Qreeus/Exchemist I am finding it difficult to appreciate, simply because inclusion of a new factor (refer to E formula at 5.3.4 with gamma etc) would increase the E value. If I recall correctly the increase in E value must be read as increase in number of field lines. I may be corrected.
Just evaluate the exact expression for E' for various θ values. Weaker along the axis, and stronger at the equator, than for a stationary charge. A properly formulated surface integral recovers the same net flux i.e net field lines passing through a closed bounding surface, irrespective of charge velocity. It's an easy exercise when working with the instantaneous present position formula used here. Quite a bit more tricky if working from a retarded fields formulation!
Once again, commendable clarity. This should knock it on the head, I think - unless tsmid wants more assistance with the maths itself.
Actually: according to Maxwell's theory of electromagnetism. The Coulomb electric field is the solution to Maxwell's equations for the special case of a stationary charge. The equations* on page 110 of the notes you linked to give the solution to Maxwell's equations for the special case of a charge moving with constant velocity. The key part here is: constant velocity. The electromagnetic field around an accelerating charge is not so simple and involves a time-delay effect (if you are 1 light second away from a moving charge, and the charge decides to change direction, then it will take another second before the change of direction makes a difference to the electromagnetic field at your location compared with if the charge hadn't change direction). And if you have a bunch of charges moving around inside some matter then they have to be constantly changing direction (accelerating) in order to remain confined to the matter. So the formula you copied doesn't apply (at least, not so simply) in that case.
True. But by setting the bounding surface to very large r from some given relatively confined source of moving charges, we can split the evaluation into the sum of 1/r² velocity fields and 1/r radiation fields. The latter are always orthogonal to the retarded radius vector, hence to the surface normal of bounding surface. Hence radiation fields make no net contribution. So in the end it still reduces to only needing to consider the field of constant velocity charge. An explicit proof for arbitrary charge motions, by direct calculation using the retarded fields has no doubt been done, but I somehow lack motivation to labour at it.
This was my reasoning also. It seems the art of knowing when some mathematical problem is too complex to waste a significant portion of your life obsessing over it has been lost, at some point. Whole lives have been wasted trying to reach the last digit of the constant pi with pencil and paper. Even if they made no mistakes on the way, a life wasted on something unworthy of the effort is still just that. Oh, but don't pay attention to that lesson.
The electrons don't "radiate", unless it is by photonic bosons interactions with the nucleus. Please tell me this is not the SM concept that has been commemorated in this mathematical mumbo jumbo by means of line integrals and Gauss's law? It is, isn't it? Gawd. No wonder quantum physics can't conceptualize its way out of a wet paper bag, even if it can calculate the line integral path all the way out. Well, that was just very instructive.
Dan, why add extraneous factors into the problem? tsmid's problem was how to avoid getting a silly answer (viz, an apparent change in net charge), applying classical special relativity to the electron and nucleus as point charges. That has now been resolved, has it not? In QM, accelerating charges in bound states can emit radiation but only certain frequencies, according to the change in eigenstate they undergo at the same time - and so long as they are not already in the ground state, yes? But I do not see that has any effect on the logic by which it is explained that charge does not, in fact, appear increased or reduced in total due to relative motion. Q-reeus explains why any radiation that may occur has no impact.
I see. Eigenstates are elements of linear algebra, not atomic structure. It's all in your head. Who has really checked this math?
I have not followed the entire thread, but Exchemist last post to Dan suggests that this is getting about "relativistic charge"?
Don't ask. Not worth it. If you are interested, read tsmid and Q-reeus contributions. These are by and large on-topic and about the science.
