Permittivity/Permeability, the speed of light, and the meter

[TRB8985]

TL;DR Summary

Is the meter defined only by the speed of light, or is it tied to deeper and more fundamental constants like the permittivity and permeability of free space?

Good evening all,

I had a question on how the standard for the meter is defined. A simple Google search tells us that since 1983, the meter has been internationally defined as the length of the path traveled by light in vacuum during a time interval of 1 / 299, 792, 458 of a second. Many other sources agree with this, directly tying the definition of the meter to only the speed of light.

From my electrodynamics course, I remember that the speed of light is equal to the following:

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

If the meter is defined in terms of c, and the above equation is true, wouldn't that really mean that the meter is more fundamentally defined in terms of the permittivity and permeability of free space? Hopefully that question makes some sense.

Thanks for your time!

 

[Ibix]

No, it means that $\epsilon_0$ and $\mu_0$ are defined in terms of $c$.

Fundamentally, $c$ is the scaling constant between the meter and the second. Since both are units of measure of "distance" through spacetime, it's only a human convenience to use different units for them - 1m of time is an inconveniently short 3ns, and 1s of distance is an inconveniently long 3×10⁸m.

 

[Orodruin]

Also note: The relation you quote is a relationship between permittivity and permeability of free space. If you know one then you know the other since the constant in this relationship is precisely defined.

No, it means that $\epsilon_0$ and $\mu_0$ are defined in terms of $c$.

I would not say that they are defined in terms of $c$. Rather, they have a relationship between them that involves $c$. In the old (pre 2019 redfinition) of SI units, $\mu_0$ used to be a precisely defined quantity based on the definition of the Ampere. It is now an experimentally determined quantity.

 

[Andrew Mason]

I would not say that they are defined in terms of $c$. Rather, they have a relationship between them that involves $c$. In the old (pre 2019 redfinition) of SI units, $\mu_0$ used to be a precisely defined quantity based on the definition of the Ampere. It is now an experimentally determined quantity

$\epsilon_0$ is essentially the proportionality constant relating an electric charge to the surface integral of electric field surrounding that charge.

$\mu_0$ is essentially the proportionality constant relating an electric charge flow rate to the line integral of the magnetic field about that flow.

But, from Special Relativity we know that the magnetic field is the relativistically transformed electric field of a moving charge.

So, ultimately, the speed of light defines the relationship between these two proportionality constants. On that basis, I would say that Ibix is not overstating it.

AM

 

[Orodruin]

So, ultimately, the speed of light defines the relationship between these two proportionality constants. On that basis, I would say that Ibix is not overstating it.

Yes, it defines the relationship between them. That does not mean it defines them individually. If I tell you to use a system of units where the speed of light takes a numerical value of 4783 (for whatever strange reason) you cannot give me the value of either without either measuring it or the other.

 

[Andrew Mason]

Yes, it defines the relationship between them. That does not mean it defines them individually. If I tell you to use a system of units where the speed of light takes a numerical value of 4783 (for whatever strange reason) you cannot give me the value of either without either measuring it or the other.

Ok. But because $\epsilon_0$ and $\mu_0$ are related through SR by the Lorentz transformation of the electric field, and because $c = \sqrt{\epsilon_0\mu_0}$, the question is essentially whether $\epsilon_0$ or c is more fundamental.

That is a matter of semantics, I suppose, rather than physics. But, since c seems to be a more general phenomenon that affects or, perhaps, defines time, space, inertia, gravity, force, energy etc and not just electric fields, it may be fair to say that c defines $\epsilon_0$ and not the other way around.

AM

 

[Orodruin]

Ok. But because $\epsilon_0$ and $\mu_0$ are related through SR by the Lorentz transformation of the electric field, and because $c = \sqrt{\epsilon_0\mu_0}$, the question is essentially whether $\epsilon_0$ or c is more fundamental.

