Electrostatic Problem Research Articles

Students’ understanding of the inverse square law in electrostatics

One problem of learning Electrostatics is that students often learn from their commonsense beliefs about electric force and electric field. This study investigated students’ conceptual understanding of finding electric force, electric field, and electric potential of point charge after learning an introductory physics course. We administered the Electrostatics Conceptual Evaluation Test to four lecture-based classes in high school. The first question was a comparison of the electric force from two-point charges at two different positions and the second question was a comparison of the electric field from a point charge at two different positions. The use of the inverse square law is required to find the electric force and the electric field at various positions. It was found that many students answered incorrectly. They described that the electric force and the electric field decrease whereas the distance increases by neglecting the inverse square law. This finding can be particularly used to suggest high school teachers to develop their effective strategy to support student learning.

Accelerating the charge inversion algorithm with hierarchical matrices for gas insulated systems

Surface charges accumulating on dielectrics during long-time operation of Gas Insulated High Voltage Direct Current (HVDC-GIS) equipments may affect the stable operation and could possibly trigger surface flashovers. In industrial applications, to quantify and identify the location of the surface charge accumulation from experimental measurements, the surface potential distribution is evaluated using, e.g., electrostatic probes, then the charge density is determined by solving an electrostatic problem based on an inversion procedure known as Charge Inversion Algorithm. The major practical limitation of such procedure is the inversion and the storage of the fully dense matrix arising from the representation via Integral Equations of the electrostatic phenomenon, resulting in O(N 3) computational complexity and O(N 2) memory requirement. In this paper it is shown how hierarchical matrices can be efficiently used to accelerate the charge inversion algorithm and, more importantly, reduce the overall memory requirement.

Bounding Formulae for Capacitance of Cylindrical Capacitor with Nonhomogeneous Dielectric Material

The paper deals with the capacitance of cylindrical capacitor which consists of nonhomogeneous dielectric materials. The infinite long cylindrical surfaces of capacitor have uniform distributed electric charges in axial direction, so that electrostatic boundary value problem which determines the electric field in the nonhomogeneous dielectric material is a twodimensional boundaryvalue problem. The derivation of the bounding formulae is based on the CauchySchwarz inequality. Examples illustrate the applications of the derived upper and lower bound formulae.

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Multiscale modeling and simulation methods for electromagnetic and multiphysics problems

Problem Solving Ability of Distance Education Student on Electrostatic Topic

In this paper, we present the result of a study on the identifying problem-solving ability on distance education students’ physics. Descriptive-qualitative is used for method of research. The subjects of the study were in-service students’ physics of distance education. Sample were 19 students from all of student taking electricity and magnetism course in term 2019/2020. Essay test of electrostatics problem solving is used for collecting data technique. Test of problem solving is given at the beginning of the course in asynchronous tutoring of electric and magnetic cores. The finding of the study indicated problem solving ability of in-service students’ physics of distance education are categorized low (44,56%) in solving the problems. Therefore, it is necessary to make efforts to improve students’ ability to solve problems, through asynchronous tutorials by applying problem-solving strategies in interactive discussion and collaborative work in groups.

A comparative analysis of two computational schemes for solving local fractional Laplace equations

In this paper, we implement the semi‐analytical schemes, namely, local fractional homotopy perturbation Sumudu transform method (LFHPSTM) and local fractional homotopy analysis Sumudu transform method (LFHASTM), for finding the approximate analytical solutions of local fractional Laplace equations under different initial conditions on Cantor sets. The novelty of the work lies in the application of these suggested schemes which were never used to solve the local fractional Laplace equation. The Laplace equation mainly contributes to electrostatistics, mechanical engineering, and theoretical physics. This equation also helps in formulation of the electrostatic problem of a rod under a torsion load. Moreover, it helps in handling potential field problems also. These aspects enhance the significance of getting a solution of the local fractional Laplace equation in fractal media. The suggested methods are copulation of local fractional homotopy perturbation method and local fractional homotopy analysis method with the local fractional Sumudu transform, respectively. The computational process indicates that the schemes are very efficient to obtain nondifferentiable solutions for given equations in a smooth way. Moreover, the solution analysis depicts that both the local fractional hybrid homotopy schemes are in a good agreement with each other. These methods depict the accuracy and efficiency of the implemented methods in view of correspondence with solutions obtained with other methods in previous works. The numerical simulations for obtained results are discussed for various values of order of a local fractional derivative. The 3D graphs on the Cantor set are constructed with the help of Matlab software.

