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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.
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It might be a good idea to study classical electrodynamics, the subjects are standard exercises.
Magnetic force on electric charges is described by the Lorentz force....
The photoelectric effect is beyond classical electrodynamics, but is, now, also, standard material.
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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.
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A revision with a more appropriate title is attached. The Conclusion section is specific about the difference between what is in this report and what is in at least some popular textbooks.
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I am interested to know the opinion of experts in this field.
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Photons are massless and therefore non-localisable (consider any typical solution of Maxwell's equations, ), i.e. there are none that stay at a fixed and specific point-like location in space. In contrast, the wavefunction of a massive particle can be so localised.
Thus I would say that photons never match the common definition of a particle (because they are not point-like localisable, even in principle). However, since they can be counted, I would, if prevailed upon to suggest a qualitative description, instead describe them as "countable waves".
This is because in QED we quantize inside "mode" solutions of Maxwell's equations (see any quantum optics text, or the paper I cite above), and can describe the quantum state within each mode in terms of combinations of photon number states.
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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?
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when I don't "see" the answer to a problem immediately, I find it sometimes worthwhile to play around with the help of a math program. For example, with integrals it replaces a heavy book like Gradshteyn, Ryzhik: Table of Integrals, Series, And Products, and if an integral cannot be solved symbolically, it can be solved at least numerically for a few cases.
Assuming a unit disk and using polar coordinates (r, φ), for a charge at a certain point (p, 0) on the surface one can split the surface into an inner disk with radius <= p - ϵ and an outer ring with inner radius > p + ϵ, for a small ϵ (and outer radius 1, of course). Then one can write an equation for the r component of the force (the φ component is zero due to symmetry) which takes into account the 1/d^2 dependence of the force (d being the distance between p and an arbitrary point on the disk) as well as the angle of the force. 2D integration over the inner disk and over the outer ring results in the forces exerted by both, and addition gives the total force. Please see the attached figure 1. Since this procedure involves no term expressing a dependence of charge density on location, it covers evenly distributed charge, and figure 1 refutes this idea: Except at the center, the force isn't zero, so the charge density would change (here, positive values stand for centrifugal forces and vice versa).
If the term to be integrated is expanded by a factor modeling a charge density dependent on r, then the results change accordingly. The correct term, 1/sqrt(1 - r^2) in the case of a unit disk, results in figure 2.
These notes by Ted Bunn might be of interest because they explain how to arrive at the correct term:
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Give an example where the electric field is zero at a point but divergence of the electric field is non zero there?
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Dear Spiros Konstantogiannis, thank you, I agree, it means the identity is independent of the type of gauge.
I just could not remember if the vectorial identity somehow could depend on the type of one of the 2 gauges for the case of electromagnetism.
Best Regards.
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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)?
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I guess that the Heisenberg uncertainty principle for energy and time holds well in experiments where the lifetime of certain quasiparticles can be measured,
so (Δ t)2 square won't give additional information in this case.
Best Regards.
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Could anyone recommend a good textbook to study about Green Function in classical electrodynamics? Thank you.
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For the behavior of refractive space-dependent 3D electromagnetic waves using Green Function formalism in continuous media, the subject was addressed in chapter 6 of the monography:
Methods of Quantum Field Theory in Statistical Physics, (1963) by Abrikosov, Gorkov, and Dzyaloshinski. Dover NY.
Best Regards.
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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.
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Dear Igor Ostanin , I think that you know about MMANA. It is the free EM solver, but non-open source solver of course
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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?
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Nice Dear Joaquin Diaz-alonso
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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?
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Dear Brian:
I am writing to you because I have just stumbled upon your discussion in the video from which I have lifted and commented upon, as follows:
- - - "Hidden Dimensions: Exploring Hyperspace" 15 January 2015, World Science Festival
Brian Greene, Physicist, Columbia University
This is file 20180521 What string theory says about possible vibrations .txt
At this place in the video: https://youtu.be/h9MS9i-CdfY?t=3339
Brian Greene (Columbia University Physicist) says:
"The extra-dimensional shape dictates the possible vibrational patterns ... the possible notes, if you will, that the strings can play ... "
Me: The extra dimensions are the overtone series permitted by all the other kinds of matter in the systems under test, which includes their resonant frequencies at various volumes and shapes. This is what I call "Symphonic Resonance" and insight is given in my results for fragment resonances at particular Wavelengths and Temperatures, under constraints of L% and E% counts (where E% maps the m of E=mc^2). One might think of this as bulk Huygenian Entrainment. It all relates to all the overtone series that lock in at different Wavelengths and Temperatures throughout C.P. Saylor's 1935 concept of the "Zone of Match", which might better be thought of as the "Regions of Symphonic Resonances".
