Modeling the electromagnetic field of distant sources in a nonhomogeneous medium

Asymptotic formulas are derived for calculating the far-zone field inside a layered medium induced by a vertical magnetic dipole located on the surface of a horizontally homogeneous layered Earth with a three-dimensional nonhomogeneity. The asymptotic solution is compared with the exact solution. The far-zone dipole field is simulated for the model of a horizontally homogeneous layered earth with a three-dimensional nonhomogeneity. The simulation results show that the 3D-nonhomogeneity is easily detected from electromagnetic measurements with an artificial dipole source on the Earth’s surface.     

Electromagnetic interaction of moving conductors with magnets: kinematic solutions for infinitely long square and cylindrical geometries

The electromagnetic drag force on a point dipole near a moving conductor caused by the induced electric currents is investigated by numerical and analytical computations. Our focus is on prototypical configurations for Lorentz force velocimetry, i.e. velocity measurement from the electromagnetic drag force on the dipole. We examine the particular cases of conducting infinite bars of square or round cross-section, which are moving with constant velocity in the field of arbitrary oriented magnetic dipole. In addition, we study the laminar liquid-metal flow in a square duct. The motion of the conductor is prescribed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On certain cases of determination of trajectories of plane conservative motion of a particle : PMM vol. 43, no. 6, 1979 pp. 1111–1113

A method is developed for obtaining an incomplete integral of the two-dimensional Hamilton— Jacobi equation for a particle when a particular solution that satisfies a special equation in partial derivatives is known. The equation for the trajectory of dipole particles moving with zero total energy in an arbitrary two-dimensional electrostatic field is obtain, as an example.     

Laminar magnetohydrodynamic duct flow in the presence of a magnetic dipole

We study magnetohydrodynamic flow of a liquid metal in a straight duct. The magnetic field is produced by an exterior magnetic dipole. This basic configuration is of fundamental interest for Lorentz force velocimetry (LFV), where the Lorentz force opposing the relative motion of conducting medium and magnetic field is measured to determine the flow velocity. The Lorentz force acts in equal strength but opposite direction on the flow as well as on the dipole. We are interested in the dependence of the velocity on the flow rate and on strength of the magnetic field as well as on geometric parameters such as distance and position of the dipole relative to the duct. To this end, we perform numerical simulations with an accurate finite-difference method in the limit of small magnetic Reynolds number, whereby the induced magnetic field is assumed to be small compared with the external applied field. The hydrodynamic Reynolds number is also assumed to be small so that the flow remains laminar. The simulations allow us to quantify the magnetic obstacle effect as a potential complication for local flow measurement with LFV. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Distortion of liquid metal flow in a square duct due to the influence of a magnetic point dipole

We consider liquid metal flow in a square duct with electrically insulating walls under the influence of a magnetic point dipole using three-dimensional direct numerical simulations with a finite-difference method. The dipole acts as a magnetic obstacle. The Lorentz force on the magnet is sensitive to the velocity distribution that is influenced by the magnetic field. The flow transformation by an inhomogeneous local magnetic field is essential for obtaining velocity information from the measured forces. In this paper we present a numerical simulation of a spatially developing flow in a duct with laminar inflow and periodic boundary conditions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vortex Generation in a Liquid Metal Duct Flow near a Magnetic Dipole

The generation of vortical structures by a strong magnetic dipole field in a liquid metal duct flow is studied by means of three-dimensional direct numerical simulations. The dipole is considered as the paradigm for a magnetic obstacle which will deviate the streamlines due to Lorentz forces which act on the fluid elements. Our model uses the quasi-static approximation applicable in the limit of small magnetic Reynolds numbers. The analysis covers the stationary flow regime at smaller flow Reynolds numbers Re as well as the fully time-dependent regimes at higher values with a turbulent flow in the wake of the magnetic obstacle. We present a systematic study of these two basic flow regimes on Re and the Hartmann number Ha, a measure of the strength of the magnetic dipole field. Furthermore, three orientations of the dipole are compared, the streamwise, spanwise and wall-normal ones. The most efficient generation of turbulence at a fixed distance above the duct occurs for the spanwise orientation in which we can observe the formation of Hartmann layers at the top plate. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wave function and the probability current distribution for a bound electron moving in a uniform magnetic field

