constant (related to our choice of time-units). The other constant is the permittivity of the vacuum, $\varepsilon_0$. So the permeability $\mu_0$ is not independent, and is rather determined by speed of light and the permittivity, or in other words its value is necessitated by SR and the permittivity.

The question now becomes whether we can similarly fix the value of the vacuum permittivity $\varepsilon_0$. Well, it's related to the way electrons influence each other, so really it can be reduced to things like coupling constants in the Standard Model.

But then the question becomes - can we explain those? Can we explain the coupling constants? And the answer to this seems to be that currently we can't, but it increasingly appears that our universe if part of a multiverse which means that there is nothing to explain. The multiverse contains universes with all coupling constant values, and we live in a part of it that has these values.

One then wonders whether the equations describing the multiverse are necessary. And again one reaches these same two options: either the equations are basic, and have no explanation, or else the multiverse encompasses all possible realities, so they need no explanation. The latter is Max Tegmark's "Mathematical Universe Hypothesis". It still leaves unexplained why the multiverse should take on this all-encompassing structure. (And, of course, is nothing but wild conjecture at this point.)

Ultimately, explanations are based on more fundamental theories. We can climb down a level, but either the whole structure is infinite or we'll reach the bottom-most theory at some point (we'll reach a brute fact, that has no explanation). Neither options seems palatable, and there is no solution for this conundrum that I know of.

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The vacuum's permeability and permittivity take the values they have by the definition of our units. While the metre's definition in terms of the second fixes $c^2=1/(\mu_0\varepsilon_0)$ in SI units, the definition of the Ampere implies $\mu_0=4\pi\times 10^{-7}\text{NA}^{-2}$.

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  • $\begingroup$ I feel, this just turns the question around. I mean, here c is the magic constant which has to be just taken as given property to calculate µ0. $\endgroup$ – user65208 Aug 6 '18 at 9:14
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Assume a 2-dimensional space (maybe square) and suppose you have two charges pinned down at diagonally opposite corners of the square and the square is filled with a certain material. The electric force felt by one of the charges due to the other charge will strongly depend on the material. Simply put, permittivity of the material is a property which decides how much force a charge will feel and this force changes if you change the material (hence permittivity). So the vacuum case is similar. It's the vacuum that decides how much force those charges will feel. You have the same reasoning for permeability too.

One of the best properties of the vacuum is that it is isotropic unlike certain materials which can have different permittivity and permeability values for different directions. Such materials are called anisotropic and the permittivity and permeability values now become tensors rather than just scalar numbers.

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