Direct derivation of Lienard Wiechert potentials, Maxwell's equations and Lorentz force from Coulomb's law

In 19th century Maxwell derived Maxwell equations from the knowledge of three experimental physical laws: the Coulomb's law, the Ampere's force law and Faraday's law of induction. However, theoretical basis for Ampere's force law and Faraday's law remains unknown to this day. Furthermore, the Lorentz force is considered as experimental phenomena, the theoretical foundation of this force is still unknown. To answer these fundamental theoretical questions, we derive Lienard Wiechert potentials, Maxwell's equations and Lorentz force from two simple postulates: (a) when all charges are at rest the Coulomb's force acts between the charges, and (b) that disturbances caused by charge in motion propagate away from the source with finite velocity. The special relativity was not used in our derivations nor the Lorentz transformation. In effect, it was shown all the electrodynamic laws, including the Lorentz force, can be derived from Coulomb's law and time retardation. This was accomplished by analysis of hypothetical experiment where test charge is at rest and where previously moving source charge stops at some time in the past. Then the generalized Helmholtz decomposition theorem, also derived in this paper, was applied to reformulate Coulomb's force acting at present time as the function of positions of source charge at previous time when the source charge was moving. From this reformulation of Coulomb's law the Lienard Wiechert potentials and Maxwell's equations were derived. In the second part of this paper, the energy conservation principle valid for moving charges is derived from the knowledge of electrostatic energy conservation principle valid for stationary charges. This again was accomplished by using generalized Helmholtz decomposition theorem. From this dynamic energy conservation principle the Lorentz force is derived

Numerical comparison of compound and extracted eye models for high frequency dosimetry

This paper compares the numerical results for the induced electric field, the specific absorption rate (SAR), and the corresponding temperature increase in two detailed models of the human eye. The first model features the human eye placed in the free space, while the second one incorporates the eye model in the realistic head model obtained from the magnetic resonance imaging (MRI) scans. The electromagnetic model is based on the hybrid FEM/BEM formulation for the biological tissue, whereas the thermal dosimetry model is based on the bioheat equation solved by using the finite element method. The preliminary analysis showed a similar distribution of the induced electric field along the pupillary axis obtained in both models, however, the numerical results for the SAR and related temperature increase showed discrepancy between the two models, which can be attributed to the high values of induced field in the corneal and scleral regions obtained in the compound eye model

On the Applicability of Numerical Quadrature for Double Surface Integrals at 5G Frequencies

The human exposure assessment to wireless communications systems including the fifth generation (5G) mobile systems is related to determining the specific absorption rate (SAR) or the absorbed power density (APD). The assessment of both quantities requires the use of various numerical techniques, including moments method (MoM). As the use of MoM results in a fully populated system matrix, a tremendous computational cost is incurred, both in terms of matrix fill time and memory allocation, as the matrix size is directly related to frequency of the problem. This paper investigates the applicability of numerical integration at frequencies related to 5G. The novelty of this work is related to the comprehensive set of tests of various combination of source and observation triangles using the developed unit cube test. A number of convergence tests were performed to investigate the effects of the increasing frequency and the discretization scheme on the numerical solution, as well as to determine how to curb the computational requirements by the proficient use of numerical integration. The results show that in the lower GHz range, lower integration orders could be used, resulting in the decrease of matrix fill time without loss of solution accuracy

Analysis Method for the Heating of the Human Eye Exposed to High Frequency Electromagnetic Fields

The paper studies the thermal rise in the human eye caused by time harmonic electromagnetic waves. An eye has been illuminated by a high frequency plane wave with powerdensity 5.0 mW/cm2. Such a problem has been considered as an electromagnetic scattering problem since part of EM energy is transmitted to the eye and part of it is reflected. The total electric field inside an eye and related Specific Absorption Rate (SAR) has been calculated in a frequency range from 0.7 to 4.4 GHz via a hybrid BEM/FEM approach. Knowing the SAR distribution inside the eye provides the calculation of related temperature rise in the human eye due to high frequency radiation by solving Bio-Heat Transfer Equation via standard finite element method

A Study on the Use of Compound and Extracted Models in the High Frequency Electromagnetic Exposure Assessment

