I guess I do not understand the point about a DC current making conducting wires ``completely transparent''. Not an objection, just a point I do not understand.

Dear Prof. Leyvraz, thank you very much for your interest!  The idea behind the transparency is: Ideal current sources have infinite impedance. If we had a one-dimensional structure made of electrically very short wires connecting constant current sources, an incident wave with its E component parallel to the axis of the structure would cause only a small additional current on the wires, and the wave would be only moderately scattered.

If we know the properties of the incident wave in advance, we can (in principle) control the current sources in such a way that the additional current in the wires is exactly compensated. In this case the structure does not scatter the wave at all.  A kind of "curtain" made from parallel structures of this kind would be transparent.

Of course, it would be hard if not impossible to exactly implement such a structure but I think it is permissible to employ it in a thought experiment. In the case at hand, I think that complete transparency of M is not a crucial feature but contributes just to the simplicity.

I guess the details of how you reach ``transparency'' are important. If your currents are truly DC and maintained by passive current sources, I would tend to agree with Remi that your purpose will not be reached: high currents iin a cylindrical conductor do not make the conductor transparent.

On the other hand, if, as you seem to suggest, the current sources could be active elements, behaving in a time-dependent manner to keep the cylindrical conductor transparent (or approximately so) then this active behavior introduces an energy source which makes all you arguments questionable.

Summarising, Poynting's theorem is an easy consequence of Maxwell's equations, and your system does not appear to go beyond Maxwell's equations, so that no paradox of the kind you imagine can arise.

Dear Prof. Cornwall, dear Prof. Leyvraz,

I will try to shed some light on the question of transparency from another angle:

If a unipolar pulse of charge and current is propagating along a straight conductor, there is a region containing accelerated charges and a region containing decelerated charges but ideally there is no radiation because the radiation fields from both regions cancel each other.

In my thought experiment, M contains many wires of, say, PEC. These wires are electrically short as well as much shorter than the distance of M to the radiator R. In a first stage, these wires are not connected. Then the incident field will sometimes accelerate and at other times decelerate charged particles in these wires, thereby causing radiation. This radiation is the scattered part of the incident field.

In the second stage, the wires are connected by controlled (ideal, active) current sources. In principle, these sources can be controlled in such a way that for every charged particle accelerated by the incident field there is a charged particle decelerated by the current sources, and vice versa. Since the incident field is known in advance, there is no need for any kind of feedback control. So, similar to the case of a pulse propagating along a wire, the total radiation field is zero, i. e. there is no reflection nor scattering of the incident field but M is completely transparent. Since the reactive E field of M is a multipole field (and M has no radiation field) there is virtually no retroactive effect by M on R. Since energy is neither “lost” by radiation nor converted into heat within the PEC wires, the current sources have to deliver only reactive power. A supply of active power is needed only for the operation of the current sources if their efficiency is less than 1 (which will always be the case), and this energy is turned into heat. One might consider M as some kind of an “active metamaterial”.

In the final stage, a DC current is superimposed on the pure AC current in the second stage. Now M is still transparent but there is an additional transfer of energy to or from the sources in M. In principle, the amount of this additional energy can be made arbitrarily large. It might be considered reactive energy but even for reactive energy the conservation of energy holds, of course. This reactive energy is transferred to or from the static H field caused by the DC current. The crucial point of the paradox is that the absorption of each radiation pulse which is superimposed on the static H field always converts the energy of the pulse into heat but leaves the total H field weaker or stronger by the amount of the additional energy mentioned above, depending on the polarity of the pulse.

I hope this clarifies my intent a little bit rather than complicating matters. I could add some graphics but only after returning from vacation, after the weekend.

@Remi Cornwall:

I agree with your point about the need to invoke quantum effects but I guess there is at least one additional case: If the difference between the classical spatial distribution of the energy of EM waves (sub-wavelength structure with zeros where E and H have zeros) and the QM energy distribution ("smeared" over several wavelengths) becomes noticeable in the measurable results. I think that is so in the case at hand: As I wrote, no problem would appear if we were able to overlook a whole period, but things become paradoxical if the space is limited to less than half a wavelength.

And I agree also with your last sentence: I was afraid nobody would take the time to read 4 or 5 pages, so I tried to compress the description into about 1 page; but obviously, my attempt wasn't quite successful.

Now there is a more detailed version of my text with equations, graphs, and some references attached.

