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

That equation has to be true because it's derived by solving Maxwell's equations for a particular set of boundary conditions. Permeability and permittivity appear in that solution if and only if you chose to write Maxwell's equations in a form in which they appear - and to make that choice is to choose a system of units.

We got into this mess in the first place because many systems of units were developed before Maxwell's equations were discovered. Using these systems in Maxwell's equations is perfectly legitimate, but it complicates the formulas without adding any new insight - just as doing relativity problems in the mks system so that the speed of light is ##2.998\times{10}^8## m/sec is harder but no more informative than using light-seconds for distance and seconds for time.
Both permittivity and permeability were first experimentally measured "electrically", that is completely independently of any speed of light concept. Units have nothing to do with the fact that vacuum has specific permittivity and permeability whose relation is directly proportional to the speed of light. You can set those numbers to equal one, but you can not change the relation they have with the speed of light.

Yes, it does, although many people go through many courses in physics before they understand units and systems of units sufficiently to understand why. I will try to help as best as I can, but in the end there is no substitute for actually working a number of problems with different sets of units, like Gaussian, English, SI, Planck, and Geometrized units.
Some equations actually do need "k", not because it is a constant and because that equation is specific, but because such equation is general and "k" is variable from case to case. Like Hooke's law with its spring constant.

Suppose that you have some arbitrary (correct) physics equation a=b. Now, it is possible, in general, to use a system of units such that a and b have the same units. Such a system of units is called "consistent" with that equation. For example, SI units are consistent with Newton's second law: ∑f=ma where f is in newtons, m is in kilograms, and a is in m/s^2.

However, it is also possible to use other systems of units which are not dimensionally consistent with a given equation. In those systems of units you need to change the equation to a=kb, where k is a constant which changes the units on the right to match the units on the left. For example, you could express Newton's second law in US customary units as: ∑f=kma where f is in pounds-force, m is in avoirdupois pounds, a is in ft/s^2 and k is the constant 32.17 lbf s^2/(ft lbm).

The constant k is present only because of the system of units and is the factor that is required to convert the units on the left to the units on the right. Now, a given system of units may have several such conversion factors. Any combination of those conversion factors with the same base units is also itself a conversion factor and will therefore necessarily have the same units and the same value.
There is a difference. Permittivity and permeability are either independent properties on their own, like spring constant in Hooke's law, or they are properties of something else. Are you saying permittivity and permeability are not properties of vacuum, but rather properties of that 'q' charge or whatever else we have in those equations?

So, in SI units, c is the conversion factor between m and s (SI units are inconsistent with E=mc^2), μ0 is the conversion factor between kg m and s^2 A^2 (SI units are inconsistent with Ampere's law), and ε0 is the conversion between s^4 A^2 and kg m^3 (SI units are inconsistent with Coulomb's law). So 1/√(μ0 ε0) is an SI conversion factor between m and s, and must therefore match all other SI conversion factors between m and s, therefore it must equal c.
c is an actual number. Unit conventions can not produce any actual numbers out of thin air, it has to be relative to experimental measurements. Conversion factor between m and s must equal to 'm/s' in this case, not actual value of c. It does not explain why would relation between two different conversion factors equal the actual measurement number for the speed of light. Relation in this equation is not a conversion factor, it's actual physical relation. Is it not?

Dale
Mentor
Units have nothing to do with the fact that vacuum has specific permittivity and permeability whose relation is directly proportional to the speed of light. You can set those numbers to equal one, but you can not change the relation they have with the speed of light.
This simply is not true. In Gaussian units ε0=1 and μ0=1 so c≠1/√(ε0 μ0). They are all merely artifacts of the system of units so their relation clearly does depend on the units.

Dale
Mentor Relation in this equation is not a conversion factor, it's actual physical relation. Is it not?
It is not. See my counterexample above.

I will try to respond to the rest tomorrow.

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This simply is not true. In Gaussian units ε0=1 and μ0=1 so c≠1/√(ε0 μ0).
In Gaussian units ε0 and μ0 are not 1, they do not even exist as they are practically attributed to be a property of something else. It is therefore invalid to compare c with ε0 and μ0 in Gaussian units.

Can you point some link about getting the speed of light from Maxwell's equations in Gaussian units?

They are all merely artifacts of the system of units so their relation clearly does depend on the units.
That's just a general equation that applies to any medium turned it into specific equation that applies only to vacuum. If you want those equations to be general, as equations should be, then you must have permittivity and permeability factors which vary from material to material.

Dale
Mentor
Some equations actually do need "k", not because it is a constant and because that equation is specific, but because such equation is general and "k" is variable from case to case. Like Hooke's law with its spring constant.
Yes, this is correct.

