Electrostatic Theorem Research Articles

A closed-form expression of electric polarizability for elliptical apertures excited by nonuniform electric field

It is well known that electric field penetration through a small aperture in a perfectly conducting plane behaviors like an electric dipole. When excited by uniform electric field, the dipole moment has analytical solution for elliptical apertures and numerical/experimental values for other shapes. In this paper, for the case of nonuniform field excitation, based on the electrostatic reciprocity theorem, the closed-form expression of electric dipole moment is derived for elliptical apertures. This expression is in integral form, and its integrand is the product of the nonuniform electric field distribution and the potential distribution when excited by uniform field in the aperture plane. Taking a point charge located on the aperture plane as an example of nonuniform field excitation, the dipole moment is calculated using the proposed closed-form expression. The validation is verified by comparisons with exact integral solution for circular apertures and with finite element simulations for elliptical apertures.

Boundary effects on ideal fluid forces and Kelvin's minimum energy theorem

The electrostatic force on a charge above a neutral conductor is generally attractive. Surprisingly, that force becomes repulsive in certain geometries (Levin & Johnson, Am. J. Phys., vol. 79, issue 8, 2011, pp. 843–849), a result that follows from an energy theorem in electrostatics. Based on the analogous minimum energy theorem of Kelvin (Camb. Dublin Math. J., vol. 4, 1849, pp. 107–112), valid in the theory of ideal fluids, we show corresponding effects on steady and unsteady fluid forces in the presence of boundaries. Two main results are presented regarding the unsteady force. First, the added mass is proven to always increase in the presence of boundaries. Second, in a model of a body approaching a boundary, where the unsteady force is typically repulsive (Lamb, Hydrodynamics, 1975, § 137, University Press), we present a geometry where the force can be attractive. As for the steady force, there is one main result: in a model of a Bernoulli suction gripper, for which the steady force is typically attractive, we show that the force becomes repulsive in some geometries. Both the unsteady and steady forces are shown to reverse sign when boundaries approximate flow streamlines, at energy minima predicted by Kelvin's theorem.

Activated diffusiophoresis.

Perturbations of fluid media can give rise to non-equilibrium dynamics, which may, in turn, cause motion of immersed inclusions or tracer particles. We consider perturbations ("activations") that are local in space and time, of a fluid density which is conserved, and study the resulting diffusiophoretic phenomena that emerge at a large distance. Specifically, we consider cases where the perturbations propagate diffusively, providing examples from passive and active matter for which this is expected to be the case. Activations can, for instance, be realized by sudden and local changes in interaction potentials of the medium or by local changes in its activity. Various analytical results are provided for the case of confinement by two parallel walls. We investigate the possibility of extracting work from inclusions, which are moving through the activated fluid. Furthermore, we show that a time-dependent density profile, created via suitable activation protocols, allows for the conveyance of inclusions along controlled and stable trajectories. In contrast, in states with a steady density, inclusions cannot be held at stable positions, reminiscent of Earnshaw's theorem of electrostatics. We expect these findings to be applicable in a range of experimental systems. The phenomena described here are argued to be distinct from other forms of phoresis such as thermophoresis.

化学键的量子化学研究(一)

阐述了化学键研究所涉及的几个基本量子化学概念。通过量子化学计算，进行分子轨道和电子密度差分析，从Hellman-Feynman静电定理的视角，考察了化学键之一共价键的本质，共价键是在核(或基团)之间共享成键区电子所引起的结合力。 Some quantum chemical concepts involved in study of chemical bonds are described. By means of quantum chemistry calculation, we conducted molecular orbital analysis and electron density difference analysis, and from view point of Hellman-Feynman electrostatic theorem, we investi-gated the nature of covalent bond, which is one typical kind of chemical bond. We showed that covalent bond is binding force caused by electrons in bonding region shared among nuclei (or groups).

