Laws Of Electrostatics Research Articles

Analytical and numerical analysis of third-order geometric and first-order chromatic aberrations in a bi-potential electrostatic lens

Glaser and Robl (Z. Angew. Math. Phys. 2 (1951) 444) first proposed a theoretical model for a bi-potential electrostatic lens, whose axial potential distribution satisfies the law of electrostatics and whose paraxial trajectory equation is soluble. In this paper, the Gaussian optical property in the bi-potential electrostatic lens has been thoroughly studied. Meanwhile, based on electron optical canonical aberration theory, analytical formulas of third-order geometrical and first-order chromatic aberration coefficients have first been derived and their computational results obtained for this bi-potential electrostatic lens. It is to be emphasized that this theoretical study can be used to estimate third-order geometric and first-order chromatic aberrations and to provide a theoretical criterion for numerical computation in a rotationally symmetric electrostatic immersion lens.

Applications of multidimensional distributions in electrostatics

The multidimensional Dirac delta function and the Heaviside unit step function are shown to be analytical tools of use in solving classical demonstrations of Gauss's Law in electrostatics. Applications in spherically symmetric and cylindrically symmetric scenarios are presented-and verified by the differential form of Gauss's Law.

Ultra electron microscopy in the SEM and STEM modes

The approach, in this paper, to the subject of ultra electron microscopy, is based on the fundamental assumption, that the classical laws of electrodynamics and electrostatics, will apply at all levels of image formation, down to spacings which approach the electron diameter itself. The electron optical principles of ultra electron microscopy, are shown to be based on a phenomenon found in cold field-emission sources, namely, a phenomenon where the paraxial emission manifests an electron density which is orders of magnitude greater than the overall density of emission. It is argued, further, that the laws of electrodynamics and electrostatics will be valid, also, in regard to the constitution of the atom, and an example which supports that point of view, is shown.

One And The Same Route: Two Outstanding Electrodynamics

We show that the same route that leads to Maxwell's electrodynamics leads also to Podolsky's electrodynamics, provided we start from Podolsky's electrostatic force law instead of the usual Coulomb's law.

Classical Coulomb systems: Screening and correlations revisited

From the laws of macroscopic electrostatics of conductors (in particular, the existence of screening), taken as given, one can deduce universal properties for the thermal fluctuations in a classical Coulomb system at equilibrium. The universality is especially apparent in the long-range correlations of the electrical potentials and fields. The charge fluctuations are derived from the field fluctuations. This is a convenient way to study the surface charge fluctuations on a conductor with boundaries. Explicit results are given for simple geometries. The potentials and the fields have Gaussian fluctuations, except for a short-distance cutoff.

The Ohm Is Where the Art Is: British Telegraph Engineers and the Development of Electrical Standards