Dear Prof. Fricke,

                              I have a very basic school level question. When we study quantum nanowires isn't it necessary that the energy gets transmitted by mathematical transitivity (please refer to Mallick, Hamburger and Mallick (2016) my RG site) by conductance otherwise there will be an information indeterminacy (Smith (2014), Mallick (2014)) which will impede the quantum mechanics of the system. I am an Econophysicist at best so this is some kind of a Classical Stock and Flow question.

SKM for SKM, NH, SM

Dear Prof. Mallick,

I'm not an expert on nanowires but since there are macroscopic dielectric waveguides (e. g. Goubau line) I guess information (and energy) can be transmitted along nanowires lacking conductance, too. At least with electrically conducting macroscopic wires, most of the energy is transmitted by the fields outside the wire anyway. I hope this answer isn't too far from your point.

I feel I should add some remarks on the described setup:

Obviously, it's about the effect of the superposition of unipolar pulses and a static field. In my opinion, the crucial point in this case is not the entry (through the transmissive M which got nearly exclusive attention so far) but the leaving of the pulse by absorption because this latter process takes place in some distance from the sources of the static field.

To illustrate this, please consider a TEM cell enhanced by an additional wire at the outside of each plate which is running in or against the direction of propagation and is bent beyond the feeding point and the terminating point in such a way as to form a coil whose H field crosses the inside ot the TEM cell parallel or antiparallel to the H component of TEM pulses propagating inside the TEM cell. A current source is feeding a DC current through this wire.

When a unipolar pulse is fed into the TEM cell whose H component weakens the static field, the weakening of the static field induces a voltage in the coil so that the energy lost in the total field is transmitted to the current source. When the pulse reaches the terminating point the energy of the static H field is restored while this increase of the H field induces a voltage in the coil which causes the current source to provide the energy for restoring the H field.

So, in this case the energy is balanced at all times.

In the case of the cylindrical setup containing R, M, and A, the same happens when a pulse passes M thereby entering the static H field: Some energy is transferred between the current sources in M and the static field. But the absorption of the pulse takes place in a distance of nearly lambda/4 from the current sources. In the framework of classical physics, there seems to be no way of transmitting the necessary energy between the current sources and the location of absorption immediately. Additionally, the retransmission of the necessary energy would require a complete reflection of the pulse instead of absorption.

Of course, this whole thing feels as if I made a stupid mistake somewhere, and this feeling was, and is, the motivation for asking for your opinion.

Dear Dr. Fricke,

If I have well understood, the paradox is about violating energy conservation principle.

I suggest this paper (Section 4):

https://www.researchgate.net/publication/242077693_Multi-Layer_Crystals_of_Metallic_Wires_Analysis_of_the_Transmission_Coefficient_for_Outside_and_Inside_Excitation

It is shown that the line source, or the plane wave source is not matched to free space. Thus, adding parasitic structures can increase the radiated power by making the source more matched to the space. The problem is that the maximum of power is not radiated from the beginning. Then it is logic that by changing the source (or its environment) we can radiate more energy.

The ideal line source or ideal plane wave sources are the worst receiving antennas because a wave can pass through them without being absorbed. Thus, there are also the worst transmitting antenna.

Article Multi-Layer Crystals of Metallic Wires: Analysis of the Tran...

Dear Dr. Boutayeb,

thank you very much for the reference to your stimulating and well readable paper! Intuitively, I was surprised by the high values of Zs in Fig. 29.

You are right, the paradox is about energy conservation. In order to avoid any misunderstanding: I do not suggest to have found a perpetuum mobile ;-) but am looking for a way to reconcile the presented setup with the principle of energy conservation.

The main reason for choosing a cylindrical form was to avoid any modification of the power flow from the radiator to the source of the static H field by this static field; a byproduct is, of course, that with infinite length the problem becomes one-dimensional.

Admittedly, in the current form the setup has no practical use but is just a thought experiment; therefore I think the bad performance of the line radiator is tolerable. If it ever came to an implementation, the UWB character of the radiation might pose a problem. In preliminary simulations I used a fundamental frequency of 67.8 MHz, and took into account harmonics up to about 4.2 GHz. Do you think that under these conditions an improvement by parasitic structures were still possible?

BTW, I learned another lesson from your paper: While I tried in this thread by various ways to make the existence of an object acceptable which simultaneously is transparent and carries a current, you handled the problem in section 4.1.1 with five words: "The source is considered transparent ..."; I will do it your way the next time. :-)

I would very much appreciate any of your further thoughts on the paradox.

Hi Dr. Fricke,

I have also analyzed a cylindrical form of the problem here:

https://www.researchgate.net/publication/4060609_Cylindrical_periodic_structures_of_metallic_wires

https://www.researchgate.net/publication/3018431_Analysis_and_Design_of_a_Cylindrical_EBG-Based_Directive_Antenna

You can see "transmission coefficients" with amplitudes larger than one. This is because the ideal source (transparent source) is not matched to free space from the beginning.