On the electrostatic equilibrium of charges and cavities in a conductor

We consider a charged conductor of arbitrary shape, in electrostatic equilibrium, with one or more cavities inside it, and with fixed charges placed outside the conductors and inside the cavities. The field inside a particular cavity is then only due to charges within that cavity itself and to the surface charge induced on the surface of the same cavity. A similar statement holds for the exterior of the conductor. Although this is an elementary property of conductors, it is not a trivial statement, as explained in this article. Undergraduate texts in electrodynamics do not discuss at length or provide a complete argument for an important problem such as this. Two simple and complete proofs are provided in this note with the help of the standard electrostatic uniqueness theorems.

A variational proof of Thomson's theorem

Thomson's theorem of electrostatics, which states the electric charge on a set of conductors distributes itself on the conductor surfaces to minimize the electrostatic energy, is reviewed in this letter. The proof of Thomson's theorem, based on a variational principle, is derived for a set of normal charged conductors, with and without the presence of external electric fields produced by fixed charge distributions. In this novel approach, the variations are performed on both the charge densities and electric potentials, by means of a local Lagrange multiplier associated with Poisson's equation, constraining the two variables.

Dispersion dipoles for coupled Drude oscillators.

We present the dispersion-induced dipole moments of coupled Drude oscillators obtained from two approaches. The first approach evaluates the dipole moment using the second-order Rayleigh-Schrödinger perturbation theory wave function allowing for dipole-dipole and dipole-quadrupole coupling. The second approach, based on response theory, employs an integral of the dipole-dipole polarizability of one oscillator and the dipole-dipole-quadrupole hyperpolarizability of the other oscillator over imaginary frequencies. The resulting dispersion dipoles exhibit an R(-7) dependence on the separation between the two oscillators and are connected to the leading-order C6/R(6) dispersion energy through the electrostatic Hellmann-Feynman theorem.

Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem

For the dynamics x '' =-∇ x V(x), an equilibrium point x ̲ are always unstable when on a neighborhood of x ̲ the non constant V satisfies P(x,∂)V=0 for a real second order elliptic P. The proof uses a result of Kozlov [6].

Arthur Melville Thompson 1917–2009

Arthur Melville (‘Mel') Thompson graduated from the University of Adelaide in 1938 with First Class Honours in Physics. After graduation he joined Australia's Council for Scientific and Industrial Research as one of the ‘founding fathers' of the National Standards Laboratory and embarked on a life-time career in metrology. His work in precision electrical measurement, ratio-arms transformer bridges and techniques for defining and measuring small capacitances is internationally renowned. He conceived the design of a calculable-capacitor, the Thompson-Lampard capacitor, which led to a new theorem in electrostatics and provided the basis for an absolute determination of the unit of resistance with an increase in accuracy of an order of magnitude. Beyond the calculable capacitor his work had a major impact in electrical impedance measurement in general and on other fields of metrology such as dilatometry and thermometry. Mel Thompson was an inspirational leader and his work facilitated the development of many scientific careers.

Application of the electrostatic Thompson–Lampard theorem to resistivity measurements

A method for measurement of resistivity of flat samples and thin layers complementary to the well-known van der Pauw technique has been proposed. The method is based on the application of the Thompson–Lampard theorem of electrostatics used in metrology for the realization of calculable capacitor, according to which a large variety of electrode systems can be designed. A prototypic electrode arrangement is shown on which the practical performance of the method was tested.

Dispersion, static correlation, and delocalisation errors in density functional theory: An electrostatic theorem perspective

Dispersion, static correlation, and delocalisation errors in density functional theory are considered from the unconventional perspective of the force on a nucleus in a stretched diatomic molecule. The electrostatic theorem of Feynman is used to relate errors in the forces to errors in the electron density distortions, which in turn are related to erroneous terms in the Kohn-Sham equations. For H(2), the exact dispersion force arises from a subtle density distortion; the static correlation error leads to an overestimated force due to an exaggerated distortion. For H(2)(+), the exact force arises from a delicate balance between attractive and repulsive components; the delocalisation error leads to an underestimated force due to an underestimated distortion. The net force in H(2)(+) can become repulsive, giving the characteristic barrier in the potential energy curve. Increasing the fraction of long-range exact orbital exchange increases the distortion, reducing delocalisation error but increasing static correlation error.

