Electrostatic Problem Research Articles

Four-dimensional reflection groups and electrostatics

We present a new class of electrostatics problems that are exactly solvable by adding finitely many image charges. Given a charge at some location inside a cavity bounded by up to four conducting grounded segments of spheres: if the spheres have a symmetry derived via a stereographic projection from a 4D finite reflection group, then this is a solvable generalization of the familiar problem of a charge inside a spherical cavity. There are 19 three-parametric families of finite groups formed by inversions relative to at most four spheres, each member of each family giving a solvable problem. We solve a sample problem that derives from the reflection group D4 and requires 191 image charges.

Uniform quantized electron gas: Radiation correction

In this paper we analyze how radiation effects influence the correlation functions, the excess energy, and in turn the electron correlation energy of the quantized electron gas at temperature T=0. To that aim we resort to a statistical mechanical description of the quantum problem of electron correlations, based on the path integral formalism. In previous works we studied and found accurate results for the usual situation with the electrostatic Coulomb interaction. Here the additional problem with radiation is taken into account. This is facilitated by the equivalence to a dielectric fluid for which correlation functions for dipolar moments are established. From these functions follows the usual density–density (or charge–charge) correlation function needed for the longitudinal electrostatic problem, and in addition the one needed for the transverse radiation problem. While electrostatic excess energy is negative, the transverse one is positive. This quantity is small and decreases rapidly for decreasing densities. However, for high densities it approaches the electrostatic contribution, eventually becoming even larger. The part of the transverse energy from induced correlations turns out to be very small. Also, the non-local longitudinal and transverse dielectric constants of the electron gas are identified from the induced correlation functions.

Four solutions of a two-cylinder electrostatic problem, and identities resulting from their equivalence

Summary Four distinct solutions exist for the potential distribution around two equal circular parallel conducting cylinders, charged to the same potential. Their equivalence is demonstrated, and the resulting analytical identities are discussed. The identities relate the Jacobi elliptic function $sn$, the Jacobi theta functions $\theta _1 ,~\theta _2 $ and infinite series over trigonometric and hyperbolic functions.

Application of the Schwarz-Christoffel Transformation to the Solution of Harmonic Dirichlet Problems in Electrostatics

In this paper, a purely conformal mapping method for efficiently solving harmonic Dirichlet problems of electrostatic in domains free of charge and with charge whose boundaries have inconvenient geometries consisting of straight-line segments is presented. The method which uses the inverse of an appropriately determined Schwarz-Christoffel transformation as the mapping function, was applied to harmonic Dirichlet problems in an infinite strip and infinite sector and the solution or electrostatic potential for the problem obtained for each case. Furthermore, the equipotential lines of the electric field were also obtained in order to show the features of the solution and the field analysed accordingly. The electric field intensity was also analysed to show its variation in the field. This method could therefore be a suitable alternative method for solving Laplace's equation in two dimensional electrostatic problems.

Cálculo de la Distribución de Voltajes en las Cadenas de Aisladores de una Línea de Transmisión de 500 kV utilizando el Método de los Elementos Finitos

En este artículo se presentan las distribuciones de voltaje a lo largo de una cadena de aisladores de una línea de transmisión de 500 kV de Ecuador. El problema electrostático relacionado a la distribución de voltajes se resolvió utilizando el Método de los Elementos Finitos. Se realiza un dibujo a escala 1:1 en una herramienta de dibujo tipo CAD para luego ser exportada al programa FEMM que resuelve el problema y del que se obtienen los resultados relacionados a los voltajes en la cadena de aisladores. Se aprecian las diferencias de voltaje en los diferentes aisladores que componen una cadena y se observa la diferencia entre las cadenas determinada por la ubicación geométrica en el espacio de solución.