Fundamental interactions in a classical Wigner system.

Wigner crystallization in macroscopic systems provides a simple avenue for studying classical many-body physics. The Coulomb repulsion of millimeter-sized conducting spheres rolling on a high potential surface enables the self-assembly of ordered structures. Prior attempts at formulating pairwise force models for Wigner islands have resulted in different functional forms with no apparent consensus. In this study, we explore fundamental particle interactions using the finite element method to solve the electrostatic problem for one-, two-, and many-sphere configurations. Analysis of charge and force shows three interaction regimes parametrized by the packing fraction, which includes many-body force behavior for compact geometries. We find that the force relationship between spheres is accurately represented by an aligned dipole model over all configurations. Both the finite element method and aligned dipole models were validated in absolute terms using an experimental setup featuring a gravity potential. Future dynamic studies, which rely on accurate force relationships, can now be treated.

Electrostatic Problem for Toroid Surfaces: Analytical Regularization

The electrostatic problem described by the Laplace equation with Dirichlet boundary condition was rigorously and efficiently solved for closed and open arbitrarily shaped isolated toroid surface of revolution. Application of Orthogonal Polynomial Method and Analytical Regularization Method allowed the problem to be reduced to a well-conditioned infinite system of linear algebraic equations of the second kind. It possesses robust numerical solution with, in principle, any desired accuracy.

Modeling nonlinear dielectric properties of laminated composites

sThe article is devoted to the development of a method for predicting the nonlinear dielectric properties of layered composites with complex dependences of the dielectric constant on the electric field strength, which consider the effects of a sharp increase and subsequent decrease in the dielectric constant with increasing field strength. The method is based on the use of the theory of asymptotic averaging proposed by N.S. Bakhvalov, E. Sanchez-Palencia, in relation to the nonlinear problem of electrostatics in a layered medium. As a result of applying this method, local nonlinear problems of electrostatics on a periodicity cell are formulated, and an algorithm for calculating effective nonlinear dielectric constants is proposed. It is shown that a layered composite is a transversely isotropic nonlinear dielectric material if its layers are isotropic materials. A numerical example of calculating the nonlinear properties of a 2-layer composite based on barium titanate and ferroelectric ceramic varicond VK4 is considered. Calculations have shown that the developed method for calculating the nonlinear-dielectric ratios of composites is quite effective and makes it possible to predict the dielectric properties of composites at various values of the electric field strength, including the regions where the dielectric permittivities of the layer materials are saturated. The developed technique can serve as a basis for designing new nonlinear dielectric composite materials with unusual properties.

Accurate Calculation of Capacitance for Three-Dimensional Axisymmetric Conductors with Arbitrary Profile

This paper presents a rigorous mathematical approach to the analysis of electrostatic problems for three-dimensional objects of revolution. A cavity can be formed by rotation of an arbitrarily shaped generating curve and has coaxial apertures on both ends. Such flexibility allows to consider a wide class of previously studied conductors and stimulate design of new ones. The accurate free space capacitance calculations for finite open cylinders and truncated cones are performed and the results are compared to the known data from the literature.

A peridynamic model for galvanic corrosion and fracture

A coupled peridynamic (PD) corrosion-fracture model is introduced, in which the local corrosion rate is determined by solving the corresponding electrostatic problem. By being able to consider variable distributions of corrosion rates along the anodic surface, the model is particularly useful for simulating galvanic corrosion. A novel analytical calibration for parameters in the corrosion model is provided. The corrosion model is verified in terms of the electric potential, current density, and corrosion depth, and validated against experimental results for AE44 (Mg alloy) – mild steel and AE44 – AA6063 (Al alloy) galvanic couples taken from the literature. The PD results are also compared with those from a COMSOL FEM-based model. It is found that an artificial initial “step-down” in geometry at the galvanic joint is required for the COMSOL model in order to provide reasonable results, but it is not needed in the PD model. A coupled galvanic corrosion-fracture problem accounting for the combined electro-chemical attack and strains induced by mechanical loadings is solved.