The paper presents the numerical results for the induced electric field in the various models of the human eye and the head. The comparison between the extracted or the single organ models and the compound organ models placed inside realistic head models obtained from the magnetic resonance imaging scans is presented. The numerical results for several frequencies and polarizations of the incident electromagnetic (EM) plane wave are obtained using the hybrid finite element method/boundary element method (FEM/BEM) formulation and the surface integral equation (SIE) based formulation featuring the use of method of moments, respectively. Although some previous analysis showed the similar distribution of the induced electric field along the pupillary axis obtained in both eye models, this study showed this not to be the case in general. The analysis showed that the compound eye model is much more suitable when taking into account the polarization of the incident EM wave. The numerical results for the brain models showed much better agreement in the maximum values and distributions of the induced surface field between detailed models, while homogeneous brain model showed better agreement with the compound model in the distribution along selected sagittal axis points. The analysis could provide some helpful insights when carrying out the dosimetric analysis of the human eye and the head/brain exposed to high frequency EM radiation

The Wave Equation for a Moving Source and a Moving Receiver

The ordinary 3D wave equation for nondissipative, homogeneous, isotropic media admits solutions where the point sources are permitted to move, but as shown in this paper, it does not admit solutions where the receiver is allowed to move. To overcome this limitation, a new wave equation that permits both the receiver and the source to move is derived in this paper. This new wave equation is a generalization of the standard wave equation, and it reduces to the standard wave equation when the receiver is at rest. To derive this new wave equation, we first mathematically define a diverging spherical wave caused by a stationary point source. From this purely mathematical definition, the wave equation for a stationary source and a moving receiver is derived, together with a corresponding free-space Green function. Utilizing the derived Green function, it is shown that unlike the standard wave equation this new wave equation also permits solutions where both the receiver and the source are permitted to move. In conclusion, this paper demonstrates that, instead of an ordinary wave equation, the wave equation for a moving source and a moving receiver governs the waves emitted by moving point sources and received by moving receivers. This new wave equation has possible applications in acoustics, electrodynamics, and other physical sciences

Galilean non-invariance of Maxwell equations revisited

It is universally accepted that Maxwell equations do not remain invariant under the Galilean transformation. This conflicts the principle of relativity which states that the physical law must remain invariant in the mathematical form in all inertial frames of reference. For this reason, the Lorentz transformation is invented, and the Galilean transformation is nowadays superseded by the Lorentz transformation. However, this paper challenges this widely held belief that the Maxwell equations are not invariant under the Galilean transformation. By applying the Galilean transformation to Lienard-Wiechert electromagnetic fields it is mathematically proven that the Maxwell equations indeed remain invariant under the Galilean transformation. In addition, the critical error in Lorentz's proof of Galilean non-invariance of Maxwell equations is pointed out, and  it turns out that Lorentz's conclusion that Maxwell equations are not invariant under the Galilean transformation is the result of a mathematical error.</p

A Method for Determining the Envelope of Induced Electric Field on a Simple Human Head Model by Peaks Detection

This paper is on the use of a hybrid boundary element method/finite element method (BEM/FEM) to determine the induced electric field in spherical human head models exposed to high-frequency plane electromagnetic (EM) wave. The geometrically simplified models include the homogeneous one and the non-homogeneous one, featuring compartments such as skin, skull, CSF, and brain. Both models are illuminated by plane EM wave at frequencies including 900 and 1800 MHz, 3500 MHz pertaining to 5G communication systems and also 6000 MHz representing the transition frequency related to the EMF safety standards. The numerical results for the electric field induced in both human head models are presented, while the emphasis is on the electric field along the propagation axis. The novelty of this work is related to the subsequent post-processing of the sampled induced field along the model axis by using two different numerical filtering techniques. It is shown that using the peak detection algorithm, the spline interpolation could be used to estimate the signal envelope. The exponentially decaying nature of the envelope allows to assess the penetration depth of the EM radiation within the biological tissue. Moreover, it is shown that the analytically calculated penetration depth, derived for the unbounded medium, is well reproduced by the numerical computation of EM field

Hybrid FEM/BEM for human heads exposed to high frequency electromagnetic radiation

In this paper the hybrid finite element/boundary element method (FEM/BEM) is used to analyze the human head exposed to plane wave radiation. The radar cross-section (RCS) of a coated sphere at interior resonance frequency obtained using the proposed method showed an excellent agreement with the analytical Mie series solution, thus verifying the validity of the formulation. High frequency electromagnetic wave incident on the human head representing an unbounded scattering problem is formulated via the Stratton-Chu expression, while the interior domain is governed by the vector Helmholtz equation. Some computational examples for the induced electric field are presented. Numerical results showed the highest values of the induced fields around the ocular region and the nose. The presented method is found to be useful in the assessment of the induced fields in the anatomically realistic human models.Peer reviewe