Electrostatic Problem Research Articles

Electrostatic boundary problems and T-matrix for the dielectric half-spheroid

We solve the electrostatic boundary problems of a dielectric or conducting hemispheroid (half-spheroid) under arbitrary excitation. The solutions are obtained by expanding the potentials as series of spheroidal harmonics, and integrating over the boundary to obtain matrix equations which can be used to solve for the coefficients. The solutions are used to derive the capacity, polarizability and spatial fields. We simplify the results to that for a hemisphere, which for specific excitation fields agrees with the literature. We make a link to the T-matrix method, and present analytic expressions for the T-matrix and auxiliary Q and P matrices in the electrostatic limit. We show that the standard T-matrix approach of the extended boundary condition method (EBCM) cannot be used for this geometry, and that the P and Q-matrices do not match the EBCM form.

Image charge effects under metal and dielectric boundary conditions.

Theimage charge(IC)effect is a fundamental problem in electrostatics. However, proper treatment at the continuum level for many-ion systems, such as electrolyte solutions or ionic liquids, remains an open theoretical question. Here, we demonstrate and systematically compare the IC effects under metal and dielectric boundary conditions (BCs), based on a renormalized Gaussian-fluctuation theory. Our calculations for a simple 1:1 symmetric electrolyte in the point-charge approximation show that the double-layer structure, capacitance, and interaction forces between like-charged plates depend strongly on the types of boundaries, even in the weak-coupling regime. Like-charge attraction is predicted for both metal and dielectric BCs. Finally, we comment on the effects of a dielectricallysaturated solvent layer on the metal surface. We provide these results to serve as a baseline for comparison with more realistic molecular dynamics simulations and experiments.

Theory for measuring electric charge density of a ring from scanning force microscopy

We consider a ring that carries an arbitrary electric charge density in the presence of a scanning force microscopy tip. We propose an algorithm that predicts this charge density from knowledge of the electrostatic ring-tip contribution to the total scanning force microscopy force–distance curves. We first solve the direct electrostatic problem of finding the electrostatic forces by the ring on a scanning force microscopy conducting probe. These forces are in the pN and nN range and, therefore, measurable with current technologies. Finally, we describe a method based on the least squares minimization method to measure the charge on the ring.

Diagnosing electrostatic problems and hazards in industrial processes: Case studies

AbstractStatic electricity is a phenomenon commonly encountered yet often misunderstood and underestimated in terms of its hazard potential; it continues to challenge the safety and reliability of industrial processes, particularly those involving flammable substances. This paper delves into the critical issue of electrostatic hazards in industrial settings by presenting two flash fire/explosion case studies, one involving biphenyl dust and the other gasoline vapor; both linked to electrostatic discharges. The first case study examines an explosion in a flaker and pack‐out hopper during biphenyl flake manufacturing. The investigation reveals the role of electrostatic charges in the incident and how well‐intentioned equipment changes created warning signs of increased risk that were missed. The second case study discusses a gasoline vapor flash fire, highlighting the common yet overlooked hazard of static electricity during the transfer of flammable liquids. It underscores how common activities involving people can generate sufficient electrostatic charge to ignite flammable vapor–air mixtures in industry. The outcomes of these studies highlight the importance of acting on early warning signs of static electricity and the crucial role of data‐driven diagnostic techniques in addressing and controlling those hazards. By dissecting the intricacies of electrostatic phenomena, safety professionals can formulate actionable strategies to adapt plant operations and prevent hazards from static electricity. The paper advocates for a proactive approach to identifying and mitigating electrostatic risks in industrial settings beginning with process hazard analyses that incorporate static electricity hazards analysis and continues through employee training to ensure that early warning signs are identified, understood, and acted upon. It stresses the need for comprehensive safety measures, including proper grounding and bonding, use of static dissipative materials, and regular maintenance of safety equipment, to prevent incidents. Through these case studies, the paper contributes to understanding of electrostatic hazards in industrial processes and highlights the importance of integrating electrostatic safety measures into routine industrial operations.