Conference Paper Cylindrical periodic structures of metallic wires

Article Analysis and Design of a Cylindrical EBG-Based Directive Antenna

Dear Dr. Fricke,

I have read your paper

On Divergence in Radiation Fields Toward highly focused neuron stimulation for neurology and HCI

I am really impressed by your theory of enhanced Maxwell's equations and the wide range of knowledge that is used. Proposed applications and the setup for the demonstration are also very interesting. I think I will need time to understand it very well.

Best regards

Hi Dr. Boutayeb,

please excuse my delayed answer, I was offline for some days.

Thank you very much for the links to your papers on cylindrical structures! Detailed reading will take some days but after a quick "scan" I'm certain to find many helpful particulars.

Thank you also for your kind remarks on my paper "On divergence ...". I guess its subject looks a little bit strange at first glance, and so I'm delighted that you were not deterred from reading. However, while thought experiments A and B are correct in my opinion, C doesn't hold water yet (as I discovered only some days ago). When I formulated the paradox this thread is about, I was still looking for the "real thing"; hopefully, I found it some days ago.

So, should you find errors or ambiguities in the paper, or if you feel some aspects should be and could be elaborated please do not hesitate to send me a notice. Since this might lead too far from the subject of this thread, you could reach me also at [email protected].

All the best

Dear Professors Fricke and Boutyab,

                                                                  At the cost of privacy in commissioned research to a certain extent I would just like to add that if you would like to create quantum wells out of quantum nanowires you will have to describe a process of geometric tranformation which is almost self evident (some work I have done for Prof. Egil Tjaland of NTNU with Reservoir Engineering Applications and is ongoing and has been published in Journal of Physics D and submitted to the Royal Swedish Academy of Sciences) and has been done  as an application of the Dbranes String Quantum Mechanical Method developed by us (see the paper on String Theory and Econophysics of Coal Mining for example) in the SO(32) Heterotic String Matching (Genetic Algorithmic) Quantum Field developed by us (Mallick, Hamburger, Mallick (2016)) contains a convolution of the continuum mechanics of quantum wells  mechanics by using Dbranes String Theory and genetic quality utility and Arrow of Time postulates to result in interaction of information and energy in resultant efficient general topological structures solvable by differential topological manifold triangulations, designing quantum wells or quantum reservoirs by statistical ensemble stock and flow transitivity and jump conductance. This entire geometric convolution of Dranes String Homologous Statistical spaces can be reduced into six meta step algorithms (like time and motion experimental studies). The convolutions can have a meta heuristic ensemble definition (H(1)) consisting of blasting potentials as in Coal Miming Systems Applications (the above papers) (Ingelman (1982) or the Alfred Nobel Dynamite Sticks Lesson or infact the Big Bang Evidence stylization).

SKM for SKM, NH, SM

I know that there is in internet a software which can write a document similar to a scientific paper (including equations and references). The software just need the name of the authors and different keywords (for example: economy, utility, Pareto, Arrow, string theory, quantum, general relativity, Alfred Nobel, genetic algorithm, etc.).

Administrators of Researchgate should check these types of papers in this website.

We completely deplore such attempts. However R&D spillovers has a very well documented research agenda in Economics at least which in a different paper (paper on NEP listed on RG site also) one of us has collaborated on and we have shown that there is a scientific preference (although weak) across a broad spectrum of research institutions all over the world for categorised and competitively published and classified research output. It goes against the research literature which shows that R&D spillovers give rise to second best allocations that also if you know the identity of the researchers. We stand by your stand.

SKM for SKM, NH, SM

Hi Dr. Fricke,

I suggest you to check eq. (7) in the document with graphs. You use the DC current IM and the radiated electric field to calculate the energy. Since the DC current is continuous and constant, it is always there and according to eq. (7) this will be interpreted as localized energy. This is not correct I think:  energy due to the DC current should be considered and subtracted when analyzing the entropy of the system. I also think that a correct formulation for the energy should use both electric and magnetic fields.

Best regards

Hi Dr. Boutayeb,

many thanks for working through the long version of the description and pointing out a problem with eq. (7)! I'm not sure whether I understand your points completely; so I will try to clarify some aspects:

The current IM contributes to the energy of the system, of course, by causing the magnetic field HM which is considered in eq. (8), superimposed by the magnetic component of the pulse in the first integral, and exclusively in the second integral. Since the observation is limited to the space between rho = 13/64 lambda and rho = lambda/2, the upper limit of the second integral is lambda/2 as well, but HM, and the energy connected to IM, reaches beyond lambda/2, of course.

