# Speed of Light (revisited). Permeability and Permittivity

I started a thread on this a log time ago asking why is the speed of light the value that it is. Unfortunately that thread degraded into a discussion of units that amounted to “it’s the speed it is in the units we measure it in” which doesn’t really answer the question.

I found out that the speed of electromagnetic wave is inversely proportional to the magnetic permeability and electric permittivity of a vacuum. OK, that makes sense and I thought I understood how permability works it is the ability of a magnetic field to be formed in a vacuum (excuse my layman explanation and correct me if I’m wrong) but I don’t understand permittivity. It is in effect the capacitence of a material right? How can a vacuum “store” electrical charge?

It’s not the vacuum storing charge; it’s how much electric field a charge produces. That’s one of the things that determines how strong a capacitor will be, but capacitors aren’t the only thing it’s relevant for.

Particles like electrons and protons, not an empty vacuum, carry electric charge, but surrounding a charged particle is an electric field, even if the particle is sitting in a vacuum. (One may ask interesting questions about how two particles separated by a vacuum can actually exert a force on each other…)

In God’s physics book, space and time are measured in the same units, the speed of light is equal to one, and the equations of physics never have the speed of light in them. We didn’t understand that when we started choosing units, so we are stuck with a number for the speed of light that is very large in our units and seems arbitrary, but the value is just a feature of the arbitrary units we chose.

God’s physics book says the vacuum electric permittivity and the vacuum magnetic permeability are reciprocals of each other, so when you multiply them together you get one.

Now we are back to the situation that you mentioned was the outcome of a previous thread. I assert, however, that it really does answer the question.

So is it fair (though simplistic) to say vacuum permittivity is a measure of how far an electric charge extends out in a vacuum?

In which may I direct you to Francis Vaughan’s at post #36, which I thinks is a very clear explanation that may help you to understand what it means to say it’s “just the units”:

https://boards.straightdope.com/sdmb/showpost.php?p=21661726&postcount=36

Also, the fine-tuning problem is an unresolved question in physics somewhat similar to what your asking, why does our universe have the parameters that it does? But if you read about that, you’ll see that for similar reasons it only makes sense to ask such a question about dimensionless physical constants, such as mass ratios, where there are no arbitrary units.

Not at all, since it extends infinitely far, though the strength drops off (inverse-square law). Also you probably meant “electric field” instead of “electric charge”; charged particles like electrons are really small.

I saw this recent thread and thought I might be permitted to ‘check my math’ before responding to a poster on a different board who’s not understanding the SoL, time dilation and relativity.

I’m using the familiar example of the ‘super-fast car with its headlights on’ that manages a speed of 0.999c, racing against a parallel light beam. The ‘course’ is a straight line over one light day distance. Probably dozens of examples like this scattered over the internet that might help, but I’d prefer it if it’s in my own words.

From the perspective of the race official at the starting line, both the car’s headlight and the light beam go forward and reach the halfway point at the same time (twelve hours), with the car itself only slightly behind. Similarly, he observes the race finishing a further twelve hours later, with the light beam and headlight arriving at the same time and the car a short way behind. He notes the car’s clock appears to be a little slow though.

From the perspective of the race driver, his headlights (and the light beam) rush off ahead at a speed of 1.0c relative to him. Once he’s nearly at half way, he sees both headlight and light beam ‘arrive’ at the finishing line, even though only twelve hours have passed on his clock.

The race driver reaches the finish line a short time after one full day according to his own clock.

Where my math is needing the fine-tuning is in getting an idea of the relative times showing on the race official’s clock according to the race driver at the halfway point. If it’s clearer to use a different course length in order to emphasise the differences, I’m ok with that.

What would be super-helpful would be some kind of animated cartoon version of this, if anyone’s seen anything like that?

Apologies for a slight derailment, but I felt the topic was close enough (and with sufficiently expert posters around) to chance my arm. Thanks.

Your problem (one car sitting still, another passing it at a substantial fraction of the speed of light) is completely symmetric, so you should be able to work it out from that (they both experience “time dilation” where the moving clock appears to tick slower).

Similarly, a stationary electric charge is moving in a different frame of reference, so one observer will detect a magnetic field where the other does not.