The effect of temperature on charging behaviour of polyimide subjected to electron beam at vacuum

Charge accumulation is one of the main factors leading to electrostatic discharge and restricting the development of spacecraft. This paper investigates the effect of temperature (243–343 K) on the charging behaviour of polyimide in vacuum using electron beam radiation test system. The surface potential distribution shows that there are two charging modes, ‘negative-peak’ mode and ‘negative-crater’ mode. For investigating the charge accumulation mechanism, the space charge experiments were carried out by the method of pulsed electro-acoustic. The thresholds of injection from the polyimide/electrode interface are affected by temperature. Under high temperature (above 323 K), the hole could be easily injected from the polyimide/electrode interface, with the extremely low injection threshold. The incident electrons and the injected holes co-existence and form the ‘negative-peak’ mode. Under low temperature (below 303 K), the area of hole injection from the electrode decreases with the decrease in temperature. The hole injected locally forms an opposing electric field, which makes the surface potential present the ‘crater’ mode. This work shows that the hole injected from the dielectric/electrode interface play a critical role in the charge accumulation process under electron radiation, which has important significance to avoid the spacecraft electrostatic discharge problem.

Designing a High-Performance Boundary Element Library With OpenCL and Numba

The Bempp boundary element library is a well-known library for the simulation of a range of electrostatic, acoustic, and electromagnetic problems in homogeneous bounded and unbounded domains. It originally started as a traditional C++ library with a Python interface. Over the last two years, we have completely redesigned Bempp as a native Python library, called Bempp-cl, that provides computational backends for OpenCL (using PyOpenCL) and Numba. The OpenCL backend implements kernels for GPUs and CPUs with SIMD optimization. In this article, we discuss the design of Bempp-cl, provide performance comparisons on different compute devices, and discuss the advantages and disadvantages of OpenCL as compared to Numba.

Numerical Modeling of Solutions of the Applied Problems for the Poisson’s Equation

Many problems of continuum physics, applied physics, electrostatics, heat engineering, and structural mechanics are described by boundary value problems for the Poisson equation. The article is devoted to the construction of a numerical algorithm for solving plane boundary value problems for the Poisson equation. The algorithm is based on the transition to a polyharmonic equation, which is solved by the method of linear boundary elements. We consider two boundary value problems with different boundary conditions: Dirichlet boundary conditions and Neumann boundary conditions. To construct a numerical solution using the linear boundary element method, the boundary of the area is replaced by an inscribed polygon. The boundary conditions are satisfied at the middle (control) points of the elements. A polyharmonic equation is reduced to a system of linear integral equations, which can be reduced to a system of linear algebraic equations. Test numerical examples confirming the effectiveness of the proposed algorithm are given.

Preparation of new conductive organic coating for the fiber reinforced polymer composite oil pipe

The fiber reinforced polymer composite oil pipe has gradually replaced the metal pipe because of its low cost, corrosion resistance and so on, while it would face the electrostatic problem, because of the insulation of composite pipe. To settle this problem, this article proposed a new method for preparing conductive organic coating of basalt fibers. The coating was the composite mainly consisting of the acidified multi-walled carbon nanotubes (A-MWCNTs) and epoxy resin. To enhancing MWCNTs' hydrophilicity, stabilization and dispersion, they were acidified with concentrated HNO3, forming A-MWCNTs. Then the basalt fiber was modified with A-MWCNTs /epoxy composite coating through a dip-coating process. The results demonstrated that the coating could enhance the mechanical properties of basalt fibers and change the insulating fibers into conductive ones.

