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Classical Electricity And Magnetism
Abstract
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... where m denotes the rest mass of the particle, c is the speed of light and Q the particle charge [11]. The mechanical momenta are denoted by p M = mγv, where v is the particle speed and γ = 1/ » 1 − v 2 /c 2 is the Lorentz factor. ...
... A further simplification can be achieved noticing that P z is decoupled from the other dynamical variables, so that its computation can be neglected if we are only interested in the dynamics of the transversal variables, reducing the number of equations (11) to the four ones associated to X, Y , P x and P y . ...
... In the case of particle motion inside a magnetic quadrupole, the ODE system is given by (11). The magnetic vector potential is written in the form (31) and, in many practical applications, only its sampled values at equally spaced locations in Z are available. ...
Magnetic quadrupoles are essential components of particle accelerators like the Large Hadron Collider. In order to study numerically the stability of the particle beam crossing a quadrupole, a large number of particle revolutions in the accelerator must be simulated, thus leading to the necessity to preserve numerically invariants of motion over a long time interval and to a substantial computational cost, mostly related to the repeated evaluation of the magnetic vector potential. In this paper, in order to reduce this cost, we first consider a specific gauge transformation that allows to reduce significantly the number of vector potential evaluations. We then analyze the sensitivity of the numerical solution to the interpolation procedure required to compute magnetic vector potential data from gridded precomputed values at the locations required by high order time integration methods. Finally, we compare several high order integration techniques, in order to assess their accuracy and efficiency for these long term simulations. Explicit high order Lie methods are considered, along with implicit high order symplectic integrators and conventional explicit Runge Kutta methods. Among symplectic methods, high order Lie integrators yield optimal results in terms of cost/accuracy ratios, but non symplectic Runge Kutta methods perform remarkably well even in very long term simulations. Furthermore, the accuracy of the field reconstruction and interpolation techniques are shown to be limiting factors for the accuracy of the particle tracking procedures.
... The 'integration over a simultaneity space slice' in (4.2) follows the insight of Fermi [22], who used this setup in his solution to the infamous 4/3 problem (see [23,24,25,26,27,28,29,30] for discussions of the 4/3 problem). In the limit of small inertial angular velocity, |ω e | ≪ 1, a MacLaurin expansion of (4.3) in powers of |ω e | gives the expected result ...
... While there is hardly any literature on the Abraham model with spin, the Abraham model without spin is discussed in most major texts since Abraham [2], e.g. [13,24,26,27,29,81], and also in the more leisurely text of Peierls [65]. However, none of these texts makes the elementary observation that the initial value problem for the purely electromagnetic Abraham model(s) is singular in the sense explained above, nor that Abraham's 'evolution' equation (A.113) preserves this singular structure in time; some of the main conclusions drawn in [26,13,24,27,81,29] are even in open conflict with the singular nature of Abraham's evolution equations. ...
... [13,24,26,27,29,81], and also in the more leisurely text of Peierls [65]. However, none of these texts makes the elementary observation that the initial value problem for the purely electromagnetic Abraham model(s) is singular in the sense explained above, nor that Abraham's 'evolution' equation (A.113) preserves this singular structure in time; some of the main conclusions drawn in [26,13,24,27,81,29] are even in open conflict with the singular nature of Abraham's evolution equations. A re-assessment of these traditional treatments of the Abraham model seems to be called for. ...
A new, relativistically covariant, massive Lorentz Electrodynamics (LED) is presented in which the bare particle has a finite positive bare rest mass and moment of inertia. The particle's electromagnetic self-interaction renormalizes its mass and spin. Most crucially, the renormalized particle is a soliton: after any scattering process its rest mass and spin magnitude are dynamically restored to their pre-scattering values. This guarantees that ``an electron remains an electron,'' poetically speaking. A renormalization flow study of the limit of vanishing bare rest mass is conducted for this model. This limit yields a purely electromagnetic classical field theory with ultra-violet cutoff at about the electron's Compton wavelength! The renormalized limit model matches the empirical electron data as orderly as one can hope for at the level of Lorentz theory. In particular, no superluminal equatorial gyration speeds occur.
... In the traditional "AT relativity" approach with the synchronous definition of length and the Lorentz contraction (see, e.g., [21]) the charges on all sides are supposed to be zero in the rest frame of a loop with current. In the IFR in which the loop with current is moving it is found that Q ′ AB = Q AB , due to the Lorentz contraction, and that the charge on the EF side is "-" of that one on the AB side. ...
... Furthermore, it is obtained that the charges on the vertical sides of a loop with current are zero for both the loop at rest and moving loop, since for vertical sides there is no Lorentz contraction. Such results obtained in the common approach led the physics community (I am not aware of any exception) to conclude that there is an electric moment P for a moving loop with current, (see [21], Eq. (18-58)). The appearance of this "relativistic" effect and its consequences are discussed in numerous papers and books. ...
... Using this result and the covariant definition of charge (10) the expressions (12) for the current density 4-vectors of a CCC were found in the ions' rest frame. Then the 4-vectors E α and B α for a CCC are determined by means of the known F αβ (15) and the relations (20) and (21), which connect the F αb and E α , B α formulations of electrodynamics. This yields Eq. (22), which is one of the main results found in this paper. ...
In this paper the physical systems consisting of relatively moving subsystems are considered in the "true transformations relativity." It is found in a manifestly covariant way that there is a second-order electric field outside stationary current-carrying conductor. It is also found that there are opposite charges on opposite sides of a square loop with current and these charges are invariant charges.
... Radiation of the long-range electromagnetic waves, emitted by a classical electric dipole, is well known. Derivation can be found in almost any textbook on electrodynamics [2,6,7,17,18]. The main assumptions used in the derivation of these results are ...
... which simplifies to old classical expression (18) under assumption λ ≫ d taken for classical charge. This current, similar to its classical counterpart (18), satisfies the charge conservation law for the upper charge. ...
... which simplifies to old classical expression (18) under assumption λ ≫ d taken for classical charge. This current, similar to its classical counterpart (18), satisfies the charge conservation law for the upper charge. In effect, (19) represents AC current wave in space and in time, "flowing" or traveling in the direction of positive charge. ...