ELECTROSTATIC LENS SYSTEM DESIGN WITH THE ARTIFICIAL NEURAL NETWORKS

Yapay zekâ algoritmalarıyla son yıllarda birçok bilim dalında başarılı uygulamalar geliştirilmektedir. Deneysel veya benzetim programlarından elde edilen veriler söz konusu algoritmalarla işlenmektedir. Tasarlanan algoritma mimarileri ile veriler işlenerek tahmin ve sınıflandırma çalışmaları yapılmaktadır. Bu algoritmalardan, amaca ve veri kümesine uygun olan algoritmanın seçilmesi büyük önem taşımaktadır. Bu kapsamda, fizik alanındaki yenilikçi çalışmalarda yapay sinir ağı algoritması kullanmak yüksek performans değerleri elde etmeyi sağlamaktadır. Biyolojik nöronlardan esinlenen yapay sinir ağı (YSA), öğrenme yeteneğine sahip paralel hesaplama sistemidir. Bu çalışmada, üç katmanlı yapay sinir ağı kullanılarak beş elemanlı elektrostatik silindir lenslerin paralel demet modu belirlenmektedir. Çalışmada kullanılan veri kümesi, yüksek doğrulukta hesaplama yapabilen CPO(Yüklü Parçacık Optiği) programı yardımıyla elde edilmiştir. Verilerin analizi Matlab R2012b programı kullanılarak gerçekleştirilmiştir. Elde edilen sonuçlara göre, yapay sinir ağının fizik alanında paralel demet modunu belirlemede yüksek performans değerlerine sahip olan ve elektrostatik problem çözümlerinde sonlu fark ve sınır eleman metoduna alternatif bir metot olduğu ortaya konulmuştur. Oluşturulan YSA algoritması, test verilerinin %85,7’sini doğru olarak sınıflandırmıştır.

On the electrostatic potential and electric field of a uniformly charged disk

We calculate the electrostatic potential and electric field of a uniformly charged disk everywhere in space. This electrostatic problem was solved long ago, and its gravitational analogue was solved even earlier. However, it seems that physics students are not aware of the solution, because it is not presented in textbooks. The purpose of the present article is to fill this gap in the pedagogical literature.

A uniformly charged circular disk with an anisotropic Coulomb interaction potential

We consider a uniformly charged circular disk containing elementary charges that interact with an anisotropic Coulomb interaction potential. Such an anisotropic Coulomb interaction potential has been recently considered within the framework of a two-dimensional system of electrons in the quantum Hall regime. The anisotropy of the potential is introduced by a phenomenological parameter that can be tuned continuously as to incorporate the standard isotropic Coulomb interaction potential as a special case. The energy of the uniformly charged circular disk is calculated exactly for the given anisotropic interaction potential as a function of the anisotropy parameter. The results obtained are related to electrostatic problems that involve uniformly charged flat elliptical plates and may be useful to understand finite two-dimensional systems of electrons in the quantum Hall regime in which all point charges interact with the anisotropic Coulomb interaction potential presently considered.

Universal behavior in non-stationary Mean Field Games

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled objects in interaction. Though these models are much simpler than the underlying differential games they describe in some limit, their behavior is still far from being fully understood. When the system is confined, a notion of “ergodic state” has been introduced that characterizes most of the dynamics for long optimization times. Here we consider a class of models without such an ergodic state, and show the existence of a scaling solution that plays a similar role. Its universality and scaling behavior can be inferred from a mapping to an electrostatic problem.