For all of those saying it’s about the units (yet again) miss the whole point of the question. The correct answer is, as I pointed out in post #1, it is the inverse of the geometric mean of the vacuum permeability and permittivity. It’s about the velocity and not how we measure it. That’s a tautology - it’s speed is 299792458 m/s because it goes 299792458 meters every second does NOT answer why that particular speed whereas if you say that it is that particular speed because as an electromagnetic wave the speed is depends on how easily magnetic and electric fields can form in a vacuum, well now that seems obvious.

Forgive me for entering a discussion about physics, but maybe in my ignorance I can help see what the others (and I) are missing from your question. I think you understand the first part, as keeps getting said, the speed of light is 1 speed of light. So there could be a question of why we measure it in meters and seconds, and I’m sure you realize those are just arbitrary units that we chose to measure ordinary things with a long time ago.

In your second question about permeability and permittivity are you asking if those properties are limitations on the speed of light that cause it to be the particular speed we measure it as? Do you mean something like the speed of light is the fastest an EM wave can propogate because of those properties?

I think you have to reword your questions some way that doesn’t sound like “why is the speed of light what it is?” If I’m not reading you at all then please forgive me, just trying to help.

You could say that it’s the inverse of the geometric mean of the electric and magnetic constants, but that’s just because those constants are the inverses of each other.

I’ll echo TriPolar.

The “it’s just the units thing” is a foundational piece of the answer, and I think it isn’t clear whether that piece is mutually agreed yet. To rehash Chronos’s example from the thread linked to by Riemann:

If we thought height and width were fundamentally different types of measurements, we might measure them in different units like centimeters for height and inches for width, and in that world, the constant c = 2.54 cm/in would show up all over physics. But obviously height and width are two dimensions in a single space, and treating them differently is clumsy and artificial. The exact same thing is true of space and time. We treat them differently, so the constant c = 3x10[sup]8[/sup] m/s shows up all over physics. But less obviously space and time are two dimensions in a single spacetime, and treating them differently is clumsy and artificial.

In the first example, why is the number “2.54”? It’s because one billionth of the distance between the north pole and equator (the height standard) is 2.54 times bigger than one twelfth the length of a typical adult male’s foot (the width standard). Clearly arbitrary.

In the second example, why is the number “3x10[sup]8[/sup]”? It’s because one ten millionth of the distance between the north pole and equator (the space standard) is 3x10[sup]8[/sup] times bigger than the time it takes the earth to complete 1/86400 of a rotation (the time standard). Clearly arbitrary.

In the first example, I compared height with width, and you’re okay with that.

In the second example, I compared spatial distance with temporal distance. Are you okay with that? There equivalence as two different directions in a single space (or spacetime) is as fundamental as height v. width. This needs to be established before the values of permittivity and permeability can be directly discussed.

No I’m not. To me all of the unit argument boil down to
We have this length here called a meter. We also have this duration of time called a second. The reason the speed of light is 299792458 m/s is because it travels 299792458 of these distance during this duration of time.

But why that distance over that time? Why not 100000 m/s?

Because that would be different units.

Different example: Suppose I ask why the Bugatti Veyron has a top speed of 267.856 mph. I’m looking for the engineering reason but y’all are saying it’s all about the units. That’s the top speed because it’s intantaneous velocity has a maximum of 267.856 mph.

No, I say, what is the physics/engineering where that is it’s particular top speed. Why does it go that fast but no faster.

You sigh and look over your glasses at me. Well you could say it goes 431.072 km/h but that’s because kilometers are different units than miles.

I persist. No, what is it about the engine, aerodynamics, tires, road surface, etc. where it’s top speed is 267.856 mph. Why can’t it go 300 mph?

Because those would be different units. :smack:
What would I have to change in the universe: permeability, permittivity, fine structure constant, magnetic monopoles, making McRibs permanent, etc. that would cause the speed of light to change independent of units - like changing this constant 4% would cause light to slow 7% from where it is now.

I think there’s a bit of miscommunication here.

If I’m reading the OP right, the question is not one of units, it’s why fundamental constants take the particular values they do.

That is, it doesn’t matter to the OP if c is 310^8 m/s or 51.6 quatloos per whatsit but why is it that particular value at all? Why isn’t it 410^8 m/s with the ‘same’ meter and second as now (if these units can even have a meaningful definition in such a universe)? Could such a universe even support life as we know it?

This is more of a philosophical question than a scientific one (see: Anthropic Principle) and there’s been a lot of thought put into it but no satisfactory answers.

