Continuous Charge Distribution Research Articles

Point-charge models and averages for electromagnetic quantities considered in two relativistic inertial frames

Electromagnetic quantities at a spacetime point have tensor Lorentz transformations between relatively-moving inertial frames. However, since the Lorentz transformation of time between inertial frames depends upon both the time and space coordinates, averages of electrodynamic quantities at a single time will in general depend upon the inertial frame, and will differ between inertial frames. Here we illustrate how the use of continuous charge and current distributions rather than point-charge distributions can lead to physically mystifying and even inaccurate results for electromagnetic quantities and physical phenomena. The discrepancy noted between the average electric field values in different inertial frames is particularly striking because it is first order in the relatative velocity between the frames.

Electrostatic potential of a uniformly charged annulus

The calculation of the electrostatic potential and/or electrostatic field due to a continuous distribution of charge is a well-covered topic in all calculus-based undergraduate physics courses. The most common approach is to consider bodies with uniform charge distribution and obtain the quantity of interest by integrating over the contributions from all the differential charges. The examples of a uniformly charged disk and ring are prominent in many physics textbooks since they illustrate well this technique at least for special points or directions of symmetry where the calculations are relatively simple. Surprisingly, the case of a uniformly charged annulus, namely, an annular disk, is largely absent from the literature. One might speculate that a uniformly charged annulus is not extremely interesting since after all, it is a uniformly charged disk with a central circular hole. However, we show in this work that the electrostatic potential created by a uniformly charged annulus has features that are much more interesting than one might have expected. A uniformly charged annulus interpolates between a uniformly charged disk and ring. However, the results of this work suggest that a uniformly charged annulus has such electrostatic features that may be essentially viewed as ring-like. The ring-like characteristics of the electrostatic potential of a uniformly charged annulus are evident as soon as a hole is present no matter how small the hole might be. The solution of this problem allows us to draw attention to the pedagogical aspects of this overlooked, but very interesting case study in electrostatics. In our opinion, the problem of a uniformly charged annulus and its electrostatic properties deserves to be treated at more depth in all calculus-based undergraduate physics courses covering electricity and magnetism.

Exact non-Abelian supertubes

Supertubes are supersymmetric configurations in string theory in which branes are extending along a closed curve. For a supertube of codimension two, its dipole charge is characterized by the duality monodromy around the closed curve. When multiple codimension-2 supertubes are present, the monodromies around different supertubes can be non-commuting, namely non-Abelian. Non-Abelian configurations of supertubes are expected to play an important role in non-perturbative physics of string theory, especially black holes. In this paper, in the framework of five-dimensional supergravity, we construct exact solutions describing codimension-2 supertubes in three-dimensional space. We use an extension formula to construct a three-dimensional solution from a two-dimensional seed solution. The two-dimensional seed is an F-theory like configuration in which a torus is nontrivially fibered over a complex plane. In the first example, there is a stack of circular supertubes around which there is a non-trivial monodromy. In some cases this can be thought of as a microstate of a black hole in AdS2 × S2. The second example is an axi-symmetric solution with two stacks of circular supertubes with non-Abelian monodromies. In addition, there is a continuous distribution of charges on the symmetry axis.

Quantum Mechanics: Keeping It Real?

It is well-known that Schrodinger wanted to interpret the quantum wavefunction ψ as representing a continuous distribution of charge traveling through three-dimensional space. What is less appreciated is that Schrodinger also originally desired that his wavefunction be represented by a real-valued function and not a complex one. This paper is an introduction to and advertisement of Schrodinger’s 4th-order real wave equation, the first published time dependent Schrodinger equation. I introduce two ways of interpreting this equation and show how this equation provides insight into quantum time reversal invariance. My hope is that readers will find other examples where “keeping it real” can help.

Zeta function regularization technique in the electrostatics context for discrete charge distributions

Spectral functions, such as the zeta functions, are widely used in quantum field theory to calculate physical quantities. In this work, we compute the electrostatic potential and field due to an infinite discrete distribution of point charges, using the zeta function regularization technique. This method allows us to remove the infinities that appear in the resulting expression. We found that the asymptotic behavior dependence of the potential and field is similar to the cases of continuous charge distribution. Finally, this exercise can be useful for graduate students to explore spectral and special functions.

