Spherical magnetic multipole moments system of currents

Concepts of spherical magnetic multipoles that represent distributions of electric currents over a spherical surface are introduced. Vector potentials of magnetic multipoles meet solenoidal- and harmonic-field conditions outside of the spherical surface and are continuous on it. Within the sphere, the vector potential of currents flowing outside of it is represented by the sum of vector potentials of basis magnetic multipoles with coefficients expressed by spherical multipole moments of system of currents. This expansion of the vector potential is in many respects analogous to the multipole expansion known from electrostatics. The first three terms of the expansion represent components of the well-known magnetic moment, the next five terms represent components of the magnetic quadrupole moment, etc. Possible applications of the magnetic spherical multipole technique are discussed. Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 66–72, October, 1999.     

Cardiac magnetic source imaging based on current multipole model

It is widely accepted that the heart current source can be reduced into a current multipole. By adopting three linear inverse methods, the cardiac magnetic imaging is achieved in this article based on the current multipole model expanded to the first order terms. This magnetic imaging is realized in a reconstruction plane in the centre of human heart, where the current dipole array is employed to represent realistic cardiac current distribution. The current multipole as testing source generates magnetic fields in the measuring plane, serving as inputs of cardiac magnetic inverse problem. In the heart-torso model constructed by boundary element method, the current multipole magnetic field distribution is compared with that in the homogeneous infinite space, and also with the single current dipole magnetic field distribution. Then the minimum-norm least-squares (MNLS) method, the optimal weighted pseudoinverse method (OWPIM), and the optimal constrained linear inverse method (OCLIM) are selected as the algorithms for inverse computation based on current multipole model innovatively, and the imaging effects of these three inverse methods are compared. Besides, two reconstructing parameters, residual and mean residual, are also discussed, and their trends under MNLS, OWPIM and OCLIM each as a function of SNR are obtained and compared.     

The two-body multipole problem of electrodynamics

A 2-body system composed of two objects having arbitrary distributions of charge and current is discussed. An expression for the velocity dependent potential between these two objects has been obtained in the non-relativistic approximation. This potential consists of two parts viz. a velocity independent scalar potential Φ_eff and another part which is linearly dependent on the relative velocity between the objects. The second part naturally suggests a vector potential A_eff. The potentials have been expanded into multipole terms. It has been found that Φ_eff is a sum of two components viz. Φ_EE and Φ_MM such that each multipole term in Φ_EE represents an interaction between the electric multipoles of the two systems, each term in Φ_MM represents an interaction between their magnetic multipoles whereas each term in A_eff represents an interaction between an electric multipole of one and a magnetic multipole of the other. The results have been applied to the interaction between an electric dipole and a magnetic dipole. The symmetry among the multipole terms in A_eff suggests vanishing vector potential between two identical objects. A corollary of this appears to be absence of spin orbit interaction between two identical particles in the same spin state.     

Elastic charge density representation of the interaction via the nematic director field

The interaction between particle-like sources of the nematic director distortions (e.g., colloids, point defects, macromolecules in nematic emulsions) allows for a useful analogy with the electrostatic multipole interaction between charged bodies. In this paper we develop this analogy to the level corresponding to the charge density and consider the general status of the pairwise approach to the nematic emulsions with finite-size colloids. It is shown that the elastic analog of the surface electric charge density is represented by the two transverse director components on the surface imposing the director distortions. The elastic multipoles of a particle are expressed as integrals over the charge density distribution on this surface. Because of the difference between the scalar electrostatics and vector nematostatics, the number of elastic multipoles of each order is doubled compared to that in the electrostatics: there are two elastic charges, two vectors of dipole moments, two quadrupolar tensors, and so on. The two-component elastic charge is expressed via the vector of external mechanical torque applied on the particle. As a result, the elastic Coulomb-like coupling between two particles is found to be proportional to the scalar product of the two external torques and does not directly depend on the particles' form and anchoring. The real-space Green function method is used to develop the pairwise approach to nematic emulsions and determine its form and restrictions. The pairwise potentials are obtained in the familiar form, but, in contrast to the electrostatics, they describe the interaction between pairs (dyads) of the elastic multipole moments. The multipole moments are shown to be uniquely determined by the single-particle director field, unperturbed by other particles. The pairwise approximation is applicable only in the leading order in the small ratio particle size-to-interparticle distance as the next order contains irreducible three-body terms.     

