Question about how two fundamental constants are measured

[dand5]

How is the permittivity of free space, $$\epsilon_0$$, and the magnetic permeability in vacuum, $$\mu_o$$, measured?

Thanks

 

[jtbell]

They're not fundamental constants and are not measured. They're sort of like unit-conversion factors, and can be defined exactly.

Because of the way the ampere is defined, $\mu_0 = 4 \pi \times 10^{-7}$ N/A^2 exactly.

The speed of light $c$ is defined as 299 792 458 m/s exactly.

From the electromagnetic wave equation,

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

from which you can calculate $\epsilon_0 = 8.8541878... \times 10^{-12}$ C^2/N.m^2 to as many decimal places as you like, in principle. (limited only by how many decimal places you have for $\pi$)

 

[dand5]

Ok, I understand how the permiability is defined now. But I thought that the permitivitty of free space existed before the speed of light was defined. Also, I always thought the reason c could be "defined" was because the permiability and the permitivitty appeared in the Maxwell's equations out of which comes that relation for vacuum? But you are saying that c is defined first, then the permitivitty is determined. Is that right?


Thanks

 

[jtbell]

Ok, I understand how the permiability is defined now. But I thought that the permitivitty of free space existed before the speed of light was defined.

I'm sorry, I don't understand this statement.

Also, I always thought the reason c could be "defined" was because the permiability and the permitivitty appeared in the Maxwell's equations out of which comes that relation for vacuum?

No, $c$ is defined as a specific constant because (a) the speed of light is constant, as a fundamental principle of relativity which has been verified repeatedly by experiment, and (b) we can thereby define the meter as the distance light travels in 1/299792458 second. (We do this rather than define the second as the time it takes light to travel 299792458 meters, because we can measure time more precisely than distance, under the "old" definitions.)

But you are saying that c is defined first, then the permitivitty is determined. Is that right?

The values of $\epsilon_0$ and $\mu_0$ are determined by our choice of units for electric charge and current, and the electric and magnetic fields. Advanced E&M textbooks, and theorists, commonly use Gaussian units, which eliminate $\epsilon_0$ and $\mu_0$ completely from electromagnetic equations. Jackson's Classical Electrodynamics has a good discussion of this.

 

[dand5]

Ok, I understand it now. Thanks. Just out of curiosity, why was 1/299792458 chosen? Did it best match the existing second?

Ok, I understand how the permiability is defined now. But I thought that the permitivitty of free space existed before the speed of light was defined.

What, I meant by this was that the permittivity was used in Maxwell's equations before the advent of relativity, i.e. before it was known that the speed of light is frame independent.

 

[Meir Achuz]

mu0 and epsilon0 have nothing to do with the permeability and permittivity of free space. Those terms are completely wrong misnomers.
Each number comes from a mismatch of units. They were introduced by an Italian engineer named Giorgi in the early 1900's and somehow became internationally recognized in SI units due to the political activity of Georgi and his followers. The number for c was first measured electrically in 1856. Its close equality to the known speed of light was an early indication that light was an EM wave. After relativity, it was recognized that c was just the conversion between the space and time axes in space-time, so even c can no longer be measured. Its value is determined by definition of the meter using the distance light travels in one second. When this was done (not too long ago), they picked the best known value of c. Too bad, they didn't just pick 3.

 

[jtbell]

Ok, I understand it now. Thanks. Just out of curiosity, why was 1/299792458 chosen? Did it best match the existing second?

Basically, yes. Before that definition was made, the speed of light was directly measurable in terms of the earlier standard definitions of the meter and the second. The new definition of the meter was chosen to agree with the most precise value of the speed of light at that time, averaging together the best existing measurements and taking into account their experimental uncertainties. Thus the new definition would not disrupt any earlier measurements.