Efficiently solving large-scale electrostatic lightning problems by integrating characteristic basis functions into boundary element method

The evaluation of the electric field near objects of complex geometries is critical in the pre-bridging regime of lightning strikes in high-voltage transmission systems, which is subject to an open domain problem and can be computationally expensive. In this paper, the characteristic basis function method (CBFM) is incorporated into the conventional boundary element method (BEM) to efficiently handle large-scale electrostatic problems. Within the CBFM, low-level basis functions have been replaced by high-level macro characteristic basis functions, resulting in a significant reduction in the number of degrees of freedom. This reduction in unknowns allows iteration-free direct solvers to approximate the matrices, further reducing the computational complexity of the problem. Numerical examples of the surface charge distribution of an actual 500 kV tower-transmission-line system and a three MW wind turbine in the lightning environment demonstrate the accuracy of the proposed CBFM, while requiring considerably much fewer computational resources compared with conventional BEM. Furthermore, CBFM is highly parallelizable, implying the feasibility of handling large electrostatic lightning problems with complex geometries.

Mesh‐free Monte Carlo method for electrostatic problems with floating potentials

AbstractNumerical simulation plays a crucial role in the analysis and design of power equipment, such as lightning protection devices, which may become inefficient using traditional grid‐based methods when handling complex geometries of large problems. The authors propose a grid‐free Monte Carlo method to handle electrostatic problems of complex geometry for both the interior and exterior domains, which is governed by the Poisson equation with a floating potential boundary condition that is neither a pure Dirichlet nor a Neumann condition. The potential and gradient at any given point can be expressed in terms of integral equations, which can be estimated recursively within the walk‐on‐sphere algorithm. Numerical examples have been demonstrated, including the evaluation of the mutual capacitance matrix of multi‐conductor structures and lighting striking near real fractal trees. The proposed method shows advantages in terms of geometric flexibility and robustness, output sensitivity, and parallelism, which may become a candidate for game‐changing numerical methods and exhibit great potential applications in high‐voltage engineering.

Development and Generalization of the Method of Reflections in Problems of Electrostatics and Thermal Conductivity of Plane-Layered Media

Development and Generalization of the Method of Reflections in Problems of Electrostatics and Thermal Conductivity of Plane-Layered Media

An adhesion model for contact electrification

Contact electrification (CE) is a universal phenomenon that occurs at the contact interface where tribo-charges transfer from one surface to another. These CE-driven charges accumulate at the interface and can significantly affect the soft adhesive contact. So far, only a few analytical and numerical models have been proposed to partially solve the CE-induced electroadhesive contact problem. In this study, we have developed a full self-consistent contact model to study the CE-induced electroadhesion between a dielectric elastic axisymmetric parabolic surface and a dielectric rigid flat. Both surfaces are initially neutralized and undergo only one loading–unloading cycle. The surface interaction is determined by both the Lennard-Jones and the electrostatic interaction laws. The electrostatic problem is numerically solved to accurately calculate the electrostatic traction due to both the tribo-charges and bound charges. The normal traction, interfacial gap, and hysteresis loop of the force curve are thoroughly investigated. The effect of non-uniform tribo-charge density on the force curve and electrostatic traction is explored. The numerical model developed in the present work can serve as the foundation for more complex electroadhesive contact models. It also sheds light on the interfacial design of CE-related devices, e.g., contact-separation triboelectric nanogenerator.

Can the Dirac deltas in dipole fields be ignored in classical interactions?

When studying (or teaching) classical electromagnetism, one is bound to deal with the electric field of an ideal electric dipole, as well as its magnetic counterpart. A careful analysis then reveals that each of those fields must include, for consistency, a term proportional to a Dirac delta function localized at the position of the dipole. However, one is usually told not to worry about those terms since, as classical interactions always involve sources which are spatially separated, the Dirac-delta terms are only relevant for quantum mechanics, where they are directly related to important phenomena. In this work, we pose and solve a purely classical problem in electrostatics in which the Dirac-delta terms in the dipole fields are indispensable. It involves the computation of the interaction energy between a conductor with a spherical cavity and an (ideal) electric dipole located at the center of that cavity. We also solve its magnetic counterpart, replacing the conductor with a superconductor and the electric dipole with a magnetic one.

Effect of Water Adsorption on Lubricating Film Stability in Slippery Coatings.

The absorption of water by slippery coatings is a ubiquitous phenomenon that arises due to small but finite water dissolution during the contact of aqueous media with lubricants. In this study, using the concept of surface forces, we have analyzed the influence of trace amounts of water in lubricants on the stability of slippery coatings for both coatings with hydrophilic porous bases, prone to form hydrogen bonds with water, and those with hydrophobic porous bases. To perform such analysis, we have considered for the first time the electrostatic problem of the distribution of the electric potential and electric field strength in stratified films that contain two thin dielectric layers imitating the lubricant and a hydrophobic layer sandwiched between the porous substrate and air or water. Based on the developed approach, the equations for the calculation of the excess free energy and the disjoining pressure, associated with water dissolution in the lubricant film, were derived. An analysis of the excess film energy and the contribution of the image interaction to the disjoining pressure isotherm indicates that the dissolved water can both stabilize and destabilize the oil film, depending on the ratio of the static and dynamic dielectric permittivities of the lubricant and the phases confining the film. The results of calculations and simple experiments carried out here indicate much better water contact stability of silicone films impregnating a hydrophobized porous substrate than films impregnating a hydrophilic porous substrate. The obtained results indicate the preference for using the hydrophobic bases for the fabrication of long-lived slippery coatings characterized by better preservation of slippery functional properties under conditions of lubricant depletion and prolonged contact with water.