Multipole structure of static continuous matter distributions in general relativity

A mass skeleton is defined for a static extended body in a gravitational field. It is a scalar-valued distribution on a tangent space, and is equivalent to that part of the reduced multipole moment structure which describes the mass density of the body. An explicit form is given for this distribution in terms of the mass density and the scalar potential of the field. It is deduced that the mass skeleton and the scalar potential are not completely independent. The smoothness of the mass distribution imposes certain weak restrictions on those scalar potentials which are compatible with a given mass skeleton.Dedicated to Achille Papapetrou on the occasion of his retirement.     

On Three Magnetic Relativistic Schr?dinger Operators and Imaginary-Time Path Integrals

Three magnetic relativistic Schr?dinger operators are considered corresponding to the classical relativistic Hamiltonian symbol with magnetic vector and electric scalar potentials. We discuss their difference in general and their coincidence in the case of constant magnetic fields, as well as whether they are covariant under gauge transformation. Then results are surveyed on path integral representations for their respective imaginary-time relativistic Schr?dinger equations, i.e. heat equations, by means of the probability path space measure coming from the Lévy process concerned.     

Angular spectrum representation of the electromagnetic multipole fields, and their reflection at a perfect conductor

Angular spectrum representations are derived for electric and magnetic multipole fields of arbitrary order. The result involves generalized spherical harmonics and generalized vector spherical harmonics, and the representations are in the form of integrals over the k_∥-plane. The representations are especially useful for the study of reflection and transmission of multipole radiation by a plane interface. As an example, we have considered the reflection at a perfect conductor. The reflected field of a multipole field could be expressed in the form of an angular spectrum with a very simple relation to the angular spectrum of the source field. The radiation pattern of a multipole near the perfect conductor is obtained with the method of stationary phase. We also introduce a method for determining the mirror image of the source of an arbitrary multipole.     

Spherically symmetric space-times transparent to scalar multipole waves

Using nonscattering potentials of Chang and Janis, a large class of spherically symmetric space-times is constructed on which all multipole solutions to the minimally coupled scalar wave equation are expressible in terms of characteristic data functions in essentially as simple a fashion as for flat space-time. The space-times are transparent to multipole waves in the same sense that flat space-time is. Both conformally flat and not conformally flat space-times are obtained. Some examples are discussed which show that the variety of transparent space-times is large even within the class of Robertson-Walker spaces.     

Light scattering by axisymmetric multilayer particles

A new recursive algorithm for solving the problem of scattering a plane electromagnetic wave by axisymmetric dielectric multilayer particles is constructed. The approach that was proposed earlier and demonstrated for uniform axisymmetric particles is used. It has the following basic features: (1) the fields are represented in the form of a sum of two terms, one of which is independent of the azimuthal angle, whereas averaging of the second term over this angle gives zero; (2) the axisymmetric problem is solved by using the scalar potentials related to the azimuthal components of electromagnetic fields; and (3) the non-axisymmetric problem is solved by using the superposition of Debye potentials and vertical components of the magnetic and electric Hertz vectors. It is of principal importance for the solution proposed here that the scattering problem is formulated in the form of surface integral equations in these scalar potentials, which are represented in the form of expansions in wave spherical functions. Infinite systems of linear algebraic equations for unknown expansion coefficients are obtained, which are rather simple in structure. The reduced systems for multilayer particles have the same dimension as the systems for identical uniform particles. In the case of multilayer spherical particles, the algorithm gives an explicit solution to the problem, and the dependence on the radial spherical functions for the layers is specified in terms of the derivative of the logarithm (i.e., the ratio of the derivative to the function itself) and the ratio of the functions of neighboring layers. Numerical calculations demonstrated the high efficiency of the algorithm.     