DETERMINATION OF THE EFFECTIVE PERMITTIVITY OF A HETEROGENEOUS MATERIAL

Purpose. To develop a two-dimensional numerical-field model for determining the effective permittivity of a multicomponent material represented by a system of homogeneous volumes with known physical characteristics. Methodology . The model is based on the solution by the finite element method of an electrostatic problem with the subsequent determination of the energy contained in the volume under consideration. Than we have compared this result with the energy of a flat capacitor with a rectangular cross-section of the plates and determined the effective permittivity of test material. We also have used Rayleigh, Odelevsky and Lichtenecker models and the model with a perpendicular arrangement of layers relative to the main electric flux. Results. Based on the developed field model, the effective permittivities for dry, wet and transformer oil-soaked insulating papers of various grades, including taking into account ash, are determined. We have proved that a macroscopically homogeneous multicomponent material is well approximated by uniformly spaced cylindrical volumes with a substance of different nature in a matrix of another substance. We have showed a significant error of the layer model and the Rayleigh model relative to the proposed model. We have showed the equivalence of models with the location of inclusions in the nodes of a rectangular and parallelogram mesh. Originality. For the first time we have proposed wet paper models with an asymmetric arrangement of a cylindrical volume of water with a circular and segment cross-section in a cylindrical pore. For the first time we have proposed models of insulating paper with evenly spaced cylindrical inclusions of different volumes. Practical value. The proposed model allows to calculate the effective permittivity of an inhomogeneous material with a given accuracy without restricting the shape of the components. Based on the proposed field model, it is possible to determine the Lichtenecker index, which allows to calculate the effective permittivity for any ratio of the volumes of the components of a heterogeneous material.

Electrostatic responses of anisotropic dielectric films

We study the electrostatic responses (i.e. retardation effects due to the propagation of electromagnetic waves are ignored) of a linear homogeneous and anisotropic (LHA) dielectric film to an arbitrary external electrostatic potential. A set of algebraic equations has been established to calculate the polarisation charges induced in the film. In our derivation, the idea is exploited that a physical boundary can be looked upon as a region of rapid variation in polarisation rather than a simple geometric separation. With this no boundary conditions are needed in solving the relevant electrostatics problem. Our approach makes it clear that the responses consist of two contributions, one arising from the very presence of surfaces while the other existing even in an infinite medium. In light of the results, we discuss graphene plasma waves under the influence of a LHA dielectric film such as a few-layer hexagonal boron nitride. It is found that the dispersion of these waves is strongly affected by the anisotropy at wavelengths comparable to the film thickness.

MODELING OF ELECTROSTATIC PROBLEM VIA A SUBPROBLEM FINITE ELEMENT METHOD

This work deals with the modeling of electrostatic problem by means of a subproblem finite element method. The aim of this paper is to present naturally the distributions of the electric scalar potential and the electric field in regions existing the various voltages or difference of voltages. The method permits to simultaneously use both node and edge elements in function spaces of the studied problem. Indeed, the distributed fields (e.g. electric scalar potential and electric field) are presented in the same function space and a week formulation written for both nodal and edge finite elements.

NUMERICAL SOLUTION OF A SYSTEM OF SINGULAR INTEGRAL EQUATION WITH HILBERT AND CAUCHY KERNELS

In the paper a specific system of first kind singular integral equations with the Hilbert and Cauchy kernels arising when solving some problems of electrostatics and electrodynamics is studied. The method of discrete singularities is applied for constructing its discrete mathematical model, which is a system of n linear algebraic equations. Under the additional smoothness assumptions on the right-hand parts of the equations of the initial system and regularity assumptions on the kernels of the integrals in them the obtained system of linear algebraic equations is proved to admit a unique solution for n sufficiently large. The rate of convergence of the solution of the discrete problem to the exact solution of the system of singular integral equations is estimated.

The Lp electrostatic q-capacitary Minkowski problem for general measures

Existence of solutions to the Lp Minkowski problem of electrostatic q-capacity for discrete measures was very recently given by G. Xiong, J. Xiong and L. Xu (JFA, 2019) when 0<p<1 and 1<q<2. In this paper, an existence result of solutions to this problem for general measures is proved when 0<p<1 and 1<q<n, which is a supplement to the previous result.

Two inequalities for convex equipotential surfaces

We establish two geometric inequalities, respectively, for harmonic functions in exterior Dirichlet problems, and for Green's functions in interior Dirichlet problems, where the boundary surfaces are smooth and convex. Both inequalities involve integrals over the mean curvature and the Gaussian curvature on an equipotential surface, and the normal derivative of the harmonic potential thereupon. These inequalities generalize a geometric conservation law for equipotential curves in dimension two, and offer solutions to two free boundary problems in three-dimensional electrostatics.

