[flr (OP)]

Which relation from Maxwell equation imply that c is constant? Can it be put in plain English?

Thanks.



Mod edit - please could you make the subject a question in future?  This makes it much easier to navigate the forum, and makes it more likely that you will get answers!

[JP]

There are some constants appearing in Maxwell's equations relating the electric and magnetic field.  The equation with the constants is
Curl(B)=μ₀ε₀∂E/∂t, if you're mathematically inclined, where E and B are the electric and magnetic fields, respectively and μ₀ and ε₀ are two constants which had, prior to Maxwell, been discovered experimentally because of their relation to the electric and magnetic fields and charges and currents.

I take it you don't want to see the math involved, but what you do is manipulate this equation using a little calculus (and you have to invoke the other three equations as well, but those don't have the constants), to find that you get two equations which happen to describe the electric and magnetic fields as waves which move with speed c=(μ₀ε₀)^-1/2.  Maxwell correctly deduced that light was therefore a wave that had to move with speed c. 

[flr (OP)]

I still can't see how c is invariant from a pure intuitive point of view.
The fact that light is a wave is sufficient to make it invariant?
Or there is still something else required?

The speed of light does not depend on the speed of the source that generate it.
Is the same true for sound wave (just because they are wave?)

[JP]

It all depends on how you derive the wave equation--i.e. what if physically represents.  In most cases, it represents some medium moving in response to an applied vibration.  For example, sound waves in air are described by an equation that specifies how sound wiggles in response to a vibration.  It's clear in these equations that the air is carrying the vibrations and that if you move with respect to the air, you see the sound waves moving faster or slower with respect to you.  You can even catch up with the sound waves if you want. 

The thing about Maxwell's equations and "c" is that the value of the speed of light came out of the equations without any reference to a medium.  Scientists had measured a bunch of properties of the electric and magnetic fields in different labs without thinking about light or waves, and somehow the equations they came up with described a wave that apparently moved with a constant speed without any reference to a medium of propagation.  What was more baffling is that light somehow traveled through a vacuum (space) without a problem.  All other known waves needed a medium and couldn't propagate through space.  In fact, it wasn't immediately obvious that light didn't need a medium and most physicists would have thought there was an invisible medium (ether) which supported the light waves. 

It wasn't until later that scientists realized that light's speed was constant, and only after Einstein's special relativity had been thoroughly tested that mainstream science accepted it. 

[lightarrow]

I still can't see how c is invariant from a pure intuitive point of view.

If it were intuitive why should Einstein have been considered as a genius? [] More "common" scientists would have discovered relativity much before.

The fact that light is a wave is sufficient to make it invariant?

Sound is a wave. Is its velocity invariant?
The fact light's speed c is invariant in Maxwell's equations come from the fact that c = 1/√μ0ε0 and that these constants are...constants in those equations! (That is, they don't depend on the frame of reference).

[Geezer]

I still can't see how c is invariant from a pure intuitive point of view.
The fact that light is a wave is sufficient to make it invariant?
Or there is still something else required?

The speed of light does not depend on the speed of the source that generate it.
Is the same true for sound wave (just because they are wave?)

Hi Flr, and welcome!

Waves travel at a speed that depends on the medium that they travel through (technical term is propagation). Sound travels at a constant speed in air (the density of the air does affect the speed though, as does the wind speed.)

You may have noticed that the frequency of a car's horn changes as it passes you. The speed of the car effects the frequency of what you hear, but the time it takes for the sound to get from the horn to your ear is simply a function of the distance between the horn and your ear.

This is because the sound does not really "travel" through the air. The horn compresses the air and the pressure wave radiates from the horn. But the radiation is simply caused by varying pressure in the static air.

Light seems to act in a somewhat similar fashion. It even exhibits a change in frequency if the source and destination are moving.

Hope this helps!

