Biot-Savart Equation Research Articles

Fully Microscopic Treatment of Magnetic Field Using Bogoliubov–De Gennes Approach

This work introduces an algorithm designed to solve the Bogoliubov–de Gennes equations of superconductivity theory. What sets this algorithm apart is its remarkable ability to precisely and consistently consider the impact of an external magnetic field, all within the microscopic approach. The computation scheme’s convergence is guaranteed by addressing the Biot–Savart equation for the field where the vector potential appears on both of its sides. To showcase the capabilities of this approach, we provide several key examples: the Abrikosov lattice, vortex core states, and the vortex structure in the intermediate mixed state of a superconductor. This method promises to offer valuable insights into the microscopic physics of intertype superconductivity.

The time-development of energy spectrum of a quantized vortex ring

The present work is devoted to the study of time dependence of the energy spectrum that is associated with quantized vortex ring movement in the superfluid helium at different temperatures. Temperature changes from 1.3 K to 1.9 K. The full Biot-Savart equation is used to calculate velocity fields. The energy spectrum is calculated by using the method of structure functions. It is established that the structure function amplitude is associated with energy dissipation, while the energy spectrum is independent of temperature and time. The relationship between the energy dissipation rate and the vortex ring length is also discussed.

Current distribution reconstruction in switching arcs by means of regularization based on GSVD

The knowledge of the variation of arc current is helpful in improving the breaking performance of a vacuum circuit breaker. As a novel non-intrusive method, arc magnetic performance testing technology has attracted attention. In view of the widely used regularization method of the standard form in arc current reconstructions cannot fully overcome the ill-posed nature of the Biot-Savart equation, the regularization approach based on generalized singular value decomposition (GSVD) is proposed to improve the situation. Moreover, a discrete 3-D arc model with the help of the finite element technique incorporating the current divergence-free and boundary conditions is introduced to improve the authenticity of the model. The performance of the truncated generalized singular value decomposition (TGSVD) and general-form Tikhonov regularization with derivative operators L 1 and L 2 are studied in numerical tests. In addition, the effectiveness of generalized cross validation criterion, quasi optimality criterion and normalized cumulative periodic graph method in selecting regularization parameters are compared. The methodology can help improve arc control and guide the design of vacuum circuit breakers with improved performance parameters.

The generalized spin-orbit interaction: A microscopic origin of the Ørsted magnetic field

This work introduces a generalization of the form of the spin-orbit interaction, the generalized spin-orbit interaction (GSOI). It expresses the magnetic field induced by two charged particles moving with a non-zero relative velocity as a field defined at all points in space, and exists in the reference frames of both particles. This is in contrast to spin-orbit interaction theory, in which the generated magnetic field is defined at only one point in space, and exists in the reference frame of one of the two particles. At the macroscopic scale, it is shown that the GSOI theory implies the same form of the Ørsted magnetic field produced by a current-carrying wire. However, the theory is incompatible with the microscopic form of the Biot-Savart equation that implies that a charged particle induces a magnetic field by having a non-zero velocity. The implications of the GSOI theory on properties of the Ørsted magnetic field in current-carrying atomically thin two-dimensional materials, such as graphene, are discussed. The framework established in this paper aims at re-imagining classical physical concepts in light of an advanced microscopic understanding.

Three-Dimensional Localization of a Robotic Capsule Endoscope Using Magnetoquasistatic Field

Recent research efforts regarding advanced Robotic Capsule Endoscopes (RCEs) have primarily focused on the development of actively locomotive endoscope capsules. However, accurate movement of an RCE inside the digestive organs remains a challenge that hinders the further development of an autonomous RCE that with applicability in clinical practice. To address this challenge, this study proposed and developed a novel three-dimensional (3D) location positioning method that is compatible with an RCE manipulated by an external magnetic actuation system. The developed localization methodology employed one embedded single-axis receiving coil (Rx) in the RCE and three external transmitting coils (Txs) placed under the clinical bed. The magnetic flux density obtained from the electromotive force at the Rx was applied to the solution of 3D nonlinear Biot–Savart equations and enabled the determination of the position of the Rx in relation to the corresponding magnetoquasistatic field source in the Tx. For implementation, this study developed: (1) an accurate mathematical model and volumetric analysis method for the magnetoquasistatic field by applying equipotential contour and surface mapping, (2) a method to determine the optimal Tx arrangement, and (3) a prototyped device and in-vitro validation of the feasibility of the 3D localization. In the helical trajectory tracking experiment, the device demonstrated an error of 2.03 ± 1.14 mm, and the feasibility in the clinical environment was verified through gastrointestinal phantom experiments. The proposed method will be further evaluated clinically for the retargeting and accurate localization of internal pathologies as well as the closed-loop control of an actively locomotive RCE.

