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

[Sreenath Skr]

Constant for all observers?

I have heard that maxwell showed that the speed of light is constant for all observers even before Einstein did. Is that true?

If not, then how can we say maxwells equation shows the speed of light is constant?

 

[Simon Bridge]

If you look up "Maxwell's equations" you will see that none of them explicitly include the speed of light in a vacuum.

However - Maxwels equations can be combined to produce a wave equation for electromagnetic waves.

That equation can be used, like any wave equation, to predict what speed EM waves should travel at in a vacuum. Do the maths, and that speed turns out to be given by: $$v=\frac{1}{\sqrt{\epsilon_0\mu_0}}$$

Since the permittivity and permiability of free space are (were presumed in Maxwell's day to be) constants, this is a big hint that the EM wave speed is also a constant.

When you crunch the numbers you get $v=c$ the speed of light.
... of course it's exact these days, but in Maxwell's day these were things that you measured separately.

For details see:
https://www.physicsforums.com/showthread.php?t=533832

 

[Bill_K]

If you look up "Maxwell's equations" you will see that none of them explicitly include the speed of light in a vacuum.

If you look them up in Gaussian units, you will find that they do.
https://en.wikipedia.org/wiki/Maxwell's_equations#Equations_in_Gaussian_units

The so-called "permittivity and permeability of free space" are merely artifacts of the SI system of units. Especially the latter, which has the exact, preordained value of 4π x 10^-7.

 

[Simon Bridge]

Hmmm... did Maxwell use gaussian units?

 

[Nugatory]

Hmmm... did Maxwell use gaussian units?

No. I think BillK's point is that now that we know what to look for, it's a lot easier to consider using units that don't obscure the physics... And obscure it they do, or OP wouldn't have needed to ask.

 

[Nugatory]

Constant for all observers?

I have heard that maxwell showed that the speed of light is constant for all observers even before Einstein did. Is that true?

If not, then how can we say maxwells equation shows the speed of light is constant?

SimonBridge's answer pretty much says it all. The only that I can add is that Maxwell's equations and the predicted speed of light were published in 1861. For almost a half-century after that, the great unsolved problem of physics was how to reconcile Maxwell's equations with Newtonian/Galilean classical physics; and it was this problem that Einstein solved with special relativity in 1905.

 

[Integral]

Constant for all observers?

I have heard that maxwell showed that the speed of light is constant for all observers even before Einstein did. Is that true?

If not, then how can we say maxwells equation shows the speed of light is constant?

Einstein did not show that the speed of light is constant. He ASSUMED is was constant, based on Maxwell's prediction, which was common knowledge at that time.

 

[bcrowell]

Maxwell's equations do not predict that the speed of light is the same for all observers. They are stated in a certain frame of reference, and they predict that light has a certain speed in that frame. They are silent on what happens in other frames. To decide what happens in other frames, you have to decide how measurements of space and time change when you change frames. In Maxwell's era, everyone believed that changing the frame of reference was defined in terms of the Galilean transformation, and that meant that Maxwell's equations would have their simplest form in a certain special frame, and change their form when you changed to some other frame. The special frame was interpreted as the frame of the ether. According to that interpretation, the speed of light would *not* be the same in all frames of reference.

 

[PhilDSP]

Another way to put this, which sort of echoes the last post and others, is that the Maxwell equations as we know them apply only for charges or bodies that are not moving with respect to the bodies that are emitting radiation.

Maxwell initially attempted to expand the equations to support moving bodies but he used rather complicated Eulerian formulations which were not successful. He didn't publish anything further on the subject. This is one reason why Einstein's 1905 article was titled "On the Electrodynamics of Moving Bodies".

 

[PAllen]

Another way to put this, which sort of echoes the last post and others, is that the Maxwell equations as we know them apply only for charges or bodies that are not moving with respect to the bodies that are emitting radiation.

