| MadSci Network: Physics |
The question: "Why do the permeability and permitivity of space coincide mathematically? I understand the nature of electromagnetic waves (self-propagating perpendicular magnetic and electric fields). I also understand how the permeability and permittivity of free space affect magnetic and electric fields, respectively. However, one divided by the square root of the product of the permeability of free space and the permittivity of free space (1/sqrt(permeability*permitivity)) is equal to the speed of light (the units, Tm/A^2 and Nm^2/C^2 even end up reducing to meters per second!) In terms of equations, why is this mathematically in relationship to electromagnetic waves? Also, physically, why does it work? It baffles even my physics teacher."
It's because spacetime seems to be built that way! Indeed, the best minds in the world have struggled with questions like this for centuries. Is it just a coincidence that two of the constants that describe the properties of the vacuum should result in the speed of electromagnetic radiation in the vacuum? The fact that the coincidence is so striking argues for an answer of "no" to the question, and points to some fundamental connection between the constants.
The theory of electromagnetic radiation formulated by James Maxwell, resting on the shoulders of the work of many people before him, results in the equation you have pointed out. There is a huge amount of information available in textbooks and on the Web about Maxwell's equations, some of the better Web sites being listed here:
The (classical) description of electromagnetic radiation described by Maxwell's equations has been so thoroughly tested over a century and a half that this set of equations must be considered one of the most successful (classical) theories ever. Can there be any question that electromagnetic radiation is described by these equations? I think not. And so, in my opinion, the spacetime of the universe must in some way be fundamentally, if incompletely, described by these equations. It is true that they are not complete in the sense that only electromagnetism is described (the weak and strong forces, and gravity, are not described by the equations), but electromagnetism is described. (Later work by many physicists has resulted in a more complete theory, quantum electrodynamics, which is also thoroughly tested, and Maxwell's equations can be considered a subset of QED theory.) So, again, is there some sort of trick in play that results in c = 1/sqrt(permeability*permittivity)? Again, in my opinion, the answer is "no", and it has to do with the very fabric of the universe.
Now, the speed of light can be expressed in many units. However, as long as the speed of light, the permeability of free space, and the permittivity of free space are expressed in the same system of units, the equation still holds in any consistent set of units. So, is there any other significance to the constant "c"? In addition, if there is some fundamental connection between some of the constants of the universe, are there others? The answer is actually "yes". It turns out that there are three constants that serve as conversion factors in each of the three major theories. "Special relativity postulates symmetry operations (Lorentz transformations) that mix space and time. However, space and time are measured in different units, so for this symmetry concept to make sense, there must be a conversion factor between them. That role is fulfilled by 'c'." [1]. Similarly (also according to the article by Frank Wilczek), general relativity needs a conversion between spacetime curvature and energy, and that conversion is fulfilled by the gravitational constant "G". In quantum theory wavelength and momentum are proportional to frequency and energy, and since those pairs of quantities are in slightly different units, its conversion is accomplished by Planck's constant "hbar" (hbar equals h/(2Pi) ).[2]. (The article by Frank Wilczek expounds greatly on this idea.) So there seems to be a very good reason that the three quantities "c", "G", and "hbar" are so important to the description of the workings of the universe.
Max Planck was one of the first scientists to expound on this subject of units and constants. In fact, he proposed some ideas that are today called "Planck Units", and you can find some information about that subject in the following Web sites. The first URL below also has some very interesting information about the "fine structure constant".
Numerology of Physical Constants
There is a huge amount of information in the pages referenced by this answer. Spend some time absorbing what the articles say, and I think you will get an idea of the fundamental importance of these constants. The article by Frank Wilczek is actually one of a series of three articles, and that series is actually a followon of a previous three-article set. They are all referenced sufficiently by the one I have shown to be able to find them all.
John Link, MadSci Physicist
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