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.

Deviation of inverse square law based on Dunkl derivative: deformed Coulomb’s law

In this paper we consider the Coulomb’s law with deviation. We use the Dunkl derivative to derive the deformed Gauss law for the electric field and the electrostatic potentialwhich gives a new deformed electrostatics called a Dunkl-deformed electrostatics. Wemodify the Dunkl derivative for the electric field for multi sources or continuous chargedistribution. We discuss some examples of the Dunkl-deformed electrostatics.

Molecular dynamics simulations on aqueous solution confined in charged nanochannels: asymmetric effect of surface charge

ABSTRACT This work aims at investigating the effect of surface charge sign on static and dynamic properties of aqueous solutions confined in the charged nanochannel. Molecular dynamics simulations of two confinement systems have been performed, in which the nanochannel walls were modelled as the structureless walls with the uniform continuous distribution of surface charge. It is found that the sign of surface charge has a noticeable impact on the aqueous solution in a region close to the wall. The aqueous solution is more strongly layered and less translational and rotational diffusive in the positively charged nanochannel. But, the water molecules are more preferably oriented in the negatively charged nanochannel. The results have been consistently explained by microscopically analysing the forces acting atoms of water molecules and counter-ions, and the contributions into the torque acting on the water molecule.

Fast Coulomb Matrix Construction via Compressing the Interactions Between Continuous Charge Distributions

The continuous fast multipole method (CFMM) is well known for its asymptotically linear complexity for constructing the Coulomb matrix in quantum chemistry. However, in practice, CFMM must evaluate...

Python script used as a simulator for the teaching of the electric field in electromagnetism course

We present in this article a Python script, based on a methodology to obtain the electric field produced by n electric charges. This tool was implemented in courses of electromagnetism and its laboratory in three institutions of higher education. The aim of this work was to incorporate information and communication technologies (ICTs) at the physics subjects, in accordance with the programs promoted by the Colombian Ministry of Education. We wanted to connect the students with sensory experiences of the physical phenomena that allow them to improve their experience of learning of subjects traditionally studied through the board. Finally, in this work, an interactive computational code was obtained, in which the electric field of the discrete and continuous charge distributions can be calculated, for the classical problems that are shown in an electrical physics course.

Investigating and improving introductory physics students’ understanding of electric field and the superposition principle: The case of a continuous charge distribution

Investigating and improving introductory physics students’ understanding of electric field and the superposition principle: The case of a continuous charge distribution

Passivation of Grain Boundary by Squaraine Zwitterions for Defect Passivation and Efficient Perovskite Solar Cells.

Unavoidable defects in grain boundaries (GBs) are detrimental and critically influence the organometal halide perovskite performance and stability. To address this issue, semiconducting molecules have been employed to passivate traps along perovskite GBs. Here, we designed and synthesized three squaraine molecules (SQ) with zwitterionic structure to interact with under-coordinated Pb2+ and passivate Pb-I antisite defects. Density functional theory calculation shows symmetric O atoms could coordinate with perovskite grains simultaneously, resulting in continuous charge distribution at the SQ-perovskite interface. The energetic traps distribution in CH3NH3PbI3 perovskite is influenced significantly by the interaction between SQ and perovskite as analyzed by thermally stimulated current, in which the deep-level defects are considerably reduced due to efficient SQ passivation. In addition, we explore how SQ molecules with different energy offset affect the charge extraction, which is suggested to facilitate exciton separation at the perovskite-SQ interface. These benefits lead to enhanced perovskite efficiency from 15.77 to 18.83% with the fill factor approaching 80%, which is among the highest efficiency reported for MAPbI3 solar cells fabricated in an ambient environment at 60% relative humidity (RH). Considerable retardation of perovskite device degradation was achieved, retaining 90% of initial efficiency when kept 600 h at 60 ± 5% RH.