In this work by using the assumptions that wavelength is much smaller than charge separation distance of an electric dipole, which in turn is much smaller than a distance up to the point of observation, the new results for radiation of an electric dipole were obtained. These results generalize and extend the standard classical solution, and they indicate that under the above assumptions the electric dipole emits both long-range longitudinal electric and transverse electromagnetic waves. For a specific values of the dipole system parameters the longitudinal and transverse electric fields are displayed. Total power emitted by electric and electromagnetic waves are calculated and compared. It was shown that under the standard assumption of charge separation distance being much smaller than wavelength: a) classical solution correctly describes the transverse electromagnetic waves only; b) longitudinal electric waves are non-negligible; c) total radiated power is proportional to the fourth degree of frequency and to the second degree of the charge separation distance; d) transverse component of our solution reduces to classical solution. In case wavelength is much smaller than charge separation distance: a) the classical solution is not valid and it overestimates the total radiated power; b) longitudinal electric waves are dominant and transverse electromagnetic waves are negligible; c) total radiated power is proportional to the third degree of frequency and to the charge separation distance; d) most of the power is emitted in a narrow beam along the dipole axis, thus emission of waves is focused as with lasers.
... where we recalled that I UH = [0, π/2] ∪ [3π/2, 2π] and introduced the total length L = 2ζ traveled by the particle, thus obtaining that Eq. (85) is a half of the standard result [73] because we are restricted to the upper hemisphere. For full details, please see Appendix D. Now, we will carry out the integrals over ϕ 1 for the interesting case when the parallel VC radiation presents total internal reflection at the upper medium, which is of practical interest for waveguides. ...
... where terms of order ∆ 4 Θ have been neglected in the second step. We appreciate that Eq. (108) is almost the remaining half of the well-known result of the standard VC radiation energy [73], which is reasonable due to our integration over the other half of the whole space. Nonetheless, there is a missing half of the energy at the lower hemisphere, because in the limit ∆ Θ = 0, the sum of Eqs. ...
... Nonetheless, there is a missing half of the energy at the lower hemisphere, because in the limit ∆ Θ = 0, the sum of Eqs. (105) and (108) must reproduce the standard VC radiated energy [73]. Despite the absence of cylindrical and conical surface waves due to constant Fresnel transmission coefficients, the integrals of the electric field (35), (36), (38) and (42) exhibit a pole when rewritten using the Sommerfeld identity (B10), owing to the presence of k x,1 in the denominator. ...
... Then the net force on the charge is calculated by an integration over the whole sphere. However, the calculations could be performed much more conveniently in the instantaneous rest frame of the charged particle, where the leading term in self-force, in the approximation of small radius of the electron (ò → 0), turns out to be m v 4 3 0 , with v as the non-relativistic acceleration [6][7][8]. ...
... In another approach, purely within classical electromagnetism itself, to do away with this apparently extraneous factor of 4/3, modifications were proposed in the standard definition of the electromagnetic field energy-momentum, purportedly to make them relativistically covariant even when the unbalanced electromagnetic forces due to charges may be present [10,[14][15][16]; modifications that have made appearance even in standard text-books of electromagnetism [6,7]. For instance, for a charge moving with a velocity v = βc with a corresponding Lorentz factor ( ) g b = --1 2 1 2 and having electromagnetic fields E and B, the energy and momentum in the modified definition are given by the volume integrals (see Jackson [6] as well as [16][17][18] ...
... It has been said that this 4/3 factor in momentum appears only in the so-called 'bound' fields, associated with electric charges, and not in 'free' fields representing electromagnetic radiation [7,40,41] and the modified definitions of electromagnetic energy-momentum, accordingly, have been proposed for the bound fields alone, to eliminate this unwanted factor of 4/3. The conventional wisdom thus is that the energy-momentum of free electromagnetic radiation, contained in a volume having no electric charges or currents, transforms as a 4-vector, without any such 4/3 factor [7]. ...
The electromagnetic energy-momentum of a moving charged spherical capacitor may be calculated by a 4-vector Lorentz transformation from the energy in the rest frame. However, energy-momentum of the moving system computed directly from electromagnetic fields yields extra terms; in particular a factor of 4/3 in momentum appears, similar to that encountered in the classical electron model, where this enigmatic factor has been a source of confusion for more than a century. There have been many attempts to eliminate this `unwanted' factor, noteworthy among them is a modification in electromagnetic field energy-momentum definition that has entered even standard textbooks. Here it is shown that in a moving charged spherical capacitor, such a factor of 4/3 in the electromagnetic momentum arise naturally from electromagnetic forces in system or equivalently from terms in the Maxwell stress tensor; contributions that do not otherwise show up in 4-vector transformations. A similar factor of 4/3 in the momentum of a perfect fluid comprising a randomly moving ultra-relativistic gas molecules or an isotropic photon gas, filling an {\em uncharged} spherical capacitor in motion, appears owing to the contribution of pressure. Thus, genesis of the ``enigmatic'' factor of 4/3 can be traced to pressure or stress whose presence in the system may even be of non-electromagnetic origin and where the proposed modifications in energy-momentum definition do not even come into picture, implying there is no justification in modifying the standard definition of electromagnetic energy-momentum.
... where we recalled that I UH = [0, π/2] ∪ [3π/2, 2π] and introduced the total length L = 2ζ traveled by the particle, thus obtaining that Eq. (82) is a half of the standard result [65] because we are restricted to the upper hemisphere. For full details, please see Appendix D. Now, we will carry out the integrals over ϕ 1 for the interesting case when the parallel VC radiation presents total internal reflection at the upper medium, which is of practical interest for waveguides. ...
... where terms of order ∆ 4 Θ have been neglected in the second step. We appreciate that Eq. (105) is almost the remaining half of the well-known result of the standard VC radiation energy [65], which is reasonable due to our integration over the other half of the whole space. Nonetheless, there is a missing half of the energy at the lower hemisphere, because in the limit ∆ Θ = 0, the sum of Eqs. ...
... Nonetheless, there is a missing half of the energy at the lower hemisphere, because in the limit ∆ Θ = 0, the sum of Eqs. (102) and (105) must reproduce the standard VC radiated energy [65]. At this stage, we do not know whether the missing half is converted into non propagating radiation or there is a fundamental reason for this result. ...
Our study delves into the modifications observed in Vavilov-\v{C}erenkov radiation when its generating charged particle moves parallel to an interface formed by two generic magnetoelectric media, focusing on topological insulators. We compute the electromagnetic field through the Green's function. Applying the far-field approximation and the steepest descent method, we derive analytical expressions for the electric field, revealing contributions from spherical and lateral waves with topological origins. Subsequently, we analyze the angular distribution of the radiation, particularly focusing on parallel motions in close proximity to the interface. Our findings indicate that the radiation along the Vavilov-\v{C}erenkov cone is inhomogeneous and asymmetric. We analyze the radiated energy at both sides of the interface. Finally, we discuss the particle's retarding force, which is notably enhanced in the ultrarelativistic regime. We illustrate these results for the topological insulator TlBiSe and the magnetoelectric TbPO.
