How do Maxwell's equations indicate that the speed of light is constant?

[joyz2008]

How do Maxwell's equations predict that the speed of light is constant? I found different answers and some people even said that they don't.
I'm still confused...

 

[Vanadium 50]

If you calculate \nabla \times \nabla \times E (or \nabla \times \nabla \times B) it pops right out.

 

[PhilDSP]

More detail:

Consider \nabla \times B = \mu_0 J + \mu_0 \epsilon_0 \frac {\partial E}{\partial t}

Here \mu_0 \epsilon_0 = \frac {1}{c^2}

In a vacuum J = 0 so that \nabla \times B = \frac {1}{c^2}\frac {\partial E}{\partial t}

Take the curl of \nabla \times E \ \ \ \ \ The result will be \frac {\partial}{\partial t}\nabla \times B = \frac {1}{c^2} \frac {\partial^2 E}{\partial t^2}

Using the vector identity \nabla \times \nabla \times E = \nabla (\nabla \cdot E) - \nabla^2 E\ \ \ \ \ and the first Maxwell equation \nabla \cdot E = 0 \ \ \ \ \ we get \nabla \times \nabla \times E = - \nabla^2 E

So that \nabla \times \nabla \times E = \frac {1}{c^2} \frac {\partial^2 E}{\partial t^2} = - \nabla^2 E \ \ \ \ \ which is the wave equation (\nabla^2 - \frac {1}{c^2} \frac {\partial^2}{\partial t^2}) E = 0 \ \ \ \ \ in d'Alembertian form

The wave equation can be reduced to first order wave equation terms traveling in opposite directions \ \ \ \ \ (\nabla^2 - \frac {1}{c^2} \frac {\partial^2}{\partial t^2}) E = (\nabla - \frac {1}{c} \frac {\partial}{\partial t}) (\nabla + \frac {1}{c} \frac {\partial}{\partial t})E

which clearly shows propagation at a constant c provided \mu_0 and \epsilon_0 are constant

 

[harrylin]

How do Maxwell's equations predict that the speed of light is constant? I found different answers and some people even said that they don't.
I'm still confused...

Different people mean different things with that sound bite...

Nowadays many people mean that the speed of light is invariant under a Lorentz transformation.

However, originally it meant that the speed of light in vacuum (as measured "in rest" in the vacuum) is assumed to be the same in all directions, independent of the kind of light source and independent of its motion. That was not a prediction of Maxwell's equations but a postulate of those equations which was maintained in Special relativity: SR is "a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies".
- http://www.fourmilab.ch/etexts/einstein/specrel/www/

Of course the equations must agree with that postulate; and that can for example be verified as shown by PhilDSP

 

[PhilDSP]

Different people mean different things with that sound bite...

Yes, and the mathematical analysis we just did is only half the story - probably the less interesting half. We considered only "free space" without the presence of matter or charge. For a real understanding of how the waves look like from particle-to-particle we need to do the far more difficult part of developing a wave equation for light interaction with the first particle (which is moving) then relate that to light interaction with the second particle (which is moving at a different relative speed)

That requires not only equations for media, but for moving media where \mu and \epsilon are not only not constants, they are tensors. However the attempt to develop equations for moving media was abandoned upon the deaths of Oliver Heavyside and Heinrich Hertz. (Heavyside did not re-transcribe Maxwell's equations for moving media into today's vector form as he did for non-moving media) The Lorentz Transformation became an alternative to using Maxwell equations for moving media. Lorentz's procedure for handling moving bodies is fairly thoroughly documented though and can be analyzed mathematically whereas Maxwell wrote next to nothing about his attempt to solve the problem.

 

[fizzle]

... (Heavyside did not re-transcribe Maxwell's equations for moving media into today's vector form as he did for non-moving media) ...

Heaviside did investigate moving media, if I understand what you're saying here. In 1902, he went as far as assigning a physical meaning to the momentum in an electromagnetic wave in Electromagnetic Theory, Vol. III. It's his "Moving Compressible Ether". I've attached a PDF I created that contains his lead-in description and the derivation using more modern notation/units (obviously it's a completely classical derivation). I've always wanted a mathematician to look at the theory and see where it would lead experimentally. The particularly interesting prediction is that two em waves will interact if strong enough.

 

[harrylin]

[..] However the attempt to develop equations for moving media was abandoned upon the deaths of Oliver Heavyside and Heinrich Hertz. (Heavyside did not re-transcribe Maxwell's equations for moving media into today's vector form as he did for non-moving media) [..]

I'm not sure what you refer to; in any case, Heaviside did predict the effect of moving charges. He did not write it in vector notation but nevertheless with directionality (cos alpha etc.). His 1889 paper in which he expands on Maxwell's theory predicts the same electromagnetic field strengths of moving charges that later also followed from relativity. And that's as it should be: Maxwell's equations are fully compatible with special relativity.

 

[PhilDSP]

Heaviside did investigate moving media, if I understand what you're saying here. In 1902, he went as far as assigning a physical meaning to the momentum in an electromagnetic wave in Electromagnetic Theory, Vol. III.