Computer homogenization of porous piezoceramics of different ferrohardness with random porous structure and inhomogeneous polarization field

The article is concerned with the homogenization problems, in which the effective moduli of porous piezoceramic composites are determined taking into account the inhomogeneity of the polarization field. The homogenization problems are solved by the finite element method in the framework of the theory of effective moduli and the Hill energy principle using the ANSYS package. To this end, in static problems of electroelasticity, the displacements and electric potential, which are linear in spatial variables, are specified on the boundary of a representative volume to provide constant stress and electric induction fields for a homogeneous reference medium. After solving a set of boundary value problems under different boundary conditions and determining the volume-averaged stress components and the electric induction vector, a complete set of effective moduli for the piezoelectric composite is calculated. A representative volume of the piezocomposite is created in the form of a regular finite element mesh consisting of cubic elements. Pores in the representative volume are assumed to be filled with a piezoelectric material with extremely small moduli. Finite elements with pore properties are selected according to a simple random algorithm. The inhomogeneous polarization field is found by solving an electrostatic problem, in which the polarization process in the representative volume is modeled based on a simplified linear formulation. The local coordinate systems for individual finite elements of the composite matrix are specified by the directions of the polarization vectors. In the following, when solving the problems of electroelasticity, these local coordinate systems associated with the elements of the piezoelectric matrix allow recalculating the material properties according to the formulas of transformation of the tensor components as the coordinate systems rotate. In addition, consideration is given to different models describing the change in the moduli of the material from an unpolarized state to a polarized one as a function of the polarization vector. Computational experiments were carried out for three types of piezoceramics: soft ferroelectric piezoceramics PZT-5H, piezoceramics PZT-4 of medium ferrohardness, and piezoceramics PZT-8 with higher degree of ferrohardness. The dependences of the effective moduli on porosity are compared for different laws of polarization inhomogeneity and different kinds of piezoceramic material of the composite matrix.

On Some Applications of the Superposition Principle in the Course of Physics and Mathematics

Key words: problem, field, property, vector, sum, geometry, proof The article is devoted to the effective applications of the superposition principle in teaching physics and mathematics. In the process of teaching any subject, including physics and mathematics, it is necessary to pay due attention to the general ideas and principles applying to almost all sections of the course. Sequential study of the principles not only contributes to a complete understanding of the subject, but also makes it possible to transform and apply the skills acquired in one section in the process of teaching another section. Moreover, sometimes such opportunities arise not only at the intra-subject level, but also at the inter-subject level, when an idea or principle taken from the “source” of one subject can, in a certain sense, be applied in teaching another subject. The search and realization of such opportunities can promote the development of interdisciplinary connections and teaching quality. In this work, after a general presentation of the main idea of the superposition principle, its specific applications in electrodynamics are considered. A set of electrostatics problems are presented, the solutions of which are accompanied by methodological instructions. Possible options for posing other similar problems of electrodynamics are also indicated. Trying to establish certain similarities, the ideas of this principle are applied in the process of solving some problems in a mathematics course, which is the main scientific and methodological novelty of the work.

On the T-Matrix in the Electrostatic Problem for the Spheroidal Particle with a Spherical Core

On the T-Matrix in the Electrostatic Problem for the Spheroidal Particle with a Spherical Core

Self-Diffusiophoresis and Symmetry-Breaking of a Janus Dimer: Analytic Solution

A self-diffusiophoretic problem is considered for a chemically active dimer consisting of two equal touching spherical colloids that are exposed to different fixed-flux and fixed-rate surface reactions. A new analytic solution for the autophoretic mobility of such a catalytic Janus dimer is presented in the limit of a small Péclet number and linearization of the resulting Robin-type boundary value problem for the harmonic solute concentration. Explicit solutions in terms of the physical parameters are first obtained for the uncoupled electrostatic and hydrodynamic problems. The dimer mobility is then found by employing the reciprocal theorem depending on the surface slip velocity and on the normal component of the shear stress acting on the inert dimer. Special attention is given to the limiting case of a Janus dimer composed of an inert sphere and a chemically active sphere where the fixed-rate reaction (Damköhler number) is infinitely large. Examples are given, comparing the numerical and approximate analytic solutions of the newly developed theory. Singular points arising in the model are discussed for a dimer with a fixed-rate reaction, and the flow field around the dimer is also analysed. The new developed theory introduces a fast way to compute the mobility of a freely suspended dimer and the induced flow field around it, and thus can also serve as a sub grid scale model for a multi-scale flow simulation.

