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The method of path integrals is used to average the magnetic-induction equation written for the vector potential over the velocity field. This approach avoids the assumption that the random field is two-scale, replacing it with the assumption of short time correlations. As a result of averaging, the classical equation of the mean field is obtained, at least for a homogeneous and isotropic medium. Based on the developed approach, the problem of the average field solenoidality is discussed, as well as a possible generalization of the method to the case of a statistically inhomogeneous and anisotropic flow.

Harmonic Cartesian polynomial and rational functions are used to obtain a semi-analytical solution to the uncoupled, two-dimensional problem of thermo-magnetoelasticity for a long, transversely isotropic elastic cylinder of nearly-circular normal cross-section carrying an axial, steady electric current. Numerical results and plots are provided and discussed. Comparison with the case of a circular cross-section allows to assess the in uence of imperfection of the cross-sectional shape on the quantities of practical interest. The effect of dependence of the magnetic permeability on strain is investigated. We use the magnetic vector potential which is parallel to the axis of the cylinder. The mathematical problem reduces to the determination of ve harmonic functions, interrelated by a set of boundary conditions. Two of these harmonic functions combine to yield Airy's stress function in the cross-sectional domain in case the mechanical problem is solved in stresses. A third harmonic function enters in the formulation of the thermal problem for temperature distribution. The two remaining harmonic functions enter in the description of the vector potential distribution inside the conductor and in the surrounding region. The results nd application in calculating the deformation of straight central sections of electric conductors in various instruments, in particular busbars in electric power stations, and the stresses induced in them due to the electric current.

Lateral wave plays an important role in electromagnetic (EM) wave propagation in layered conducting media. The full utilization of lateral wave in marine layered media may provide an innovative way of transmedia wireless communication. However, the existing approaches for solving lateral wave are tedious and complicated. In this letter, a simple model was presented to obtain lateral wave and other EM components in marine layered media. The model based on vector potential has simple process, definite physical meaning, and high accuracy. The theoretical computations were compared with the experimental data on sea. The good agreement of the two results verified the rationality and reliability of the model. In addition, the lateral wave near the sea surface was investigated theoretically and experimentally. The results show that lateral wave has a good performance of propagation from seawater to air. The present approach is more concise and the results can provide more physical insight compared to applying asymptotic expansion methods.

In this paper, numerical calculations were made for Ar loop-type inductively coupled thermal plasma (loop-ICTP). The loop-ICTP was developed originally by the authors’ group for rapid surface modification of large areas. Loop-ICTP is sustained with a unique three-dimensional (3D) configuration inside a circular loop quartz tube and on the substrate. A 3D and two-temperature thermofluid thermal plasma model was adopted for this calculation. Mass, momentum, and energy conservation equations were solved using a Maxwell equation for vector potential, an electron energy transport equation, and Saha’s equation in the 3D space. Results indicate that Ar loop-ICTP can be sustained and formed in the loop tube and also on the substrate. Moreover, the heavy particle temperatures reaches 1800–2000 K, whereas the electron temperature is about 10,000 K. Loop size effects on the gas temperature and gas flow field were also investigated using the developed model. Results show that adoption of a larger loop tube can be expected to improve the plasma uniformity on the substrate when applied to rapid surface modification.

In this paper, the spontaneous emission rate of an excited atom between the two dispersive and dissipative dielectric slabs is determined. It is assumed that the atom has a definite transition electric dipole moment, and the dielectric function of the slabs is an arbitrarily complex function of frequency satisfying Kramers–Kronig relations. In this research, by noting to the connection between the spontaneous emission rate and the imaginary part of the vector potential Green function, only by evaluating the vector potential Green function tensor of the system of the two slabs, an expression for the spontaneous emission rate is introduced. In this formalism, it is needless to derive the vacuum electric field operator. The emission rate is obtained in terms of the dielectric function, the thickness and the distance between the slabs. By considering a Lorentz model for the dielectric function of the slabs, the typical spatial variations of the spontaneous emission rate between the two slabs are demonstrated.

The article deals with the dynamic equations of the linear micropolar theory of elasticity. They are related with respect to vector fields of displacements and microrotations. Coupled equations are also obtained for vortex vector potentials of displacements and microrotations. Using (or not) calibration conditions, it is possible to formulate uncoupled equations that include the same main differential operator. Conditions for the hyperbolicity of the main operator and a condition connecting the defining constants that ensures its factorizability, that is, representation as a product of simpler differential operators commuting with each other. The actual construction of the operator factors in the factorized form is carried out.

We generalize our previous unification of the Schrödinger and guidance equations in a single inhomogeneous Schrödinger equation to a Riemannian manifold with an external vector potential. A special case yields the unified theory for a spin $$\frac{1}{2}$$ 1 2 rigid rotator. The theory is proved to be symmetrical under the Galileo group, the unified field that integrates the particle and guiding wave being a 2-spinor.