WNYC Journalist John Hockenberry Gets it! "So describing this sort of interactivity, if you're going back to the French Horn Model, by simply looking at what is coming out of the bell, you're going to get one small picture, but you have to understand that there is music and interaction that is directly corresponding to the piece that you're hearing in the concert hall, taking place at every point in the French Horn itself." [And at every point in the Concert Hall! --SL, 21 May 2018]
I have tried but already failed several times to publish my work, which is again under peer review for what I hope will be soon publication.
Thank you for your interesting questions, Nancy! I hope this helps. :-)
In the spirit of the Holiday Season, I am attaching a piece of art that derives from some graphical play with my data.
-Steve-
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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
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Dear Stam,
I'm sorry to say that you don't know what is a magnetic monopole at all and let me to say that Maxwell equations cannot tell anything about this issue as the magnetic (or electric ) dipole. Respect to the different aspects of the magnetic monopole, although is not the best reference let me to recommend you the basic
where you have the topological interpretion.
More explicitally(just two simple references )the magnetic monopole in the Standard Model as I have told you
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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
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If you can solve
1) ∇2 ϕ1 = 0
ϕ1 = { V/2 , - < x < 0 & y = 0
V , 0 <= x < +∞ & y = mx };
2) ∇2 ϕ2 = 0
ϕ2 = { V/2 , 0 < =x < + & y = 0
V , -< x <= 0 & y = -mx },
then ϕ(x,y)=ϕ1 (x-L/2,y)+ϕ2 (x+L/2,y).
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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?
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more precisely:
" The net change regarding momentum and energy exchange with EM field
is identical to SPONTANEOUS emission, isn't it?"
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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.
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Today I realized that due to the "curvature effect" (I do not know the proper English name of the effect for sure) the charge density is larger where the curvature of a metal surface is larger, thus I guess the equation is not valid for a general surface, unless the curvature is uniform.
But I'm still full of uncertainty about this question...
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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?
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Dear Joerg,
The formula is given in a "Collection of problems on classical electrodynamics" by Alekseyev, problem #318, on page 98 (see attached file). There it is asked to calculate intensity of radiation from a homogeneous, charged sphere. In the end of the book there is an answer to this problem (page 225). Also, I found a "tutorial" with detailed solution to this problem on page 12 (see second attached file).
Unfortunately, both files are in Russian. But I hope you can make sense out of formulas.
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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?
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Dear Gotleyb,
Your question is very interesting. I've asked the same question to my teachers during my doctorate and nobody could respond it. And you're right: changes in electric field in electrochemistry changes the electrochemical systems
On the other hand, we have to think about the electrochemical system as a whole: the electric field will induce the ion's mobilities in solution, and the double layer will change its composition. That's why we use a supporting electrolyte, making the electric field between electrodes constant, and the transport number for analyte is so small if compared with electrolyte's transport number, and this analyte does not suffer any influence from electric field.
We have also another question: we cannot assume a electrostatic arguments for electrochemistry (even from steady state experiments such impedance spectroscopy and potentiometry) because the electrical charges in all electrode changes all the time. Maybe some arguments from electrodynamics can help.
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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?
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An observer cannot move at the velocity of light never. This was a worry of Einstein, what would happen for an observer on an electromagnetic wave? Let us to have one imaginary massless observer on the electromagnetic wave, he should see that a magnetic field were arising and disappearing substitud by a complementary electric field. The curious is that this observer cannot distinguish any kind of velocity for everybody and everything was at rest for him, but another massless observer placed in another light beam would be exactly at the velocity of the light with respect to him. Very crazzy behaviour which shows that our hipothesis of assuming a physical observer on an electromagnetic wave is wrong.
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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.
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This is actually a very interesting question and has analogs in kinetic theory and statistical physics. I would look at the wikipedia page on the Einstein relation: https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory)
The proof does a good job of explaining what is going on.
Essentially, to summarize: there is a detailed balance between drift and diffusion currents under equilibrium. Under non-equilibrium steady state, this can be used to derive how the current is related to potential gradients.
The more fundamental version of this is called the fluctuation dissipation theorem.
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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?
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Dear colleague,
inside of metal conductors is free of negative charges just in the statical case when all the free electrons are gathered on the conductor surface.