This is just a guess, but if you change any one of those things the universe goes kablooey. If you could change all of them so that the speed of light was measured at 100000 m/s without the universe going kablooey then you would find that meters are just bigger in this new universe you have defined.

And this may be wrong, but it’s the only way I can conceive of this stuff, but the speed of light just “is”, and everything else works because that’s the speed it is.

No, as I mentioned up thread, this type of question - the fine tuning problem - is only meaningful for dimensionless constants. These constants define the relationship between things in our universe, where the presence of any arbitrary units cancels out.

It remains true that the primary answer to the OP’s question is “it’s just the units”.

Thanks. This makes it clearer where the miscommunications are coming from. The thing is, 4-dimensional spacetime really is a single thing, and distances in any direction in that 4-d space are in the same units fundamentally. It really is, under the hood, just like the height v. width case. This is the heart of Special Relativity and the heart of electromagnetism.

A tangent, in case it is new and/or helpful: Electric fields and magnetic fields are not really different things. They are two ways of looking at one thing, electromagnetic fields. If you have an electric charge (say an electron) and you are moving relative to it, that electric charge creates both electric and magnetic fields. Mathematically, we talk about the electromagnetic field (or potential), a single beast, but humans have chosen to also give names to different aspects of this beast. In particular, the piece that shows up when you are stationary relative to the charge is the “electric” field and the piece that shows up when you are moving relative to the charge is the “magnetic” field (plus still a piece that looks a like an electric field).

If you start with basic electrostatics and you do nothing else except introduce Special Relativity (i.e., the very concept that says that time and space are two directions in a single spacetime), then voila, you have magnetism. Magnetism is not a separate thing. It is an emergent phenomenon that happens when there is relative motion between electric charges and observers.*

With this in mind, it is clear that physical constants that relate to the not-at-all-fundamentally distinct E and M will be related. In the cgs (or Gaussian-cgs) unit system, there is no such thing as permittivity or permeability constants. Coulomb’s law reads F=q[sub]1[/sub]q[sub]2[/sub]/r[sup]2[/sup] and the Lorentz force reads F=q(v/c x B). No misleading constants hither and yonder. Electric fields and magnetic fields are measured in the same units. If one jams in historical choices for some units like electric charge, then those choices need to be canceled out elsewhere in the math. This is why permittivity and permeability show up in SI units and why they are reciprocals of one another. The fact that c also shows up in SI units reflects the other piece of historical gumming-up, namely that special relativity was not considered when choosing units for measuring distances (spatial and temporal).

Another aside: In F = q(v/c x B), notice the quantity “v/c”. Physicists give this ratio a different symbol often: [symbol]b[/symbol] (beta). This is because [symbol]b[/symbol] is the fundamental quantity, free from human artifacts. When [symbol]b[/symbol]=0, you are moving through time at 1 second per second. When [symbol]b[/symbol]=1, you are moving through space at 1 second per second. (Notice that I’ve used “second” as the fundamental distance unit here.) When [symbol]b[/symbol] is between 0 and 1, you are moving through space and time.

Trying to have permittivity and permeability change to influence how c comes out is the tail wagging the dog. Distances in space and distances in time are the same type of thing, full stop, and if you choose to measure them in different units, then you automatically introduce arbitrary, non-fundamental choices that lead to physical constants like c, [symbol]e[/symbol][sub]0[/sub], and [symbol]m[/symbol][sub]0[/sub]. The presence of all these constants is not fundamental. They over-specify the system. If you change one, you must change others to compensate or the math breaks. Much better is to rationalize the unit system (as is done regularly, just not in SI) to remove the “false” constants.
[sub]* Purely classical picture here. Quantum mechanics would be an irrelevant distraction.[/sub]

A related question that may (or may not) shed some light on the OP’s question.

Is the speed of light constant in an accelerating frame? If I take a tube that bounces light back and forth, and flashes at either end when it reaches it, will I, or will an observer, see those flashing lights slow down, as I am accelerating?

If I take an atomic clock and put it under my feet, (in a gravity field, like on Earth) it will tick slower than if I lift it over my head. Would the same hold true for my light tubes?

You always measure the speed of light right next to you to be the same, but if you are accelerating, your notion of simultaneity is constantly changing so you have to say what it means to measure the speed of light somewhere distant. In some sense you may be able to say that light travels “faster” near the ceiling compared to near the floor.