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D. Funaro, A Full Review of the Theory of Electromagnetism, (2005). arXiv:0505068. Preliminary version of the book: Electromagnetism and the Structure of Matter, world-scientific link .

We will provide detailed arguments showing that the set of Maxwell equations, and the corresponding wave equations, do not properly describe the evolution of electromagnetic wave-fronts. We propose a nonlinear corrected version that is proven to be far more appropriate for the modellization of electromagnetic phenomena. The suitability of this approach will soon be evident to the reader, through a sequence of astonishing congruences, making the model as elegant as Maxwell’s, but with increased chances of development. Actually, the new set of equations will allow us to explain many open questions, and find links between electromagnetism and other theories that have been searched for a long time, or not even imagined.

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D. Funaro, The Fractal Structure of Matter and the Casimir Effect, Preprint (2009). arXiv:0906.1874v1 .

The zero-point radiation is an electromagnetic form of energy pervading the universe. Its existence is granted by standard quantum theories. We provide here an explanation based on deterministic classical electrodynamics, by associating to particles and nuclei a series of shells, made of constrained photons, with frequencies decaying with the distance. Such photons are part of a pre-existing background, evolving in vacuum even at zero temperature, and are captured by stable subatomic particles to form very distinctive quantized patterns. The evolving shells bring, for instance, to the creation of a fractal-type structure of electromagnetic layers around a conductive body. This property is then used to justify, both qualitatively and quantitatively, the attractive Casimir force of two metal plates. The analysis is carried out by standard arguments, except that here the surrounding zero-point energy is finite and, albeit with a very complicated appearance, very well-organized.

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D. Funaro, Electromagnetic Radiations as a Fluid Flow, Preprint (2009). arXiv:0911.4848v1 .

We combine Maxwell’s equations with Eulers’s equation, related to a velocity field of an immaterial fluid, where the density of mass is replaced by a charge density. We come out with a differential system able to describe a relevant quantity of electromagnetic phenomena, ranging from classical dipole waves to solitary wave-packets with compact support. The clue is the construction of an energy tensor summing up both the electromagnetic stress and a suitable mass tensor. With this right-hand side, explicit solutions of the full Einstein’s equation are computed for a wide class of wave phenomena. Since our electromagnetic waves may behave and interact exactly as a material fluid, they can create vortex structures. We then explicitly analyze some vortex ring configurations and examine the possibility to build a model for the electron.

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C. Chinosi, L. Della Croce, D. Funaro, Rotating Electromagnetic Waves in Toroid-Shaped Regions, International Journal of Modern Physics C, Vol. 21, n. 1 (2010), pp. 11-32. DOI: 10.1142/S0129183110014926, arXiv:1002.1206v1 .

Electromagnetic waves, solving the full set of Maxwell equations in vacuum, are numerically computed. These waves occupy a fixed bounded region of the three dimensional space, topologically equivalent to a toroid. Thus, their fluid dynamics analogs are vortex rings. An analysis of the shape of the sections of the rings, depending on the angular speed of rotation and the major diameter, is carried out. Successively, spherical electromagnetic vortex rings of Hill’s type are taken into consideration. For some interesting peculiar configurations, explicit numerical solutions are exhibited.

We show some animations related to electromagnetic waves, solving the set of Maxwell’s equations and trapped in bounded regions of space having a toroid topology. These vortex rings are of Hill’s type; the complete 3-D geometry is obtained by rotating the following images around the vertical axis. The external boundary turns out to be spherical, while the internal boundary is a ring in the first and the second case, another sphere in the last case. According to the results of the paper, in order to get a stationary outer boundary, the size of the internal boundary cannot be arbitrary, but must be precisely assigned.

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D. Funaro, Numerical Simulation of Electromagnetic Solitons and their Interaction with Matter, J. Sci. Comput., Vol. 45, 1 (2010), p. 259. DOI 10.1007/s10915-009-9338-5 , arXiv:0912.2639v1 .

A suitable correction of the Maxwell model brings to an enlargement of the space of solutions, allowing for the existence of solitons in vacuum. We review the basic achievements of the theory and discuss some approximation results based on an explicit finite-difference technique. The experiments in two dimensions simulate travelling solitary electromagnetic waves, and show their interaction with conductive walls. In particular, the classical dispersion, exhibited by the passage of a photon through a small aperture, is examined.