Prelude to molecular dynamics: Topography‐driven gaussian charge models

AbstractA new strategy to develop Gaussian charge models (GCMs) for molecules like ammonia, water, ethene, hydrogen sulfide, formaldehyde and benzene is presented. These molecular models comprising of positive point charges and negative Gaussian charge distributions (GCDs), which represent nuclei and continuous electron charge distribution, are found to correctly represent the ab initio Molecular Electrostatic Potential (MESP) and reproduce its essential topographical features of corresponding molecules. The models use optimized parameters: positive charges at nuclei, negative charges on GCDs, Gaussian exponent and centers. The Potential Energy Surface (PES) of water dimer has been explored using water GCMs. A good agreement has been found between PES obtained using GCMs and wave function. The Gaussian models correctly predict structure of benzene‐water complex. It is thus recommended to use GCMs for molecular dynamic simulations.

Enhancing the supercapacitive performance of Nickel based metal organic framework-carbon nanofibers composite by changing the ligands

The present work offers a facile, cost-effective and green strategy to efficiently regulate the pore dimensions and surface morphology of Nickel based Metal organic frameworks (MOFs) using three carboxylic acids of different molecular lengths. The material characterization studies revealed that Ni MOF with longer organic linker i.e., 4, 4‘-Biphenyl dicarboxylic acid (Ni-BPDC) exhibits large surface area and large pore dimensions which enables swift movement of electrolyte ions through the framework. Also, the interconnected network of Ni-BPDC nanoplates results into continuous charge distribution, thus delivering better electrochemical performance amongst three Ni MOFs. Furthermore, to enhance the electrochemical performance of Ni-BPDC, it was combined with carbon nanofibers (CNF) and the electrochemical characterization studies of Ni-BPDC/CNF as a single electrode material shows its high specific capacity of 250.6 mA h g−1 at current density of 1 A g−1. To further explore real time application of Ni-BPDC/CNF composite, a symmetric supercapacitor device was fabricated in a Swagelok cell and it is found to deliver excellent energy density of 48.1 Wh kg−1 and power density of 1064.7 W kg−1 at an operating voltage of 1.2 V. Besides, it manifests a long cycle life with only 8% deterioration of initial specific capacity after 5000 cycles. We believe that these encouraging results not only suggests the possibility for tuning properties of existing MOFs using different organic linkers but also provides an efficient way to develop hybrid MOF composites for future energy storage applications.

Equilibrium charge distribution on a finite straight one-dimensional wire

The electrostatic properties of uniformly charged regular bodies are prominently discussed on college-level electromagnetism courses. However, one of the most basic problems of electrostatics that deals with how a continuous charge distribution reaches equilibrium is rarely mentioned at this level. In this work we revisit the problem of equilibrium charge distribution on a straight one-dimensional (1D) wire with finite length. The majority of existing treatments in the literature deal with the 1D wire as a limiting case of a higher-dimensional structure that can be treated analytically for a Coulomb interaction potential between point charges. Surprisingly, different models (for instance, an ellipsoid or a cylinder model) may lead to different results, thus there is even some ambiguity on whether the problem is well-posed. In this work we adopt a different approach where we do not start with any higher-dimensional body that reduces to a 1D wire in the appropriate limit. Instead, our starting point is the obvious one, a finite straight 1D wire that contains charge. However, the new tweak in the model is the assumption that point charges interact with each other via a non-Coulomb power-law interaction potential. This potential is well-behaved, allows exact analytical results and approaches the standard Coulomb interaction potential as a limit. The results originating from this approach suggest that the equilibrium charge distribution for a finite straight 1D wire is a uniform charge density when the power-law interaction potential approaches the Coulomb interaction potential as a suitable limit. We contrast such a finding to results obtained using a different regularised logarithmic interaction potential which allows exact treatment in 1D. The present self-contained material may be of interest to instructors teaching electromagnetism as well as students who will discover that simple-looking problems may sometimes pose important scientific challenges.