Math, all by itself, divorced from the physical world isn't science. It's not subject to falsification other than in a very limited range of operations. Neither does linear algebra render all there is for science to know about atomic structure. Not even close. We learned enough with simple, classical physics math applied to uniform circular motion to understand that no net electric charge occasioned from any relativistic effects of the electron clouds around an atom. What math genius was it who decided they needed linear algebra, supercomputers and a mathematical model to average the behavior of a swarm of electrons to convince themselves this was the case? I've attended and viewed more physics lectures than most people here. I don't think I've ever seen any work from anyone that was more mathematically OC challenged than this. You are as entitled to your opinions of this model as I am to mine, exchemist. This smells pretty bad from where I'm sitting. I wouldn't give a wooden nickel for a dozen papers about this kind of analysis. As I have already said, it was a most instructive example. Thanks again, Q-reeus.
It's all in your diseased mind Dan. Q-reeus explained it without reference to any of that. You are rambling again, I'm afraid. Why not get some sleep and try again tomorrow, eh?
There is obviously no need to consider electronic radiation wrt ground state orbitals in atoms/molecules. However, for metals at any temperature above 0 K unpaired conduction electrons are thermally promoted into effectively continuum energy level bands, where they freely couple to photons/phonons thus continually radiate. The fact such electrons (also holes for semiconductors/semi-metals) have a Fermi velocity of ~ 10^6 m/s and finite mean-free-paths in the sub micron range guarantees they must experience accelerations/decelerations continuously. Even for insulators at say room temp, polariton collective oscillations of lattice ions must occur as an interaction with ever present photons and phonons. I'm assuming you are aware of what the following implies wrt OP topic: https://en.wikipedia.org/wiki/Stefan–Boltzmann_law https://en.wikipedia.org/wiki/Thermal_radiation So, while only a small fraction of electrons in a typical solid engage in radiative exchange, it is present and thus legitimate to take into account. That a gas or liquid heats up when exposed to purely radiant energy likewise guarantees such atoms/molecules engage in radiative exchange, although the details may be quite complex. See above. Take a deep breath Dan and just relax a bit.
I have studied thermodynamics, of course. I have a tendency to set aside any issues of charge distributions for most thermodynamic problems of this variety, but I'll take your (and exchemist's) word for it that any changes of state of the electron clouds produce photons and phonos and that these distributions are important in predicting things like blackbody radiation, Hawking radiation, etc. But it's a fact; accelerations mean exactly zero because of relativity's clock postulate. Only instantaneous velocities are important. When someone shows me a bit of quantum physics trying to calculate such things while completly ignoring or neglecting the clock postulate (which took over 90 years to make it into the scientific knowledge base associated with relativity), well I think of that as a rather severe mathematical oversight of the kind only possible by being completely ignorant of the latest math associated with relativity. If you allude to a problem with residual Coulomb charge based on relativity, either do it right or don't pretend you have actually taken relativity into account. Don't just pile a load of boost matrices into your math, calculate everything (particularly radiative processes) with accelerations of the electron cloud and think you have anything describing physical reality. The electrons will simply decay and you get electron degeneracy, and yeah, then you've found some considerable residual charge issues. The electrons in an electron cloud surrounding an atom experience no accelerations unless they are absorbing or emitting radiation. If they did, they would radiate energy away until they reach the ground state, or perhaps even degenerate into even lower energy states. Chandrasakar's neutron stars, for example, would acquire a rather strong and unstable amount of positive electrical charge when electrons and protons degenerated into neutrons. The electrons and protons don't all degenerate at once when it collapses, does anyone imagine? It would most likely blow itself apart before it finished collapsing. Got a calculation or a model for that process? Don't be shy; if you have confidence in your math, give it a REAL test. I should stop this. We're not going anywhere anyone else here wants to go, and I know this. I'll just chalk it up to being one of the bad ideas a lazy mathematics professor somewhere inflicted on some hapless group of physics graduate students one day, who had no better sense than to try solving the poorly defined problem he gave them.
Until you brought them up in #58, afaik no-one here thought clock postulate, Hawking radiation, classical prediction of electron inspiral to nucleus, bizarre NS scenarios etc. had any connection to topic. Honestly. But I wholeheartedly agree with that in red. Wholeheartedly. Bye for now.
Although it appears to be a decent explanation that Gaussian integration will take care of the issue raised by OP. But let's complicate it, if we can have relativistic mass then why not relativistic charge?