... Usually these solutions are given by the so-called Jefimenko's equations. However, it will be shown how by following a similar method as in Ref. [11] for the far field approximations, it is possible to express the electric field mostly in terms of the current density. Next we will discuss some of the efforts that have been made in order to represent the EM field created by and oscillating charge. ...
... The academic problem of describing the E and B fields generated by a charged particle that moves along a given trajectory u(t), which are called the Liénard-Wiechert fields, has been studied in many books [6,7,10,15,16,11]. The basic idea is to consider a charge density ρ and an electric current j given by: ...
... And from here there are many ways to tackle the problem of obtaining the E and B generated fields: Jackson [6] and Landau [10] consider an elegant formalism using quadrivector approach. Panofsky [11] and Heald [15] use the so called Liénard-Wiechert potentials which can be obtained by direct substitution on equations (22)(23) and then carrying all the necessary derivatives. The deduction of the fields E and B following this procedure can be seen in Ref. [7]. ...
In this paper, the electromagnetic radiation from an oscillating particle placed in the vicinity of an object of size comparable to the wavelength is studied. Although this problem may seem academic at first sight, the details of the calculations are presented throughout without any detail left under the carpet. A polyharmonic decomposition of the radiation sources allows the diffraction problem to be fully characterised while satisfying energy conservation. Finally, the source expressions obtained are suitable for use in a numerical code. A 3D illustration using finite elements is provided.
... According to the "classical view" [8], the generation of oscillations in the gyrotron [9][10][11][12][13][14] is caused by the beam instability that is brought about by the rotation of electrons in the magnetic field when there is an electromagnetic wave at the resonant frequency. As a result, a phase grouping of particles and the intensification of the wave appear. ...
... A similar ansatz was used by Stanyukovich [47] for studying astrophysical oscillations, and Amiranashvili et al. [48] and Dubin [49] for studying nonlinear oscillations in non-neutral oblate and regular spheroidal plasmas. However, with allowance for the beam's self-fields as well as external electric and magnetic fields (see Equations (4), (8) and (14)), in the case under consideration, this model describes a redistribution of momentum from the azimuthal to the axial components. Moreover, proceeding from the results [50], one may try to expand the Brillouin model for asymmetric beams. ...
Based on the cold-fluid hydrodynamic description, the interaction of a non-relativistic charged-particle beam with crossed magnetic fields is studied. This process results in the transfer of energy/momentum from the field to the beam, which, in turn, enhances the beam’s own electrostatic oscillations. This paper investigates the development features of such coupled axial and radial oscillations near resonant frequencies. The necessary conditions for the resonant amplification of this beam’s natural oscillations are identified. Such a process may be used for the creation of effective radiation sources.
... Obviously, the multipliers in the curly brackets of formulas (32) and (33) for any values of current strengths that are actually encountered in practical electrical engineering, differ very little from unity. For example, if current of ∼120 A flows through two copper conductors with a diameter of 2 mm, then the drift velocity of electrons in each conductor is ≈ 0.3 cm/s. ...
... Among theoretical arguments in favour of the correctness of some relations, it is worth pointing to the requirement of their relativistic invariance [7,32,44]. The principle of relativity leads to completely defined laws of transformation of physical quantities during the transition from one inertial reference frame to another. ...
This paper extends our relativistic framework for teaching electrodynamics in higher educational institutions. Building upon our previous work on deriving Maxwell's equations from first principles - the principle of relativity and Coulomb's law - we examine persistent contradictions in conventional electrodynamics teaching regarding conductors with constant current. We analyze the stationary electric field of current-carrying conductors, resolve contradictions concerning its potentiality, and explain the experimental non-observability of non-potential components through relativistic compensation effects. The paper addresses and resolves inconsistencies in the literature regarding the condition of neutrality for conductors with current, proposing a physically consistent condition: ρ⁰₊ = -ρ⁰₋. Within this framework, we develop a relativistic description of the interaction between conductors with current that satisfies both the principle of relativity and physical adaptation requirements. This approach aligns with the fundamentalization of physics education, providing a theoretically robust alternative to traditional empirical methods of teaching electrodynamics. The proposed methodology creates a conceptually unified framework that better reflects modern physics while addressing existing inconsistencies in pedagogical literature, transforming how electromagnetism is taught in higher educational institutions.
... Substituting the values of L and C with the vacuum's intrinsic electromagnetic constants µ 0 (the magnetic permeability) and ϵ 0 (the electric permittivity), we obtain the well-known expression for the speed of light in a vacuum [16]: ...
... In an idealized scenario, an inductor stores energy solely in its magnetic field and releases it back to the circuit without any losses. However, real inductors always experience some degree of energy dissipation due to these inherent resistances and other factors, meaning that not all stored energy returns to the circuit [16]. ...
This paper presents a novel cosmological framework interpreting the vacuum as a system of harmonic oscillators, resonating at relativistic scales and manifesting properties that unify aspects of quantum mechanics and general relativity. By modeling the vacuum through an equivalent RLC circuit, fundamental constants, including the speed of light c, gravitational constant G, and fine-structure constant α, are derived as emergent properties of this oscillatory vacuum structure, revealing a dynamic structure within the vacuum, and linking oscillatory vacuum states to the emergence of gravitational and electromagnetic phenomena.
Based on this framework, they are postulated explanations for the cosmological constant, observable gravitational phenomena, and large-scale structure, proposing a resonance-based expansion model of the universe consistent with current cosmological observations. By re-envisioning the vacuum as an active, resonant medium, this model offers a unified theoretical basis that could integrate quantum mechanics, relativity, and cosmology, with implications for both fundamental theory and potential observational validation.
Finally, the model further explores energy exchange across a hypothesized matter-antimatter boundary, conceptualized as a "quantum black hole" network, which would induce spacetime curvature and give rise to gravitational and electromagnetic interactions, postulating itself as a significant step toward a complete and consistent "Theory of everything".
... The four fundamental equations governing electromagnetic fields in Maxwell's equations are extraordinarily accurate in material systems where the charge density and current density are well understood [34][35][36]. The distribution of charges and their dynamical properties are typically inferred experimentally or computed for specific thermodynamic states and then homogenized using a variety of approximation procedures [32,33,37,38]. ...