Previous articleNext article No AccessInstruments & AuthorityThe Ohm Is Where the Art Is: British Telegraph Engineers and the Development of Electrical StandardsBruce J. HuntBruce J. Hunt Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Osiris Volume 9, Number 11994Instruments Published for the History of Science Society Article DOIhttps://doi.org/10.1086/368729 Views: 21Total views on this site Citations: 32Citations are reported from Crossref Copyright 1994 The History of Science Society, Inc.PDF download Crossref reports the following articles citing this article:Rebecca L. Jackson “The Uncertain Method of Drops”: How a Non-Uniform Unit Survived the Century of Standardization, Perspectives on Science 29, no.66 (Nov 2021): 802–841.https://doi.org/10.1162/posc_a_00395JOHN HANDEL The Material Politics of Finance: The Ticker Tape and the London Stock Exchange, 1860s–1890s, Enterprise & Society 80 (Mar 2021): 1–31.https://doi.org/10.1017/eso.2021.3Peter M. Grant, John S. Thompson Standardization of the Ohm as a Unit of Electrical Resistance, 1861–1867, Proceedings of the IEEE 107, no.1111 (Nov 2019): 2281–2289.https://doi.org/10.1109/JPROC.2019.2945495Richard Noakes Physics and Psychics, 23 (Sep 2019).https://doi.org/10.1017/9781316882436Peter M Grant, John S Thompson Standardisation of the Unit of Electrical Resistance, (Sep 2019): 1–4.https://doi.org/10.1109/HISTELCON47851.2019.9039959Daniel Jon Mitchell “The Etherealization of Common Sense?” Arithmetical and Algebraic Modes of Intelligibility in Late Victorian Mathematics of Measurement, Archive for History of Exact Sciences 73, no.22 (Nov 2018): 125–180.https://doi.org/10.1007/s00407-018-0218-yJan Gyllenbok A–Z of Scientific and Informal Measures, (Apr 2018): 33–238.https://doi.org/10.1007/978-3-319-57598-8_3Isobel Falconer No actual measurement … was required: Maxwell and Cavendish's null method for the inverse square law of electrostatics, Studies in History and Philosophy of Science Part A 65-66 (Oct 2017): 74–86.https://doi.org/10.1016/j.shpsa.2017.05.001Terry Quinn From artefacts to atoms - A new SI for 2018 to be based on fundamental constants, Studies in History and Philosophy of Science Part A (Aug 2017).https://doi.org/10.1016/j.shpsa.2017.07.003Daniel Jon Mitchell Making sense of absolute measurement: James Clerk Maxwell, William Thomson, Fleeming Jenkin, and the invention of the dimensional formula, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 58 (May 2017): 63–79.https://doi.org/10.1016/j.shpsb.2016.08.004Sybil G. de Clark The dimensions of the magnetic pole: a controversy at the heart of early dimensional analysis, Archive for History of Exact Sciences 70, no.33 (Oct 2015): 293–324.https://doi.org/10.1007/s00407-015-0168-6Simon Reif-Acherman Georg Simon Ohm and the First Comprehensive Theory of Electrical Conductivity in Metals [Scanning our Past], Proceedings of the IEEE 104, no.11 (Jan 2016): 198–209.https://doi.org/10.1109/JPROC.2015.2503461Simon Reif-Acherman Augustus Matthiessen: His Studies on Electrical Conductivities and the Origins of his “Rule” [Scanning Our Past], Proceedings of the IEEE 103, no.44 (Apr 2015): 713–721.https://doi.org/10.1109/JPROC.2015.2414487MICHAEL KERSHAW ‘A thorn in the side of European geodesy’: measuring Paris–Greenwich longitude by electric telegraph, The British Journal for the History of Science 47, no.44 (Feb 2014): 637–660.https://doi.org/10.1017/S0007087413000988RICHARD NOAKES Industrial research at the Eastern Telegraph Company, 1872–1929, The British Journal for the History of Science 47, no.11 (Apr 2013): 119–146.https://doi.org/10.1017/S0007087413000174Simone Müller-Pohl Working the Nation State: Submarine Cable Actors, Cable Transnationalism and the Governance of the Global Media System, 1858–1914, (Sep 2012): 101–123.https://doi.org/10.1007/978-3-642-32934-0_6Aashish Velkar Globalization and Voluntary Consensus Standardization in the British Wire Industry, 1880, (Jan 2013): 45–70.https://doi.org/10.1057/9781137263636_3William P. Fisher Measurement, Reduced Transaction Costs, and the Ethics of Efficient Markets for Human, Social, and Natural Capital, SSRN Electronic Journal (Jan 2013).https://doi.org/10.2139/ssrn.2340674FALK MÜLLER Johann Wilhelm Hittorf and the material culture of nineteenth-century gas discharge research, The British Journal for the History of Science 44, no.22 (Nov 2010): 211–244.https://doi.org/10.1017/S0007087410001329Aashish Velkar Transactions, standardisation and competition: Establishing uniform sizes in the British wire industry c.1880, Business History 51, no.22 (Mar 2009): 222–247.https://doi.org/10.1080/00076790902726582William P. Fisher Distinguishing between Consistency and Error in Reliability Coefficients: Improving the Estimation and Interpretation of Information on Measurement Precision, SSRN Electronic Journal (Jan 2009).https://doi.org/10.2139/ssrn.1685556Iwan Rhys Morus Working out in the nineteenth century, Studies in History and Philosophy of Science Part A 38, no.33 (Sep 2007): 605–609.https://doi.org/10.1016/j.shpsa.2007.06.008Iwan Rhys Morus The Two Cultures of Electricity: Between Entertainment and Edification in Victorian Science, Science & Education 16, no.66 (Sep 2014): 593–602.https://doi.org/10.1007/s11191-006-9023-0Michael Kershaw The international electrical units: a failure in standardisation?, Studies in History and Philosophy of Science Part A 38, no.11 (Mar 2007): 108–131.https://doi.org/10.1016/j.shpsa.2006.12.012 The Philosophical Magazine, (Aug 2003): 261–274.https://doi.org/10.1201/9781482272611-12Crosbie Smith Force, Energy, and Thermodynamics, (Jan 2001): 289–310.https://doi.org/10.1017/CHOL9780521571999.017Bruce J. Hunt Electrical Therory and Practice in the Nineteenth Century, (Jan 2001): 311–328.https://doi.org/10.1017/CHOL9780521571999.018Bruce J. Hunt Scientists, engineers and Wildman Whitehouse: measurement and credibility in early cable telegraphy, The British Journal for the History of Science 29, no.22 (Jan 2009): 155–169.https://doi.org/10.1017/S0007087400034208Kathryn M. Olesko Precision, Tolerance, and Consensus: Local Cultures in German and British Resistance Standards, (Jan 1996): 117–156.https://doi.org/10.1007/978-94-009-1784-2_4Sungook Hong Forging Scientific Electrical Engineering: John Ambrose Fleming and the Ferranti Effect, Isis 86, no.11 (Oct 2015): 30–51.https://doi.org/10.1086/357074 John Neu Current Bibliography of the History of Science and Its Cultural Influences, 1995, Isis 86 (Oct 2015): 1–331.https://doi.org/10.1086/357399Andrew L. Russell Ideological Origins of Open Standards I: Telegraph and Engineering Standards, 1860s–1900s, (): 25–57.https://doi.org/10.1017/CBO9781139856553.004