We study the effects of electromagnetic fields on nonrelativistic charged spinning particles bound by a short-range potential. We analyze the exact solution of the Pauli equation for an electron moving in the potential field determined by the three-dimensional δ-well in the presence of a strong magnetic field. We obtain asymptotic expressions for this solution for different values of the problem parameters. In addition, we consider electron probability currents and their dependence on the magnetic field. We show that including the spin in the framework of the nonrelativistic approach allows correctly taking the effect of the magnetic field on the electric current into account. The obtained dependences of the current distribution, which is an experimentally observable quantity, can be manifested directly in scattering processes, for example.     

An algorithm to determine a dipole current in a sphere

Suppose a spherical conductor has a single dipole. If it is assumed that the direction of current is perpendicular to the location of dipole, then we prove that the dipole can be obtained as an exact form from the induced magnetic field and provide an easy algorithm to locate the dipole from the samples of a magnetic field outside the sphere without using any series approximation. A numerical implementation is also presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Allowed regions in the problem concerning dynamics of a charged particle in a superposition of dipole and uniform magnetic fields

The motion of charged particles in the Earth’s magnetic field has been of interest to mathematicians and physicists in connection with the study of the polar aurora and cosmic rays. In 1907, Norwegian mathematician Stromer gave the mathematical formulation of this problem. It became the problem of great importance with the discovery of the Van Allen radiation. As is known, the Earth’s magnetic field can be considered approximately as a superposition of dipole and uniform magnetic fields, and the dipole’s magnetic moment is either parallel or antiparallel to the induction of the uniform field. Thus, the problem concerning the dynamics of the charged particle in the magnetic field of the Earth is reduced to that of charged particle dynamics in the composed field. The paper is devoted to the construction and investigation of the allowed regions in a superposition of dipole and uniform magnetic fields for positive values of Stormer’s constant γ and the same orientation of magnetic moment and uniform field.     

Irradiation of a cone by a magnetic dipole

Th electromagnetic field produced by a magnetic dipole in thepresence of a perfectly conducting cone of arbitrary cross-sectionis determined. The solution is used to find out how a currenton the cone travelling towards the apex is reflected. Some valuesof the reflection coefficient are calculated. In particular,it is shown that there is a sort of resonance with the reflectionincreasing significantly as the cone approaches a plane.     

Point force in a stratified fluid in a cylinder under a magnetic field

The flow induced by a body moving in an inviscid incompressible density stratified fluid in an infinite circular cylinder under the influence of a uniform axial magnetic field is studied using the method of replacing the body by an isolated point force. This method was adopted by Childress and others in discussing the body effects in a viscous fluid. The solution is obtained using the Fourier transformation and the Lighthill’s radiation condition. The cases of weak and strong magnetic fields are discussed.     

Nonlinear electrodynamic lensing of electromagnetic waves in a magnetic dipole field

We consider propagation of electromagnetic waves in magnetic dipole and gravitational fields proceeding in accordance with the nonlinear vacuum electrodynamics laws. We derive formulas describing the effect of nonlinear electrodynamic lensing of electromagnetic waves in the magnetic dipole field. We show that rotation of the magnetic dipole moment about an axis noncoincident with this moment leads to a nonlinear electrodynamic modulation of the electromagnetic radiation intensity by frequencies that are multiples of the dipole rotation frequency. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 85–94, January, 2007.     

