Classical Electrodynamics - Science topic

In the Standard Model, if we ignore the unverifiable property of colour charge and consider neutrinos as ‘dark matter particles’ for the time being [1], then we can consider fermions to have the signature properties of electric charge, spin magnetic moment and mass. We consider the electron as a representative, which differs from other fermions only by its mass size, stability, and position in the composite particle.

‘Charge’ was one of the first properties of particles to be discovered, and it appears to correspond to “mass-charge”, which has a similar behaviour [Weyl][Heaverside]. While we have paid a great deal of attention to the existence of an origin of mass [2] and introduced the Higgs mechanism [9], no one seems to have paid much attention to the existence of an origin of electric charge since the beginning of the last century. In order to establish an electromagnetic worldview [3], physicists at that time worked on determining the electron model [4][5][6] : is it rigid? What is its radius? A most crucial question is how should the charge in it be distributed? To this day, physics still does not know the structure of the electron, and what the charge is, except that there exists e+e- ↔ γ γ . Then，

1) Does electric charge have an origin? The fact that it is capable of annihilation and creation, there must be a process of generation. What determines this process? Doesn't a process need to be described, even if it is vacuum-excited generation?

2) Is electric charge an independent entity? We have never seen a ‘charge’, only electrons.

3) A charge cannot be a ‘point’, how does it manage not to repel itself? Poincaré once postulated the existence of a non-electromagnetic reaction force that balances the repulsion between distributed charges to keep them from splitting [7].

4) Does the electric field of a charge act on itself? Why do we see this as a problem? [10]

5) Why is the charge a discrete (quantised) value?1 or 1/3 ^‡. Is the discrete nature of energy related to the discrete nature of charge? Or furthermore, do all discretisations originate from the discrete nature of energy? 〠

6) How can charges be positive and negative and perfectly equal? What is the physical pathway by which charge is created? How can different positive and negative charges be created at the same time in the same physical picture? And positive and negative charges can cause annihilation of positive and negative electrons, not just positive and negative charges.

7) Is there a relationship between electric and magnetic charge? According to Dirac [8], the electric charge e and the magnetic charge g must co-exist, hc/eg=2 ^*. Why can the spin-magnetic moment (the inner discreative magnetic moment of the electron) [11] not be considered as a result of ‘magnetic charge’? The magnetic charge must be a magnetic monopole [12], can't it be a magnetic dipole ^**? We are looking for magnetic monopoles, why not electric charges? [13]

8) Charge appears to be independent of mass. How can particles with different masses (e, μ, τ; u, c, t; d, s, b) have the same charge? But when e+e- → γ γ occurs, the charge disappears and so does the mass.

9) How can electric charge share a particle with magnetic charge and mass? † Wouldn't this be a good answer if they were all the result of spin [14]?

10) U(1) symmetry produces conserved charge [15]; charge is conserved when interacting. Is conservation of charge independent of conservation of energy? What will it mean if they are not conserved? 〠

11) What should the charge of a black hole be if it is one of its characteristics? Will the charge of the ultimate black hole eventually be the same as that of an electron?

12) The more important question is this: all of these questions, mentioned above, must be answered at the same time for the problem to be truly solved.

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Supplement (2024.8.28)

【NO.46】Phenomena Related to Electric Charge，and Remembering Nobel Laureate Tsung-Dao (T.D.) Lee;

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* Note in particular that the relationship between electric and magnetic charge is related solely to Planck's constant h and the speed of light c. This implies that their roots are the same.

** “If Magnetic Monopoles Would Annihilate Like Positive and Negative Electrons, Would Magnetism Still Exist?”https://www.researchgate.net/post/NO23If_Magnetic_Monopoles_Would_Annihilate_Like_Positive_and_Negative_Electrons_Would_Magnetism_Still_Exist

† The central question of interest here is why should fermions have multiple properties and only these properties? Where do these properties come from? What must be the relationship between these properties? How do they fit together?

‡ Dirac asked, "the reason for the existence of a smallest electric charge."

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[1] Adhikari, R., Agostini, M., Ky, N. A., Araki, T., Archidiacono, M., Bahr, M., Baur, J., Behrens, J., Bezrukov, F., & Dev, P. B. (2017). A white paper on keV sterile neutrino dark matter. Journal of Cosmology and Astroparticle Physics, 2017(01), 025.

[2] Wilczek, F. (2006). The origin of mass. Modern Physics Letters A, 21(9), 701-712.

[3] Battimelli, G. (2005). Dreams of a final theory: the failed electromagnetic unification and the origins of relativity. European Journal of Physics, 26(6), S111.