... We now split J into fractal divergence and curl-free components. Let J = J T + J L , where J T is transversal and J L solenoidal parts of J such that ∇ β · J T = 0 and ∇ β × J L = 0. Assuming the Coulomb gauge condition, Equation (35) can be split into two equations ...
A fractal-order entropy dynamics model is developed to create a modified form of Maxwell’s time-dependent electromagnetic equations. The approach uses an information-theoretic method by combining Shannon’s entropy with fractional moment constraints in time and space. Optimization of the cost function leads to a time-dependent Bayesian posterior density that is used to homogenize the electromagnetic fields. Self-consistency between maximizing entropy, inference of Bayesian posterior densities, and a fractal-order version of Maxwell’s equations are developed. We first give a set of relationships for fractal derivative definitions and their relationship to divergence, curl, and Laplacian operators. The fractal-order entropy dynamic framework is then introduced to infer the Bayesian posterior and its application to modeling homogenized electromagnetic fields in solids. The results provide a methodology to help understand complexity from limited electromagnetic data using maximum entropy by formulating a fractal form of Maxwell’s electromagnetic equations.
... and ∇ k acts on the generic q k , not on the actual q k (t). In (28), p c,k defined by ...
... Incidentally, this invalidates the derivation of the Newtonian equation of motion from the purely electromagnetic Abraham model(s) in[15,21,28,29,30]. In[31] a "more conservative position" is taken and the spinless Abraham model with bare mass considered. ...
We discuss a class of quantum Abraham models in which the N-particle spinor wave function of N electrons solves a Pauli respectively Schroedinger equation, featuring regularized classical electromagnetic potentials which solve the semi-relativistic Maxwell-Lorentz equations for regularized point charges, which move according to some de Broglie-Bohm law of quantum motion. Thus there is a feedback loop from the actual particle motions to the wave function. The electrons have a bare charge and positive bare mass different from their empirical charge and mass due to renormalization by the self-fields. In the classical limit the various models reduce to the Hamilton-Jacobi version of corresponding Abraham models of classical electron theory.
... The more general results are written earlier, see equation [5]. The above results are consistent with the findings of the text books [31,32,33] and also Nussenzveig [24], Toll [21], Landau and Lifshitz [17] and Rauch and Rohrlich [25], all of which assume a physical model for the material. None of these texts are referring specifically to the vacuum as the medium. ...
... More recent tutorial accounts of the dispersion relations which include a discussion on causality can be found, for example see [29,30]. Text book accounts [31,32,33], will mention that n(ω) → 1 as ω → ∞ but will only argue based on the physical model for the refractive index. We have shown in a previous work [1,2] that if [n(ω) − 1] satisfies the dispersion relations then, regardless of how it was derived, ...
In this paper we show that the Scharnhorst effect (Vacuum with boundaries or a Casimir type vacuum) cannot be used to generate signals showing measurable faster-than-c speeds. Furthermore, we aim to show that the Scharnhorst effect would violate special relativity, by allowing for a variable speed of light in vacuum, unless one can specify a small invariant length scale. This invariant length scale would be agreed upon by all inertial observers. We hypothesize the approximate scale of the invariant length.
... are the time-dependent electric Maxwell and electrostriction stress tensors, respectively, γ ijkl = −(1/2) 0 n 4 p ijkl , where n being the optical refractive index and p ijkl being the photoelastic tensor. The minus sign used in the definition of the electrostrictive force follows the conservative force convention [1,41]. Using our field expansion eq. ...
... These two contributions can be obtained by integrating by parts eq. (S25) and disregarding the electrostriction surface pressure term [18,41], ...
The simultaneous control of optical and mechanical waves has enabled a range of fundamental and technological breakthroughs, from the demonstration of ultra-stable frequency reference devices to the exploration of the quantum-classical boundaries in laser-cooling experiments. More recently, such an opto-mechanical interaction has been observed in integrated nano-waveguides and microcavities in the Brillouin regime, where short-wavelength mechanical modes scatters light at several GHz. Here we engineer coupled optical microcavities spectra to enable a low threshold excitation of mechanical travelling-wave modes through backward stimulated Brillouin scattering. Exploring the backward scattering we propose microcavity designs supporting super high frequency modes ( GHz) an large optomechanical coupling rates ( kHz).
... This type of optical field interacts with the propagation media inducing local polarization effects through charge redistribution. This last statement can be justified by analyzing the differential equation associated with the irradiance equation, where its critical points are related to the charge features [19]. These points are obtained when the derivative of the irradiance function becomes zero. ...
... Conforming the compression is increasing, all the dipoles become parallel and the propagation media is polarized generating a tandem array of dipoles along z-coordinate. This is very important because all the dipoles generate a singularity of infinite order corresponding to a tandem array of charges as was described in [19]. These charges are the necessary condition to generate non-homogeneous wave. ...
We analyze the evolution of the speckle pattern subjected to a compression showing that the spatial evolution of the probability density function satisfies a non-linear diffusion equation. Its asymptotic solution corresponds to a non-homogeneous wave with string-shape. During the compression, an irradiance interaction is generated between the speckles. The interaction satisfies a type-logistic equation. The electric field components associated with the speckles present a random behavior, however, during the compression process, the transversal components cancel each other, and the resulting light presents a state of polarization parallel to the propagation coordinate, justifying the non-homogeneous behavior. Computer simulations are presented.
... [6][7][8][9] do not appear. However, there are other expressions of Maxwell's equations and the constitutive relations, for example, that proposed by Panofsky and Phillips [15], ...
There is disagreement about which is the correct expression for the force on a magnetic dipole, and at least two expressions for this force have been proposed, generating a controversy between Vaidman [1985] and Franklin [2018]. Our view here exposed is that the macroscopic Maxwell equations and the constitutive relations imply, via electromagnetic momentum balance equations, several force densities which include those proposed by some authors. Therefore, the question is not which one is correct, since all are legitimate deductions of Maxwell´s equations, but under what conditions they may be useful to explain some phenomena. The discussion of conceptual problems of electromagnetism is very useful to both graduate students and researchers.
... The electric volumetric force in a dielectric fluid can be represented by (Panofsky and Phillips 2005;Landau et al. 2013): where ρ e , E, ε R , ε 0 , and ρ denote the free charge density, the electric field intensity, the relative dielectric permittivity, the vacuum permittivity, and the fluid density. The Coulomb force (the first term on the right-hand side of Eq. (1)) is considered to be disregarded for a clean dielectric liquid. ...
