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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...
I'm still confused...
Different people mean different things with that sound bite...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...
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)Different people mean different things with that sound bite...
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.... (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.[..] 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) [..]
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.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.
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.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.
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.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.
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.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.
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.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").
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".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.