Maxwell initially attempted to expand the equations to support moving bodies but he used rather complicated Eulerian formulations which were not successful. He didn't publish anything further on the subject. This is one reason why Einstein's 1905 article was titled "On the Electrodynamics of Moving Bodies".

No, this isn't right. Maxwell's equations apply to charges in all states of motion (else there would be no magnetism, no wave equation!), as long as the measurements are made in the special type of frame in which they are valid. Circa 1880, that special frame was assumed to be the aether frame. SR established that it was any inertial frame.

 

[PhilDSP]

I meant, of course, the use of the Maxwell equations without the application of the Lorentz transform. The LT effectively performs the moving body calculations. The Maxwell equations do handle non-relativistic motion of non-radiating charges (near field effects) also, but not motion involved with radiation between source and sink.

 

[WannabeNewton]

The Maxwell equations do handle non-relativistic motion of non-radiating charges (near field effects) also, but not motion involved with radiation.

What? The Larmor formula is exactly for non-relativistic radiating charges and it is obtained directly from Maxwell's equations.

 

[PhilDSP]

What? The Larmor formula is exactly for non-relativistic radiating charges and it is obtained directly from Maxwell's equations.

Interesting! On a first look it seems to invoke near field effects and acceleration of the charge in the near field. This is worth looking at in much deeper detail. But it also implicitly involves more than one application of the Maxwell equations at one time to calculate retarded potentials. So some assumptions are being made in how the Maxwell equations apply to different times rather than a single instance of time.

 

[WannabeNewton]

But it also implicitly involves more than one application of the Maxwell equations at one time to calculate retarded potentials. So some assumptions are being made in how the Maxwell equations apply to different times rather than a single instance of time.

Sorry I don't understand; what do you mean by this?

 

[PAllen]

I still don't get what there is to debate. Maxwell's equations are (classically) exactly correct for any motion of any charges in any given inertial frame. You can have 10 charges oscillating in different ways with speeds near c, and (in principle) describe all radiation an field strength anywhere using Maxwell's equations. The only change from 1880 understanding is that it was, at that time, assumed they only applied (exactly, all cases) in the aether frame, and to apply them in some other frame you would have to figure out how they transform under a Galilean frame change. SR changed this understanding that they apply exactly in all inertial frames, and the the LT was the right transform of the description in one frame to the description in another.

[edit: of course, in 1880s understanding, when it came to describing the response of matter to field, you would have an issue of what classical dynamics to use - Newtonian or relativistic. But I see this as a separable issue from the description of field evolution given some specified motion of charge distribution.]

 

[PhilDSP]

Maybe the best way of putting it is that the Maxwell equations accurately describe field values based on the position(s) and motion(s) of charged particles. But they don't describe the position(s) and motion(s) of the charged particles given the field values.

That means that other equations such as the Lorentz force law must be used to determine the effect of changing field values on bodies. So there is a missing description (in the Maxwell equations) of how radiation from a moving particle affects other charged particles. A naive application of the Lorentz force law isn't sufficient because the charged particle motion is affected by the fields which are affected by the particle and so on recursively...

That description in the form of extensions or modifications to the Maxwell equations had been given by a number of early physicists: Maxwell (who aborted the attempt), Heaviside, Hertz, Cohn, Bucherer and Ritz. Each of those solutions are somewhat unique but all, I believe, employ material derivatives rather than partial derivatives for changing field values.

 

[PhilDSP]

Sorry I don't understand; what do you mean by this?

I mean that, to solve the problem that the Larmor formula addresses, the Maxwell equations need to be invoked iteratively. In most problems where iterative techniques are used to solve differential equations, such as the finite element method for example, the overlap of time and position initial values for each invocation is pretty straightforward. But in this case, since it involves retarded times as initial values analytically, it's not quite as straightforward.

We can also notice that the Larmor formula is single-ended. It gives the power of the radiation of an emitting particle. Or it gives the power of radiation absorbed by another particle. But it doesn't give us a description of the coupling between those 2 events.