This paper experimentally investigated the impact of the electric field strength (E), electrode installation heights (H), and the electrode shape on enhanced pool boiling heat transfer performance under a downward heating surface with an electric field. It is observed that the critical heat flux (CHF) generally increases with increasing electric field strength. For instance, for the mesh electrode, the CHF is increased by 100.0%, 240.0%, 340.0%, and 440.0% at E = 0.35 × 10⁶ V/m, 0.70 × 10⁶ V/m, 1.05 × 10⁶ V/m, and 1.40 × 10⁶ V/m, respectively, compared to E = 0 V/m. Furthermore, the electrodes hinder the detachment of vapor bubbles, which becomes more pronounced when the electrode installation height is low. At the same time, the more micro-ribs of the electrodes and the denser the distribution, the more uniform the electric field generated. Under this condition, the “pinch-off effect” caused by the non-uniform electric field is reduced, which is more conducive to enhancing boiling heat transfer performance. Ultimately, at H = 5.0 mm and E = 1.40 × 10⁶ V/m, the CHF with grid electrodes improved by 101.1% compared with the horizontally upward without the electric field, which is a superior combination of working conditions and suggests that a more optimistic boiling heat transfer performance can be obtained in microgravity. This work provides guidance for enhancing boiling heat transfer in microgravity by an electric field.
... J. L. I came in contact there with some very good teachers, in particular the physics teacher, Melba Phillips. She wrote a famous book, together with Wolfgang K. H. Panofski, on electricity and magnetism [26]. She had been a student of Robert Oppenheimer -there is a famous Oppenheimer-Phillips effect. ...
This is the first part of an oral history interview on the lifelong involvement of Joel Lebowitz in the development of statistical mechanics. Here the covered topics include the formative years, which overlapped the tragic period of Nazi power and World War II in Europe, the emigration to the United States in 1946 and the schooling there. It also includes the beginnings and early scientific works with Peter Bergmann, Oliver Penrose and many others. The second part will appear in a forthcoming issue of European Physical Journal H.
... In line with modern particle physics, often a structureless point charge model is adopted, leading to the well-known Lorentz-Abraham-Dirac (LAD) equation of motion [2,3] for elementary charged particles. As is also well known [4], however, the LAD equation is plagued by irreconcilable deficiencies manifested by runaway solutions, in which the particle momentum grows exponentially toward infinity, or by preacceleration solutions, where the particle starts to accelerate even before the onset of any external force. A popular remedy is to express in the LAD equation the particle acceleration perturbatively in terms of the external force [3], which renders the equation stable. ...
Several noncovariant formulations of the electromagnetic self-force of extended charged bodies, as have been developed in the context of classical models of charged particles, are compared. The mathematical equivalence of the various dissimilar self-force expressions is demonstrated explicitly by deriving these expressions directly from one another. The applicability of the self-force formulations and their significance in the wider context of classical charged particle models are discussed.
... where E 0 is its amplitude, k is the wave vector, ω is the frequency of the wave, t is time,ẑ is a unit vector in the z-direction, and i is the square-root of minus one. We may assume that the electrons have non-relativistic motions and thus ignore the effects of the magnetic fields of the incoming electromagnetic waves as the magnitude of the magnetic force on an electron will be negligible since it depends directly on the electron's speed [3]. Using equation (1) and taking the electrons to act as classical oscillators, we can find the equation governing the motion of an electron under the influence of the electric field E of a passing electromagnetic wave. ...
The standard explanations for the phenomena of transmission and refraction of electromagnetic waves of optical wavelengths upon entering a transparent medium is in terms of the speed of the waves being altered. However, these phenomena are explicable without assuming that the speed of electromagnetic waves changes. The actual speed does not alter in various media but is customarily assumed to do so as this facilitates straight-forward calculations and allows easily understood explanations to be advanced. An underlying account of transmission and refraction at the level of individual atoms is presented in which the speed of electromagnetic waves remains constant.
... The angular momentum of the radiation field dL incident on the area ds during the time dt is calculated through the Poynting vector P [22,23] ...
The angular momentum of radiation from an arbitrarily moving relativistic charge is studied. The angular momentum is presented as the sum of the angular momentum relative to the point where the charge is located at a retarded moment of time and the angular momentum relative to an arbitrary stationary center. In particular, the instantaneous center of curvature of the trajectory is considered as such a center. Explicit expressions for the angular distribution of these components of angular momentum fluxes are obtained and studied. It is shown that the angular momentum of the field relative to the position of the charge is determined only by the properties of the electromagnetic radiation field, and the angular momentum relative to an arbitrarily distant point is the vector product of the displacement of this point and the force corresponding to radiation pressure. It is shown that in the ultrarelativistic limit, the canonical angular momentum of the radiation coincides with the angular momentum following from the symmetrized energy-momentum tensor of the electromagnetic field.
... The dipole solution for the oscillating charge has been presented by many authors using Maxwells equations (Hertz, 1893), (W. Panofsky, 1962), (Jones, 1964), (Jackson, 1975). Note that the solutions for an oscillating magnetic dipole would be identical with the E and B fields interchanged. ...
The speed of light is not a constant speed as once thought, and this has now been proved by Electrodynamic theory and by Experiments done by many independent researchers. The results clearly show that light propagates instantaneously when it is created by a source, and reduces to approximately the speed of light in the farfield, about one wavelength from the source, and never becomes equal to exactly c. This corresponds the phase speed, group speed, and information speed. Any theory assuming the speed of light is a constant, such as Special Relativity and General Relativity are wrong, and it has implications to Quantum theories as well. So this fact about the speed of light affects all of Modern Physics. Often it is stated that Relativity has been verified by so many experiments, how can it be wrong. Well no experiment can prove a theory, and can only provide evidence that a theory is correct. But one experiment can absolutely disprove a theory, and the new speed of light experiments proving the speed of light is not a constant is such a proof. So what does it mean? Well a derivation of Relativity using instantaneous nearfield light yields Galilean Relativity. This can easily seen by inserting c=infinity into the Lorentz Transform, yielding the Galilean Transform, where time is the same in all inertial frames. So a moving object observed with instantaneous nearfield light will yield no Relativistic effects, whereas by changing the frequency of the light such that farfield light is used will observe Relativistic effects. But since time and space are real and independent of the frequency of light used to measure its effects, then one must conclude the effects of Relativity are just an optical illusion.