Currents from the Underworld: Electricity and the Technology of Display in Early Victorian England

Previous articleNext article No AccessCurrents from the Underworld: Electricity and the Technology of Display in Early Victorian EnglandIwan Rhys MorusIwan Rhys Morus Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Isis Volume 84, Number 1Mar., 1993 Publication of the History of Science Society Article DOIhttps://doi.org/10.1086/356373 Views: 20Total views on this site Citations: 10Citations are reported from Crossref Copyright 1993 History of Science Society, Inc.PDF download Crossref reports the following articles citing this article:R. A. Martins, A. P. B. Silva Joule’s Experiments on the Heat Evolved by Metallic Conductors of Electricity, Foundations of Science 26, no.33 (Jun 2020): 625–701.https://doi.org/10.1007/s10699-020-09681-1Isobel Falconer No actual measurement … was required: Maxwell and Cavendish's null method for the inverse square law of electrostatics, Studies in History and Philosophy of Science Part A 65-66 (Oct 2017): 74–86.https://doi.org/10.1016/j.shpsa.2017.05.001Koen Vermeir Electricity and Imagination: Post-romantic Electrified Experience and the Gendered Body. An Introduction, Centaurus 57, no.33 (Apr 2016): 131–155.https://doi.org/10.1111/1600-0498.12097Lissa Roberts Orienting natural knowledge: the complex career of Hiraga Gennai, Endeavour 33, no.22 (Jun 2009): 65–69.https://doi.org/10.1016/j.endeavour.2009.04.009James A. Secord Science, technology and mathematics, (Mar 2009): 443–474.https://doi.org/10.1017/CHOL9780521866248.014Richard Noakes Cromwell Varley FRS, electrical discharge and Victorian spiritualism, Notes and Records of the Royal Society 61, no.11 (Dec 2006): 5–21.https://doi.org/10.1098/rsnr.2006.0161Alexander Rueger Aesthetic appreciation of experiments: The case of 18th-century mimetic experiments, International Studies in the Philosophy of Science 16, no.11 (Mar 2002): 49–59.https://doi.org/10.1080/02698590120118828 Lissa Roberts Science Becomes Electric: Dutch Interaction with the Electrical Machine during the Eighteenth Century, Isis 90, no.44 (Oct 2015): 680–714.https://doi.org/10.1086/384507Iwan Rhys Morus Manufacturing nature: science, technology and Victorian consumer culture, The British Journal for the History of Science 29, no.44 (Jan 2009): 403–434.https://doi.org/10.1017/S0007087400034725David Gooding Re-presenting Faraday, The British Journal for the History of Science 27, no.33 (Jan 2009): 361–364.https://doi.org/10.1017/S0007087400032246

On Coulomb’s inverse square law

The famous inverse square law in electrostatics, first published in 1785 by C. A. Coulomb, was strongly contested during the next 40 years, especially in Germany. Therefore, at the Carl von Ossietzky University Oldenburg a replication of the apparatus described by Coulomb was made, and the reasons were investigated why none of his contemporaries succeeded in reproducing his results. In addition the respective theoretical concepts of Coulomb, his supporters, and his opponents were analyzed.

