How do maxwell's equations show that speed of light is constant

[Dale]

I see the problem now, c can not be derived from Maxwell equations written in Gaussian units because they already contain it. That's like a chicken growing old to become an egg, it's a reversed causality paradox. So anyway, given Maxwell's equations in Gaussian units, what is c equal to?

There is no paradox involved, simply a recognition of how different systems of units work, which unfortunately is not something that is taught in most physics courses due to the adoption of SI units only.

In Gaussian units c is ~3E10 cm/s, which is in the material already provided. Please read it.

 

[carrz]

In Gaussian units c is ~3E10 cm/s, which is in the material already provided.

I'm not asking about the actual number, but symbolic relation. When we take Maxwell equations in Gaussian units and isolate c on the left side, what is it we get on the right side?

I don't know how to deal with those curl operators but I can see we can not get any defined values on the right side, those are all empty container variables without any actual numbers in them. That's the paradox.

You were right, unit conventions can indeed create numbers out if thin air, apparently, but that's not a good thing. It's bad, very bad thing.

 

[Dale]

I'm not asking about the actual number, but symbolic relation.

I think that the answer to the question you are asking is c=c, but I am not sure why you want that relation.

When we take Maxwell equations in Gaussian units and isolate c on the left side, what is it we get on the right side?

I don't know how to deal with those curl operators but I can see we can not get any defined values on the right side, those are all empty container variables without any actual numbers in them. That's the paradox.

Huh? You don't solve differential equations that way. I have no idea what you are trying to do here. What you seem to be describing is certainly not something that you would do in SI units either.

The most that you would do is to derive the wave equation from Maxwell's equations in vacuum. If you do that then in SI units you get c=1/√(ε0 μ0) and in Gaussian units you get c=c, where the left hand is the c in the wave equation and the right hand is the parameters in Maxwell's equations.

Are you perhaps asking how Gaussian units treats Maxwell's equations in matter?

You were right, unit conventions can indeed create numbers out if thin air, apparently, but that's not a good thing. It's bad, very bad thing.

Which is why I prefer Gaussian units over SI units for EM problems. It reduces the number of "bad, very bad things" from 3 to 1 for EM.

 

[carrz]

Huh? You don't solve differential equations that way. I have no idea what you are trying to do here. What you seem to be describing is certainly not something that you would do in SI units either.

I'm saying that those equations have different meaning in SI and Gaussian units. In SI you get the speed of light limit as a consequence of permittivity and permeability. In Gaussian units you get some limits in E and B field caused by the speed of light. The cause and effect are shifted. The speed of light limit is not a cause, it's an effect.

The most that you would do is to derive the wave equation from Maxwell's equations in vacuum. If you do that then in SI units you get c=1/√(ε0 μ0) and in Gaussian units you get c=c, where the left hand is the c in the wave equation and the right hand is the parameters in Maxwell's equations.

You can not possibly get c = c in any units. I don't think electromagnetic wave equation can be derived from Maxwell's equations in Gaussian units to begin with. You'd have to split c for E and B field, and I don't see how can you do that without involving either permittivity or permeability, or both.

 

[Dale]

I'm saying that those equations have different meaning in SI and Gaussian units. In SI you get the speed of light limit as a consequence of permittivity and permeability. In Gaussian units you get some limits in E and B field caused by the speed of light. The cause and effect are shifted. The speed of light limit is not a cause, it's an effect.

This cause and effect relationship you are talking about simply does not exist even in SI units. A cause has to physically come before an effect (in time). In SI units there is no physical temporal order to ε0, μ0, and c.

You can not possibly get c = c in any units.

Of course you can. It is a tautology. How could you not get c=c? The question isn't whether or not it is possible, obviously it is. The question is why did you want that?

I don't think electromagnetic wave equation can be derived from Maxwell's equations in Gaussian units to begin with.

Sure it can. Do you know how to derive it from Maxwell's equations in SI units? Follow the same process for Gaussian units. The units don't change this derivation.

 

[carrz]

This cause and effect relationship you are talking about simply does not exist even in SI units. A cause has to physically come before an effect (in time). In SI units there is no physical temporal order to ε0, μ0, and c.

Cause and effect don't need to be a temporal sequence, they can be simultaneous. Gravity force is the cause for water draining out of a kitchen sink, and so is the drain hole. You can not say it is the speed of water flowing out which causes and defines gravity or width of the drain hole, it is the other way around.

I'm not yet sure how to explain that in terms of E and B fields and the speed of light, but you have to agree the speed of light limit can not be a cause, it can only be an effect.

Of course you can. It is a tautology. How could you not get c=c? The question isn't whether or not it is possible, obviously it is. The question is why did you want that?

You can not get c = c from Maxwell's equations just like you can not get m = m from F= m*a. When you isolate c on the left side, on the right side you get a ratio between the time and E and B fields, and these further include ratios between charge magnitude and its velocity vector.

Sure it can. Do you know how to derive it from Maxwell's equations in SI units? Follow the same process for Gaussian units. The units don't change this derivation.

Have you ever seen it? Can you point a link?

You see E and B have their own separate and different factors, both of which have their special proportion and relation with the speed of light. I don't see a way to rewrite that without referring to permittivity or permeability. Do you?

 

[carrz]

Ooops, I made a mistake above, those two equations do have the same factor. And that's very odd. I have to look at the whole derivation more closely.