Since General Relativity is based on Special Relativity, then it has the same problem. A better theory of Gravity is Gravitoelectromagnetism which assumes gravity can be mathematically described by 4 Maxwell equations, similar to to those of electromagnetic theory. It is well known that General Relativity reduces to Gravitoelectromagnetism for weak fields, which is all that we observe. Using this theory, analysis of an oscillating mass yields a wave equation set equal to a source term. Analysis of this equation shows that the phase speed, group speed, and information speed are instantaneous in the nearfield and reduce to the speed of light in the farfield. This theory then accounts for all the observed gravitational effects including instantaneous nearfield and the speed of light farfield. The main difference is that this theory is a field theory, and not a geometrical theory like General Relativity. Because it is a field theory, Gravity can be then be quantized as the Graviton.
Lastly it should be mentioned that this research shows that the Pilot Wave interpretation of Quantum Mechanics can no longer be criticized for requiring instantaneous interaction of the pilot wave, thereby violating Relativity. It should also be noted that nearfield electromagnetic fields can be explained by quantum mechanics using the Pilot Wave interpretation of quantum mechanics and the Heisenberg uncertainty principle (HUP), where Δx and Δp are interpreted as averages, and not the uncertainty in the values as in other interpretations of quantum mechanics. So in HUP: Δx Δp = h, where Δp=mΔv, and m is an effective mass due to momentum, thus HUP becomes: Δx Δv = h/m. In the nearfield where the field is created, Δx=0, therefore Δv=infinity. In the farfield, HUP: Δx Δp = h, where p = h/λ. HUP then becomes: Δx h/λ = h, or Δx=λ. Also in the farfield HUP becomes: λmΔv=h, thus Δv=h/(mλ). Since p=h/λ, then Δv=p/m. Also since p=mc, then Δv=c. So in summary, in the nearfield Δv=infinity, and in the farfield Δv=c, where Δv is the average velocity of the photon according to Pilot Wave theory. Consequently the Pilot wave interpretation should become the preferred interpretation of Quantum Mechanics. It should also be noted that this argument can be applied to all fields, including the graviton. Hence all fields should exhibit instantaneous nearfield and speed c farfield behavior, and this can explain the non-local effects observed in quantum entangled particles.
*YouTube presentation of above arguments: https://www.youtube.com/watch?v=sePdJ7vSQvQ&t=0s
... Teniendo en cuenta que Faraday se encontraba inmerso en la filosofía de la simetría de la naturaleza, entendía que según la teoría de Ampère sobre las corrientes en el imán, estas corrientes tendrían por simetría sobre la acción-reacción, crear una corriente en un conductor circular de cobre. La realización práctica la realiza mediante un conductor espiral que acerca a un polo de un imán y observa la creación de una corriente en la espira 82 en sus memorias sobre los experimentos realizados el 28 de noviembre de 1825. Aunque el concepto de inducción ya lo tenía en mente desde hacía años. ...
En esta tesis se desarrollan diversos conceptos relacionados con el magnetismo y la relatividad especial. Desde el disco de Arago, el disco de Faraday y las diversas modificaciones que hemos realizado. Iniciando una nueva tecnología sobre la construcción de nuevos motores eléctricos que hemos denominado Turbinas Electrodinámicas.
... In a waveguide, the dispersion relationship is given by k 2 g + k 2 s = k 2 0 , where k 0 is wavenumber of the wave in free space, k g is the guide propagation constant (corresponding to longitudinal component k z in free space), and k s is associated with the cutoff frequency of modes (corresponding to transverse component k r in free space). Under the condition that k s is determined by the section size and refractive index of the waveguide, i.e., ∂k s /∂ω = 0, the relation equation is derived as v φ v g = c 2 [36]. This means that at a subluminal group velocity, the phase velocity of spatially structured beams would be superluminal. ...
The structured beams especially with spatially varying phase distribution have attracted tremendous attention in both physics and engineering. Recently, studies have shown that the transverse spatial confinement of optical fields or photons leads to a modification of the group velocity but the phase velocity of propagating structured beams is revealed insufficiently in the experiments. In this work, we provide the theoretical model and experimental observation of propagation phase of structured beams. The analysis suggests that the spatially structured beams with a definite axial component of wavevector kr carry a so called “lagging propagation phase”, which can be considered as a generalized Gouy phase that originally appears within a focal region. Taking the higher-order Bessel beam as an example, the propagation phase difference is demonstrated by mapping to the rotating angle of intensity patterns superposed with different radial and angular phase gradients. Physically, the lagging propagation phase may provide an interpretation for the dynamic evolution of complex structured beams or interfering fringes upon propagation such as the vortex knots or braids. From the application aspect, the lagging propagation phase would facilitate a promising way for structured beams in optical sensing and metrology.
... Thus, effectively, the observer perceives retarded charge and current densities in-access or less than the actual, depending upon the particle's velocity vector. [5] However, when the same logic were extended to the accelerating rod, the picture gets distorted further; as the observer would see different parts of the rod travelling with different velocities. This may be visualized from figure-2 below: Assuming the rod moving with low enough velocities ( v << c ) and acceleration ( a∆t << c ), nonrelativistic (Newtonian) equations of motion would hold good. ...
Lienard-Wiechert potentials have been derived for a non-relativistically moving and 'classically' spinning point-charge; assuming it to be a rigid, moving and spinning charged-sphere, and subsequently reducing its dimensions to "point-particle" limit. The paper demonstrates that when the effect of "rotational-acceleration" were considered, together with causality, expressions for the LW potentials accompany additional correction terms that contain spin-angular-momentum (or simply the "classical spin") of the point-charge.
... To investigate the transition to synchronization, we refer to the work by Sarasola et al. in 2004(Sarasola et al., 2004Sarasola, d'Anjou, Torrealdea and Graña, 2005) on the energy approach. Having this in mind, we consider that the dynamic of the coupled systems is strictly connected to the time derivative of the energy as shown in (Panofsky and Phillips, 1955). We will considerḢ for our investigations. ...
