Meaning Of Permeability Of Free Space

[LikesIntuition]

When we talk about the permeability of free space, are we talking about something with physical meaning on its own? Or is it simply a useful constant?

If it does have meaning on its own, what exactly is that meaning?

 

[Dale]

The permeability of free space is just a constant which is needed for conversion between different units in the SI system of units and other similar systems of units. It other systems of units it doesn't even exist.

 

[LikesIntuition]

Alright thanks!

 

[abitslow]

I don't know the difference between a "physical constant" and "something that has meaning on its own".
The speed of light, c, is a physical constant. Does it have meaning? Your question is really not capable of a simple answer. Depending on your level of knowledge (interest) you might be ok with someone telling you it is "just" a conversion factor. Its not that it isn't, it is. But take a look at the last equation in this section of wikipedia:https://en.wikipedia.org/wiki/Covar...agnetism#Electromagnetic_stress-energy_tensor
εₒµₒc² = 1... the permittivity and permeability of the vacuum are two aspects of the same thing and from them you can calculate (should I say "specify"??) the speed of light. So, I wouldn't say "just". εₒ and µₒ are the conversion factors between electricity and magnetism. Here is another wiki example of the laws of electromagetism...which you would agree, I hope, are important:https://en.wikipedia.org/wiki/Class...lativity#Maxwell.27s_equations_in_tensor_form
You don't have to UNDERSTAND the equations to see that in both references µₒ appears all over the place. Sure, you can define some unit so that it's value is 1, and even make it unitless. BUT if you do that, you won't be able to speak about (for example) velocity being in units of distance ÷ time. I think that would be a step over the line. Maybe it's just me? It is a whack-a-mole (corn hole) question. If you got rid of it (by setting its value to 1 (unitless), many other normal physical quantities would have to use conversion factors to convert them into distance, time, charge, force,... you really can't get rid of it: it WILL pop up somewhere (else).

 

[Dale]

Sure, you can define some unit so that it's value is 1, and even make it unitless. BUT if you do that, you won't be able to speak about (for example) velocity being in units of distance ÷ time.

That isn't necessarily correct. Again, it all depends on the system of units being used.

In SI units you have the permeability of free space and you have velocity in units of distance/time. In Gaussian units the permeability of free space doesn't exist (it is a dimensionless 1) and you have velocity in units of distance/time. In Geometrized units the permeability of free space doesn't exist and velocity is unitless.

So it depends on the system of units. Gaussian units are common in the EM literature. Geometrized units are common in the GR literature. SI units are common in most of the rest of the literature. English units are common in engineering literature. You should be familiar with many systems of units and know how to handle them.

 

[LikesIntuition]

Does changing units allow us to say permeability is gone? Or is it more accurate to say it's just 1? Although I guess you could take any equation and multiply it by 1, stating that 1 is a constant.

 

[Dale]

I guess that would be more accurate. But since you can always divide by 1 without changing anything you can always divide it out completely.

For example, in Newton's writings the second law is not ∑F=ma, but rather ∑F=kma. Newton wasn't using SI units, he was using the units of the time, which would be closer to English units than SI. If you write it in some systems of units you still need to add that unit conversion factor back in, but since we generally teach using SI we usually think of Newton's law without the k at all.

 

[gburkhard]

Hi folks,
It was always my understanding that the 'meaning' behind these numbers is relatively simple: when we talk about permittivity or permeability, we're setting the scale for the electric or magnetic polarizability of the vacuum. Ie, the general electric field is the displacement field, D=epsilon*E (and for magnetism, H=mu*B). epsilon0 and mu0 are the polarizabilities of the empty vacuum and we measure the polarizabilities of materials relative to that. However, you can change the units and call the vacuum polarizability something else if you prefer -- then the materials have different relative values in that system of units. But at the end of the day the permittivity and permeability of the vacuum is just the extent to which the vacuum is polarized when you apply a field to it. It is the reference value that we set, since we can't have an absolute field in absence of vacuum (because what would that mean?), but that's what it is. Basically the same as asking what is the meaning of setting the lorentz gauge such that the potentials go to 0 at infinity. It's just sets a reference from which we can measure everything else.

Does this make sense?

 

[LikesIntuition]

What does it mean to polarize the vacuum? Is it to give a point in the vacuum the ability to exert a force on something (so set up a field in it)?

