Biot-Savart Research Articles

Numerical and Analytical Analysis of the Low-Frequency Magnetic Fields Generated by Three-Phase Underground Power Cables with Solid Bonding

There is a special concern for measuring and simulating low-frequency magnetic fields generated by underground power cables, particularly in human exposure studies. In the present study, an accurate 2D finite element model for computing magnetic fields generated by three-phase underground power cables with solid bonding is proposed. The model is developed in ANSYS Maxwell 2D low-frequency electromagnetic field simulation software for a typical 12/20 kV (medium-voltage) three-phase underground power cable in both trefoil and flat formations, but it can be adapted to any cable system. Model validation is achieved by analytical computations conducted with a software tool based on the Biot–Savart law and the superposition principle. RMS magnetic flux density profiles calculated at various heights above the ground with these two methods correlate very well. This is also true for induced shield currents. The application of the finite element model to multiple three-phase power cables laid together is also considered.

Axisymmetric self-similar finite-time singularity solution of the Euler equations

Self-similar finite-time singularity solutions of the axisymmetric Euler equations in an infinite system with a swirl are provided. Using the Elgindi approximation of the Biot–Savart kernel for the velocity in terms of vorticity, we show that an axisymmetric incompressible and inviscid flow presents a self-similar finite-time singularity of second specie, with a critical exponent ν. Contrary to the recent findings by Hou and collaborators, the current singularity solution occurs at the origin of the coordinate system, not at the system’s boundaries or on an annular rim at a finite distance. Finally, assisted by a numerical calculation, we sketch an approximate solution and find the respective values of ν. These solutions may be a starting point for rigorous mathematical proofs.

A theoretical and experimental investigation on a volume coil with slotted end-rings for rat MRI at 7 T

ObjectiveA volume coil with squared slots-end ring was developed to attain improved sensitivity for imaging of rat’s brain at 7 T.Material and MethodsThe principles of the high cavity resonator for the low-pass case and the law of Biot-Savart were used to derive a theoretical expression of B_1/i. The slotted-end ring resonator showed a theoretical 2.22-fold sensitivity improvement over the standard birdcage coil with similar dimensions. Numerical studies were carried out for the electromagnetic fields and specific absorption rates for our coil and a birdcage coil loaded with a saline-filled spherical phantom and a digital brain of a rat.ResultsAn improvement of the signal-to-noise ratio (SNR) can be observed for the slotted volume coil over the birdcage regardless of the load used in the electromagnetic simulations. The specific absorption rate simulations show a decrement for the digital brain and quite similar values with the saline solution phantom. Phantom and rat’s brain images were acquired at 7 T to prove the viability of the coil design. The experimental noise figure of our coil design was four times less than the standard birdcage with similar dimensions, which showed a 44.5% increase in experimental SNR.DiscussionThere is remarkable agreement among the theoretical, numerical and experimental sensitivity values, which all demonstrate that the coil performance for MR imaging of small rodents can be improved using slotted end-rings.

A right inverse of curl which is divergence‐free invariant and some applications to generalized Vekua‐type problems

In this work, we investigate the system formed by the equations and in bounded star‐shaped domains of . A Helmholtz‐type decomposition theorem is established based on a general solution of the above‐mentioned div‐curl system. When , we obtain a bounded right inverse of which is a divergence‐free invariant. The restriction of this right inverse to the subspace of divergence‐free vector fields with vanishing normal trace is the well‐known Biot–Savart operator. This right inverse will be restricted to guarantee its compactness and satisfy suitable boundary value problems. Applications to Beltrami fields, Vekua‐type problems, as well as Maxwell's equations in inhomogeneous media are proposed.

A Control Method of Projectile Spinning Speed Based on Magnetic Field Vector Transformation in a Railgun

This article studies the spinning speed control method of projectile in a railgun. The core idea is to adjust the current in extra rails to produce a controllable magnetic field and rotational torque. Then, the spinning speed and direction can be accurately controlled. Based on the Biot–Savart law, a model is established to analyze the rotating characteristics affected by the current waveform in the extra rails. Results show that this method is feasible since the spinning speed is linear and controllable. It can achieve a high speed to meet the fly spinning stability. It is also found that the increase of spinning speed is at the expense of muzzle velocity, and the launching system cannot achieve the maximum spinning speed and maximum muzzle velocity at the same time. The greater the energy given to the inner rails, the higher the muzzle velocity, and the faster the spinning speed. Without a large muzzle velocity, it may not achieve a high spinning speed. Finally, the influence of different parameters on the consistency of spinning speed is analyzed. The method proposed in this article can control the spinning speed accurately, which is of reference significance for the development of electromagnetic launch technology.