The main objective of this thesis is the investigation and the control of the dynamics of networks. The resolution of this problem required three key concepts that we have adopted: the dead zone, amplification and optimal control. Due to its numerous applications in telecommunication as well as in human relations, the concept of dead zone could be a better way to model the coupling between two or more systems. For example, in systems using the magnetic field for the coupling, the magnetic field may vary with the distance between systems, thus making the coupling between these systems intermittent. The studies carried out in this thesis show that this dead zone not only improves the transition to synchronization in the case of two coupled systems, but can also allow the transition from one behaviour to another (chimera states, multi-chimera states, clusters, synchronization). An implementation in Pspice using electronic components allowed us to design a circuit modeling this dead zone and thus showed the feasibility of the synchronization obtained by the numerical resolution in MATLAB. The existence of synchronization in certain devices such as gear systems, pulley-belt systems improves efficiency. The improvement of the speed in these devices is done through the transmission ratio considered as the amplification coefficient in this work. Thus, the concept of amplification is a key concept that is highly sought-after in many electronic, mechanical and even electrical systems. Obtaining this amplification in linear systems is more obvious. However, most real systems are non-linear and therefore, to highlight this amplification becomes a titanic task. In this work, the study of the influence of amplification in the case of two similar coupled systems shows that it is possible to switch from phase-flip to phase synchronisation or phase-lock. In the case of networks, this amplification also makes it possible to switch from desynchronization to synchronization via clusters, splays etc. The stability of synchronization can be accessed via the master stability function. This convenient tool allows for treating the node dynamics and the network topology in two separated steps, and, thus, allows for a quite general treatment of different network topologies. In this thesis, the network formed by the Rössler systems used gave us a type III MSF allowing us to highlight a synchronization zone and two desynchronization zones. Later, the application of the Hamilton-Bellman-Jacobi method enabled us to
manufacture an optimal controller that minimizes the transition time to a precise state based on the cost administered. Thus, the resulting controller allowed us to achieve synchronization with minimal time.
... In electromagnetic theory radiation pressure results from time averages of the Maxwell stress tensor (Panofsky and Phillips, 1962). Consequently, in the general case there is a tensor of radiation pressure which is not an isotropic pressure at all. ...
... In these media, the velocities of EM waves are different. Further, if any of these media are in relative motion with respect to the stationary Ether, then the velocity suffers from Fresnel Drag[23,46]. ...
... If F L(AB) r is positive (or negative), then the electrostatic force is repulsive (or attractive). In other words, F L(AB) is repulsive if F L(AB) r > 0, that isq (A)q(B) > 0 and is attractive if F L(AB) r < 0, that isq (A)q(B) < 0. Therefore, the Coulomb law (see, e.g., [21,38]) stating that charges with opposite sign attract each other (Coulomb attraction) and charges with same sign repel each other (Coulomb repulsion) is reproduced also for line charges using the concept of the J (AB) -integral. ...
In this work, the J-, M- and L-integrals of two line charges and two line forces in generalized plane strain are derived in the framework of three-dimensional (3D) electrostatics and three-dimensional compatible linear elasticity, respectively, in order to study the interaction between them. The key point in this derivation is achieved by expressing the J-, M- and L-integrals of point charges and point forces in three dimensions in terms of the corresponding three-dimensional Green functions, that is, the three-dimensional Green function of the Laplace operator and the three-dimensional Green tensor of the Navier operator, respectively. The major mathematical tool used in deriving the J-, M- and L_3-integrals of line charges and line forces from the corresponding J-, M- and L-integrals of point charges and point forces is the method of embedding or method of descent of Green functions to two dimensions (2D) from the corresponding Green functions in 3D. The analytical expressions of J-, M- and L_3-integrals of line charges and line forces in antiplane and plane strain are derived and discussed. The J-integral is the electrostatic part of the Lorentz force (electrostatic interaction force) between two line charges in electrostatics and the Cherepanov force between two line forces in elasticity. The M-integral of two line sources (charges or forces) equals half the electrostatic interaction energy in electrostatics and half the elastic interaction energy in elasticity of these two line sources, respectively. The L_3-integral of two line sources (charges or forces) is the z-component of the configurational vector moment or the rotational moment representing the total torque about the z-axis caused by the interaction of the two line sources. An important outcome is that the J-integral is twice the negative gradient of the M-integral, leading to the result that the J-integral for line charges and line forces is a conservative force (with the corresponding interaction energy playing the role of the potential energy) and consequently an irrotational vector field. Finally, the obtained J-, M-and L_3-integrals being functions of the distance and of the angle, are able to describe the physical behaviour of the interaction of two line sources (charges and forces), since they represent the fundamental and necessary quantities, that is, the interaction force, interaction energy and total torque produced by the interaction of these two line sources.
This paper offers an informal instructive introduction to some of the main notions of geometric continuum mechanics for the case of smooth fields. We use a metric invariant stress theory of continuum mechanics to formulate a simple generalization of the fields of electrodynamics and Maxwell’s equations to general differentiable manifolds of any dimension, thus viewing generalized electrodynamics as a special case of continuum mechanics. The basic kinematic variable is the potential, which is represented as a p-form in an n-dimensional spacetime. The stress for the case of generalized electrodynamics is assumed to be represented by an -form, a generalization of the Maxwell 2-form.
This book examines advances in bioinspired materials and metamaterials and investigates how their tailor-engineered properties enable the design of devices and ultimately the ability to solve complex societal problems that are, in principle, impossible with traditional materials. The aim is to survey the scientific foundations of the design and properties of bioinspired materials and metamaterials and the way they enter engineering applications. The text is intended for materials engineering students who have completed courses in general physics, chemistry, and calculus, as well as researchers in materials science and engineering.
Electromagnetism is a fundamental pillar of physics, underlying numerous natural phenomena and modern technologies. This study analyzes the evolution of electromagnetic theory from the classical era of Maxwell, through the relativistic contributions of Einstein, to the quantum developments of Feynman. The theory has undergone three main phases: classical, relativistic, and quantum, each significantly advancing scientific understanding and technological innovation, including radio communication, lasers, fiber optics, and medical imaging technologies. The study also explores future challenges, such as addressing modern physics anomalies like dark matter and dark energy, and prospects for interdisciplinary integration to drive further advancements. These findings affirm the enduring relevance of electromagnetic theory in scientific and technological progress, establishing it as a critical foundation for future exploration and applications
Traditional nonlinear optics emphasizes processes driven by the electric field of light at moderately high intensities while generally ignoring dynamic magnetic effects. High frequency magnetism is generally associated with metamaterials or bulk magneto-electric solids. However, magneto-electric interactions can achieve magnetic response at the molecular level in essentially all dielectric materials. Classical and quantum models of nonlinear interactions driven by the combined forces of optical electric and magnetic fields are reviewed in this paper. Experimentalm conditions are also identified under which electric and magnetic field-driven interactions induce enhanced magnetic dipole response as well as a longitudinal Hall effect. Several mechanisms that account for dynamic enhancement of magnetic response are identified, including a torque-driven exchange of orbital angular momentum for rotational angular momentum. Experiments on this
topic are summarized, and connections are established between electric and magneto-electric susceptibilities. The review concludes by anticipating novel photonic technology reliant on dynamic magneto-electric effects.
