Measuring the magnetic and electric constants

The permeability of free space is defined as u₀ = 4 pi x 10^-7 Henrys per meter in the mks system of units. Because the speed of light is a defined quantity, and 1/sqrt(u₀ e₀) = c = speed of light, e₀ is then derived as 8.85 x 10^-12 Farads per meter. So the measurement of the permittiviy depends on the the speed of light, which is now an exact defined quantity, because the meter is based on the speed of light: The meter is the distance light travels in 1/299792458 seconds.

 

The permeability and permittivity of free space are now defined by other quantities, such as the length of a meter and the speed of light. Suppose you measured the capacitance of two one-square-meter plates separated by 1 millimeter in vacuum. Because the meter is defined by c, and c = 1/sqrt(u₀e₀), the measurement is not an independent measurement. the value of all the fundamental constants are based on a variety of interdependent quantities, every value is over determined, and least square fits are used to get the most probable values. In the 2002 table of the fundamental constants (which I am now looking at), both the permeability and permittivity of free space are "exact" defined quantities, meaning no uncertainty in value.

So you cannot go out into intergallactic space and measure them, because of their dependence on other quantities. On the other hand, dimensionless quantities, like alpha, are dimensionless and measurable.

 

Added detail- Look at the Particle Data Group page
http://pdg.lbl.gov/2008/reviews/contents_sports.html
Click ion the first catergory (constants), then look at about lines 13 and 14 for permeability and permittivity of free space. They are shown as EXACT quantities, because they are based on a) definition of the permeability of free space and b) the speed of light.

 

Why do you need to go into outer space to measure a quantity in a vacuum? Got a bell jar?

 

No, because you can measure the speed of light directly without involving these constants, in the simplest case just by measuring the time it takes for a signal to travel between point A an B.
Of course there are more sophisticated ways of doing it, but the point is that back when they were doing this (before the speed of light was defined to be constant, nowadays they instead measure the length of the meter) they used methods where the result only depended on a time and a length (and therefore indirecly on the definition of the meter, which back then was defined using an artifact).

Constants like these are used to "glue" the SI system together. Some constants are always needed in a system and choice of which ones to include is to some extent arbitrary. In e.g the CGS system the permeability of vacuum is a dimensionless number equal to one. Hence, in equations where the CGS systems is used (nowadys mainly in the area of magnetism) there are fewer constants when dealing with magnetism (the fact that the equations are different depending on the system used can be quite confusing)