[4] Waite, T., Barut, A. O., & Zeni, J. R. (1997). The Purely Electromagnetic Electron Re-visited. In J. P. Dowling (Ed.), Electron Theory and Quantum Electrodynamics: 100 Years Later (pp. 223-239). Springer US. https://doi.org/10.1007/978-1-4899-0081-4_18

[5] Williamson, J., & Van der Mark, M. (1997). Is the electron a photon with toroidal topology. Annales de la Fondation Louis de Broglie,

[6] Damour, T. (2017). Poincaré, the dynamics of the electron, and relativity. Comptes Rendus Physique, 18(9), 551-562. https://doi.org/https://doi.org/10.1016/j.crhy.2017.10.006

[7] Poincaré, H. (1905). Sur les Invariants Arithmétiques (On the dynamics of the electron). http://poincare.univ-lorraine.fr/fr/fonds-et-archives； http://www.academie-sciences.fr/fr/Colloques-conferences-et-debats/henri-poincare.html；

[8] Dirac, P. A. M. (1931). Quantised singularities in the electromagnetic field. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 133(821), 60-72. Dirac, P. A. M. (1948). The theory of magnetic poles. Physical Review, 74(7), 817.

[9] Higgs, P. W. (2014). Nobel lecture: evading the Goldstone theorem. Reviews of Modern Physics, 86(3), 851.

[10] Wheeler, J. A., & Feynman, R. P. (1949). Classical electrodynamics in terms of direct interparticle action. Reviews of Modern Physics, 21(3), 425.

[11] Ohanian, H. C. (1986). What is spin? American Journal of Physics, 54(6), 500-505.

Yang, C. N. (1983). The spin. AIP Conference Proceedings,

Sasabe, S., & Tsuchiya, K.-i. (2008). What is spin-magnetic moment of electron? Physics Letters A, 372(4), 381-386.

[12] Rajantie, A. (2012). Introduction to magnetic monopoles. Contemporary Physics, 53(3), 195-211.

Rajantie, A. (2016). The search for magnetic monopoles. Physics Today, 69(10), 40-46.

[13] Aad, G., Abbott, B., Abbott, D. C., Abud, A. A., Abeling, K., Abhayasinghe, D., Abidi, S., AbouZeid, O., Abraham, N., & Abramowicz, H. (2020). Search for magnetic monopoles and stable high-electric-charge objects in 13 TeV proton-proton collisions with the ATLAS detector. Physical Review Letters, 124(3), 031802.

[14] Yang, C. N. (1983). The spin. AIP Conference Proceedings,

[15] Lancaster, T., & Blundell, S. J. (2014). Quantum field theory for the gifted amateur. OUP Oxford.

In cases where the rotational of the magnetic field H is zero, we can define this field as the gradient of a scalar function defined as the magnetic scalar potential (similar to the electric potential). What is the physical meaning of this magnitude?

Are annihilation and pair production mutually inverse processes?

p+p− → γ γ'

“Annihilation can happen when all the quantum numbers of two colliding particles add up to zero. It might be electron on positron, proton on antiproton, neutron on antineutron, quark on antiquark etc. The force responsible depends on the possible interactions of the annihilating particles.” “Annihilation does not require the presence of other fields.”[x]

“In particular, one concludes that the two photons resulting from the annihilation of slow positrons in matter always have their planes of polarization perpendicular to each other. This has been pointed out by Wheeler who also proposed a possible experimental verification.”[2]

γ γ' →p+p−

It is often assumed that the concept of pair generation was first introduced by Breit and Wheeler, ω1+ω2→e+e-; however, in their paper [1], "pair generation" appears as an old term and cites the paper by Weizsäcker, CF, Z (1934), and Williams' formula。

Perrin (1933) (in French) was probably the first to introduce the concept of 'pair production'. He had a paper entitled "The possibility of materialization by the interaction of photons and electrons."

Regarding pair production: 1）At first sight light-light scattering seems to be impossible because in classical electrodynamics (linear Maxwell equations) the process does not occur. The resulting wave is everywhere given by the sum of the two incoming waves. 2）In quantum mechanics however the situation is quite different. Due to the uncertainty principle a photon of energy E can fluctuate into states of charged particle pairs (with mass mpair.）Experimentally it is very difficult to collide high energy photon beams. A very elegant way of avoiding this difficulty is again to use virtual particles, this time the quantum fluctuation of an electron into an electron photon state.[3]

The identification of pairs is usually a result of statistical findings[4][5][7][8][9]. e.g.

The identification of γ γ → pp events is mainly based on three artificial neural networks, used to separate antiprotons from e−, µ− and h−, where h− represents either a π− or K−[4]

QCD predictions for large-momentum transfer cross sections of the type ‘γγ→ BB' are given, for B and B' any members of the baryon octet or decuplet, and all possible helicity combinations for photons and baryons[8].

An electron enters the laser beam from the left, and collides with a laser photon to produce a high-energy gamma ray. The electron is deflected downwards. The gamma ray then collides with four or more laser photons to produce an electron-positron pair [9].