Motivating the Biot-Savart Law

Introducing students to a new law is pedagogically challenging. One can just declare it as fact, but it is easier on (at least some) students if we can derive it when possible, or, if not, at least motivate it somehow. The Biot-Savart law is particularly challenging in this respect, because it is only derivable from more advanced formulations (such as Maxwell’s equations), which are in their turn merely stipulated (this is the approach of Feynman et al., for example). The other possibility is simply to declare its empirical truth, without details (see, for example, Griffiths).

Biot-Savart Law for Determination of Speed of Particle Beyond the Speed of Light

This paper presents the modification of Biot-Savart law that thoroughly put to calculate magnetic field at any point in the space due to motion of charge in a complete circuit and further applies to determine velocity of particle faster than the speed of light. The modified Biot-Savart law provides complete mathematical result as given by ordinary Biot-Savart law, but most importantly, do not involve the need of derivative, integration of line element. In order to derive the relation ( among electric field ( magnetic field (( of electromagnetic wave, Biot-Savart law is given by formula B= , Where be angle between direction of velocity v of moving charge and position vector of given point where magnetic field has to be determined. This formula is applicable to address the question of how is it possible to achieve velocity of particle in excess of C and to open a discussion towards a new way of acquiring speed of particle beyond speed of light, namely C , Although, there is popular statement "nothing can move faster than speed of light". In this highly controversial statement, the specific purpose is not to comprises the merits of existing Biot-Savart law, but rather to introduce a concise and careful reasoned account of Biot-Savart law to propagate particle faster than speed of light.

On the Magnetic Field of a Finite Solenoid

The magnetostatic field of a finite solenoid with infinitely thin walls carrying a dc current oriented in the azimuthal direction is calculated everywhere in space in terms of complete elliptic integrals by direct integration of the Biot–Savart law. The solution is particularized near the solenoid axis and in the midplane perpendicular to the axis obtaining expressions that agree with some typical approximations that are made in introductory courses of electromagnetism or in the technical literature. The range of validity of these approximations has been studied comparing them with the obtained general expression.

Nucleus Independent Chemical Shift (NICS) at Small Distances from the Molecular Plane: The Effect of Electron Density.

When close to the molecular plane, the behavior of nucleus independent chemical shift (NICS) as a function of the distance from the molecular plane deviates from its behavior at larger distances. By using a dense grid of NICS-probes (BQs) it is shown that, when close to the molecular plane, maximal (absolute) NICS values are obtained above the atoms. These maxima move towards the center as the grid is elevated until the (absolute) maximum NICS is obtained at the center and stay there when the grid is further elevated. It is shown that this behavior is a result of the current density, which is influenced by the electron density, according to the Biot-Savart law, which, in turn, causes the induced magnetic field measured by the NICS. It is thus concluded that if magnetic aromaticity is studied, the NICS calculations should be carried out at a large enough distance so that only the π-ring current affects the NICS. At distances ≥2 Å, NICS(r)π,zz =A+B*Cr . Using non-linear correlation for obtaining A, B and C and extrapolate to NICS(1)π,zz and NICS(1.7)π,zz is recommended as measures for aromaticity.

The method of multi-objective parametric design of magnetic field active canceling robust system for residential multy-story buildings closed to double-circuit overhead power lines

Aim. Development the method of multi-objective parametric design for robust system of active canceling of magnetic field based on binary preference relations of local objective for multi-objective minimax optimization problem. Methodology. Spatial location coordinates of the compensating winding and the current in the shielding winding were determined during the preference-based multi-objective parametric design of systems of active canceling based on solution of the vector minimax optimization, in whith the vector objective function calculated based on Biot-Savart's law. The solution of this vector minimax optimization problem calculated based on nonlinear Archimedes algorithm. Components of Jacobi matrix and Hesse matrix calculated based on multi-swarm multi-agent optimization. Results. Theoretically and experimentally confirmed the effectiveness of reducing the level of the magnetic field in residential multy-storey old building of a double-circuit overhead power transmission lines with a barrel-type arrangement of wires by means of active shielding with two compensation winding. Originality. The method of multi-objective parametric design for robust system of active canceling of magnetic field based on binary preference relations of local objective for multi-objective minimax optimization problem is developed. Practical value. It is shown the possibility to reduce the level of magnetic field in residential multy-storey old building closed to double-circuit overhead power transmission lines with a barrel-type arrangement of wires by means of system of active canceling with two canceling winding to a level safe for the population with an induction of 0.5 μT.