On the electrostatic potential for the two-hyperboloid and double-cone of a single sheet with elliptic cross-section

Summary The study of the response of divergence-free electric fields near corners and edges, resembling singularities that accumulate charges, is significant in modern engineering technology. A sharp point can mathematically be modelled with respect to the tip of the one sheet of a double cone. Here, we investigate the behaviour of the generated harmonic potential function close to the apex of a single-sheeted two-hyperboloid with elliptic cross-section, whose asymptote is the corresponding elliptic double cone with one sheet present. Hence, the electrostatic potential problem, involving a single sheet of a two-hyperboloid, is developed using the theory of ellipsoidal-hyperboloidal harmonics, wherein the particular consideration enforces as solution in terms of generalised Lamé functions of non-integer order. A numerical method to determine these functions is outlined and tested. We demonstrate our technique to the solution of a classical boundary value problem in electrostatics, referring to a metallic and charged single-sheeted elliptic two-hyperboloid and its double-cone limit. Semi-analytical expressions for the related fields are derived, all cases being accompanied by the necessary numerical implementation.

Soft dielectric elastomer tubes in an electric field

Soft dielectric elastomers with high relative permittivity, very low modulus and high electric breakdown strength have emerged as promising materials for various applications as sensors, actuators and in energy harvesting and soft robotics. We study the intricate deformation behaviour of a soft dielectric elastomer tube of finite length and closed ends, that carries a dead load, is internally pressurised by an injected fluid and has a high electric potential applied across its walls. As for soft tubes in the absence of a potential, this electro-hyperelastic problem involving very large deformations, exhibits a multitude of possibilities, including homogeneous deformation, inhomogeneous bifurcation, snap-through instabilities and post bifurcation behaviour in the form of propagation of axisymmetric bulges. We develop a coupled Finite Element procedure for the situation where the domains of the mechanical and electrostatic problems coincide. The procedure can handle volume flow rate controlled electro-elastic problems. We use it to study the many aspects of the deformation behaviour and limitations placed by competing failure mechanisms on practical utilisation of the large areal strains and axial actuations that can be produced. If electrical breakdown can be avoided by ingenious design of loading sequences, the large volume increase in the tube due to the triggering of instabilities can be harnessed and has far-reaching technological implications.

New solution method for the problem of a uniformly charged straight wire

A common problem in electrostatics is determining the electrostatic potential due to a uniformly charged straight wire. The solution of this problem illustrates well the types of calculations that one must perform in order to obtain the electrostatic potential or field of a given continuous charge distribution. In this work, we reconsider and solve the problem of a uniformly charged straight wire via a new method that is different from the popular direct integration approach found in the majority of physics textbooks. The outcomes of the two methods are compared and the results suggest several interesting mathematical formulas involving special functions.

Relations between Spheroidal Harmonics and the Rayleigh Approximation for Multilayered Nonconfocal Spheroids

Relations between Laplace’s spheroidal harmonics associated with different spheroidal coordinates are derived. The transition matrices for the functions of the 1st kind are lower triangular and are related by inversion. The matrices for the functions of the 2nd kind are the transposed ones for the functions of the 1st kind. The series for the functions of the 1st kind are finite, and those for the 2nd kind are infinite. In the latter case the region of convergence is considered. Using the derived relations, the rigid solution to the electrostatic problem for the multi-layered scatterers with nonconfocal spheroidal boundaries of the layers is obtained and the Rayleigh approximation is constructed, as well as an approximate approach to a similar light scattering problem, which provides reliable results far beyond the range of applicability of the Rayleigh approximation, is suggested.

ON THE APPLICATION OF CONFORMAL MAPS IN THE SOLUTION AND ANALYSIS OF SOME ELECTROSTATIC FIELDS

In this paper, we have employed appropriate conformal maps to solve and analyze some harmonic Dirichlet problems in electrostatics. The technique involves the transformation of a given problem in the w plane onto one in the upper half of the z plane where it is identified as the imaginary part of the logarithmic function with branch cut at the ray -π/2 and efficiently solved or directly solving it in the w plane where it is posed. The equipotential lines and flux lines of the electric field were also generated to show the nature of the field and the features of the solution. The method gave exact general analytical solutions in closed form to the problems considered and could therefore be a useful alternative method for solving Laplace’s equation for two dimensional electrostatic problems.