This chapter highlights the literature on electroreception before 1974 by Faraday, Lissmann & Machin, Murray, Parker &Van Heusen, and Dijkgraaf & Kalmijn. The author discusses the literature from 1974 to 2021 that conflicts with his own scientific writings. Included are the author’s criteria for establishing an electric and a magnetic sense in sharks and rays. That the ampullae of Lorenzini have the physical ability to provide the animals with navigational information cannot be properly grasped without a comprehensive understanding of the physical concepts from the animals’ perspective. The lack of the necessary understanding of the physical principles has led to erroneous physical interpretations and related unsound behavior experiments. Through step-by-step explanations of the underlying physics, this chapter addresses the inaccuracies, misunderstandings, and shortcomings of alternative theories that have been put forward in the literature.
In this paper, we derive the solution of anti-plane deformation for two circular cylindrical holes embedded in an elastic matrix of infinite extent under remote shears. The two holes may have different radii and distance between each other. The matrix is subjected to arbitrary direction of remote shears. The solution is obtained via the boundary integral formulation in conjunction with the degenerate kernel of bipolar coordinates instead of Möbius transformation using complex variables. Several examples are revisited to compare with those data of Honein et al. Three differences between the present result and those of Morse and Feshbach and Lebedev et al. are discussed.
Fluid flows coupled with electrical phenomena represent a fascinating and highly interdisciplinary scientific field. Recently, a remarkable success of electrospinning in producing polymer nanofibers has led to extensive research aimed at understanding the behavior of viscoelastic jets affected by the applied electric and aerodynamic forces, such as those imposed by the surrounding gas flows. Theoretical models have uncovered various unique aspects of the underlying physics of polymer solutions in these jets, offering valuable insights for experimental platforms. This chapter explores the progress made in the theoretical description and numerical simulations of polymer solution jets in electrospinning. It emphasizes the instability phenomena arising from both electric and hydrodynamic factors, which are pivotal for understanding the flow physics. The chapter also outlines specifications for creating accurate and computationally feasible models. Topics covered include electrohydrodynamic modeling, theories describing jet bending instability, recent advancements in Lagrangian approaches for jet flow description, strategies for dynamic refinement of simulations, and the effects of intense elongational flow on polymer networks. In addition, the present chapter discusses current challenges and future prospects in this field, which encompasses the physics of jet flows, non-trivial material properties, and the development of multiscale techniques for modeling viscoelastic jets.
This study introduces a polyharmonic framework for analyzing the electromagnetic (EM) field generated by an oscillating point charge near a dispersive bulk of size comparable to the wavelength under study. We critically evaluate traditional approaches such as Liénard-Wiechert, Landau, and Raimond, and propose a Fourier representation of sources that simplifies numerical implementation and enhances analytical clarity. Our method effectively addresses the limitations of conventional models and is applicable to both relativistic and non-relativistic scenarios. It includes the oscillating point dipole fields, providing a comprehensive understanding of the EM field behavior. The Finite Element Method (FEM) is employed for numerical analysis, demonstrating the method’s adaptability to complex geometries. While offering significant insights, this study acknowledges certain limitations and outlines directions for future research.
Displacement current is the last piece of the puzzle of electromagnetic theory. Its existence implies that electromagnetic disturbance can propagate at the speed of light and finally it led to the discovery of Hertzian waves. On the other hand, since magnetic fields can be calculated only with conduction currents using Biot-Savart's law, a popular belief that displacement current does not produce magnetic fields has started to circulate. But some people think if this is correct, what is the displacement current introduced for. The controversy over the meaning of displacement currents has been going on for more than hundred years. Such confusion is caused by forgetting the fact that in the case of non-stationary currents, neither magnetic fields created by conduction currents nor those created by displacement currents can be defined. It is also forgotten that the effect of displacement current is automatically incorporated in the magnetic field calculated by Biot-Savart's law. In this paper, mainly with the help of Helmholtz decomposition, we would like to clarify the confusion surrounding displacement currents and provide an opportunity to end the long standing controversy.
Stored energies of radiating systems have generated research interest for several decades due to their relationship with Q-factor and fractional bandwidth. In this paper material derivatives are used to provide a new interpretation of widely used stored energy expressions. This provides a fundamental relationship between stored energies of radiating systems and their electromagnetic material properties. It is shown that electric stored energy is related to the electric material parameter (permittivity) derivative. Similarly, magnetic stored energy is linked to the derivative of the permeability. Further, as an extension Cauchy-Riemann equations are used to relate stored energy to dissipation.
The reflection tensor of a metafilm with bianisotropic elements on a dielectric interface is analytically derived. First, using the dipole approximation, the nanoparticles are replaced with electric and magnetic dipoles. Next, the local fields are obtained as the superposition of the background field and the interaction fields (sum of fields by all dipoles and their images). The latter is written in a form that directly results in either fast-converging series or series with closed-form solutions. Utilizing the local fields, the collective polarizability tensor and the dipole intensities are determined. Then, assuming the array to be dense, the dipole array is replaced with current sheets, for which the radiated fields in the presence of a substrate are well-known. Next, the reflection tensor is derived, validated, and discussed. Moreover, using a numerical scheme, the polarizability tensor of a bianisotropic nanoparticle in free space is calculated. Using the computed polarizability tensor and the derived analytical expressions, reflection from a metafilm with bianisotropic nanoparticles on an interface is obtained. Finally, the more complex problems of embedded particles and oblique illumination are briefly addressed.
This study deals with the exact solution of Einstein's field equations for a sphere of incompressible liquid without the additional limitation initially introduced in 1916 by Schwarzschild, by which the space-time metric must have no singularities. The obtained exact solution is then applied to the Universe, the Sun, and the planets, by the assumption that these objects can be approximated as spheres of incompressible liquid. It is shown that gravitational collapse of such a sphere is permitted for an object whose characteristics (mass, density, and size) are close to the Universe. Meanwhile, there is a spatial break associated with any of the mentioned stellar objects: the~break is determined as the approaching to infinity of one of the spatial components of the metric tensor. In particular, the break of the Sun's space meets the Asteroid strip, while Jupiter's space break meets the Asteroid strip from the outer side. Also, the space breaks of Mercury, Venus, Earth, and Mars are located inside the Asteroid strip (inside the Sun's space break).
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