1) The process of "pair production" and the process of annihilation of positive and negative particles are not mutually invertible. Just as the mass-energy equation is not reciprocal (E=mc^2, which is irreversible for photons), p+p- → γ γ' and γ γ' → p+p- are not γ γ' = p+p-. This is one of the differences between the mathematical equations and the physical equations.

(2) The process of "annihilation" does not require special conditions, while the process of " pair production" must require auxiliary conditions, the presence of other particles being necessary. What is the essential function of these auxiliary conditions? What are the conditions under which photons can "collide" and not just interfere?

3) Is the process of "pair production" one or two processes? Must the " pair of particles" be produced in pairs at the same time, or with equal probability for positive and negative particles? Or is it both. The literature [6] describes pairs of positive and negative particles as being produced simultaneously. This question is very important because it determines the mechanism of the "photon-particle" transition and even their structure.

(4) The colliding positive and negative particles do not necessarily annihilate into photons, but essentially depend on whether the magnitude of the energy reaches the energy value of a certain particle, e+e-→µ+µ-. Here is the root of the problem of the level difference of the three generations of particles implied, just as the energy level difference of orbiting electrons. Can quantum field theory give a concrete, or directional, explanation?

5) Where do the properties of the original positive and negative particles go after annihilation occurs? Charge, spin-magnetic moment, mass, and the spacetime field of the elementary particle. Can the origin of the properties be inferred from this? That is, if the properties are somehow conserved, then by reversibility, do the annihilated photons imply all the properties of the elementary particles. The total charge is conserved after the annihilation of the positive and negative electrons. But where does the positive charge go and where does the negative charge go? The following issues are involved here: https://www.researchgate.net/post/How_Fermions_combine_four_properties_in_one

[1]【Breit, G. and J. A. Wheeler (1934). "Collision of two light quanta." Physical Review 46(12): 1087】

[2]【Yang, C.-N. (1950). "Selection rules for the dematerialization of a particle into two photons." Physical Review 77(2): 242】

[3]【Berger, C. and W. Wagner (1987). "Photon photon reactions." Physics Reports 146(1-2): 1-134】

[4]【Achard, P., O. Adriani, M. Aguilar-Benitez and etl. (2003). "Proton–antiproton pair production in two-photon collisions at LEP." Physics Letters B 571(1-2): 11-20】

[5]【de Jeneret, J., V. Lemaitre, Y. Liu, S. Ovyn, T. Pierzchala, K. Piotrzkowski, X. Rouby, N. Schul and M. V. Donckt (2009). "High energy photon interactions at the LHC." arXiv preprint arXiv:0908.2020.】

[6]【Michaud, A. (2013). "The Mechanics of Electron-Positron Pair Creation in the 3-Spaces Model." International Journal of Engineering Research and Development 6: 2278-2800】* Researchgate Link：

Minimum mass issues are involved here:

https://www.researchgate.net/post/Is_there_a_minimum_value_of_m_in_the_mass-energy_equation_Emc2

[7]【Klein, S. R. and P. Steinberg (2020). "Photonuclear and two-photon interactions at high-energy nuclear colliders." Annual Review of Nuclear and Particle Science 70: 323-354.】

[8]【Farrar, G. R., E. Maina and F. Neri (1985). "QCD Predictions for γγ Annihilation to Baryons." Nuclear Physics B 259(4): 702-720】

[9]【SLAC. (1970). "SLAC Experiment E144 Home Page." from https://www.slac.stanford.edu/exp/e144/.】

【Burke, D. L., R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, A. W. Weidemann, C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D. A. Reis and W. Ragg (1997). "Positron Production in Multiphoton Light-by-Light Scattering." Physical Review Letters 79(9): 1626-1629】

【Schwarzschild, B. (1998). "Gamma Rays Create Matter Just by Plowing into Laser Light." Physics Today 51(2): 17-18】

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2023-06-25

For the "pair production" experiment, the 2021 STAR Collaboration collectively published a paper "Measurement of e+ e- momentum and angular distributions from linearly polarized photon collisions" [4].

"At RHIC, scientists accelerate gold ions to 99.995% of the speed of light in two accelerator rings. If the speed is high enough, the strength of the circular magnetic field can be equal to the strength of the perpendicular electric field," Xu said. perpendicular electric and magnetic fields of equal strength is exactly what a photon is-a quantized "particle "So, when the ions are moving close to the speed of light, there are a bunch of photons surrounding the gold nucleus. As the ions pass one another without colliding, two photons (γ) from the electromagnetic cloud surrounding the ions can interact with each other to create a matter-antimatter pair: an electron (e-) and positron (e+) [5]. [The headline of the media report is more interesting [5][6][7]]

The history of the discovery of the physics of particle production and annihilation is presented in paper [1]; paper [3] is an analysis of the experimental phenomena by Anderson, the discoverer of positrons, in which four possibilities are proposed for each result, "pair production" being one of them. He finally determined that "pair production" was the real case. The results provided by André Michaud [9] should be similar [see his replies for details].