Using Fractal Calculus to Express Electric Potential and Electric Field in Terms of Staircase and Characteristic Functions

The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, $\chi_{C_{n}}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha}$-integration and $F^{\alpha}$-derivative of the product of characteristic function and function of staircase function, namely $f(S^{\alpha}_{C_{n}}(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F$^{\alpha}$-calculus, has been developed which involve F$^{\alpha}$-integral and F$^{\alpha}$-derivative, of orders $\alpha$, $0&lt;\alpha&lt;1$, where $\alpha$ is dimension of $F$. In F$^{\alpha}$-calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.

Using Fractal Calculus to Express Electric Potential and Electric Field in Terms of Staircase and Characteristic Functions

The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, $\chi_{C_{n}}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha}$-integration and $F^{\alpha}$-derivative of the product of characteristic function and function of staircase function, namely $f(S^{\alpha}_{C_{n}}(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F$^{\alpha}$-calculus, has been developed which involve F$^{\alpha}$-integral and F$^{\alpha}$-derivative, of orders $\alpha$, $0&lt;\alpha&lt;1$, where $\alpha$ is dimension of $F$. In F$^{\alpha}$-calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.

Fast calculation of steady‐state charge distribution in high voltage power cables

AbstractModern high voltage direct current (HVDC) power cables consist of nonlinear electric field and temperature depending insulation materials, eg, polyethylene (PE) or mass‐impregnated paper (MI). Within the insulation material, slowly time varying electric fields are the result of accumulated space charges. Due to the long operation time of power cables, the stationary charge distribution and the resulting additional field stresses inside the insulation material need to be taken into account, to ensure a stable operation of the cable system. The long accumulation time of charges in comparison to fast charge dynamics usually requires extended computation times within transient electro‐quasistatic simulation models. An alternative approach to calculate the final steady‐state charge distribution arises from the simultaneous solution of both, the stationary current and the electrostatic potential problem, which are nonlinearly coupled by Ohm's law. Each of the two potential equations are discretized using the finite difference method. Utilizing a nonlinear fixed point iteration for the electric potential, the steady‐state solution of the charge distribution is computed until convergence is obtained. During this pseudo time stepping process, oscillations may occur that yield incorrect results of the charge distribution. These oscillations are eliminated utilizing underrelaxation strategies. Numerical tests on a cylindrical and a plane parallel problem, using both approaches for the calculation of the steady‐state charge distribution, show a faster computation with the nonlinear fixed point iteration, reducing the computation time by at least one order of magnitude.

Modeling Floating Potential Conductors Using Discontinuous Galerkin Method

Isolated conductors appear in various electrostatic problems. In simulations, an equipotential condition with an undefined/floating potential value is enforced on the surface of isolated conductors. In this work, a numerical scheme making use of the discontinuous Galerkin (DG) method is proposed to model such conductors in electrostatic problems. A floating-potential boundary condition, which involves the equipotential condition together with a total charge condition, is “weakly” enforced on the conductor surfaces through the numerical flux of the DG method. Compared to adaptations of the finite element method used for modeling conductors, this proposed method is more accurate, capable of imposing charge conditions, and simpler to implement. Numerical results, which demonstrate the accuracy and applicability of the proposed method, are presented.

Electrostatic field of angular-dependent surface electrodes

We present an analytic strategy to find the electric field generated by surface electrode SE with angular dependent potential. This system is a planar region $\mathcal{A}$ kept at a fixed but non-uniform electric potential $V(\phi)$ with an arbitrary angular dependence. We show that the generated electric field is due to the contribution of two fields: one that depends on the circulation on the contour of the planar region ---in a Biot-Savart-Like (BSL) term---, and another one that accounts for the angular variations of the potential in $\mathcal{A}$. This approach can be used to find exact solutions of the BSL electric field for circular or polygonal contours of the planar region with periodic distributions of the electric potential. Analytic results are validated with numerical computations and the Finite Element Method. Keywords: Surface-electrode, Biot-Savart law, electrostatic problems, exactly solvable models.