Flashover in wide-band-gap high-purity insulators: Methodology and mechanisms

The electron beam of a scanning electron microscope (SEM) is used to charge an unmetallized insulator (Al2O3, Y2O3, SiO2) in vacuum. The charging is found to be stable in time after the e-beam is switched off. The SEM is also used to measure the implanted charge by measuring the resulting electrostatic potential. The distribution of potential around the trapped charges is determined by the classical laws of electrostatics. The electrostatic energy stored in the polarized dielectric can thus be determined. The implanted charge can be removed by the introduction of carriers into the polarized sample by flooding the surface with electron beams of varying energies. Slow or rapid relaxation occurs depending on the operating conditions of the flood gun (energy, intensity, etc.) which introduces the electrons. When the relaxation kinetics are slow, the electrostatic charge decreases slowly as a function of time. On the other hand, a rapid relaxation of the dielectric leads to the appearance of a high-density plasma which spreads over the insulator surface, resulting in treeing on the insulator surface. Along the path of the arc, the transfer of energy to the lattice triggers the sublimation of the ceramic and its mechanical fracture through thermal shock. At particular flood gun setup conditions, we have observed the formation of parallel fracture lines along the ceramic surface, away from the treeing region. These results constitute the basis for a new approach to understanding flashover along ceramics-vacuum interfaces. The important step is the plasma initiation, which we interpret on the basis of dielectric relaxation mechanisms. The parameter of the insulator that determines its breakdown initiation is its complex dielectric constant. It is concluded that the insulator’s band gap, the nature and density of defects localized in the band gap, and the dipolar relaxation induced by a variation of the electric field and connected with the presence of defects determine the holdoff level of insulators.

Low-high homojunction in the stationary state

This work develops an analytical description of the electrostatic (stationary state) properties of an abrupt low-high homojunction within the context of Maxwell–Boltzmann statistics (nondegenerate semiconductors). The considerations are based on an application of the electrostatic law of abrupt junctions and summarized by a generalized approximate solution of Poisson’s equation as an analytical representation combined with adapted numerical procedures. Low-field calculations give a set of universal curves which may be used to find direct relationships between the different distributions at any point within the free carrier space charge. This approach, having a relatively simple behavior as well as very high accuracy, shows especially the macroscopic features of the accumulated space charge on the lightly doped side and is especially useful in junction modelization and numerical simulations.

Charging phenomena on insulating materials: Mechanisms and applications

AbstractWe are interested in the physical origin of the electric charge that is produced in insulators by electron bombardment, by an electric field, or by mechanical stress. The charge is due to the trapping of carriers on the defects formed in the course of the dissipation of energy by the lattice. The characteristics of the charge (potential, electric field) result from the laws of electrostatics. The stability of the charge under an electron beam is interpreted on the basis of the laws governing the behavior of the discharges in gases and in solids. The stability of the charge after shutting off the beam depends on the position of the traps with respect to the surface and the experimental conditions (gas pressure, temperature, time). The charge decay kinetics are then controlled by adsorption‐diffusion. These studies allow us to explain some of the electromigration phenomena that had not been interpreted and to determine the optimal experimental conditions for the observation of local variations in the dielectric constant; they also allow the use of the charge phenomenon as an indicator of the mechanical and dielectric properties of insulators.