That is a matter of semantics, I suppose, rather than physics. But, since c seems to be a more general phenomenon that affects or, perhaps, defines time, space, inertia, gravity, force, energy etc and not just electric fields, it may be fair to say that c defines $\epsilon_0$ and not the other way around.

AM

It is not a matter of being more fundamental or not. It is a question of how our unit system has been defined and that has (for a long time) been that c is defined to be a particular value through the definition of a meter. Prior to the SI redefinition in 2019, the definition of the Ampere implied that also $\mu_0$ was a defined parameter and therefore that the value of $\epsilon_0$ was also fixed by these two definitions. After the SI redefinition, this is no longer the case. Instead, the Ampere is defined such that the elementary charge obtains a particular value when given in Ampere seconds. This makes the value of $\mu_0$ an experimentally determined one (and thus the same for $\epsilon_0$).

That c is a defined quantity does have its roots in it being rather fundamental in the sense of being the invariant speed (regardless of EM waves or not, that EM waves moves at this speed follows essentially from the need for EM to be a relativistic field theory). The thing is that permittivity and permeability are not two independent quantities as it may seem at first glance. If they were then electromagnetism would not be a relativistic field theory.

 

[Vanadium 50]

more fundamental

We get questions on "which is more fundamental" all the time. Since we don't have a fundamentalness meter, these go in circles.

In a formerly used set of units, c was exact and μ0 was exact as well. That made ε0 exact too. (Now 4π is a measured quantity)

 

[TRB8985]

Thank you to all of you for chiming in and redirecting my thinking on this one. I'm sorry if the "more fundamental" inquiry led us down a semantics game or anything - definitely not my intention! My original thought was that perhaps the meter is really related to those two vacuum quantities through the transitive property in some loose way (meter based on c --> c based on e_0 & μ_0 --> meter based on e_0 & μ_0).

I think I can safely say that might have been a misguided conclusion for some of the reasons mentioned above. Thanks again, and have a great afternoon!

 

[Andrew Mason]

Thank you to all of you for chiming in and redirecting my thinking on this one. I'm sorry if the "more fundamental" inquiry led us down a semantics game or anything - definitely not my intention! My original thought was that perhaps the meter is really related to those two vacuum quantities through the transitive property in some loose way (meter based on c --> c based on e_0 & μ_0 --> meter based on e_0 & μ_0).

I think I can safely say that might have been a misguided conclusion for some of the reasons mentioned above. Thanks again, and have a great afternoon!

As you can see from the above posts, when you ask what is more fundamental, you get into metaphysics, not physics.

It seems that all of our units are tied to the electro-magnetic phenomena and the speed of light. For example, the metre is defined as the distance light travels in one 299,792,458^th of a second. The second is defined as 9,192,631,770 periods of a certain electro-magnetic radiation emitted by a Cesium 133 atom The kilogram is defined in terms of a meter and second and by fixing the Planck constant at a particular value (6.62607015×10⁻³⁴ kg m²/s). Of course, the Planck constant is intimately tied to electro-magnetism which is, again, tied to c. The Ampere is defined as the flow of 1/602,176,349 x 10^-19 electrons per second (I am not even sure that this an integer number of electrons!). So it is tied to the definition of a second, which is tied to c. Oddly enough, there is no SI definition of electric charge - it has to be derived from the Ampere and second.

AM

 

[Orodruin]

Oddly enough, there is no SI definition of electric charge - it has to be derived from the Ampere and second.

In SI units the Ampere is one of the seven base units. You could have a system of units where the base dimension was charge instead of current, it would not matter much.

Of course, the Planck constant is intimately tied to electro-magnetism

There is no Planck constant appearing anywhere in classical electro-magnetism.

 

[Andrew Mason]

There is no Planck constant appearing anywhere in classical electro-magnetism.

No, but there is in quantum electrodynamics.

AM

 

[Orodruin]

No, but there is in quantum electrodynamics.

AM

The Planck constant is not restricted to QED. It is relevant to all quantum physics so I would say it is not related to electromagnetism as a phenomenon but to quantization.