Comparing the STAR experiment [5] and the E114 experimental method [8], they produce photon "collisions" in a very different way. These two experiments are in turn different from experiments [2] and [3]. It is commonly believed that there are three possible interactions [4]: the collisions of two virtual photons (as calculated by Landau and Lifshitz, giving the total cross section for e+e- production predominantly at the pair threshold), of one virtual and one real photon (Bethe-Heitler process ), or of two real photons-the Breit-Wheeler process.

Question: Yang[1] and Andeson considered that Chao [2] and Anderson [3] are both electron pair generation processes, so is this a "photon-photon" collision "γγ → e+e- " process? If so, are the photons real or virtual, and what is the difference between them and the experiments [4][8]? If not, then there are no "photon-photon" collisions in the experiments of Chao [2] and Anderson [3], but only "photon-particle" collisions?

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Reference：

[1] Li, B. A. and C. N. Yang (1989). "CY Chao, Pair creation and Pair Annihilation." International Journal of Modern Physics A 4(17): 4325-4335.

[2] Chao, C.-Y. (1930). "The absorption coefficient of hard γ-rays." Proceedings of the National Academy of Sciences 16(6): 431-433.

[3] Anderson, C. D. (1932). "The apparent existence of easily deflectable positives." Science 76(1967): 238-239.

[4] Adam, J., L. Adamczyk and etl. (2021). "Measurement of e+ e− momentum and angular distributions from linearly polarized photon collisions." Physical Review Letters 127(5): 052302:

[5] "Collisions of Light Produce Matter/Antimatter from Pure Energy": https://www.bnl.gov/newsroom/news.php?a=119023

[6] "Colliding photons were spotted making matter. But are the photons 'real' ? ": https://www.sciencenews.org/article/colliding-photons-matter-particle-physics#:~:text=In%20a%20demonstration%20of%20Einstein%E2%80%99s%20E%3Dmc%202%2C%20collisions,colliding%20particles%20of%20light%20create%20matter%20and%20antimatter.

[7] "Scientists Generate Matter Directly From Light – Physics Phenomena Predicted More Than 80 Years Ago": https://scitechdaily.com/scientists-generate-matter-directly-from-light-physics-phenomena-predicted-more-than-80-years-ago/?expand_article=1

[8] SLAC. (1970). "SLAC Experiment E144 Home Page." from https://www.slac.stanford.edu/exp/e144/.

[9] the FERMILAB experiment E632 bubble chamber picture;

Start with a purely classical case to define vocabulary. A charged marble (marble instead of a point particle to avoid some singularities) is exposed to an external electromagnetic (E&M) field. "External" means that the field is created by all charges and currents in the universe except the marble. The marble is small enough for the external field to be regarded as uniform within the marble's interior. The external field causes the marble to accelerate and that acceleration causes the marble to create its own E&M field. The recoil of the marble from the momentum carried by its own field is the self force. (One piece of the charged marble exerts an E&M force on another piece and, contrary to Newton's assumption of equal but opposite reactions, these forces do not cancel with each other if the emitted radiation carries away energy and momentum.) The self force can be neglected if the energy carried by the marble's field is negligible compared to the work done by the external field on the marble. Stated another way, the self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy. Also, an analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone. The key points from this paragraph are the last two sentences repeated below:

(A) An analysis that neglects self force is one in which the total force on the marble is taken to be the force produced by external fields alone.

(B) The self force can be neglected if and only if the energy carried by the marble's field is negligible compared to the change in the marble's energy.

Now consider the semi-classical quantum mechanical (QM) treatment. The marble is now a particle and is treated by QM (Schrodinger's equation) but its environment is an E&M field treated as a classical field (Maxwell's equations). Schrodinger's equation is the QM analog for the equation of force on the particle and, at least in the textbooks I studied from, the E&M field is taken to be the external field. Therefore, from Item (A) above, I do not expect this analysis to predict a self force. However, my expectation is inconsistent with a conclusion from this analysis. The conclusion, regarding induced emission, is that the energy of a photon emitted by the particle is equal to all of the energy lost by the particle. We conclude from Item (B) above that the self force is profoundly significant.

My problem is that the analysis starts with assumptions (the field is entirely external in Schrodinger's equation) that should exclude a self force, and then reaches a conclusion (change in particle energy is carried by its own emitted photon) that implies a self force. Is there a way to reconcile this apparent contradiction?

According to Dirac, theoretically, each Magnetic Monopole (North mm) is connected with their counter partner in space, the Magnetic Antimonopole (South mm) via an infinite thin and possible long undetectable string called the Dirac String. By this Dirac actually acknowledges the dipole nature of magnetism even concerning magnetic monopoles which is essential for the Dirac magnetic monopoles to be compatible with Maxwell Theory.

I am wondering what the existing theory predicts of what would happen, if a monopole and antimonopole pair, a large distance apart, and that is "apparently" due to the Dirac string (i.e infinitely thin), isolated from each other, would come close together and collide?