Surface properties of finite classical Coulomb systems: Debye-H�ckel approximation and computer simulations

We report analytical and numerical studies of surface correlations in finite, homogeneously polarizable, classical Coulomb systems placed in an insulating or conducting environment. Their purpose is to understand the phenomenological, shape-dependent laws of electrostatics, from the point of view of statistical mechanics; we focus on the knowledge of the dielectric susceptibility of the system, a quantity proportional to the equilibrium fluctuation of the system's instantaneous polarization per unit volume. This goal has been achieved for a system in a conducting state. The picture is that the shape-dependent part of the susceptibilities results from the action of unbounded observables (the second moments of the instantaneous polarization of the system) on long-range surface correlations and that the relations of electrostatics are verified by means of shape-dependent thermodynamic limits. This picture is supported (i) by exact solutions and asymptotic analysis of the Debye-Huckel approximation of multicomponent plasmas in disks and spheres with insulating and conducting environment and also in ellipses in a vacuum, and (ii) by computer simulations of a one-component plasma in a disk with different environments, notably a conducting environment with permeable and impermeable wall. These observations have revealed for the first time the reason why the susceptibility of a conducting disk in a conductor with impermeable walls diverges linearly with the radius of the disk: this is due to the occurrence of long-range radial correlations in the conductor. These findings are quantitatively interpreted in terms of a novel “canonical” Debye-Huckel approximation as contrasted to the ordinary “grand canonical” version. Lastly a fresh look at the problem of the surface correlations of a conductor in a vacuum, which places the observer close to the surface of the conductor but in the vacuum, is presented and applied to the disk, the ellipse, the cylinder, the sphere, and the wedge.

On the Convergence of Particle Methods for Multidimensional Vlasov–Poisson Systems

For Vlasov–Poisson systems, particle methods are numerical techniques that simulate the behavior of a plasma by a large set of charged superparticles, which obey the classical laws of electrostatics. The trajectories of these charged particles are then followed. Estimates for the errors incurred for a “semi-discrete” approximation to the underlying Vlasov–Poisson system are given by first superimposing a rectangular grid or mesh on all of phase space and then replacing the initial continuous distribution of charges or masses by discrete charges or masses located at the centroid of each grid cell. Our analysis, on one hand, generalizes that of G. H. Cottet and P. A. Raviart (SIAM J. Numer. Anal., 21 (1984), pp. 52–76) to higher-dimensional Vlasov–Poisson systems, and, on the other, those fundamental results of Ole Hald (SIAM J. Numer. Anal., 16 (1979), pp. 726–755) and of J. T. Beale and A. Majda (Math. Comp., 39 (1982), pp. 1–52) on vortex methods for two- and three-dimensional Euler equations, to particle-in-cell methods for multidimensional Vlasov–Poisson settings.

Generalization of Coulomb’s law to Maxwell’s equations using special relativity

Maxwell’s equations are obtained by generalizing the laws of electrostatics, which follow from Coulomb’s law and the principle of superposition, so that they are consistent with special relativity. In addition, it is necessary to assume that electric charge is a conserved scalar. The Lorentz force on a charged particle and its energy conservation condition are obtained by making Newton’s second law for the particle in an electrostatic field consistent with special relativity. Magnetic monopoles can be introduced into Maxwell’s theory in a way consistent with special relativity.

On a semiclassical interpretation of inter‐ and intramolecular interactions

AbstractThe semiclassical models considered here are composed by charge distributions coming from ab initio quantum‐mechanical calculations on actual molecular systems. These charge distributions interact with one another according to the laws of classical electrostatics. This article describes some results of a systematic examination of the performances of this model in a variety of cases, with the aim of putting in evidence the usefulness and the limits of this inherently approximate representation of chemical interactions. Intermolecular interactions are examined first; the test cases are interactions of neutral molecules with H+, Li+, and C1−, and the formation of H‐bonded complexes. Attention is paid mainly to the energetics of the processes; each interacting molecule is considered as a unique entity and classical molecular reactivity indexes (electrostatic potential V, polarization term P) are introduced to compute the interaction energy, to interpret the details of the interaction process, and then to elaborate on less expensive computational procedures. Intramolecular interactions are considered. Attention is paid to the question of defining chemical groups starting from SCF molecular wavefunctions. The transferability and conservation degree of groups derived from localized orbitals of actual molecules is examined in detail, taking as tests their ability to reproduce charge distribution, one‐electron observables, and energy. The effect of classical fields on these groups is then examined, taking into consideration external fields originated either by a point charge or by a solvent, and internal fields deriving from substitution of chemical groups. The intergroup analysis is then extended to the case of bimolecular reaction acts by considering the whole system as a supermolecule. Approximate computational procedures able to reproduce the main features of these interactions are proposed and tested. All through the article the performances of the classical models are compared with ab initio SCF calculations (mainly of low or intermediate quality).