Derivation of the Field Due to a Magnetic Dipole Without Use of the Vector Potential

The mathematical form of the magnetic field due to a current loop, and the fact that it is identical to the electric field due to an electric dipole in the far field, are fundamental to our understanding of electromagnetism. While undergraduate level electromagnetism textbooks usually derive the electric field from an electric dipole, few derive the magnetic field from a current loop. Most simply state it without proof, or perform the derivation for simpler cases such as the on-axis field. Those that perform the derivation use the magnetic vector potential, a relatively advanced concept that most undergraduate students would not encounter until their final year of study, if at all. Here, a simple derivation to obtain the magnetic field due to a current loop in the far-field approximation is presented. The derivation begins from the Biot–Savart law and does not require the vector potential. The problem is set up so that only a single integration is necessary (from angle α = 0 to α = 2π around the current loop), and the result is compared with that for the electric field surrounding an electric dipole.

Эффективная реализация быстрого метода мультиполей для взаимодействия частиц с ньютоновским потенциалом

We consider rotation-based fast multipole method for the Laplace equation and its application in cases where particle interactions are governed by the Biot—Savart law. The paper presents the necessary formulas for algorithm implementation and addresses less frequently discussed topics, such as the normalization of spherical harmonics and the associated Wigner matrices normalization. The main focus of the paper is devoted to describing the details of the software implementation that significantly accelerate the performance of the code both on CPU and GPU (using CUDA technology). We provide a comprehensive explanation of proposed techniques and include code examples. We have implemented a C++ program using these methods and conducted a comparative analysis with open-source implementations of the fast multipole method, confirming the high efficiency of our approach.

A Shielding System Proposal for the Cabling of Electric Glass Melters

This paper analysesthe impact on workers of the electromagnetic field produced by large furnaces for glass production. The paper also propose a solution for the reduction and the containment of magnetic induction levels below the limits imposed by national and international directives. Using a three-dimensional numerical calculation code based on the hypothesis of filiform conductors in the air (Biot-savart law). The choice of the solution is based on the technique named highly coupled magnetically passive loop technique (HMCPL). The solution was then implemented and a comparison between the magnetic induction values before and after the installation of the shielding solution is presented.

A Unified Definition of Electrostatic and Magnetic Fields

This article originates from the observation that field lines are drawn using distinctive rules in magnetic field and electrostatic fields. It aims at reconciliating the definitions of these fields and thus reaching a consensus on the interpretation of field lines. Our unified field definition combines three orthogonal vectors and a unique scalar value. Field lines are then defined as isovalue lines of the scalar value, rendering it simpler to interpret in both field types. Specific to our field definition is the use of square root of vector’s cross product so that all vectors have the same physical unit. This enhanced field definition also enables a more efficient calculation of Biot-Savart law. This article is the first of a series allowing the drawing of isovalue contour lines.

Comparison between Motion Effect of Electric Field and Gravitational Field

It is pointed out that the magnetic field is not an independent real substance and it is considered that the electromagnetic field containing non-material magnetic field cannot establish a unified field theory with the gravitational field of the real substance. Through analysis, it is concluded that the so-called magnetic field is essentially a motion state of the electric field. The magnetic field is defined as the electric field contracting in motion. According to this definition, the Biot-Savart, Lorentz force, electromagnetic induction, displacement current and Ampere loop laws are deduced and proved. Through the experimental results of electrostatic field around the permanent magnet, it is proved that the magnetic field is indeed the electric field contracting in motion. It is believed that the electrostatic field is similar to the static gravitational field, the electric field contracting in motion can be defined as a magnetic field, and the gravitational field contracting in motion can also be defined as a similar contraction field, so that the electromagnetic field is completely similar to the gravitational contraction field, and the unification of the two fields is an inevitable result.

Electric vector potential and the Biot-Savart like law in electrostatics

A vector potential formulation is shown in this article to compute the electric field of planar surface electrodes. The electric field is derived from from the solution of the Laplace’s equation in the free-charge space. Neumann-boundary conditions must be set on the region between planar metallic sheets as the separation goes to zero. It is shown that the electric field can be written via a Biot Savart- like integral. The strategy enables to generalize the analytical result for its application in the gaped surface electrodes description.