Will these two join forming an elementary magnetic dipole or else called the Quantum Magnet?

Please notice also that the term Quantum Magnet refers to the electron.

Also, simulations and experiments with artificially emulated in spin ice mm (i.e. skyrmions) have shown that under the right conditions a monopole-antimonopole pair does not annihilate but rather joins close-up in a dipole configuration forming a tightly connected Dirac monopole-antimonopole pair:

arxiv.org/abs/1904.02257 (see fig. 4 in arxiv link).

During the BB extreme conditions this process could have actually have lead to the creation of electrons (i.e. quantum elementary magnet). This would also explain why there are today no more mm monopoles to be found in nature since they all were converted to e during the early creation of the Universe..

Next read this paper here,

https://tinyurl.com/2p8p4ve6

of why magnetic monopoles should actually regarded as fermions thus have spin 1/2 making them a strong candidate for the quantization of the electron charge and therefore also implying these to be responsible for the creation of the first electrons during the Big Bang (or anyway other possible process other than the BB of early creation of the Universe for those who don't entirely or partially albeit with the BB theory.There must have been a beginning anyway).

I am much interested here in a discussion of if this proposal is possible and probable and for any arguments against it?

Thank You!

image copyright: ©E. Markoulakis Hellenic Mediterranean University 2022.

Of course, as the frequency of electromagnetic waves increases, the magnetic force increases which people are not yet aware of. This fact is also responsible for creating the photoelectric effect. It's a mystery why people didn't realize this for the past 100 years.

This fact cannot be explained by classical electrodynamics as well as quantum mechanics.

For those that have the seventh printing of Goldstein's "Classical Mechanics" so I don't have to write any equations here. The Lagrangian for electromagnetic fields (expressed in terms of scalar and vector potentials) for a given charge density and current density that creates the fields is the spatial volume integral of the Lagrangian density listed in Goldstein's book as Eq. (11-65) (page 366 in my edition of the book). Goldstein then considers the case (page 369 in my edition of the book) in which the charges and currents are carried by point charges. The charge density (for example) is taken to be a Dirac delta function of the spatial coordinates. This is utilized in the evaluation of one of the integrals used to construct the Lagrangian. This integral is the spatial volume integral of charge density multiplied by the scalar potential. What is giving me trouble is as follows.

In the discussion below, a "particle" refers to an object that is small in some sense but has a greater-than-zero size. It becomes a point as a limiting case as the size shrinks to zero. In order for the charge density of a particle, regardless of how small the particle is, to be represented by a delta function in the volume integral of charge density multiplied by potential, it is necessary for the potential to be nearly constant over distances equal to the particle size. This is true (when the particle is sufficiently small) for external potentials evaluated at the location of the particle of interest, where the external potential as seen by the particle of interest is defined to be the potential created by all particles except the particle of interest. However, total potential, which includes the potential created by the particle of interest, is not slowly varying over the dimensions of the particle of interest regardless of how small the particle is. The charge density cannot be represented by a delta function in the integral of charge density times potential, when the potential is total potential, regardless of how small the particle is. If we imagine the particles to be charged marbles (greater than zero size and having finite charge densities) the potential that should be multiplying the charge density in the integral is total potential. As the marble size shrinks to zero the potential is still total potential and the marble charge density cannot be represented by a delta function. Yet textbooks do use this representation, as if the potential is external potential instead of total potential. How do we justify replacing total potential with external potential in this integral?

I won't be surprised if the answers get into the issues of self forces (the forces producing the recoil of a particle from its own emitted electromagnetic radiation). I am happy with using the simple textbook approach and ignoring self forces if some justification can be given for replacing total potential with external potential. But without that justification being given, I don't see how the textbooks reach the conclusions they reach with or without self forces being ignored.

I am interested to know the opinion of experts in this field.

A thin, circular disc of radius R is made up of a conducting material. A charge Q is given to it, which spreads on the two surfaces.

Will the surface charge density be uniform? If not, where will it be minimum?

Give an example where the electric field is zero at a point but divergence of the electric field is non zero there?

According to classical electrodynamics theory, an accelerating charged particle emits an electromagnetic radiation. Unruh on the other hand found that an accelerating observer (charge) will find itself immersed in a black body radiation from the vacuum. How the de Broglie wave interacts with the electromagnetic radiation created by the particle? Is there a relation between wavelength and acceleration for an accelerated charged particle? Does the black hole evaporate completely or end in a finite mass (e.g., planck mass)?

Could anyone recommend a good textbook to study about Green Function in classical electrodynamics? Thank you.

What are the most efficient open source solvers for classical electrodynamics problems? For example, solve for scattering of a steady-state TE or TM wave on a heterogeneous medium. I'd expect there should be FMM-accelerated Integral Equation solvers that are more efficient than, say, Ansys FEM.