Determination of the unit of resistance and the von Klitzing constant R k based on a calculable capacitor

The calculable cross capacitor according to Thompson and Lampard allows the unit of capacitance, the farad, and thus the unit of resistance, the ohm, and the von Klitzing constant Rk to be determined in SI units with high accuracy. The underlying electrostatic theorem and the design criteria of a cross capacitor are described, and the influence of imperfections of the electrode system are discussed. The main components of the measuring chain from the farad to the ohm are illustrated.

“Barbed nanowires” from polymers via electrospinning

AbstractElectrospinning is a highly versatile technique that allows producing fibers with diameters down to a few nanometers not only from polymers but also from metals, metal oxides, or ceramics. Fiber formation in electrospinning differs strongly from other fiber producing methods such as extrusion in that it is basically governed by self‐assembly processes induced by specific electrostatic interactions following the Earnshaw theorem of electrostatics. This allows the production of nanofibers with very peculiar shapes. Here, we report the one step fabrication of barbed nanofibers due to a particular choice of the spinning conditions. Such barbed fibers allow, among others, to control the total porosity of nanofiber nonwovens and to reduce the tendency of linear nano‐objects towards aggregation. POLYM. ENG. SCI., 2009. © 2008 Society of Plastics Engineers

Proof of Thomson's theorem of electrostatics

A 100 years old formula that was given by Thomson [Footnote on p. 154 of J.J. Thomson, Maxwell's Treatise on Electricity and Magnetism, Dover, New York, NY, 1954.] recently found numerous applications in computational electrostatics and electromagnetics. Thomson himself never gave a proof for the formula; but a proof based on Differential Geometry was suggested by Jackson [Classical Electrodynamics, Wiley, New York, NY, 1962, p. 51] and later published by Pappas [Differential geometric solution of a problem in electrostatics, SIAM Rev. 28 (1986) 225]. Unfortunately, Differential Geometry, being a specialized branch of mathematics, is normally inaccessible to the majority of scientists and engineers. This paper provides for the first time a proof that does not depend on Differential Geometry.

Generation of basis sets with high degree of fulfillment of the Hellmann‐Feynman theorem

A direct relationship is established between the degree of fulfillment of the Hellman-Feynman (electrostatic) theorem, measured as the difference between energy derivatives and electrostatic forces, and the stability of the basis set, measured from the indices that characterize the distance of the space generated by the basis functions to the space of their derivatives with respect to the nuclear coordinates. On the basis of this relationship, a criterion for obtaining basis sets of moderate size with a high degree of fulfillment of the theorem is proposed. As an illustrative application, previously reported Slater basis sets are extended by using this criterion. The resulting augmented basis sets are tested on several molecules finding that the differences between energy gradient and electrostatic forces are reduced by at least one order of magnitude.

Investigation of Donnan equilibrium in charged porous materials—a scale transition analysis

We propose a new theory describing how the macroscopic Donnan equilibrium potential can be derived from the microscale by a scale transition analysis. Knowledge of the location and magnitude of the charge density, together with the morphology of the pore space allows one to calculate the Donnan potential, characterizing ion exclusion in charged porous materials. Use of the electrochemical potential together with Gauss' electrostatic theorem allows the computation of the ion and voltage distribution at the microscale. On the other hand, commonly used macroscopic counterparts of these equations allow the estimation of the Donnan potential and ion concentration on the macroscale. However, the classical macroscopic equations describing phase equilibrium do not account for the non-homogeneous distribution of ions and voltage at the microscale, leading to inconsistencies in determining the Donnan potential (at the macroscale). A new generalized macroscopic equilibrium equation is derived by means of volume averaging of the microscale electrochemical potential. These equations show that the macroscopic voltage is linked to so-called "effective ion concentrations", which for ideal solutions are related to logarithmic volume averages of the ion concentration at the microscale. The effective ion concentrations must be linked to an effective fixed charge concentration by means of a generalized Poisson equation in order to deliver the correct Donnan potential. The theory is verified analytically and numerically for the case of two monovalent electrolytic solutions separated by a charged porous material. For the numerical analysis a hierarchical modeling approach is employed using a one-dimensional (1D)macroscale model and a two-dimensional (2D)microscale model. The influence of various parameters such as surface charge density and ion concentration on the Donnan potential are investigated.