The time dependences of the main properties of the vortex loops just after reconnection in superfluid helium

We investigate a single vortex reconnection event in superfluid helium at zero temperature, when mutual friction is absent, and at 1.9 K temperature, when mutual friction essentially changes the evolution of lines. The vortex filament method with full Biot-Savart equation is used for numerical investigation of the time dependences of the main properties of the vortex loops just after reconnection. In this paper, we present the obtained dependencies of the average curvature of the loop, Lamb momentum, angular momentum, and energy from time. The effect of mutual friction force and initial conditions on these properties is studied.

Switching arc inversion based on analysis of electromagnetic characteristics

Abstract It is difficult to describe the characteristics of an arc in low-voltage switching equipment in terms of its most essential features using conventional arc models. In this paper, arc inversion is introduced to explore a new research approach to examine the nature of low-voltage switching arcs. Based on electromagnetic analysis, the arc is equivalent to a group of threadlike current segments. Then, the arc parameters are obtained by inverting the magnetic data calculated at the sampling point array. The multi-population genetic algorithm (MPGA) is adopted to solve the Biot-Savart operator equations to search the arc position, with the current density inverted by the truncated generalized singular value decomposition (TGSVD), in which the regularization parameter is chosen by the generalized cross-validation (GCV) criterion. The result shows that, by combining the MPGA with TGSVD, both the position and current distribution of an arc can be reconstructed accurately, which can help realize arc control and guide the design of low-voltage switches with improved performance parameters.

Reconnection scaling in quantum fluids

Fundamental to classical and quantum vortices, superconductors, magnetic flux tubes, liquid crystals, cosmic strings, and DNA is the phenomenon of reconnection of line-like singularities. We visualize reconnection of quantum vortices in superfluid 4He, using submicrometer frozen air tracers. Compared with previous work, the fluid was almost at rest, leading to fewer, straighter, and slower-moving vortices. For distances that are large compared with vortex diameter but small compared with those from other nonparticipating vortices and solid boundaries (called here the intermediate asymptotic region), we find a robust 1/2-power scaling of the intervortex separation with time and characterize the influence of the intervortex angle on the evolution of the recoiling vortices. The agreement of the experimental data with the analytical and numerical models suggests that the dynamics of reconnection of long straight vortices can be described by self-similar solutions of the local induction approximation or Biot-Savart equations. Reconnection dynamics for straight vortices in the intermediate asymptotic region are substantially different from those in a vortex tangle or on distances of the order of the vortex diameter.

Dynamics of vortex loops after reconnection in superfluid helium at different temperatures

This work addresses the numerical modeling of vortex loops in superfluid helium after loop reconnection at various temperatures. Modeling was conducted within the vortex filament method using the full Biot-Savart equation. It was found that the dynamics of vortex loops substantially depend on the initial position of the loops, and that temperature affects the number and size of the formed loops, the amplitude of the Kelvin waves and their decay behavior as well as the velocity of motion of the loops and their separate elements. The time dependence of the distance between the nearest elements of the emergent vortex loop is described by the power law, with the power of 1/2 before and after reconnection.

Subproblems Applied to a 3-D Magnetostatic Facet FEM Formulation

The Facet finite-element method (FFEM) requires a source field solution that is commonly obtained by the Biot–Savart (BS) equation that can be time-consuming. The subproblem modeling is applied to the FFEM in order to minimize the computational effort and time necessary to obtain the solution. The volume sources, which allow solving the BS equation only along the active regions, are presented, as well as the BS boundary condition correction.