A hybrid spatial–spectral method for rapid and broadband calculation of the vector potential normal modes and dyadic Green’s function (GF) inside the cavity of irregular shape is presented. In the broadband method of simulation, the normal modes of the irregular cavity, which are independent of operation frequency, are first computed through a linear eigenvalue problem based on a hybrid representation of the regular (rectangular) cavity. The method is based on the extraction of the singularity of the GF that results in a rapidly convergent hybrid spatial–spectral expansion of the GF. The method is applied to a V-grooved cavity to obtain the vector potential GF inside the cavity, and the results are shown over a broad range of frequencies. The results of the hybrid method are in good agreement with that of direct method of moment (MoM), while the hybrid method is much more efficient than direct MoM as the latter requires the computation at one frequency at a time.

The Proca equation is one of the fundamental relativistic wave equations of massive vector (spin 1) boson fields. Proca equation is used to explain a relatively small particle that has a speed approaching the speed of light. The effect of photon imaginary mass to the quantum system that was coupled by noncentral potential consisting of radial, polar, and azimuthal modified Manning–Rosen potentials was investigated. The vector potential was coupled to the photon relativistic energy and the quadratic scalar potential was coupled directly to the quadratic photon mass. In the case of free space and when the vector potential was equal to scalar potential, therefore, the time-independent Proca equation reduced to a three-dimensional variable separable Schrodinger-like equation. By using variable separation method, the three-dimensional Schrodinger-like equation was resolved into three one-dimensional radial, polar, and azimuthal parts of Schrodinger-like equations. Each part of one-dimensional Schrodinger-like equation was solved using the hypergeometric differential equation through variables and wave function substitutions and resulted in the energy and wave function equations of massive photons. The relativistic energy equation of massive photons was a function of imaginary mass photon, the potential parameters, and the quantum numbers in radial, polar, and azimuthal variables. By using the Maxwell field approach, we got the correction factor for the energy function of this spin-1 particle.

Neodymium magnets were independently discovered in 1984 by General Motors and Sumitomo. Today, they are the strongest type of permanent magnets commercially available. They are the most widely used industrial magnets with many applications, including in hard disk drives, cordless tools and magnetic fasteners. We use a vector potential approach, rather than the more usual magnetic potential approach, to derive the three-dimensional (3D) magnetic field for a neodymium magnet, assuming an idealized block geometry and uniform magnetization. For each field or observation point, the 3D solution involves 24 nondimensional quantities, arising from the eight vertex positions of the magnet and the three components of the magnetic field. The only unknown in the model is the value of magnetization, with all other model quantities defined in terms of field position and magnet location. The longitudinal magnetic field component in the direction of magnetization is bounded everywhere, but discontinuous across the magnet faces parallel to the magnetization direction. The transverse magnetic fields are logarithmically unbounded on approaching a vertex of the magnet. doi:10.1017/S1446181120000097

A variant of the experiment to observe the optical Aharonov—Bohm effect has been discussed. In the experiment, a unipolar subcycle light pulse has been proposed as a source of the vector potential acting on electrons and having zero electric field strength in the region of a nonzero vector potential. Such a relation between the field and potential appears because the unipolar pulse has a nonzero electric area. An unusual situation where the fact of the passage of the pulse in one of the arms of two-beam interferometer is stored for a very long time after the passage and can be recorded in a two-beam electronic interferometer by a shift of fringes has been discussed. Detailed analysis of the phase shift under the influence of the vector potential created by the inhomogeneous unipolar pulse removes this contradiction and shows the equality of phase incursions in both arms that is unobvious at first glance.

A three-dimensional (3-D) derivation of an analytical model for an electrodynamic wheel suspension (EDWS) device is proposed in this article. The equations for source field generated by a Halbach wheel are derived by the superposition principle of magnetic charge sheets. The eddy current field distribution within the semicircular pipe track is modeled using the second-order vector potential (SOVP) theory. The analytical models describing fields and forces are validated by both finite element simulations and experimental results. In addition, the characteristics of the EDWS device with two-degree-of-freedom (DOF) motions are studied based on the proposed equations.

We proposed a model for a spherical vector-potential coil and investigated the properties of the coil, including the uniform distribution of the vector potential. We modeled a structure in which thin toroidal coils are superposed along a spherical surface. Rather than arranging the toroidal coils in close contact with each other along the surface of the sphere, the layer spacing of the toroidal coils should be constant so that the projection onto the axis of the sphere is equally spaced. When a current was applied to a toroidal coil, no magnetic field was created outside the toroidal coils; instead, a vector potential was present. When the same current was passed through each toroidal coil, the vector potential in the sphere achieved a uniform distribution by superposition. We presented the results of our numerical simulation of the distribution of the vector potential and the induced voltage when a secondary conductor was passed through the coil.