 

[gburkhard]

Yep, that's it! I'm just talking about a generalization of the concept here, not implying that there are physical dipoles aligning with the field in a vacuum (because then you would have to ask what the field would look like "outside" of the vacuum, which is not a concept). But just what you said -- if you have a field, say from a dipole, the field propagates through the vacuum, and the shape that the field lines take (curvature, and extent) depends on the vacuum polarizability. You can see that IF you could change epsilon, you can make the field lines shrink or expand just like you would if you were inside of a dielectric. So the value that we choose (e0) is the value in the dimensionality system of our choice that expresses the true value of the fields everywhere. Maybe in a different universe that value is something different! But in our universe it is what it is...

 

[LikesIntuition]

Yep, that's it! I'm just talking about a generalization of the concept here, not implying that there are physical dipoles aligning with the field in a vacuum (because then you would have to ask what the field would look like "outside" of the vacuum, which is not a concept). But just what you said -- if you have a field, say from a dipole, the field propagates through the vacuum, and the shape that the field lines take (curvature, and extent) depends on the vacuum polarizability. You can see that IF you could change epsilon, you can make the field lines shrink or expand just like you would if you were inside of a dielectric. So the value that we choose (e0) is the value in the dimensionality system of our choice that expresses the true value of the fields everywhere. Maybe in a different universe that value is something different! But in our universe it is what it is...

Oh I see now! So we have a field being set up of which we can describe changes (such as the fact that from a point source it drops off like the inverse of the distance squared). But that doesn't tell us how strong the field will be at each point without some scaling factor. We could just as well have a force that changes through space in the same way, but is stronger or weaker at all those points. So space has some kind of *seemingly* arbitrary value for how strong the fields are that get set up in it?

 

[Dale]

So space has some kind of *seemingly* arbitrary value for how strong the fields are that get set up in it?

It is not just seemingly arbitrary. It is arbitrary. It depends entirely on your units, which are arbitrary. The value of the permeability of free space doesn't tell you anything about free space, it tells you about your units.

The only non-arbitrary constants in physics are the dimensionless ones.

http://math.ucr.edu/home/baez/constants.html

 

[LikesIntuition]

It is not just seemingly arbitrary. It is arbitrary. It depends entirely on your units, which are arbitrary. The value of the permeability of free space doesn't tell you anything about free space, it tells you about your units.

The only non-arbitrary constants in physics are the dimensionless ones.

Well I'm speaking in a less mathematical sense. I'm not talking about the "number" of the field. I mean the literal force (not a number) a charge would feel in that field. It's the same physical force no matter what units we choose to describe it with. The units change the number we end up assigning to that given amount of force, but the force is the same magnitude regardless. And the actual, physical strength of our field (regardless of how we are describing it) is dependent on how our world works, not what units we're using. Right?

 

[Dale]

I mean the literal force (not a number) a charge would feel in that field. It's the same physical force no matter what units we choose to describe it with. The units change the number we end up assigning to that given amount of force, but the force is the same magnitude regardless.

But what does it even mean, to have a "physical force" that is independent of the units used? What is the "physical force" of 3 N?

All you can do is compare it to other forces, such as a certain spring compressed a certain distance or something similar. Those comparisons are dimensionless, and they don't depend on dimensionful constants like the permeability of free space, they depend only on the dimensionless constants like the fine structure constant.

Did you read the link I posted? I may have posted it after you saw my response.

 

[LikesIntuition]

But what does it even mean, to have a "physical force" that is independent of the units used? What is the "physical force" of 3 N?

All you can do is compare it to other forces, such as a certain spring compressed a certain distance or something similar. Those comparisons are dimensionless, and they don't depend on dimensionful constants like the permeability of free space, they depend only on the dimensionless constants like the fine structure constant.

Did you read the link I posted? I may have posted it after you saw my response.

Oh! I didn't see the link. I'll check that out.

Also, even if we change units, isn't the relationship between our different physical concepts still the same? So using force as an example, the relationship between mass and acceleration (which is a relationship between velocity and time, velocity being its own relationship between distance and time) is the same no matter what units we use. Yes, we'll have different numbers for our concepts in each unit system, and I guess we could have different numbers for our rates of change, too. But the relationship is still the same, right? If you watch a certain force be placed on a certain mass, some phenomena will happen. It doesn't matter what the units we write down for what we observe, that phenomena is determined by an arbitrary physical relationship between our concepts like force and mass, right?