For the case of direct current we have uniform current density, i.e. uniform distribution of electrons (charges) in conductor's cross-section. For the case of alternating current the distribution of electrons (charges) is determined by skin-effect. Thus, in the presence of a voltage source compensating for Joule losses in the directed motion of electrons, charges inside the conductor will exist.
You may see the corresponding chapters in the courses of Landau or Feynman.
Regards
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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?.
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Dear Rahul,
Your write:
"Is there anyway to describe classical light-matter interaction purely as field-field interaction (theoretically). Becoz particle picture itself is problematic. Here I am referring to particle as localised object with definite momentum."
You raise a very good point here.
You do have an obvious fields vs localized EM particle interaction with the Lorentz equation F= q(E + v x B).
This equation allows establishing a precise trajectory for a single localized electron in straight line when the density of both ambient fields are equal, or on a curved trajectory if the B field is more intense than the ambient E field.
This is often given as an example in intro textbooks to explain the triple orthogonal electromagnetic relation between both fields with respect to the direction of motion of a charge.
Considering that the electron is an electromagnetic particle, it must by definition also have internal electric and magnetic fields corresponding to its mass, and that it would be these internal fields that interact with the ambient fields that can be calculated with the Lorentz equation.
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Coherence of two electromagnetic waves takes place if their phase difference is: (i)Constant in time (ii) Constant in space (iii)Constant in time and space
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iii
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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!
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Hi Halim,
I'm looking forward to the results of this exercise! Please keep me updated. About the DC component in the radiation: Obviously, there can be no radiation with "global" DC but a crucial point of this thought experiment is that there is at most one unipolar pulse in the space between radiating antenna and absorber (distance less than lambda/2); so the radiation has kind of a temporary "local" DC component. (Of course, in this short distance the fields are a mix of radiation and reactive fields.)
The equation for the magnetic field HM is based on Maxwell's equation for curl H, and the rotational symmetry of M. Of course, it could be exactly true near M only if M were a homogeneous cylindrical sheet of current. A ring of discrete current lines would cause a different HM in the immediate neighborhood of the lines, inside M as well as outside.
Best regards
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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.
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Dear Arkady and Stam,
Physics is about exploring Nature, mathematically, to find out what the unknowns are, not about accepting what have been assumed, in terms of the so-called principles. If we just accept what we have learnt then we will not be able to realise that there are many problems in physics that cannot be explained within the present formulation of quantum physics. Arkady, can you clearly explain to me what a quantum of energy is? You also mentioned about simple but not too simple! Basically, quantum mechanics is a wave version of Newtonian physics. Isn't that simple? If you read my work then you will see that I have generalised both of them and derived few equations that I could not solve because they involve Fractional Laplacians. Is that simpler? Furthermore, one of these equations will tell you what a quantum of energy is.  I have tried to seek help but so far no results. If you can help me solve these equations then I will give you a personal reward (you buy anything you want to the maximum of AUS $1,000 and I will pay for it).
Dear Stefano,
Did you mean a mixed potential?
KInd regards,
Vu.
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Are the Integral and Differential forms of Maxwell's Equations position dependent?
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Yes, electrostatic fields depends only on the position while electromagnetic fields depend on both position and time.
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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.
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Maxwell's equations describe the electromagnetic field at the classical level. The full description is quantum electrodynamics. The problem you are referring to is the famous wave-particle duality. The electrodynamic field is a quantum system, for some states, a particle is a better approximation, for some others, waves.
A few photons behave mostly like particles, whereas many photons (coherent states) like waves. The solutions of Maxwell's equations in the one photon case can be used to calculate probabilities of the photon hitting a certain part of the screen after the two slits.
As for propagation in all directions: that is Huygens' principle applied to the wave. But the wave is used to calculate probabilities in this case. For one photon, you cannot tell if it travelled in all directions or just one direction, only that it did hit the screen at a given point. To the amplitudes used to calculate the probability of hitting that point, all possible paths contribute.
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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.
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Fruitful information. Thanks.
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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 Ex2/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 Ex-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.
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Hi Stefano,
I'm not totally familiar with the module of COMSOL that you're using, but, if you can obtain the solution for the transverse components of both the electric and magnetic fields (Ex,Ey,Hx,Hy) then you can compute the z-component of the Poynting vector directly.
Alternatively, if you know both transverse components of the electric field (Ex and Ey), then you can compute the plane-wave spectrum of each of these components (using a Fourier transform). You can then compute all the other magnetic and electric field components from these two plane-wave spectra by invoking Maxwell's equations. The Poynting vector can then be computed. In fact, I think you may be able to get the Poynting vector directly from the squared modulus of the plane-wave spectra via Parseval's Theorem.