Numerical Method for 1D Poisson-Boltzmann Equation

Poisson-Boltzmann is an equation which is used frequently to model a variety of electrostatic interactions problems in biomolecular computation. A value of atomic energy at the molecular movement becomes a singular value on the PBE. It affects the difficulty of solving this equation both analytically and numerically. In this article, this equation will be solved numerically. In this case, this equation is discretized using finite element method. The resulted system of equation is then solved with inexact backtracking Newton.

Modeling the electrostatic field of a charged ring located inside an infinite cylinder in the presence of a torus

Objectives. Analytical solution of the boundary value problem of electrostatics for modeling the electrostatic field of a charged ring located inside a grounded infinite circular cylinder in the presence of a perfectly conducting torus is considered. The field source is a thin charged ring located on a plane perpendicular to the axis of the cylindrical screen.Methods. To solve the problem, the method of addition theorems is used. The potential of the initial electrostatic field is presented in the form of spherical harmonic functions and in the form of a superposition of cylindrical and toroidal harmonic functions, using addition theorems relating spherical, cylindrical and toroidal harmonic functions. The secondary potential of the electrostatic field is also represented as a superposition of cylindrical and toroidal harmonic functions.Results. The solution of the formulated boundary problem is reduced to the solution of an infinite system of linear algebraic equations of the second kind with respect to the coefficients included in the representation of the secondary field. The influence of some parameters of the problem on the value of the electrostatic potential inside a grounded cylindrical shield in the presence of a toroidal inclusion is numerically studied. The calculation results are presented in the form of graphs.Conclusion. The proposed technique and the developed software can find practical application in the development and design of screens in various fields of technology.

Sensitivity Analysis Based New Numerical Approach with Green Function for Optimization

In this study, sensitivity analysis based on a new numerical approach is developed for the three-dimensional structural optimization of high voltage system devices. It has been developed using Green’s theorem in a closed form to satisfy all boundary conditions and necessary conditions, including material and geometric dimensions. The developed sensitivity function and objective function are suitable for structural optimization of all electrostatic problems. The suitability and application of the developed function were made on the bushing. The changing geometry information was obtained from the developed sensitivity function as a result of the optimization. The analysis made with the classical design and the developed function were compared with each other. Optimization with the developed function has provided insulation and conductor size reduction. Experimental test devices before and after optimization process produced and tested. These results are presented in the study.

Development of Methods for Solving Problems of Electrostatics and Thermal Conductivity of Plane-Layered Media

Development of Methods for Solving Problems of Electrostatics and Thermal Conductivity of Plane-Layered Media

Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation

We study the sequence of monic polynomials {Sn}n⩾0, orthogonal with respect to the Jacobi-Sobolev inner product ⟨f,g⟩s=∫−11f(x)g(x)dμα,β(x)+∑j=1N∑k=0djλj,kf(k)(cj)g(k)(cj), where N,dj∈Z+, λj,k⩾0, dμα,β(x)=(1−x)α(1+x)βdx, α,β>−1, and cj∈R∖(−1,1). A connection formula that relates the Sobolev polynomials Sn with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence {Sn}n⩾0 and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of n unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation.

Introducing corrugated surfaces in electrostatic problems via a perturbative approach

In electromagnetism courses, students often solve Poisson's equation for a point charge in the presence of an infinitely large perfectly conducting planar surface, usually by the method of images. However, no surface is perfectly flat; so at some level, corrugations must be introduced to model the real world. Clinton et al. [Phys. Rev. B 31, 7540 (1985)] solved the problem, including corrugations, using a perturbative calculation of the corresponding Green's function. We provide a detailed pedagogical review of this calculation and extend it in order to solve for the electrostatic potential of a corrugated neutral conducting cylinder in the presence of a uniform electric field. These calculations can be used as pedagogical examples of this perturbative approach in electromagnetism courses.