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D. Funaro, A Lagrangian for Electromagnetic Solitary Waves in Vacuum, Preprint (2010). arXiv:1008.2103v1 .

A system of equations, describing the evolution of electromagnetic fields, is introduced and discussed. The model is strictly related to Maxwell’s equations. As a matter of fact, the Lagrangian is the same, but the variations are subjected to a suitable constraint. This allows to enlarge the space of solutions, including for example solitary waves with compact support. In this way, without altering the physics, one is able to deal with vector waves as they were massless particles. The main properties of the model, together with numerous exact explicit solutions are presented.

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D. Funaro, On the Near-field of an Antenna and the Development of New Devices, Preprint (2012). arXiv:1203.1229v1 .

Dipole antennas have been invented a hundred years ago. Nevertheless, what really happens in their proximity during emission (the so-called near-field) is still an open question and the many explanations put forth are not fully convincing. Subject to specific conditions the signal present on the conductor assumes the properties of a pure electromagnetic wave. We would like to give our point of view on the modality of this transition, with the hope of providing suggestions for ameliorating or projecting new devices.

We provide an animation showing the evolution of the electric field emanated by a classical (non infinitesimal) dipole antenna. These lines, which are the level sets of the tangential component of the electric field (thus, a scalar), should not be confused with the lines of force of the field, which are instead distributed on perfect spherical surfaces. The situation differs from that of the infinitesimal Hertzian dipole, where the Poynting vector is not purely radial.

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D. Funaro, From Photons to Atoms – The Electromagnetic Nature of Matter, Preprint (2012). arXiv:1206.3110v1 .

Motivated by a revision of the classical equations of electromagnetism that allow for the inclusion of solitary waves in the solution space, the material collected in these notes examines the consequences of adopting the modified model in the description of atomic structures. The possibility of handling “photons” in a deterministic way opens indeed a chance for reviewing the foundations of quantum physics. Atoms and molecules are described as aggregations of nuclei and electrons joined through organized photon layers resonating at various frequencies, explaining how matter can absorb or emit light quanta. Some established viewpoints are subverted, offering an alternative scenario. The analysis seeks to provide an answer to many technical problems in physical chemistry and, at the same time, to raise epistemological questions.

Here below there is an animation showing the evolution of a rotating electric field in the section of a cylinder as described in Appendix F (see figure 4.2 at page 250). The magnetic field (not shown) oscillates perpendicularly to the screen. The entire set of Maxwell’s equations is satisfied in this case. At the boundary, the magnetic field is zero and the electric field is tangential, though other type of conditions may be assigned.

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D.Funaro, Charging Capacitors According to Maxwell’s Equations: Impossible, Les Annales de la Fondation Louis de Broglie, Vol. 39 (2014). ISSN 0182-4295arXiv:1412.6005v1 .

The charge of an ideal parallel capacitor leads to the resolution of the wave equation for the electric field with prescribed initial conditions and boundary constraints. Independently of the capacitor’s shape and the applied voltage, none of the corresponding solutions is compatible with the full set of Maxwell’s equations. The paradoxical situation persists even by weakening boundary conditions, resulting in the impossibility to describe a trivial phenomenon such as the capacitor’s charging process, by means of the standard Maxwellian theory.

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D. Funaro, Trapping Electromagnetic Solitons in Cylinders, Mathematical Modelling and Analysis, Volume 1, Issue 1(2014). DOI10.3846/13926292.2014.892904, arXiv:1304.2287v1 .

Electromagnetic waves, in vacuum or dielectrics, can be confined in unbounded cylinders in such a way that they turn around the main axis. For particular choices of the cylinder’s section, interesting stationary configurations may be assumed. By refining some results obtained in previous papers, additional more complex situations are examined here. For such peculiar guided waves an explicit expression is given in terms of Bessel’s functions. Possible applications are in the development of whispering gallery resonators.

Here is the animation relative to Fig.5 of the paper.

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D. Funaro, E. Kashdan, Simulation of Electromagnetic Scattering with Stationary or Accelerating Targets, International Journal of Modern Physics C, Vol. 26, n. 7 (2015), pp. 1-16. DOI: 10.1142/S0129183115500758 , arXiv:1305.5116v1 .