Contribution on the Source Field Calculation through the Biot-Savart Equation Using Curvilinear Elements and an Adaptive Process

Contribution on the Source Field Calculation through the Biot-Savart Equation Using Curvilinear Elements and an Adaptive Process

Kelvin wave and knot dynamics on entangled vortices

In this paper, starting from Biot–Savart mechanics for entangled vortex-membranes, a new theory — knot physics — is developed to explore the underlying physics of quantum mechanics. Owning to the conservation conditions of the volume of knots on vortices in incompressible fluid, the shape of knots will never be changed and the corresponding Kelvin waves cannot evolve smoothly. Instead, the knot can only be split. The knot-pieces evolve following the equation of motion of Biot–Savart equation that becomes Schrödinger equation for probability waves of knots. The classical functions for Kelvin waves become wave-functions for knots. The effective theory of perturbative entangled vortex-membranes becomes a traditional model of relativistic quantum field theory — a massive Dirac model. As a result, this work would help researchers to understand the mystery in quantum mechanics.

Dynamics of Nonplanar Quantized Vortex Rings Before Reconnection at Finite Temperatures

The paper presents a numerical study of the dynamics of nonplanar quantized vortices at finite temperatures on their route to reconnection. We perform numerical simulations using the vortex filament method, solving the full Biot–Savart equation at a wide range of temperatures and initial conditions. We consider the dynamics of the two rings, lying initially in different planes and at different distances. The angles between planes are taken as equal to 30\(^{\circ }\), 45\(^{\circ }\), 60\(^{\circ }\), and 90\(^{\circ }\). It is observed that the temperature and the initial position of the vortices strongly affect the dynamics of the vortices on their route to reconnection. However, when the distances between the vertices of the vortices become smaller than the distances satisfying the Schwarz reconnection criterion, the dynamics of the system change drastically, and this trend is universal. The universality is expressed in the shapes and velocities of the vertices of the vortices.

Vortex kinematics around a submerged plate under water waves. Part II: Numerical computations

This paper presents numerical computations of the flow generated by a horizontal plate immersed in a regular wave field, and associated loads acting on the plate. This numerical work is the continuation of the Poupardin et al. (2012) experimental study. This numerical study is original in the way that the vortical aspects of the flow are not neglected. Therefore, a 2D Lagrangian Vortex method is used as a numerical scheme. These methods are particularly well suited for the computation of unsteady and highly rotational flows in an open domain. The velocity field is decomposed into rotational and potential components, using the Helmholtz decomposition. The rotational part of the velocity is calculated by the Biot–Savart equation using vortex particles. The plate is modelled by a distribution of normal dipoles and the wave field is taken into account by means of a Stokes formulation, which completes the potential part of the velocity.Firstly, the numerical code is validated by means of comparison with the experimental results of Poupardin et al. (2012). In particular, the complex vortical activity and the mean flow velocity field are well reproduced and physically analysed. Secondly, forces acting on the plate are analysed on a wide range of parameters by varying the plate immersions and lengths. In the end, a new scaling is found for the lift forces acting on the plate based on the modified Stokes velocity (i.e. the bottom Stokes velocity for a water depth equals to the plate immersion) and the square of the plate length.

Dynamics of Quantized Vortices Before Reconnection

The main goal of this paper is to investigate numerically the dynamics of quantized vortex loops, just before the reconnection at finite temperature, when mutual friction essentially changes evolution of lines. Modeling is performed on the base of vortex filament method with using the full Biot-Savart equation. It was discovered that initial position of vortices and the temperature strongly affect the dependence on time of the minimum distance $\delta (t)$ between tips of two vortex loops. In particular, in some cases the shrinking and collapse of vortex loops due to mutual friction occur earlier than reconnection, thereby cancelling the latter. However, this relationship takes a universal square-root form $\delta\left(t\right)=\sqrt{\left(\kappa/2\pi\right)\left(t_{*}-t\right)}$ at distances smaller than the one, satisfying the Schwarz criterion, when nonlocal contribution to the Biot-Savart equation becomes about equal to local contribution. In the "universal" the nearest parts of vortices form a pyramid-like structure with angles which are also don't depend both on the initial configuration of filaments and on the temperature.