[Soul Surfer]

flr  as has been explained above the speed of light is determined by the electrical and magnetic susceptibilities of fee space  ε0 and  μ0 mentioned above.  These are found to be constant with space and time by observation there is also no obvious reason why they should vary much within the observable universe (although under very extreme and non observable conditions this may not be so)  and therefore the speed of light is clearly constant for stationary observers.  Maxwells equations do not obviously include relativity and considerably predated it. however relativity implied the constancy of the velocity of light from our choice of fixed points to measure things and definition of how events were seen to be simultaneous and looking back into Maxwell's equations with this knowledge it can then be seen that magnetism comes from the action of relativistic effects on the electric field together with motion through space/time.  So yes once you understand relativity maxwells equations do show the constancy of the velocity of light. 

Taking this a bit further gravitation shows similar effects when motion is included in gravitational fields and there is a gravitomagnetic effect just like the electromagnetic effect but this is very difficult to observe using normal experimentation although a space probe of incredible precision  (gravity probe B) is currently trying to observe this effect in the earth's gravitational field.

[flr (OP)]

 Thank you all for answers. It is apparent to me now that this particular issue somehow evade my everyday intuition but perhaps justifiably so.

 Another question that I would like to ask is regarding vacuum fluctuations and luminiferous aether.
 Aren't these vacuum fluctuations in some sense equivalent with the concept of aether? Except that they do not "flow" in some net direction,the vacuum fluctuations appear to be "something" in the nothingness of space, just like aether filled the empty space. Also, if I understood correctly (but I can be wrong) there are theories which attempt to explain the photon motion as a result of interactions with vacuum fluctuations. 
   

[flr (OP)]

.. and looking back into Maxwell's equations with this knowledge it can then be seen that magnetism comes from the action of relativistic effects on the electric field together with motion through space/time. 

So, if I have a magnet which attract iron or another magnet, what is moving relativistic to generate magnetism? Certainly not the magnets (relative to each other)

[Geezer]

Hi Flr: I would encourage you to start a new topic with your new questions. That way you are more likely to attract a wider audience and other people who have similar questions are more likely to be able to find the topic. It also helps people identify what posts answer which questions.

Thanks!
Geezer (Mod)

[Farsight]

flr: I can't answer your first question, but I would encourage you to read what Maxwell actually said. See the wiki entry at http://en.wikipedia.org/wiki/James_Clerk_Maxwell where you can access On physical Lines of Force and A dynamical theory of the electromagnetic field. What you read is surprisingly different to the way it's commonly described. For example "Maxwell's Equations" aren't his equations, because Oliver Heaviside recast them into vector form.

No, vacuum fluctuations aren't equivalent to "the aether". Think of them as the random wavelets on the surface of the sea, whilst a light wave is an ordinary wave, like the sort that breaks on a beach. In both cases you need something to "wave", and that something is the vacuum itself. But note that space isn't "filled" with a medium, the nothingness of space isn't nothing because space has the property of distance, and distances can change. For example a gravitational field results in radial length contraction. 

In a magnet, electrons are moving to generate magnetism. Have a browse on google to find out more:   

[flr (OP)]

In a magnet, electrons are moving to generate magnetism.

So if the electrons would had moved according to Galilei relativity, there would had been no magnetism? Is it that true?

[Farsight]

I'm not quite sure what you mean, but I'll describe a scenario and see if it fits:

If you could examine an electron sitting there in front of you, you would say it had an electric field.

If you could examine an electron that was moving past you, you would say it had a magnetic field too.

If you chased after the electron and caught up with it so that you were both moving along together, you would say it only had an electric field again.

The reason for this is that the electron has an electromagnetic field. How you perceive it depends on your relative motion. See http://en.wikipedia.org/wiki/Electromagnetic_field and note where it says:

"In the past, electrically charged objects were thought to produce two types of field associated with their charge property. An electric field is produced when the charge is stationary with respect to an observer measuring the properties of the charge, and a magnetic field (as well as an electric field) is produced when the charge moves (creating an electric current) with respect to this observer. Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field."

Relativity is involved, in that this was mentioned by Minkowski towards the back of Space and Time:

"Then in the description of the field produced by the electron we see that the separation of the field into electric and magnetic force is a relative one with regard to the underlying time axis; the most perspicious way of describing the two forces together is on a certain analogy with the wrench in mechanics, though the analogy is not complete".