Diffusion Monte Carlo study of correlation in the hydrogen molecule

AbstractThe diffusion Monte Carlo (DMC) method shows that correlation in H2 produces a set of three spatial changes: (i) an enhancement in the electron density distribution n(r) in the left and right anti‐binding regions that include separately the immediate vicinity of each of the two nuclei, (ii) a reduction in n(r) in the binding region intervening between the two nuclei as a counterbalance, and (iii) a concomitant increase in the equilibrium internuclear separation. It is stressed that the correlation energy Ec (= Tc + Vc) for diatomic molecules be defined by the difference in the total energy between the exact and the Hartree–Fock (HF) variational calculations that are performed at individually optimized internuclear separations. It is this definition that makes it possible to involve a significant contribution from a correlation‐induced change in the equilibrium internuclear separation as part of the correlation energy and to relate (i) and (ii) to (iii) in consistency with the electrostatic theorem. The present calculations fulfill the virial theorem to an accuracy of −V/T = 2.00 for DMC and −VHF/THF = 2.000 for HF. The present correlation energy Ec = −0.0408 hartree is not only in good agreement with the most accurate value previously reported, but also can be analyzed into all its components in accordance with the correlational virial theorem 2Tc + Vc = 0. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Investigation of nanoscale electrohydrodynamic transport phenomena in charged porous materials

Depending on the permeability of porous materials, different mass transport mechanisms have to be distinguished. Whereas mass transport through porous media characterized by low permeabilities is governed by diffusion, mass transport through highly permeable materials is governed by advection. Additionally a large number of porous materials are characterized by the presence of surface charge which affects the permeability of the porous medium. Depending on the ion transport mechanism various phenomena such as co-ion exclusion, development of diffusion–exclusion potentials, and streaming potentials may be encountered. Whereas these various phenomena are commonly described by means of different transport models, a unified description of these phenomena can be made within the framework of electrohydrodynamics. In this paper the fundamental equations describing nanoscale multi-ion transport are given. These equations comprise the generalized Nernst–Planck equation, Gauss' theorem of electrostatics, and the Navier–Stokes equation. Various phenomena such as the development of exclusion potentials, diffusion–exclusion potentials, and streaming potentials are investigated by means of finite element analyses. Furthermore, the influence of the surface charge on permeability and ion transport are studied in detail for transient and steady-state problems. The nanoscale findings provide insight into events observed at larger scales in charged porous materials. Copyright © 2005 John Wiley Sons, Ltd.

Accuracy of the electrostatic theorem for high‐quality Slater and Gaussian basis sets

AbstractThe fulfillment of the Hellmann–Feynman electrostatic theorem is examined for the sequences of cc‐pVxZ and cc‐pCVxZ Gaussian basis sets as well as for the VBx and CVBx basis sets of Slater‐type orbitals. The difference between the energy gradient and electrostatic forces is large in small Gaussian basis sets of the two types, but decreases quickly as the basis sets improve. In VBx Slater basis sets these differences are small but the improvement is irregular, whereas in CVBx basis sets the fulfillment of the electrostatic theorem is very satisfactory. For the high‐quality basis sets (cc‐pV5Z, cc‐pCVQZ, cc‐pCV5Z, CVB2, and CVB3) the energy gradient can be replaced by the electrostatic force in most practical applications. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Density and binding forces: Rotational barrier of ethane

The possibility of extending the relationships between density, binding forces, and bonding energies to fine chemical effects is tested taking as an example the rotational barrier of ethane. Electron densities that reasonably fulfill the electrostatic theorem were obtained for several conformations using a Slater basis set. The analysis of these densities shows that the barrier is due to the internal forces acting on the H nuclei. Out of the staggered and eclipsed conformations, the clouds of the hydrogen atoms have a nonskeletal distortion that pull their nuclei toward the staggered conformation.