Derivation of the Biot-Savart equation from the nonlinear Schrödinger equation.

We present a systematic derivation of the Biot-Savart equation from the nonlinear Schrödinger equation, in the limit when the curvature radius of vortex lines and the intervortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines,H=κ(2)/8π∫(|s-s'|>ξ(*))(ds·ds')/|s-s'|,with cutoff length ξ(*)≈0.3416293/√(ρ(0)), where ρ(0) is the background condensate density far from the vortex lines and κ is the quantum of circulation.

Kelvin-wave cascade in the vortex filament model

The small-scale energy-transfer mechanism in zero-temperature superfluid turbulence of helium-4 is still a widely debated topic. Currently, the main hypothesis is that weakly nonlinear interacting Kelvin waves (KWs) transfer energy to sufficiently small scales such that energy is dissipated as heat via phonon excitations. Theoretically, there are at least two proposed theories for Kelvin-wave interactions. We perform the most comprehensive numerical simulation of weakly nonlinear interacting KWs to date and show, using a specially designed numerical algorithm incorporating the full Biot-Savart equation, that our results are consistent with the nonlocal six-wave KW interactions as proposed by L'vov and Nazarenko.

Structure of a quantum vortex tangle in4He counterflow turbulence

The paper presents a comprehensive characterization of well-developed vortex tangles in a turbulent counterflow in quantum fluids (with a laminar normal fluid component). We perform and analyze extensive numerical simulations using the vortex filament method, solving the full Biot-Savart equations for the vortex dynamics in a wide range of temperatures and counterflow velocities. We start with the analysis of the macroscopic characteristics of the quantum vortex tangle such as vortex line density, its mean anisotropic and curvature parameters, the mean friction force between normal and superfluid components, the drift velocity of the vortex tangle, etc. Next we proceed to the main goal of the paper and move from the traditional macroscopic approach in terms of mean characteristics of the vortex tangle to the microscopic statistical and kinetic levels of description of quantum turbulence. These include objects that are much less studied or even totally neglected such as the vortex reconnection rates, the correlations and probability distribution functions (PDFs) of the vortex loop lengths, of the line curvature, of the mean curvatures of individual loops, the cross-correlation function between the loop length and its mean curvature, and the autocorrelation function of the vortex-line orientations. This detailed statistical information is required for a deeper understanding of quantum turbulence and for the development of its advanced theoretical description. In addition, we identify which of the studied properties are strongly affected by the choice of the reconnection criteria that are traditionally used in the vortex filament method and which of them are practically insensitive to the reconnection procedure. We conclude that the vortex filament method is sufficiently robust and well-suited for the description of the steady-state vortex tangle in the quantum counterflow.

Influence of High-Frequency Near-Field Coupling Between Magnetic Components on EMI Filter Design

This study presents a detailed analysis of magnetic component coupling and its influence on electromagnetic interference filter design. In contrast with other literature, the novelty of this paper reveals that magnetic coupling should be divided into two categories: low-frequency coupling and high-frequency coupling. It is proven that coupling is frequency related and high-frequency near-field stray flux distribution can be differ dramatically from low-frequency condition. The change of the near-field stray flux distribution is caused by displacement current from stray capacitors. By using the Biot-Savart equation, high-frequency near-field distribution can be well predicted and matched with experimental results. In addition, single stage differential mode (DM) LC filters are utilized to demonstrate the influence of high-frequency coupling.

Plasma column displacement measurements by modified Rogowski sine-coil and Biot–Savart/magnetic flux equation solution on IR-T1 tokamak

The modified Rogowski sine-coil (MRSC) has been designed and implemented for the plasma column horizontal displacement measurements on small IR-T1 tokamak. MRSC operation has been examined on test assembly and tokamak. Obtained results show high sensitivity to the plasma column horizontal displacement and negligible sensitivity to the vertical displacement; linearity in wide, +/-0.1 m, range of the displacements; and excellent, 1.5%, agreement with the results of numerical solution of Biot-Savart and magnetic flux equations.