Thanks for providing the file. It's difficult to put that small snippet into the larger context though. One difficulty with Heavyside is that he defines and uses a lot of unique variables that no one else seems to use and you need to wade through hundreds of pages in his 3 volume series to go back and find their definitions.

 

[PhilDSP]

I'm not sure what you refer to; in any case, Heaviside did predict the effect of moving charges. He did not write it in vector notation but nevertheless with directionality (cos alpha etc.). His 1889 paper in which he expands on Maxwell's theory predicts the same electromagnetic field strengths of moving charges that later also followed from relativity. And that's as it should be: Maxwell's equations are fully compatible with special relativity.

Yes, Heavyside not only investigated moving charges but developed his own version of the Maxwell equations for moving charges because, as I believe Author-Historian Bruce Hunt put it, Heavyside believed Maxwell's version was faulty. Maxwell's version seems to have been based on Helmholtz's theory as Author-Historian Olivier Darrigol very thoroughly has investigated. Neither version produces all electrical or optical parameters correctly though.

 

[fizzle]

Thanks for providing the file. It's difficult to put that small snippet into the larger context though. One difficulty with Heavyside is that he defines and uses a lot of unique variables that no one else seems to use and you need to wade through hundreds of pages in his 3 volume series to go back and find their definitions.

Remember, when you're reading Heaviside (and his contemporaries), you're essentially reading history in the making. It's like watching a live news report from a major event - initial information is fragmented, sometimes wrong, changes over time, names change, etc.

The snippet I put in my PDF is just the beginning of his theory. See the referenced pages in EMT Vol. III for the rest of it. What it says, and what I like about it, is that em waves are not linear. For example, if I send extremely strong waves down a transmission line from both ends, when they overlap I'm left with a region of increased "density of space" (this is his "m"). Note that Heaviside has both permeability and permittivity always change proportionately, so there is no reflection. If I shoot a weak test em wave through that region, it won't move at speed c but at a lower speed. On the other hand, if I'm completely inside that region and use em waves for measuring distance, I won't detect the increased density. That's about as far as I can go with my limited mathematical chops!

 

[PhilDSP]

Remember, when you're reading Heaviside (and his contemporaries), you're essentially reading history in the making. It's like watching a live news report from a major event - initial information is fragmented, sometimes wrong, changes over time, names change, etc.

Nice way of putting it. It could be said that a lot of the issues Maxwell, Heaviside, FitzGerald and Hertz (among others) struggled with were simply discarded or ignored by later theorists in a quest for simpler and more immediate answers. So the mystery remains about certain aspects of their work and whether some of the later simplifications have haunted us for the past 110 years.

The snippet I put in my PDF is just the beginning of his theory. See the referenced pages in EMT Vol. III for the rest of it. What it says, and what I like about it, is that em waves are not linear. For example, if I send extremely strong waves down a transmission line from both ends, when they overlap I'm left with a region of increased "density of space" (this is his "m").

Jackson's introductory description of the use of the Maxwell equations for media on page 16 of "Classical Electrodynamics" mentions briefly how non-linear effects arise when wave or field amplitudes become large. We could interpret them as being generated by the inability of the media to respond to the energy of the waves in a way that instantaneously keeps the ratio of potential and kinetic energies conserved. That sounds related to what you describe.

 

[fizzle]

Nice way of putting it. It could be said that a lot of the issues Maxwell, Heaviside, FitzGerald and Hertz (among others) struggled with were simply discarded or ignored by later theorists in a quest for simpler and more immediate answers. So the mystery remains about certain aspects of their work and whether some of the later simplifications have haunted us for the past 110 years.

Wow, I'm beginning to wonder if I have a split personality and also post here under the handle "PhilDSP"! For a perfect example of your "simplifications", look at Compton Scattering. Compton noted in his original paper that you can get the correct answers from a semiclassical analysis of the experiment. However, it's much easier to solve the problem as a high-level simple particle interaction; which is what I would call an "engineering solution", i.e. one where you're looking for workable assumptions that you can use to accomplish a task. To me, physics is more about examining the fundamentals to produce and/or validate those "engineering assumptions".

One other example. We all know how to analyze an RC circuit using simple lumped components for engineering purposes. Still, it's somewhat mysterious how the capacitor actually works and that causes confusion (there are threads here showing that confusion). A physicist wouldn't or shouldn't be satisfied that lumped component analysis, at a minimum because it requires instantaneous-action-at-a-distance, and looks at the fundamentals. He notes that a capacitor is fundamentally just a pair of closely-spaced parallel plates, which is equivalent to a low Z transmission line terminated at its far end with an open circuit. So, he exaggerates this aspect by using a long, low Z transmission line connected to a high Z feedline and sends a step wave into the feed. There's an initial partial reflection off the interface between the feed and the capacitor followed by a stepped exponential response from the "capacitor". The steps are spaced at the time for an em wave to travel back and forth along the capacitor transmission line. The exponential response is due to the reflection and transmission coefficients at the interface. Fundamentally, a capacitor is a delay line that let's a percentage of the em wave in and out at its interface to the outside world. As you might expect, an inductor is somewhat of a mirror image: a high Z transmission line terminated with a short circuit. Due to this fundamental analysis, the engineer now knows the limits of applicability of his lumped-circuit model.