 

[Dale]

Also, even if we change units, isn't the relationship between our different physical concepts still the same?

If you change units only dimensionless quantities remain the same. Luckily, if you think about what you intuitively mean by "physical concepts" they are usually dimensionless anyway.

For example, the speed of light is a big number in SI units but 1 in Planck units (same dimensions in both cases). But regardless of the units, the dimensionless ratio between the speed of light and your maximum running speed is very large. Therefore, you intuitively think of light as being very fast. Not because of the large SI value and despite the small Planck value. You think of it as fast because of the dimensionless ratio between it and something familiar: your own body's motion.

using force as an example, the relationship between mass and acceleration (which is a relationship between velocity and time, velocity being its own relationship between distance and time) is the same no matter what units we use

The relationship will be the same in all systems of units EXCEPT for the scaling factor. In some systems of units the scaling factor will be entirely absent (dimensionless 1) and in others it will be present and will have different units and dimensions.

Yes, we'll have different numbers for our concepts in each unit system, and I guess we could have different numbers for our rates of change, too. But the relationship is still the same, right? If you watch a certain force be placed on a certain mass, some phenomena will happen. It doesn't matter what the units we write down for what we observe, that phenomena is determined by an arbitrary physical relationship between our concepts like force and mass, right?

Sit down and think about what you intuitively mean by "a certain force" and "a certain mass". I bet you will find that you are mentally making dimensionless quantities.

 

[LikesIntuition]

Sit down and think about what you intuitively mean by "a certain force" and "a certain mass". I bet you will find that you are mentally making dimensionless quantities.

I see what you're saying, but I'm still having trouble with this idea. We invented math and numbers, but we didn't "invent" the physical phenomena we use math to describe. The laws of nature aren't dependent on how we choose to describe them, are they?

 

[Dale]

I don't know if we invented math and numbers or if we discovered it. I don't think the distinction matters too much.

I agree that the laws of nature do not depend on how we choose to describe them. Therefore, anything which does depend on our choice of description is part of the description, not part of the laws of nature. This includes things like the permittivity of free space.

 

[LikesIntuition]

I don't know if we invented math and numbers or if we discovered it. I don't think the distinction matters too much.

I agree that the laws of nature do not depend on how we choose to describe them. Therefore, anything which does depend on our choice of description is part of the description, not part of the laws of nature. This includes things like the permittivity of free space.

I see what you mean. How "much" field is set up in a vacuum is determined by permittivity. But the value for permittivity depends on how we choose to describe our forces, right?

 

[Dale]

Yes. That is well said.

 

[Khashishi]

The laws of physics are generally written as mathematical relations which happen to describe nature. But it is easy to change a mathematical equation into another one that describes the same phenomena. If you add the same value to both sides, or multiply, or break it up into two equations, etc. There's a lot of freedom in how we choose to express some physics as a mathematical model. By changing the units, we can add or remove constants. For example, take the basic law:
$F = ma$
In SI units, we measure force in Newtons, mass in kilograms, and acceleration in meters per second squared. But since a Newton equals kg*m/s^2, we don't need any extra conversion constant in the law. But if we measure force in lb-force, then we need to add a constant.
$F = kma$
where k is a conversion factor from kg*m/s^2 to lb-force. It doesn't have any physical meaning. It is just a conversion factor. The permeability of free space can be thought of as a conversion factor.
If we take Ampere's law and ignore the changing electric field contribution
$B = \nabla \times \mu_0 J + \dots$
In SI units, B has units of Tesla and J has units of ampere/meter^2. Since taking the curl puts in another factor of meter^-1, that means that $\mu_0$ is a conversion constant from ampere/meter^3 to Tesla.

If we instead measured B in ampere/meter^3 (similar to what we did for the force law), then we wouldn't need an extra conversion constant.

It is useful to put extra conversion constants into our laws if we want to emphasize the difference between two different kinds of quantities. For example, temperature can be measured in units of energy, but since temperature is different from energy, we measure it in Kelvin, so we don't confuse the two. But in reality, we generally want to reduce the number of extra conversion constants to simplify our laws, and the extra units exist for historical reasons.