Electrostatic energy when dielectrics are present: A general integral approach

A detailed discussion of electric energy storage in the presence of dielectrics is presented. The discussion is general and is aimed at the introductory–intermediate levels, when an integral representation of the laws of electrostatics is more familiar to the student. The case where a discrete set of point charges is slowly transported into the vicinity of an arbitrary dielectric is first considered and then an extension to the continuous distribution is presented. A simple model of atomic polarization is introduced and used to illustrate dielectric nonlinearities and the concept of self-interaction. It is shown that the energy density E⋅D/2 is restricted to the case of linear dielectrics and a novel way of introducing Maxwell’s displacement current term at a more advanced level is suggested as a natural extension of the electrostatic case.

Electromagnetic waves without Maxwell’s equations

Elementary experimental information about magnetostatic field production by turned-on current sources suggests that such fields are nonpropagating superpositions of electromagnetic waves. It is shown that the local structure of these waves may be deduced from the Lorentz force law and the transformations of special relativity, without the intervention of Maxwell’s equations. The electromagnetic energy density formula is identified in the context of field production by a model ‘‘circuit’’ consisting of opposed parallel current sheets. Coulomb’s law of electrostatics is deduced by transforming the magnetostatic field of this circuit into the rest frames of the source particles.

Electrostatic law for interfaces in abrupt semiconductor homostructures

The electrostatic law for interfaces in abrupt semiconductor structures (by which we mean structures in which an abrupt change in impurity concentration occurs) is elaborated and proved. Using this law and knowing only the temperature, the dielectric constant and the carrier concentration, we can determine easily and accurately various important values characterizing the electrical properties of the given structure, such as the electrostatic potential at the interface, the charge carrier concentration at the interface, the maximum internal electric field, the total electric charge of the double layer at the interface, and the thickness of the space charge region in the double layer.

Les decharges frappant les surfaces isolantes et l'electrostatique

When an unidirectional voltage is applied to a point-to-plane (point I, plane B) electrode system with an insulating plate inserted between them, a corona effect takes place at the point I when the voltage increases, and another one of the opposite polarity occurs when the voltage decreases. The ions released by these corona effects are trapped at the air—insulating plate interface, where they form a circular layer. In order to know the characteristics of this layer, two methods are used: (i) an induction probe, which must contact the insulating surface F, measures the charge density at different distances from the layer centre; (ii) the dust patterns method is used also in order to materialize the regions of each polarity. When the plane electrode B remains attached to the back side of the insulating plate during the detection, the two methods notably contradict each other. These contradictions are explained by the fact that the probe contacting F detects only the charge trapped on F; however, the powder detects the algebraic sum of the trapped charge on F and of the induced charge on B. Finally, it may be shown that the powder materializes the edge effect of the pseudo-condenser formed by the plane electrode B and the circular charged region on F. The study of the characteristics of the charged layer trapped on F shows that as long as the electric field near I tends to exceed the critical value ± g, the ions in the air distribute on F in such a manner that an electrostatic equilibrium is obtained as with conductors; because g is not zero, such an equilibrium cannot be realized completely. The contradictions observed between the two detection methods disappear if the charges are detected after B has been removed far from the insulating plate. The phenomena are disturbed, however, by ionisation of the air space that forms between the plate and B during their separation. This ionisation causes the rather imperfect transfer of the charge layer, previously induced on B, to the back side of the plate where it is trapped. The study performed here shows that gas discharge phenomena on striking insulating walls allow many experimental demonstrations of the classical Laws of Electrostatics. Before the availability of films and plates of highly polymeric materials, simple, practical applications of these Laws were very scarce.

Electrostatics in the plane

Electrostatic laws are obtained for charges constrained to lie in a plane. The corresponding electric field E in the plane obeys neither Poisson’s equation nor Gauss’ law. A vector field Λ is constructed which does satisfy these equations and which is a functional of the charge density. Various limiting forms of the coupling dyad M are obtained. where E=MΛ.