There is a difference. Permittivity and permeability are either independent properties on their own, like spring constant in Hooke's law, or they are properties of something else. Are you saying permittivity and permeability are not properties of vacuum, but rather properties of that 'q' charge or whatever else we have in those equations?
I think that I have been very clear and consistent in saying that the vacuum permittivity and permeability are properties of the system of units.

c is an actual number. Unit conventions can not produce any actual numbers out of thin air, it has to be relative to experimental measurements.
Unit conventions most certainly can and do produce actual numbers "out of thin air". They are conventions. In modern SI units c, ε0, and μ0 are not experimentally measured quantities, they are defined exact constants according to the conventions. There are good historical reasons behind the definitions, but nonetheless, in modern SI they are defined not measured.

Consider the example I gave above of Newton's 2nd law in customary units: ∑f=kma where f was in lbf, k was 32.17 lbf s^2/(ft lbm), m was in lbm, and a was in ft/s^2. Suppose that we measure lbf using a standardized spring and lbm using a standardized balance scale and a using standardized rods and clocks.

Now, if I wanted to consider k to be a constant of the universe and perform experiments to measure it then I could certainly do so. Every time I measure it I would get some number and an associated experimental error. I could look into my experiments to find out what the source of the variability was and gradually improve them to measure k with less and less error. At some point, my experimental technique would be so well-refined that the dominant source of error is my ability to physically realize my units, i.e. variability in my standards.

At that point, we can switch to a new system of units where k is defined and use the definition of k to define the unit with the most variable standard in terms of the other standards. This is, in fact, how SI units treat k, ε0, μ0, and c. The only difference between them is that k is defined as a dimensionless 1 (SI is consistent with Newton's 2nd law) and the others are defined as dimensionful constants (SI is inconsistent with Maxwell's equations), but they are all defined in SI, not measured.

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Dale
Mentor
In Gaussian units ε0 and μ0 are not 1, they do not even exist as they are practically attributed to be a property of something else. It is therefore invalid to compare c with ε0 and μ0 in Gaussian units.
That is fine. If you want to consider them as not existing or as existing with dimensionless values of 1, either way, the relationship depends on the system of units, contrary to your assertions. In SI units it holds, and in Gaussian units it does not. Whether it doesn't hold in Gaussian units because the left hand side is all dimensionless 1 or because the left hand side doesn't exist, either way it doesn't hold.

Can you point some link about getting the speed of light from Maxwell's equations in Gaussian units?
I would start here: http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf

I recommend reading the whole thing. At the end, you should understand the differences between SI and Gaussian units well and be able to derive the speed of light in material using Gaussian units. If you have trouble with the derivation then I will help, but only after you have read the material.
http://electron9.phys.utk.edu/phys514/modules/module2/electrodynamics.htm [Broken]
http://electron9.phys.utk.edu/phys514/modules/module3/electromagnetic_waves.htm [Broken]
That's just a general equation that applies to any medium turned it into specific equation that applies only to vacuum. If you want those equations to be general, as equations should be, then you must have permittivity and permeability factors which vary from material to material.
See above. It is not general, it applies only to SI units.

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I think that I have been very clear and consistent in saying that the vacuum permittivity and permeability are properties of the system of units.
Do you also think impedance of free space is not actual physical property, but only unit convention artifact, even though it can be experimentally measured for vacuum just like for any material dielectric? http://en.wikipedia.org/wiki/Impedance_of_free_space

Unit conventions most certainly can and do produce actual numbers "out of thin air".
Unit conventions just shift the values around the same equation, they must preserve original relations which are always experimentally established first.

I would start here: http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf

I recommend reading the whole thing. At the end, you should understand the differences between SI and Gaussian units well and be able to derive the speed of light in material using Gaussian units. If you have trouble with the derivation then I will help, but only after you have read the material.
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?

Dale
Mentor
Do you also think impedance of free space is not actual physical property, but only unit convention artifact,
Yes, this is mentioned in the material I posted above as well as discussed in more detail here:
http://web.mit.edu/pshanth/www/cgs.pdf
(see section 5)

Unit conventions just shift the values around the same equation, they must preserve original relations which are always experimentally established first.
Again, Gaussian units are a counter-example that prove this statement to be false.

Dale
Mentor
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.

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
Mentor
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.

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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
Mentor
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.

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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?

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
Mentor
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.

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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.

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BruceW
Homework Helper
$$\frac{1}{\sqrt{\epsilon_0 \mu_0}} = c$$
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
Mentor
Closed pending moderation.

Dale
Mentor
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.