STUDIES OF THE FIELD DISTRIBUTION OF THE MODULATING WIDTH MULTIPOLE PERMANENT MAGNETS

In this paper,the magnetic field distribution of the modulating width multipole permanent magnets is analyzed. Only radially magnetized permanent magnet segments are used and the required magnetic field is formed by modulating the widths of the segments in this project. In 2-dimensional ideal case pure 2N-pole magnetic field can be formed. The spatial expressions of the scalar potential for the ideal and real 2-dimensional cases are given. And the methods for reducing the undesirable higher harmonics of the magnetic field in the real case are also discussed.     

A faster aggregation for 3D fast evanescent wave solvers using rotations

A novel technique to accelerate the aggregation and disaggregation stages in evanescent plane wave methods is presented. The new method calculates the six plane wave radiation patterns from a multipole expansion (aggregation) and calculates the multipole expansion of an incoming field from the six plane wave incoming field patterns. It is faster than the direct approach for multipole orders larger than one, and becomes six times faster for large multipole orders. The method relies on a connection between the discretizations of the six integral representations, and on the fact that the Wigner D-matrices become diagonal for rotations around the z-axis. The proposed technique can also be extended to the vectorial case in two different ways, one of which is very similar to the scalar case. The other method relies on a Beltrami decomposition of the fields and is faster than the direct approach for any multipole order. This decomposition is also not limited to evanescent wave solvers, but can be used in any vectorial multilevel fast multipole algorithm.     

电矩张量与旋转带电体的磁矩

基于旋转带电体的磁矩与刚体转动惯量之间的类比关系,引入带电体的一个不同于电四极矩的电矩张量的概念,进而引入标量电矩二次曲面及电矩主轴的概念,借助正交变换及电矩张量矩阵的本征值理论,推导出沿任意方向定轴旋转带电体的磁矩的计算公式及电矩张量的若干性质,并举例说明.     

Scalar and Pseudo-Scalar Form Factors in Electro- and Weak-Production of Pion Near Threshold

We investigated pion production near threshold by the weak current in terms of multipole amplitudes. By exploiting the chiral Ward identity based on the QCD Lagrangian, we derived relevant multipole amplitudes in closed forms and presented their numerical results. In the amplitudes, scalar and pseudo scalar (PS) form factors, which represent the scalar and the PS quark density distributions, manifest by themselves. We applied these amplitudes to the cross sections for the weak- and electro-production near threshold. Both pion and PS form factor contributions are shown to account for the t-channel contribution in the charged pion electro-production near threshold. The asymmetry on the pion production by the neutrino and anti-neutrino is also discussed with their longitudinal and transverse cross sections.     

On the Frenkel problem of equivalent steady currents in a sphere

The Frenkel problem of substituting the 3D system of steady currents given in one of two concentric spherical regions by an equivalent system of currents (i.e., by that inducing the same external magnetic field) that is distributed over the surface of the other region is considered. A method of multipole moments providing the direct solution (without calculating the fields) of the problem is described. The case of currents with the density components represented by cubic polynomials of the Cartesian coordinates is considered as an example.     

Sources,potentials and fields in Lorenz and Coulomb gauge: Cancellation of instantaneous interactions for moving point charges

We investigate the coupling of the electromagnetic sources (charge and current densities) to the scalar and vector potentials in classical electrodynamics, using Green function techniques. As is well known, the scalar potential shows an action-at-a-distance behavior in Coulomb gauge. The conundrum generated by the instantaneous interaction has intrigued physicists for a long time. Starting from the differential equations that couple the sources to the potentials, we here show in a concise derivation, using the retarded Green function, how the instantaneous interaction cancels in the calculation of the electric field. The time derivative of a specific additional term in the vector potential, present only in Coulomb gauge, yields a supplementary contribution to the electric field which cancels the gradient of the instantaneous Coulomb gauge scalar potential, as required by gauge invariance. This completely eliminates the contribution of the instantaneous interaction from the electric field. It turns out that a careful formulation of the retarded Green function, inspired by field theory, is required in order to correctly treat boundary terms in partial integrations. Finally, compact integral representations are derived for the Liénard–Wiechert potentials (scalar and vector) in Coulomb gauge which manifestly contain two compensating action-at-a-distance terms.     