Electromagnetic interaction of a conducting cylinder with a magnetic dipole caused by steady rotation

The motion of a conductor in a magnetic field induces eddy currents whose interaction with the field produces Lorentz forces opposing the motion. One can determine the velocity of the conductor from the force on the magnet system since the latter is equal but opposite to the Lorentz force on the conductor. This contactless method is known as Lorentz force velocimetry (LFV). We study an idealized configuration of LFV, i.e. a rotating solid cylinder interacting with a point dipole. The understanding of parameter influences in this setup can be helpful for more realistic configurations. We use a purely kinematic approach appropriate for low magnetic Reynolds numbers. Numerical results for small and large distances between dipole and cylinder have been obtained with the commercial software COMSOL Multiphysics. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The interaction between sound and circulatory fluid motion as a problem in matched expansions

A circular cylinder is at rest in a compressible fluid witha given circulation K'. At time t=0 the cylinder is made tomove with low Mach number along a straight line perpendicularto its axis. A modified matching argument is used to describethe sound field induced by the lifting body. The velocity potentialin the sound field can be represented, to leading order, interms of a moving line dipole aligned along the direction ofmotion together with a transverse dipole sheet that extendsfrom the starting location of the centre of the cylinder toits current location. The next-order term is that of a movingline source. The line dipole accounts for the motion of thecylinder. The dipole sheet represents the sound field due tothe circulatory motion. If the circulation is constant, thenso is the strength of the dipole sheet and the time dependencearises from the changing length of the layer. In a more realisticcase, where vorticity is shed to form a wake behind the movingbody, there is a corresponding change in the circulation andin the strength of the evolving dipole sheet.     

Neutral Fermion Having Electric and Magnetic Moments in an External Electromagnetic Field

We show that in 2+1 dimensions, the Dirac equation for a neutral fermion possessing electric and magnetic dipole moments in an external electromagnetic field reduces to the Dirac equation for a charged fermion in a external field characterized by a certain 3-pseudo-vector potential. The effective charge of the neutral fermion is determined by its dipole moments. The effects of coupling electric and magnetic moments of the neutral fermion to the external electromagnetic field seem to be inseparable in physical experiments of any type. We find an exact solution of the Dirac equation for a massive neutral fermion with electric and magnetic dipole moments in a external plane-wave electromagnetic field. We derive expressions for the fermionic vacuum current induced by neutral fermions in the presence of external electromagnetic fields.     

The electromagnetic field of a dipole radiation source in a slowly moving conducting medium

We solve the problem of determining the electromagnetic field created by a dipole radiation source in a conducting, homogeneous, isotropic, slowly moving medium. The investigations are carried out depending on the orientation of the dipole source and the velocity vector of the medium.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 31–35.     

Electric dipole in a magnetic field: Bound states without classical turning points

We show that both rigid and nonrigid dipoles can be trapped by an uniform external magnetic field in classical mechanics. The trapped states of the dipole present a nontrivial example of classical bound states embedded in a continuum (BSEC) that can be treated as analogues of quantum BSECs. For example, the classical motion of the dipole is confined to a finite region in space although there are no classical turning points. We also examine the quantum motion of the dipole in a magnetic field and show that for the most natural choices of the parameters, no quantum BSEC solutions exist. The possibilities of experimental investigations of BSECs are discussed. Deceased. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp. 189–205, November, 1998.     

The ABCs of Magnetic Monopole Dynamics

We revisit the explicit solution of Newton's equation of motion (mass times acceleration equals force) for a (pointlike, massive) magnetic monopole moving in the Coulomb field of a (fixed, pointlike) electric charge or, equivalently, of a (pointlike, massive) electric charge moving in the magnetic field of a (fixed) magnetic monopole.     

Influence of moving magnetic field on three dimensional Couette flow

A three dimensional steady fully developed MHD Couette flow of a viscous incompressible electrically conducting fluid is analysed. The lower stationary porous plate is subjected to a periodic injection velocity and the upper porous plate in uniform motion to a constant suction velocity. A magnetic field of uniform strength applied normal to the planes of the plates is fixed with the moving plate. Neglecting the induced magnetic field, an approximate solution for the flow field is obtained and discussed with the help of graphs.     

Influence of moving magnetic field on three dimensional Couette flow

A three dimensional steady fully developed MHD Couette flow of a viscous incompressible electrically conducting fluid is analysed. The lower stationary porous plate is subjected to a periodic injection velocity and the upper porous plate in uniform motion to a constant suction velocity. A magnetic field of uniform strength applied normal to the planes of the plates is fixed with the moving plate. Neglecting the induced magnetic field, an approximate solution for the flow field is obtained and discussed with the help of graphs.