Hope this helps:)
Ray
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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
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@Andrew: thank you for your question, i believe you have integrity to ponder the following reasoning. I have asked a new question regarding possibility to discuss a classicla version of QCD, as well as posting my last paper on my view on this issue. But let me say it to you again: yes, classical neutrino is part of my searching toward classical electrodynamics version of QCD. This question went back to a lecture that i followed in 2009, the lecture was about Yang-Mills and QCD. I was unable to follow entire complicated language of QCD, but I recalled that Yang-Mills theory is a classical theory, and since QCD is based on SU(3) Yang-Mills theory, then does it seem possible to look for and express QCD in classical language? I asked this question to my professor, and he answered with a mysterious smile: "Of course." but since then i never found any paper or book discussing this issue, except a book by Kousyakov discussing classical Yang-Mills theory. Only a few days ago i foung two papers by Valerii Temnenko and also one more paper discussing Maxwell equations for Yang-Mills theory. 
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For example if there are two atoms ( A, B) with magnetic momentum, the total magnetic momentum follows √(µA2+µB2 ) or √(µA2-µB2 )?
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I agree with Daniel Baldomir. However, in a collinear magnetic structure Gorter's model is normally used, and the net magnetic moment is a simple vectror sum of the moments of the two sublattices. Low temperature measurements on uniaxial magnets were reported to agree with Gorter's collinear model, and the net magnetic moment was found to be consistent with the difference between spin-up and spin-down moments in a ferrimagnetic material like the hexaferrite for example (For reference, see our publication in J. Appl. Physics, (2013))
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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?
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    They are used as density Lagrangian for obtaining all the equations of motion of Electrodynamics.
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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?
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The eddy current is not purely resistive.
To check if the eddy current is resistive, you should calculate the skin depth (due to the skin effect) of the material subject to eddy current. (look up on the Internet about skin effect.)
I did this for electrical steel, and then I come at a skin depth of 4.2 mm. Its the inductive contribution of the eddy current that causes the skin effect. As electrical steel is usually laminated in thickness of about 0.3 mm, and this is way less than twice the skin depth, you can see that indeed for an electrical machine, neglecting the inductive part is acceptable.
This method is only an approximation, as the (resistive) eddy current distribution is different from the (resistive) current flowing in a conductor wherefor the skin effect is calculated. However, I am quite sure that if you want to calculate eddy current crowding due to inductive part (skin effect), you need to use same calculation techniques that are used for skin effect.
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As we know this equation in electrostatics is based on Coulomb experiments, but this is not obvious in electrodynamics. 
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Dear Alberto Díaz
Thank you for your descriptions. I am interested in to know whether the physicist prior to Maxwell recognized this equation valid in electrodynamics as in electrostatics? If yes, on what basis? If no how did Maxwell extend the law for electrostatics to electrodynamics?
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Please see the attached for details.
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Although I can't prove it, I'm starting to believe that there is no simple expression for either (1) or (3). It looks to me like the simple expression for (2) is very special.  Any comments?
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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. 
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Gentlemen. I'm afraid that much of what I said in my previous posts on this subject was wrong. Thinking about it over the weekend, at last, now I understand. The key point is that the Lienard-Wiechert potentials are retarded i.e. they are not calculated using the position of the source now, but its position when it emitted the EM field that is being measured now. For source  Q1, it makes no difference, because it is stationary (in frame S). For source Q2, it is simple to show that the field felt now by Q1, was emitted by Q2 at time d/ sqrt (c^2-v^2) , when Q2 was at vector position  (-gamma d v/c, d).
Using the formulae in the Wikipedia article, the unit vector from the source                   n= (v/c ,-1/gamma), and 1-n.beta s = 1/gamma ^2. Substitute in to the formula given,   E(r,t) = -Q2 gamma/ (4 pi eps d^2), giving a force exerted by Q2 on Q1 *exactly equal and opposite* to the force exerted by Q1 on Q2. (Note that  here   F = Q1E simply, because Q1 is stationary).
The reason why Sir Isaac is right, in this particular case, is that dA/dt =0: all the electric field is due to Del phi, so there is no field momentum, and mass dilation is only involved in the sense that the instantaneous acceleration of Q2 is lower by a factor gamma than that of Q1, when measured in S, contrary to what I said before!
All this applies to objects with charge, but no intrinsic magnetic moment. For things like electrons, which have  one, things are different.
Thanks for a very interesting discussion (and apologies for my earlier nonsense!)
Adam