 

[homeomorphic]

Volume II of Feynman's lectures in physics has a nice intuitive explanation of the speed of light for a certain plane wave, generated by a sheet of current that is suddenly turned on.

Here's a summary, which will be sort of vague because I don't want to take the time to describe the set-up in detail. If you just had that current from the start, all you'd get is a constant magnetic field parallel to the sheet. But when you suddenly turn the current on, it will take a while to get that magnetic field set up. The field is zero initially, but then attains its final value in a region spreading out from the current sheet. To calculate the speed of the resulting wave, you can just do two line integrals near the edge of the wave front to see what is going on there. Because the magnetic field is being switched on along the front, it is producing a changing magnetic flux through a loop perpendicular to it, near the wave front. According to Faraday's law that means there is an electric field. You can do another line integral for the electric field and the displacement current kicks into tell you how the magnetic field is changing. Both the line integrals are equal to the changing flux, which depends on the velocity of the wave front. So, E = vB and B is proportional to vE, and from there, it's easy to calculate the speed of light, v = c.

 

[carrz]

The so-called "permittivity and permeability of free space" are merely artifacts of the SI system of units.

Weren't permittivity and permeability experimentally measured first, and how else could they have gotten into any of those equations to begin with?

 

[Simon Bridge]

The constants get into those equations because the dimensions do not match otherwise ... something has to carry the extra units. So you measure, say, and electric force and range for a given pair of charges and you get a particular relationship.

You can choose units to make the constants any value you like.

I'm not sure that "merely" is a good description though.
Is the speed of light in a vacuum "merely" an artifact of the units chosen?
The speed of light can be any number we like after all.

 

[carrz]

The constants get into those equations because the dimensions do not match otherwise ... something has to carry the extra units. So you measure, say, and electric force and range for a given pair of charges and you get a particular relationship.

I'm pretty sure permittivity and permeability were first experimentally measured before they became a part of any equation, long before Maxwell. If they were arbitrary, how do you explain they just happen to hold the value of c?

 

[Dale]

If they were arbitrary, how do you explain they just happen to hold the value of c?

Because all three depend on the same thing: the system of units. Once you have chosen your system of units then you have fixed all of the conversion factors. Any combination of conversion factors with the same dimensions is necessarily the same.

 

[carrz]

Because all three depend on the same thing: the system of units. Once you have chosen your system of units then you have fixed all of the conversion factors. Any combination of conversion factors with the same dimensions is necessarily the same.

Dependance on "system of units" does not explain how did permittivity and permeability become a part of any equation in the first place, nor it explains why would this equation be true:

By the way, it was not Maxwell who first realized connection between permittivity, permeability, and the speed of light. It was Kohlrausch and Weber in 1854 with their Leyden jar experiment who demonstrated that the ratio of electrostatic to electromagnetic units produced a number that matched the value of the then known speed of light.

http://en.wikipedia.org/wiki/Rudolf_Kohlrausch
http://en.wikipedia.org/wiki/Wilhelm_Eduard_Weber
http://en.wikipedia.org/wiki/On_Physical_Lines_of_Force

 

[Nugatory]

Dependance on "system of units" does not explain how did permittivity and permeability become a part of any equation in the first place, nor it explains why would this equation be true:

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.

 

[Dale]

Dependance on "system of units" does not explain how did permittivity and permeability become a part of any equation in the first place, nor it explains why would this equation be true:

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.

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.

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.

 

[carrz]

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.

 

[carrz]

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]

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]

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.

 

[carrz]

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]

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.

 

[Dale]

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.

Spoiler

http://electron9.phys.utk.edu/phys514/modules/module2/electrodynamics.htm
http://electron9.phys.utk.edu/phys514/modules/module3/electromagnetic_waves.htm

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.

 

[carrz]

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.

 

[carrz]

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]

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]

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.