Dear Sirs,

Everybody knows plane and spherical wave solutions of Maxwell equations, e.g for decaying plane wave E=E0*exp(-kx)*cos(w(t-x/v)). But seems to me they give the unreal situation that the wave amplitude is nonzero at different points of space at given time moment. Could you advise the experiment or natural phenomenon which produces such a wave in nature?

Maybe we have infinte speed of the EM interaction? Do you know any real solution of Maxwel equations which exists only in one space point at the given time moment? Maybe using delta function? Or maybe there is my mistake?

Earth has a North Pole and a South Pole. Is this a paradigm or a model that we need to understand before we can begin to understand the structural (or some other) design of the universe? If so, point to one other example of polarity. If not, why not?

When Dirac introduced his magnetic monopole for explaining the quantization of the electric charge he left the mass as a free parameter of such particles, but nowadays we have many different kind of models for such particle. My question is what is the value of the mass employed for trying to look for this particle scattering processes in particle detectors or cosmological measurements

In general by strict mathematical definition of conservative fields, no magnetic vector field in any case even static can be conservative thus path-independent since it has no zero curl which is necessary for a field to be conservative [1]. In addition all conservative vector fields must be also irrotational (i.e. vortex, spiral). Even if a magnetic or other field special case, is found to be with zero curl that does not mean necessarily that it is conservative if it does not satisfy the condition in 3D space,

F:R³→R³ is continuously differentiable in a simply connected domain W∈R³ and its curl is zero:

https://mathinsight.org/path_dependent_zero_curl

Nevertheless, it is a mystery why the static magnetic field of magnet for example exhibits all the effects of a conservative field without having its properties?

No energy is consumed when a single charge particle is introduced forcing it to a circulation where equal amount of potential energy is converted to kinetic energy and vise versa. Energy is conserved thus no real work is done by the field thus conservative in effect!

Therefore a correct answer of time invariant static magnetic fields being conservative or not? Is I believe that they are virtual conservative fields by absence of any better explanation of this phenomenon and contradiction.

Emmanouil

p.s The above virtual description of static conservative magnetic field begs a definitive better answer I believe and is a mystery proving how much more we have to investigate on this matter of Electromagnetism.

[1] https://www.quora.com/Is-magnetic-field-conservative-or-non-conservative

An electron has an intrinsic spin magnetic dipole moment, and nothing is actually spinning!

The spin magnetic dipole moment, is approximately equal to a μB Bohr magneton.

Classical Electrodynamics can not explain this and therefore Quantum Electrodynamics comes the rescue... Did it?

Emmanouil Markoulakis

Technological Educational Institute of Crete

________________________________

The name spin magnetic dipole moment (https://en.wikipedia.org/wiki/Electron_magnetic_moment) was given due to the initial attempt to explain the rest magnetic dipole moment of the electron using the initial assumed spin of the electron which is not real, around it axis (like a planet spin) and since then we are stuck with this name.

The electron is not physically spinning.

Besides this theoretical model was resulting in a spin velocity with speed greater than the speed of light ! The theoretical calculation of the rest magnetic dipole moment came out short by a factor of 2!! (i.e. x2 less than that was found experimentally).

In order to rectify this "small error" hahahaha! :) a dimensionless correction g-factor about x2 was forced into the equation to match theoretical calculations with experimental data.

This alone tells me that their schizophrennic sphere electron electric monopole - magnetic dipole model is not working in this case and there is a gap in the theory in general and they don't have a clue what the electron really is and their model does not translates to the physical 100% correct and so also for electricity and magnetism.

https://www.youtube.com/watch?v=qsSI0sQpWmI (video)

Emmanouil

Dear Fellow Applied Physicists,

I have come across a simple problem turned complicated, where I attempt to solve a stationary 2D Laplace Equation (L.E) in order to better understand the effects of sharp edges on the solutions.

Starting from a simple setup, I solved, using a classic separation of variables technique, the L.E in 2D for a infinitely extended edge (with an opening angle of 120°) for the simple Dirichlet Boundary Condition of constant potential across the edge's infinite contour. The solution of this setup is easily found using 2D polar coordinates. However, once I add an additional geometric constraint to my problem, the Separation of Variables hypothesis no longer holds; thus I require some help here.

I have added one finite boundary to one of the edge's lengths, a finite value of length L, where this new boundary (at x = 0) behaves like a symmetry boundary condition.

The Laplace Equation (2 dimensional):

∇² ϕ = 0

The boundary conditions in Cartesian coordinates are as follows:

1) ϕ = { V=const 0 < x < L & y = 0

L < x < +∞ & y = mx - mL for m = slope of boundary

}

2) ∇ ϕ|_{(x = 0 , y)} = 0

I switched to Cartesian coordinates because the 2nd BC removes the separation of variables property when using polar coordinates, thus rendering it no more useful than a Cartesian formulation.