The scattering of electromagnetic waves by an obstacle is analyzed through a set of partial differential equations combining the Maxwell’s model with the mechanics of fluids. Solitary type EM waves, having compact support, may easily be modeled in this context since they turn out to be explicit solutions. From the numerical viewpoint, the interaction of these waves with a material body is examined. Computations are carried out via a parallel high-order finite-differences code. Due to the presence of a gradient of pressure in the model equations, waves hitting the obstacle may impart acceleration to it. Some explicative 2D dynamical configurations are then studied, enabling the study of photon-particle iterations through classical arguments.

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D. Funaro, A Model for Ball Lightning Derived from an Extension of the Electrodynamics Equations,  Proceedings of VI International Conference on ATMOSPHERE, IONOSPHERE, SAFETY, Kaliningrad 2018, ISBN 978-5-9971-0491-7 , arXiv:1806.05555

Ball lightning is an impressive natural electromagnetic phenomenon occurring in atmosphere under suitable circumstances. Its origin, composition and stability issues are a matter of debate, due to presence of many evidences still unexplained. An attempt to provide a model, in alternative to the ones already available, is here presented. The aim is to interpret the phenomenon through a fluid plasma self-trapped in very stable toroid-shaped regions. To this end, a suitable adaptation of the equations ruling electrodynamics is taken into account.

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D. Funaro, High Frequency Electrical Oscillations in Cavities, Mathematical Modelling and Analysis, Vol. 23, n.3 (2018), pp. 345-358, DOI: 10.10386/mma.2018.021 , arXiv:1807.06421 .

If the interior of a conducting cavity (such as a capacitor or a coaxial cable) is supplied with a very high-frequency electric signal, the information between the walls propagates with an appreciable delay, due to the finiteness of the speed of light. The configuration is typical of cavities having size larger than the wavelength of the injected signal. Such a non rare situation, in practice, may cause a break down of the performances of the device. We show that the classical Coulomb’s law and Maxwell’s equations do not correctly predict this behavior. Therefore, we provide an extension of the modeling equations that allows for a more reliable determination of the electromagnetic field during the evolution process. The main issue is that, even in vacuum (no dielectric inside the device), the fast variation of the signal produces sinks and sources in the electric field, giving rise to zones where the divergence is not zero. These regions are well balanced, so that their average in the domain is zero. However, this behavior escapes the usual treatment with classical electromagnetism.

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D. Funaro, Electromagnetic Waves in Annular Regions, Appl. Sci., MDPI, Vol. 10, n.5 (2020), p. 1780, DOI: 10.3390/app10051780.

In suitable bounded regions immersed in vacuum, time periodic wave solutions solving a full set of electrodynamics equations can be explicitly computed. Analytical expressions are available in special cases, whereas numerical simulations are necessary in more complex situations. The attention here is given to selected three-dimensional geometries, which are topologically equivalent to a toroid, where the behavior of the waves is similar to that of fluid-dynamics vortex rings. The results show that the shape of the sections of these rings depends on the behavior of the eigenvalues of a certain elliptic differential operator. Time-periodic solutions are obtained when at least two of such eigenvalues attain the same value. The solutions obtained are discussed in view of possible applications in electromagnetic whispering galleries or plasma physics. 

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D. Funaro, Electromagnetic Fields Simulating a Rotating Sphere and its ExteriorarXiv:2106.05851 .

Vector displacements expressed in spherical coordinates are proposed. They correspond to electromagnetic fields in vacuum that globally rotate about an axis and display many circular patterns on the surface of a sphere. The fields basically satisfy the set of Maxwell’s equations, but enjoy further properties that allow them to be suitably interpreted as solutions of a plasma model that combines electrodynamics with the Euler’s equation for fluids. Connection with magnetohydrodynamics can also be established. The fields are extended with continuity outside the sphere in a very peculiar manner. In order to avoid peripheral velocities of arbitrary magnitude, as it may happen for a rigid rotating body, they are organized to form successive encapsulated shells, with substructures recalling successive ball-bearing assemblies. A recipe for the construction of these solutions is provided by playing with the eigenfunctions of the vector Laplace operator. Some applications relative to astronomy are finally discussed.