A Monte Carlo Method for Fluid Simulation

We present a novel Monte Carlo-based fluid simulation approach capable of pointwise and stochastic estimation of fluid motion. Drawing on the Feynman-Kac representation of the vorticity transport equation, we propose a recursive Monte Carlo estimator of the Biot-Savart law and extend it with a stream function formulation that allows us to treat free-slip boundary conditions using a Walk-on-Spheres algorithm. Inspired by the Monte Carlo literature in rendering, we design and compare variance reduction schemes suited to a fluid simulation context for the first time, show its applicability to complex boundary settings, and detail a simple and practical implementation with temporal grid caching. We validate the correctness of our approach via quantitative and qualitative evaluations - across a range of settings and domain geometries - and thoroughly explore its parameters' design space. Finally, we provide an in-depth discussion of several axes of future work building on this new numerical simulation modality.

Frequency-Domain Lifting-Line Aerodynamic Modelling for Wing Aeroelasticity

A frequency-domain lifting-line solution algorithm for the prediction of the unsteady aerodynamics of wings is presented. The Biot–Savart law is applied to determine the normalwash generated by the wake vorticity distribution, whereas steady and unsteady airfoil theories (Glauert’s and Theodorsen’s, respectively) are used to evaluate the sectional aerodynamic loads, namely the lift and pitching moment. The wake vorticity released at the trailing edge derives from the bound circulation through the Kutta condition and is convected downstream with the velocity of the undisturbed flow. The local bound circulation is obtained by the application of the Kutta–Joukowski theorem, extended to unsteady flows. Assuming a bending and torsion wing, this paper provides the aerodynamic matrix of the transfer functions, relating the generalised aerodynamic loads to the Lagrangian coordinates of the elastic deformation. Its rational approximation yields a reduced-order state-space aerodynamic model suitable for an aeroelastic stability analysis and control purposes. The numerical investigation examines the influence of both the wake shed/trailed vorticity modelling and different approximations of the Kutta–Joukowski theorem for unsteady flows on the aerodynamic transfer functions given by the developed frequency-domain lifting-line solver. The accuracy of the solver is assessed by comparison with the predictions obtained by a three-dimensional boundary-element-method solver for potential flows. It is shown that, at least for the frequency range considered, regardless of the approximation of the Kutta–Joukowski theorem applied, the formulation based on the Theodorsen theory provides predictions that are in very good agreement with the results from the boundary element method for a slender wing. This agreement worsens as the wing aspect ratio decreases. A lower level of accuracy is obtained by the application of the sectional loads given by the Glauert theory. In this case, the predictions are more sensitive to the approximation used to express the Kutta–Joukowski theorem for unsteady flows.

Magnetic and vibrational amplitude dependences of MRE grid composite sandwich plates

In this work, both magnetic and vibration amplitude dependent properties of magnetorheological elastomer (MRE) grid composite sandwich plates (MREGCSPs) are investigated. Initially, to prove such a nonlinearly dependent phenomenon, a series of characterization tests are performed on the MREGCSP specimens with different magnetic field intensities and base vibrational amplitudes. Then, using the Jones-Nelson nonlinear material theory, the strain energy density function method, the complex modulus approach, and the Biot-Savart law, the nonlinear material assumption of MRE is defined. A theoretical model consisting of an MRE grid core and two fiber-reinforced polymer (FRP) skins is also proposed to obtain the solutions for the nonlinear frequency, damping, and vibration response parameters, which is based on the modified first-order shear deformation theory, the energy principle, the eigenvalue increment method, the Newmark- beta approach, etc. Finally, a detailed comparison of magnetic and vibrational amplitude dependent natural frequencies, loss factors, and vibration responses is performed to confirm the effectiveness and superiority of the current model over a linear model. This provides a solid basis to reveal the nonlinear dynamic mechanism of the studied smart structure subjected to complex excitation loads. Also, some practical suggestions are summarized for improving its active vibration control capability.

Mechanical response of magneto-active elastic hemispherical shells

We develop a theoretical model that can predict the mechanical behavior of magneto-active elastic hemispherical shells under arbitrary magnetic fields and verify it with experiments and finite element simulations. We use Koiter’s shell theory and consider all components of three basis vectors of an external magnetic field, unlike previous studies that have considered only components in one direction. In addition, we modify the conventional application of the Biot–Savart law to analyze the magnetic field generated by the non-ideal Maxwell coil considering different radii and heights of each winding loop, insulation gaps, magnetostatic effects, and winding imperfections. In experiments, we fabricate magneto-active shells by varying the weight percentage of magnetic fillers and the thickness of the shells and measure the displacement of the pole under various magnetic fields. The theoretical model follows the experimental results more closely than the finite element simulations. Moreover, we find the relation between the pole displacement and the variables and that the pole displacement is independent of the shell thickness. Beyond the simple hemispherical shells, the theoretical model can be used in various application studies, such as drug delivery pumps, soft grippers, and soft robotics.