The wrench analogy is referring to something like a screw thread. Maxwell talked about this in On Physical Lines of Force, see this page:

http://en.wikipedia.org/w/index.php?title=File:On_Physical_Lines_of_Force.pdf&page=53

[flr (OP)]

If you could examine an electron sitting there in front of you, you would say it had an electric field.

If you could examine an electron that was moving past you, you would say it had a magnetic field too.

If you chased after the electron and caught up with it so that you were both moving along together, you would say it only had an electric field again.

OK, I understand that, my question was: is Special relativity is a requirement for this kind of unification of electric and magnetic fields or it can also be explained strictly from Galileo Galilei relativity.

Or to rephrase it: Is the invariance of the speed of light with SR and its finite value a requirement in order to have electric and magnetic fields as above described (the magnetic effect is due to relative motion)?

[JP]

If you want Maxwell's equations so still work exactly, no matter how fast you're moving, and if you want to get rid of the aether model, then yes, they basically require special relativity.  In fact, that was a motivation for Einstein to come up with special relativity.  However, Maxwell didn't realize this when he formulated the theory.

[Farsight]

Is the invariance of the speed of light with SR and its finite value a requirement in order to have electric and magnetic fields as above described (the magnetic effect is due to relative motion)?

Strictly speaking, no. Special relativity is required to explain the constant speed of light, not the dualism of the electromagnetic field. If you are stationary with respect to the source of an electromagnetic field you observe what you call an electric field, and if you're not, you observe what you call a magnetic field. This is experimental fact.
 
To back up what JP said in another way, Maxwell said electromagnetic waves would move at a certain speed, which appeared to be the speed of light. The Michelson-Morley experiment then looked for expected directional variations in the speed of light through "the luminiferous aether", but didn't find any. Einstein then said speed is distance divided by time, so if the speed and the distance remained the same, the time had to vary. This added a new layer to the mathematics, wherein the speed of electromagnetic waves and all other electromagnetic effects described by Maxwell's equations are subject to Lorentz Invariance. Hence regardless of how fast you're moving, if the electron is moving along with you, its properties appear unchanged.

[lightarrow]

In a magnet, electrons are moving to generate magnetism.

So if the electrons would had moved according to Galilei relativity, there would had been no magnetism? Is it that true?

Exactly, in the sense that you have to apply special relativity; then you will see that the simple coulombian force in relativity becomes something different: in moving frames of reference, new electric fields appears, to which we give the name of "magnetic field" in the stationary frame. It is a consequence of Lorentz contraction of lenghts: if you have a piece of electric cable long l with some charge q, from a moving frame of reference you see it long l' < l, with the same charge q, so the charge per unit lenght is greater, and this means that the electric force measured is greater; the additional force is the magnetic field.

[Farsight]

Sorry lightarrow, but I don't think it's to do with length contraction. You see an electromagnetic field as an electric field when you have no relative motion, and as a magnetic field when you do. It's the Lorentz force law rather than Lorentz contraction, see http://en.wikipedia.org/wiki/Lorentz_force_law.

[lightarrow]

Sorry lightarrow, but I don't think it's to do with length contraction. You see an electromagnetic field as an electric field when you have no relative motion, and as a magnetic field when you do. It's the Lorentz force law rather than Lorentz contraction, see http://en.wikipedia.org/wiki/Lorentz_force_law.

And where does the magnetic field B of the Lorentz_force come from? B is nothing else than a relativistic effect of coulombian interaction.

Example:
two equal charges q are still at a distance l. Which is the force F of one on the other?

Now observe the situation from a frame of reference moving with respect the two charges, at constant speed v pependicular to the line that joins the charges: every charge now generates a magnetic field attracting the other charge, so that the total force is less than before. How is this possible?

http://physics-quest.org/Magnetism_from_ElectroStatics_and_SR.pdf

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

http://www.thenakedscientists.com/forum/index.php?topic=14958.0

http://www.thenakedscientists.com/forum/index.php?topic=26025.0