We consider the presence of odd powers of the speed of light $c$ in the covariant nonrelativistic expansion of General Relativity (GR). The term of order $c$ in the relativistic metric is a vector potential that contributes at leading order in this expansion and describes strong gravitational effects outside the (post-)Newtonian regime. The nonrelativistic theory of the leading order potentials contains the full non-linear dynamics of the stationary sector of GR., Comment: 24 pages + appendices. Version accepted for publication by EPJC. Subsection 4.4 on possible phenomenological applications added

We deal with a class of magnetic pseudo-relativistic Hartree type equation (−i∇−A(x))2+m2u+W(x)u=(Iα∗|u|p)|u|p−2u,x∈RN, where N≥3, m>0, A:RN→RN is a continuous vector potential, W:RN→R is an extern...

In this paper, a 3D analytical model of permanent magnet (PM) and electromagnetic hybrid Halbach array electrodynamic suspension (EDS) system for a conductive plate is presented. When the hybrid Halbach array moves to cut the conductive plate, the 3D eddy current forces are derived using the second-order vector potential (SOVP) based on y-component of the source magnetic flux density, which is derived using the Biot–Savart law and surface current method. The accuracy of the derived equations is verified by a 3D finite-element analysis (FEA) model. Furthermore, the effects of the main characteristic parameters on the levitation force are analysed.

The calculation of 1D electromagnetic (EM) responses of dipole sources is considered as a fundamental computation kernel in several geophysical applications, ranging from EM sensitivity analysis to high-dimensional EM forward modeling and inversion. To solve the EM responses caused by electric and magnetic dipoles with different orientations, we have developed an open-source MATLAB code for a complete treatment, which allows for an arbitrary transmitter-receiver geometry in the multilayered earth. A complete derivation of the EM field expressions with simplified exponent terms for all traditional dipole sources is accomplished based on the vector potential differential equations. The quadrature-with-extrapolation algorithm is adapted for calculating the integration of Bessel functions to produce EM responses with a desired accuracy. The MATLAB digit functions are used to alleviate the instability problem caused by the exponent terms. The accuracy of our code is verified by comparing the results with the published data and self-checks on an infinite thin sheet. A layered model is designed to test the capability of our code for all traditional dipole sources by placing the receiver at different depths. In the end, we apply our code to marine and airborne controlled-source EM explorations. The results indicate that our code is accurate and can potentially be used to calculate 1D EM responses for general EM problems.

A Feynman path integral formula for the Schr\"odinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich integral in the path integral formula for both the Schr\"odinger and heat equation with magnetic field., Comment: 43 pages

Hybrid schemes that combine the Finite Element Method (FEM) and the Boundary Element Method (BEM) have been extensively used for the solution of nonlinear magnetostatic problems. The majority of those schemes rely either on reduced and total scalar or vector potential formulations. One main issue regarding the scalar potential formulations is the cancelation errors and difficulties on the definition of potential jump condition on the surface of a magnet, while in the case of vector potential formulations, the size of the simulated problem increases drastically especially in three dimensions. An alternative formulation has been proposed by Mayergoyz et al. (1987), which is based on scalar potentials and provides assistance in ameliorating the aforementioned problems. In the present work that formulation is implemented by employing a BEM/FEM scheme, for the solution of nonlinear magnetostatic problems. The proposed BEM/FEM formulation is employed for the solution of representative magnetostatic problems dealing with different types of superconducting accelerator magnets, while the provided accuracy is assessed through comparisons made with corresponding results obtained by a well-known commercial FEM package.

In a previous work, a hysteresis modeling, called G model, was proposed with the goal of describing this phenomenon by simple equations. In that work, we developed the direct model which uses the magnetic field as the independent variable. However, with the magnetic vector potential 2-D classical formulation, the magnetic induction is the independent quantity. The main contribution of this work is to present the inverse version of the G model adapted to vector potential formulation. In addition, a Mayergoyz vector hysteresis generalization is applied in order to consider rotational fields and couple the model with a 2-D vector potential finite-element method (FEM) formulation. In this approach, the anisotropy is inserted with the orientation distribution function (ODF).

In this work, the interaction potential between an atom (with definite Static Polarizability) and a perfect neutral conductive wall is evaluated in two approaches. First, using, the explicit form of the singlet and doublet parts of the vacuum vector potential operator in atomic quadratic stark shift formula and taking the expectation value in vacuum state. Second, writing, the quadratic stark shift formula in terms of electric field correlation function, relating the correlation function to the imaginary parts of the vector potential Green function tensor components (via fluctuation dissipation theorem in statistical mechanics) and the evaluation of the required vector potential Green function tensor components. The consistency of the two methods is shown.