A Toroidal Magnetic Field Theorem

In the framework of magnetohydrodynamics the kinematic dynamo problem in a spherical fluid volume as well as in a plane layer is considered. On the premises of a purely toroidal magnetic field a nonlinear evolution equation for the toroidal scalar is derived. In this equation the flow field is constrained in such a way that no poloidal magnetic field can arise, but is otherwise arbitrary; the magnetic diffusivity is assumed to be spherically (horizontally, resp.) symmetric. Solutions of this problem are of particular interest since the magnetic field is confined to the fluid volume and therefore invisible to an external observer. It is proved in this paper that the maximum norm of smooth solutions of this equation decays exponentially fast to zero. Thus, dynamo solutions, i.e. nondecaying solutions, of this type do not exist.     

Solution of the Dirichlet problem for the Laplace equation for a multiply connected region with point symmetry

A method is proposed which uses an expansion of the potential in irreducible representations of the symmetry group of the field-defining elements of a system. A boundary-value problem is solved for multipole systems with planar plate electrodes for the C _nv symmetry group. A quadrature expression is obtained for the field potential of these systems. Constraints imposed on the electrode potentials, under which such a solution is possible, are determined. Results of calculations of the potential distribution are presented for various specific systems. Zh. Tekh. Fiz. 69, 1–9 (March 1999)     

Numerical simulation of the field distribution produced by the SP-40 magnet of the MARUSYA setup and comparison of simulation results with experimental data

In terms of the magnetostatic problem in the differential statement, a mathematical simulation of the 3D distribution of the magnetic field generated by the SP-40 magnet of the MARUSYA experimental setup (Laboratory of High Energies, Joint Institute for Nuclear Research, Dubna) is carried out. The mathematical statement of the direct magnetostatic problem is given. Computational procedures and field calculation algorithms with one vector and two scalar potentials are described. The simulated magnetic field distribution is compared with experimental data for the field of the modified SP-40 magnet with a pole spacing of 0.207 m. The results of this work are used in experimental data processing and are also helpful in simulation of magnetooptic systems.     

Class of Einstein–Maxwell phantom fields: rotating and magnetized wormholes

Using a new ansatz for solving the Einstein equations with a scalar field with inverted sign in the kinetic term (phantom field), I find a series of formulae to derive axial symmetric stationary exact solutions of the phantom fields in general relativity. I focus on the solutions which represent wormholes. The procedure presented in this work allows to derive new exact solutions up to very simple integrations. Among other results, I find exact rotating solutions containing magnetic monopoles, dipoles, etc., coupled to phantom scalar and to gravitational multipole fields.     

线性瞬态涡流电磁场定解问题解的唯一性和稳定性

线性瞬态涡流电磁场定解问题的主要特点是边界条件使用磁感应强度的法向分量边界条件代替了电场强度的切向分量边界条件，约束方程中忽略了位移电流.这种具有特殊性的定解问题的解是否唯一和稳定对于求解瞬态涡流电磁场而言是一个基本问题.本文在非涡流区引入标量位函数，证明了在推导过程中起重要作用的辅助函数的存在性.通过推导线性瞬态涡流电磁场定解问题的能量估计式，证明了该定解问题的解是唯一的，并且关于初始条件和外源项是稳定的.本结果对于线性瞬态涡流电磁场的求解有一定的指导意义.作为应用，给出了通有单脉冲电流的单匝圆环线圈与球形导体共轴的涡流问题的解析解. 关键词： 瞬态涡流电磁场 能量估计式 唯一性 稳定性