On deriving the Maxwell stress tensor method for calculating the optical force and torque on an object in harmonic electromagnetic fields

Though extensively used in calculating optical force and torque acting on a material object illuminated by laser, the Maxwell stress tensor (MST) method follows the electromagnetic linear and angular momentum balance that is usually derived in most textbooks for a continuous volume charge distribution in free space, if not resorting to the application of Noether’s theorem in electrodynamics. To cast the conservation laws into a physically appealing form involving the current densities of linear and angular momentum, on which the MST method is based, the divergence theorem is employed to transform a volume integral into a surface integral. When a material object of finite volume is put into the field, it brings about a discontinuity of field across its surface, due to the presence of induced surface charge and surface current. Ambiguity arises among students in whether the divergence theorem can still be directly used without any justification. By taking into account the effect of the induced surface charge and current, we present a simple pedagogical derivation for the MST method for calculating the optical force and torque on an object immersed in monochromatic optical field, without resorting to Noether’s theorem. Although the results turn out to be identical to those given in the standard textbooks, our derivation avoids the direct use of the divergence theorem on a discontinuous function.

ELECTRIC POTENTIAL AND FIELD CALCULATION OF CHARGED BEM TRIANGLES AND RECTANGLES BY GAUSSIAN CUBATURE

It is a widely held view that analytical integration is more accurate than the numerical one. In some special cases, however, numerical integration can be more advantageous than analytical integration. In our paper we show this benefit for the case of electric potential and field computation of charged triangles and rectangles applied in the boundary element method (BEM). Analytical potential and field formulas are rather complicated (even in the simplest case of constant charge densities), they have usually large computation times, and at field points far from the elements they suffer from large rounding errors. On the other hand, Gaussian cubature, which is an efficient numerical integration method, yields simple and fast potential and field formulas that are very accurate far from the elements. The simplicity of the method is demonstrated by the physical picture: the triangles and rectangles with their continuous charge distributions are replaced by discrete point charges, whose simple potential and field formulas explain the higher accuracy and speed of this method. We implemented the Gaussian cubature method for the purpose of BEM computations both with CPU and GPU, and we compare its performance with two different analytical integration methods. The ten different Gaussian cubature formulas presented in our paper can be used for arbitrary high-precision and fast integrations over triangles and rectangles.

Parameter Tuning for the NFFT Based Fast Ewald Summation

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P$^2$NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. Typically B-splines are applied in the scope of particle mesh methods, as for instance within the well known particle-particle particle-mesh (P$^3$M) algorithm. The publicly available P$^2$NFFT algorithm allows the application of an oversampled FFT as well as the usage of different window functions. We consider for the first time also an approximation by Bessel functions and show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that, if the parameters are tuned appropriately, the Bessel window function is in many cases even the better choice in terms of computational costs. Moreover, the results indicate that it is often advantageous in terms of efficiency to spend some oversampling within the NFFT while using a window function with a smaller support.

Charged line segments and ellipsoidal equipotentials

This is a survey of the electrostatic potentials produced by charged straight-line segments, in various numbers of spatial dimensions, with comparisons between uniformly charged segments and those having non-uniform linear charge distributions that give rise to ellipsoidal equipotentials surrounding the segments. A uniform linear distribution of charge is compatible with ellipsoidal equipotentials only for three-dimensions. In higher dimensions, the linear charge density giving rise to ellipsoidal equipotentials is counter-intuitive—the charge distribution has a maximum at the centre of the segment and vanishes at the ends of the segment. Only in two-dimensions is the continuous charge distribution intuitive—for that one case of ellipsoidal equipotentials, the charge is peaked at the ends of the segment and minimised at the centre.

Electric field and space charge distribution measurement in transformer oil struck by impulsive high voltage

Transformer oil is widely used in power systems because of its excellent insulation properties. The accurate measurement of electric field and space charge distribution in transformer oil under high voltage impulse has important theoretical and practical significance, but still remains challenging to date because of its low Kerr constant. In this study, the continuous electric field and space charge distribution over time between parallel-plate electrodes in high-voltage pulsed transformer oil based on the Kerr effect is directly measured using a linear array photoelectrical detector. Experimental results demonstrate the applicability and reliability of this method. This study provides a feasible approach to further study the space charge effects and breakdown mechanisms in transformer oil.