While eq. (6) and eq. (8) deal with the energy of both electric and magnetic fields, eq. (7) describes the energy transfer to or from the current sources in M according to WM = U * IM * t. Since the length of the setup in z-dimension under observation is 1 m, the integral in (7) gives the value of U * t. I think WM is indeed localized while the energy of HM is not.

I agree that by subtracting the energy of HM from the actual energy Wo (eq. (8)), one would see more directly whether the superposition of a pulse causes an increase or decrease of the field energy; I just thought that would become apparent by the graphs of Wo.

When the energy of a pulse is turned into heat by A, the entropy of the system is increasing. But do you consider it favorable to describe the process in terms of entropy? If so could you please explain why?

Does this cover your objections, or did I miss something?

Thanks again and best regards

Hi Dr. Fricke

I think the formula UIT is only valid for electrostatic. Please see the section generalisation to electrodynamics:

https://en.wikipedia.org/wiki/Electric_potential

also you miss the ac current due to the electromagnetic field. Energy calculated using electric and magnetic field is the only way to avoid error.

You can also represent a circuit model of the problem replacing the Radiated field by ac current (without resistance) and IM by dc current (also without resistance).

Best regards 

Hi Dr. Boutayeb,

I guess we are talking about different details, and I wish we could have half an hour at the same table with paper and pencil between us: I'm sure the problems would vanish. :-)

I think P = U * I is valid in every case, with U (voltage) and I being the momentaneous values (doesn't work with effective values or amplitudes, of course). Obviously, the multiplication by t in W = P * t has to be replaced by integration over t if U and/or I are time-dependent.

The Wkipedia article you refer to deals with potentials, and these are most helpful in finding the fields of a given charge and current distribution; but in the case at hand we know E and H already (eq. (1) and (2) resp. (4) and (5)), and those equations satisfy the Maxwell equations identical.

I'm at a loss how you would use the magnetic field to calculate the energy change in the sources of M. This energy change is effected via the charge carriers in the short conductors between the sources, so the basis is the Lorentz force. But according to the Lorentz force there can be no energy transfer to or from a carrier moving in a B field (exclusively). (And the effect of a changing B field is accounted for by the accompanying E field.) Could you please provide a formula or a link to a formula which uses both fields in order to calculate the energy of charge carriers?

I agree that I'm neglecting the AC current(s). The reasoning behind this is: For the moment, assume DC current = 0. There are two causes of AC currents in M: the incident field and the AC current provided by the current sources in M. The prerequisite for transparency of M is that both AC currents compensate each other in such a way that the reflectivity of M is zero. According to the Maxwell equations, a non-radiating system cannot exchange energy with an incident field, so there is no total turnover of energy during a period of the incident field. But without DC current, the energy exchange during the incidence of a halfwave is independent of the polarity of this halfwave. So, there is even no total turnover of energy during half a period. Since the energy exchange is a linear function of the current in M, the AC currents still don't contribute when the DC current 0.

The simplest circuit model of R and M which comes to mind were an AC voltage source R for the radiation and a DC current source M for the modulator. If voltage and current at M are parallel => energy transfer from R to M, and vice versa. But the energy transfer in this circuit differs crucially from that in the 3D setup: In the circuit, M really receives energy from R, while in the 3D setup the radiation (and hence its energy) is not altered by M. Instead, the energy received by M is (temporally) lost by the static field HM, and this is the basis for the paradox that finally occurs when the radiation is absorbed by A.

Do you plan to visit Europe within the next months? I would dearly love to implement the meeting outlined at the outset. ;-)

Best regards

Hi Joerg,

Unfortunately, I am too busy with my projects. However, I am intending to propose this exercise as a project  for my students in my FDTD course (they will have to write a code in Matlab). The FDTD (Finite Difference Time Domain) method is an exact method based on the direct discretization of Maxwell's equations in time and space domains. The next session for the course will be in September 2017. I think they can do the analysis with either cylindrical waves or with plane waves. With plane wave, they may get a better visualization. For this case, they will use Floquet Boundary conditions. For the RF excitation, they may use a Gaussian pulse or a Gaussian-derived pulse (to remove the DC from the RF excitation). The objective of this exercise will be to verify your statement.

Best regards

I have just noticed that there is also a continuity problem with your magnetic field due to the dc current. I am not sure that applying Biot and Savart law to a ring of current lines will give the distribution that you present.

Best regards 

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