В статье показано, что векторный потенциал в электродинамике является вспомогательной функцией, а в некоторых случаях может включать в себя одновременно не силовое магнитное поле. В связи с этим в эффекте Ааронова-Бома обладающая зарядом квантовая частица взаимодействует не с потенциалом, а с не силовым магнитным полем. При этом теорема Стокса, примененная к векторному потенциалу, остается верной и применимой для объяснения эффекта Ааронова-Бома за счет присутствия в векторном потенциале не силового тороидального магнитного поля. Электродинамика в токамаке находит объяснение также в терминах силовых и не силовых магнитных полей.

Purpose The purpose of this study is the introduction of finite permeability of ferromagnetic core in analytical approach for slotted spoke-type permanent magnet machine. Design/methodology/approach A two-dimensional analytical approach of magnetic field distribution is established for spoke-type permanent magnet machine to calculate the flux density distribution in the middle of airgap. The paper presents an analytical subdomain model accounting for stator slotting effect. The governing equations are obtained from Maxwell’s equations by using vector potential in all regions of the machine, i.e. magnet, airgap, stator slots and rotor/stator yoke. The finite element analysis is used to validate the analytical results. Findings It is found that the developed subdomain model including finite permeability of ferromagnetic core is accurate and is applicable for spoke-type permanent magnet machine for no-load and on-load condition. The analytical results are in accurate agreement with the numerical simulation. Originality/value Some assumptions and conditions are presented to improve and simplify the analytical method for analyzing the global saturation for spoke-type permanent magnet machine.

A correct theory of transformers with rod magnetic circuits is presented, which is based on the idea according to which the power from the primary winding is transferred to the secondary one by means of the Poynting vector, and on a preliminary analysis of the vector potential distribution in them with ideal magnetic permeability of steel. An adaptive equivalent circuit is constructed, which is called so because in addition to reflecting the magnetic fluxes in the steel sections, core winding window, and fringing flow, all its nodes correspond to some boundary surface of the magnetic circuit or windings. Owing to this topological property, the circuit allows one to switch to a model that directly reflects such physical quantities as the vector potential, flow function, and electric field strength at the boundary surfaces of steel and windings. The transition to the equivalent circuit with real steel is done by adding nonlinear branches that take into account saturation and iron losses. In contrast to the traditional theory, their total current is considered as a leakage current due to steel non-ideality and not as a magnetization current. All of the presented adaptive equivalent circuits are suitable for analyzing steady-state and transient processes with identifying over- and anti-fluxes in comparison with idling fluxes, a feature which puts the analysis of transformer electrodynamic stability to emergency situations to a higher level.

The article compares the electrodynamics of toroidal non-forceful andpoloidal electromagnetic force fields developed by the author in the light of thewell-known Maxwell equations. It is proved that the standard Maxwell equationsdo not include toroidal currents, which are the source of toroidal non-forcefulmagnetic fields. Therefore, a theorem on the limits of applicability of the standardMaxwell equations to the electrodynamics of the natural electromagnetic fieldon Earth is formulated and proved. The sources of toroidal magnetic fields onEarth are indicated. A theorem on the restoration of toroidal magnetic fields onEarth is proved based on data on the normal component of the total magneticfield on a regular surface (analogous to the Helmholtz theorem). New paradigms(stationary and quasi-stationary) are proposed for studying the electromagneticfield in geomagnetism instead of the static paradigm of K.F. Gauss. The existenceof a zero Lorentz force in a natural electromagnetic field is proved.

We consider the problem of recovering the divergence-free velocity field UϵL2 (Ω) of a given vorticity F = curl U on a bounded Lipschitz domain Ω = R3. To that end, we solve the ‘div–curl problem’ for a given F ϵ H-1 (Ω).The solution is expressed in terms of a vector potential (or stream function) A ϵ H1 (Ω) such that U = curlA. After discussing existence and uniqueness of solutions and associated vector potentials, we propose a well-posed construction for the stream function. A numerical method based on this construction is presented, and experiments confirm that the resulting approximations display higher regularity than those of another common approach, Mathematical Models and Methods in Applied Sciences, 45 (3), ISSN:0218-2025, ISSN:1793-6314

Nondipole effects in processes assisted by a THz field having the strength of a few MV/cm can be significant due to its long wavelength. We illustrate this for strong-laser-field-induced ionization assisted by a THz field. To this end, we generalize our strong-field-approximation theory so that it includes the first-order term in a 1/c expansion of the vector potential. We show that in this case, in addition to a shift of the maximum of the photoelectron momentum distribution, the differential ionization probability as well as the cutoff energy can be significantly increased. For an explanation of these unexpected results we use the saddle-point method adjusted to include nondipole effects.