If you guys can, please either refer me to texts that tackle similar problems, or advise me on how to properly select solution methods for this problem. Separation of Variables doesn't work as far as I can tell, due to the BC.1 having y = f(x).

Please inform me if more details are needed.

Best,

Christopher

The recoil force of radiation is known for spontaneous emission (for the radiation of an accelerating charge or dipole), when the photon field is empty. Is there any difference when stimulated emission is considered? Would it be enough to add an external force to the original radiative reaction-force without changing the original form of the radiative reaction?

I'm looking for the electric equivalence of the (second order linear parabolic) heat equation in order to calculate transient processes.

At the phenomenological level, there is a strict analogy between conductive heat flow, the diffusive motion of particles and the electric current. Fourier’s law, Fick’s 1st law and Ohm’s law are equivalent, etc. However, I can’t find the general equation for the electric potential which is analogous to the so called heat equation or diffusion equation (without sources) One reason can be that it is not valid for a general 3D body, because electrons tend to go to the surface of the body.

I wonder if one could experimentally detect electromagnetic radiation emitted from the charged sphere, if it spins with w = 30 Hz - 30 kHz frequency, hence, emit in the low radio diapason and diameter of the sphere approach micrometer range (let’s say r = 25 um)?

From classics electrodynamics one can estimate mean intensity emitted by accelerated charges from the surface of the sphere. Which is, from electric dipole moment, I = (2/3)q²r²w²/c³, where r – radius of the sphere, w – frequency of rotation, q – electric charge and c – speed of light. Or I = (q²w²/600c)(qrB/mc²)⁴, from magnetic momentum of the sphere, if external magnetic field B is applied and m – is mass of the sphere.

Certainly, presence of magnetic field makes situation better. But anyway, what would it take to perform such detection in most simple way possible?

The electric potential must be continuous, otherwise the electric field will be infinity (E=-dv/dx).

But, is there a a physical law that requireing the electrochemical potential to be continuous?

Why "nickels" (the U.S. 5 cents coin) are not attracted by a magnet ?And why a metal pipe made out from melted "nickels" will not exhibit the Lenz effect by dropping through the pipe a magnet (magnet falls fast through the pipe) as it would be with an aluminium or copper pipe (magnet falls slowly through the pipe) ?

special theory of relativity says;;;;;;an observer at rest with respect to a system of static free charges will see no magnetic field. however, a moving observer looking at the same set of charges does perceive a current and thus a magnetic field.

in a similar way does observer travelling with a speed of light perceive magnetic field?

In electrostatics, inside a metal there are no charges (electric field equals zero, thus one can find the potential using Laplace equation (together with the proper boundary conditions). Is it still true when I apply current through the metal? Are the charges still moving only at the surface (in order to keep zero electric field insode?) or do I need now to solve the poisson equation instead (or some other equations) and to determine somehow the charges?

I saw in a paper that the current in a metal is sigma (conductivity=e*mu*n) multiplied by dk/dx where k is the electrochemical energy (or fermi energy).

It seems to me that it is some kind of a generalized ohm's law (j=sigma*E) where the electric field is 1/e*dk/dx.

My questions are these:

# Is it truely a generalized ohm's law or is it comes from other more fundamental law?

# Is this law valid for semiconductors and/or outside equilibrium (steady state, external applied voltage)?

# I couldn't find anything on this equation and I'll be grateful if someone could direct me to some books referring this equation.

As generally stated that quantum mechanics is the wider theory and it contains classical theory as a special case. We can get classical results from its quantum counter part in the limit of planck constant tends to zero. Can we get classical maxwell electrodynamics as a similar limiting case from quantum electrodynamics?.

..

Update Oct 18: Attached is now a description enhanced by equations and graphs.

The attached text (about 1.3 pages) describes a paradox which seems to have no solution in classical physics. Two solutions in the framework of quantum mechanics are outlined.

Do you think a classical solution is possible? If so, could you please provide a sketch of your solution? If not, what do you think about the suggested QM solutions, and do you know of a better (resp. the true) solution? And finally, this might be a well known paradox; if so, could you please point me to relevant literature?

Many thanks for all helpful comments in advance!

Contrary to classical electrodynamics, the electron does not radiate when it orbits the nucleus in stationary orbits. This inconsistency may be the result of the use of Coulomb potential to describe the dynamics of a hydrogen-like atom. In order to resolve this problem we need a potential that can produce a zero net force when the electron moves in stationary orbits. It can be shown that general relativity can be used to modify the Coulomb potential in this case. Please refer to my works A TEMPORAL DYNAMICS: A GENERALISED NEWTONIAN AND WAVE MECHANICS and ON THE STATIONARY ORBITS OF A HYDROGEN-LIKE ATOM on RG for more details.

In fact, it is possible to show that a classical potential is directly related to a geometric object which is the Ricci scalar curvature and the Schrodinger wavefunctions are simply mathematical objects that can be used to construct spacetime structures of quantum particles. For this new development, please refer to my works SPACETIME STRUCTURES OF QUANTUM PARTICLES and A DERIVATION OF THE RICCI FLOW for more details.