 

[Dale]

Cause and effect don't need to be a temporal sequence

Yes, they do. It is one of the defining characteristics of a causal relationship.

This site is for learning and discussing mainstream science, not for personal speculation.

you have to agree the speed of light limit can not be a cause, it can only be an effect.

I don't even remotely agree with that. First, it is inconsistent with the definition of cause and effect. Second, I think that I have been extremely clear and consistent that all three (c, ε0, and μ0) are artifacts of the system of units, not each other.

Have you ever seen it? Can you point a link?

It should be in any freshman physics textbook (mine was by Serway), but there is a step by step derivation in SI units on Wikipedia also. Simply change the -1 in Faraday's law to -1/c and change the μ0 ε0 in Ampere's law to 1/c and follow the same steps to get the derivation in Gaussian units. The steps are the same.

http://en.wikipedia.org/wiki/Electr...e_origin_of_the_electromagnetic_wave_equation

You can not get c = c from Maxwell's equations just like you can not get m = m from F= m*a.

Obviously you can get m=m from F=ma:
F=ma by proposition
ma=ma by substitution
m=m by division

It is a tautology so you can get it starting from any consistent set of premises. I don't know why you would make a claim like this.

You see E and B have their own separate and different factors, both of which have their special proportion and relation with the speed of light. I don't see a way to rewrite that without referring to permittivity or permeability. Do you?

Obviously the factors are not and cannot be different. If they were then magnetic waves and electric waves would travel at different speeds.

You clearly do not know this material, which is fine, we are here to help you learn. But it will require you to ditch this argumentative attitude and adopt a learning attitude. Please study the material already provided, and then come back with questions about points that you do not understand. Further arguments or personal speculation will result in a closure of the thread.

 

[carrz]

Obviously you can get m=m from F=ma:
F=ma by proposition
ma=ma by substitution

You are not supposed to lose the starting relation which involves 'F'. There is no point to ma=ma expression, it's useless repetition without any practical meaning or implication. It does not answer the question, which is how one of the symbols relates to all the rest in a given equation.

I don't even remotely agree with that. First, it is inconsistent with the definition of cause and effect. Second, I think that I have been extremely clear and consistent that all three (c, ε0, and μ0) are artifacts of the system of units, not each other.

So now even c doesn't actually exist? Is that personal speculation? I didn't see any of the papers you kindly posted here says anything like that about the speed of light. They do say ε0 and μ0 are unit convention artifacts and not actual physical properties, but no one says anything like that about c.

You can ignore all those equations apply to different materials just the same as for vacuum, and you can shift values around to completely get rid of ε0 and μ0 for vacuum specific equations. But you can not get rid of the speed of light and impedance of free space, they do have, and must have, actual values in any units convention system, because they are actually real.

Obviously the factors are not and cannot be different.

They are not, in those particular equations, rather than "cannot be". In any case it's surprising and I'm giving it a closer look.

If they were then magnetic waves and electric waves would travel at different speeds.

Magnetic and electric waves are not separate waves with their individual speeds, it's one wave consisting of both electric and magnetic fields, which limited by combination of their permittivity and permeability in vacuum are constricted to moving at the speed of light.

That's the original Maxwell's interpretation and practical meaning of electromagnetic wave equation. So then the speed of light comes out from Newton's equation for the speed of sound, which similarly works for transverse waves traveling along a string:

...where K is a coefficient of stiffness/tension (permittivity), and p is density (permeability), and then you know exactly what is the cause and what is the effect.

http://en.wikipedia.org/wiki/On_physical_lines_of_force

You clearly do not know this material, which is fine, we are here to help you learn. But it will require you to ditch this argumentative attitude and adopt a learning attitude. Please study the material already provided, and then come back with questions about points that you do not understand. Further arguments or personal speculation will result in a closure of the thread.

I understand this material in the form of Coulomb's law, Biot-Savart law, and Lorentz force, which I deem is exactly sufficient. Of course, that I, just like you, think that it is me who actually understands better, is irrelevant. That's what we are supposed to find out, and the more we learn on the way, the better. It's a win-win situation any way it turns around.

 

[BruceW]

$$\frac{1}{\sqrt{\epsilon_0 \mu_0}} = c$$
Is quite striking, in how simple it is. But, we should not be surprised that it happened to be fairly simple. To begin with, we had equations like Coulombs law, and Ampere's force law, in which of course, we would choose $\epsilon_0$ and $\mu_0$ so that the equations could be written in a simple way. And then, Maxwell's equations came along, which related things like Coulomb's law and Ampere's law in a very simple way. AND Maxwell's laws also related the physics of light waves to both of those equations.

So, what I'm saying is, that since light and Coulomb's Law and Ampere's law are related in a simple way via Maxwell's laws, it is of course true that simple physical constants in each of those laws will also be related simply to each other. So, I'm saying that if
$$\frac{1}{\sqrt{\epsilon_0 \mu_0}} = c$$
is remarkably simple, then that is only because Maxwell's Laws relate Coulomb's law, Ampere's law and light in a remarkably simple way. And I would agree that Maxwell's laws are remarkably simple, given that they tie together so many physical phenomena that were previously thought to be unrelated.

 

[Evo]

Closed pending moderation.

 

[Dale]

I understand this material in the form of Coulomb's law, Biot-Savart law, and Lorentz force, which I deem is exactly sufficient.

That certainly explains a lot. If you decide that you would like to learn more, please let us know. Until then, there is nothing more to do here.

This thread will remain closed.