Are the Integral and Differential forms of Maxwell's Equations position dependent?

This is not something you can find clearly stated.

If photon is a particle then it should be not applicable, but in the double slit experiment a photon interferes with itself. But if this is the case it must be propagating in all directions but it clearly doesn't.

I am interested in finding the Green's functions for Poisson (or Laplace) differential equations for an open surface, for example a hollow cylinder with radius R and length L, or a hemisphere with radius R.

Please be informed that you can find some Green's functions for closed surfaces in Classical Electrodynamics (J. D. Jackson), but no problem is found for an open surface.

The design of an optical slot waveguide typically aims to achieve an intensity of a x-component of the E-field that is much higher than the other components, leading to a quasi TE-condition. In most of publications, when optimizing the optical field confinement factor (Gamma) into the slot, the power term, i.e. the Poynting vector, is often approximated in terms of integral of E_x²/Z, being Z the wave impedance of the mode. Nevertheless, the approximation of Z as Z₀/n, being n the real refractive index of the slot material, is valid only for TEM modes, whereas for TE and TM modes Z is not depending only on material properties, leading to a surface distribution.

Therefore, using the approximation in COMSOL Multiphysics (Wave Optics) leads to meaningless values of the Gamma factor, with respect to what achieved using the Poynting vector expression.

Anyway, I guess that for my application of a slot waveguide photonic modulator, the estimation using the E_x-field leads to a more useful Gamma, as it is referred to the same direction of the TE RF field whose overlap to the optical field must be optimized. I expect that in such a way a proper estimation of Gamma can be achieved, even if slightly lower than what achieved using the Poynting vector. 

Unfortunately, I saw that the wave impedance is not included among the variables of Wave Optics. From what read in Classic Electrodynamics books, maybe the surface distribution of the wave vector can be used to derive the wave impedance, but not even the wave vector is included as a variable in COMSOL. 

Has anyone any suggestions to solve this ?

I attached a slide trying to resume this.

In his recent paper, Valerii Temnenko proposes a new classical model of neutrino. He wrote the abstract as follows: "The theory contains a number of wave states, both one-sector (singlet or triplet waves) and compound two-sector ones (singlet-triplet waves). Wave states differ in number of currents: zero-current waves (free singlet or free triplet waves), one current, two-current, three-current and four-current ones. Wave states also differ in character of four-dimensional wave vector (the waves with time-like and space-like wave vector). Some forms of waves may have negative density of energy. Some wave states can be treated as classical models of a neutrino. Neutrino states are classified in accordance with the character of the current which forms the state: singlet (maxwellian) neutrino, Yang-Mills triplet neutrino, Maxwell-Yang-Mills singlet-triplet neutrino."

Do you think that such a Classical model of Neutrino is possible? What is your opinion? Thanks

For example if there are two atoms ( A, B) with magnetic momentum, the total magnetic momentum follows √(µA²+µB² ) or √(µA²-µB² )?

Everyone who is familiar with classical electrodynamics knows about Lorentz invariant quantities (E² - B²) and (E*B). 

Is there any application for these invariants in physics?

In transformer core,etc the eddy current path is assumed to be purely resistive in nature, but there must be some inductance  due to self linkage of the eddy current generated flux with the core itself.The value of this inductance must be very less .Thats why purely resistive  eddy current path is assumed.My question is why this inductance is very less?

As we know this equation in electrostatics is based on Coulomb experiments, but this is not obvious in electrodynamics. 

Please see the attached for details.

We have two charged bodies with charges q1 and q2 with the same mass where q1= q2.Imagine now a reference frame S so that q1 is at rest at the origin of S. Q2 is moving with velocity constant v in x direction. Imagine another reference frame S' attached to q2 so that q2 is at its origin. Let's assume that at time t = 0, q2 is passing near q1 where the distance between them is d. Namely the X coordinate of q2 is 0 and the y coordinate of q2 is d at time t = 0.As known when observed in S, the magnitude of the electric field of q2 perpendicular to its direction of motion namely the component in y direction is slightly greater (the actual value being a function of its velocity) when compared to the field of the same static charge. Consequently the electrostatic force on q1 is slightly greater then the force of a static charge at the same distance. The electric field and therefore the repulsive force q1 excerts on q2 is however the same as the force created by static charge. Thus it seems the principle actio = reactio and therefore the momentum conservation is violated at the instant t= 0. Q1 is accelerated towards q2 more then q2 is accelerated to q1. If we observe things in S' then the opposite is true. I know that in electrodynamics not only material bodies but the electromagnetic field itself carries a momentum ExB and there is momentum exchange between the subsystems so that only total momentum is conserved. But if one considers the arrangement of the charges it seems to me that this doesn't solve the problem.