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The Electrostatic Potential Is the Coulomb Integral Transform of the Electric Charge Density

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ENSENANZA˜ REVISTA MEXICANA DE FISICA´ E 54 (2) 153–159 DICIEMBRE 2008 Mathematics motivated by physics: the electrostatic potential is the Coulomb integral transform of the electric charge density L. Medinaa and E. Ley Koo a;b aFacultad de Ciencias, Universidad Nacional Autonoma´ de Mexico,´ Ciudad Universitaria, Mexico´ D.F., 04510, Mexico,´ e-mail: [email protected], bInstituto de F´ısica, Universidad Nacional Autonoma´ de Mexico,´ Apartado Postal 20-364, 01000 Mexico D.F., Mexico, e-mail: eleykoo@fisica.unam.mx Recibido el 19 de octubre de 2007; aceptado el 11 de marzo de 2007 This article illustrates a practical way to connect and coordinate the teaching and learning of physics and mathematics. The starting point is the electrostatic potential, which is obtained in any introductory course of electromagnetism from the Coulomb potential and the superposition principle for any charge distribution. The necessity to develop solutions to the Laplace and Poisson differential equations is also recognized, identifying the Coulomb potential as the generating function of harmonic functions. Correspondingly, the convenience of expressing the electrostatic potential in terms of its multipole expansion in spherical coordinates, or as integral transforms based on harmonic functions in different coordinate systems, is also established. These connections provide a motivation for teachers and students to acquire the necessary mathematics as a basic tool in the study of electromagnetic theory, optics and quantum mechanics. Keywords: Electrostatics; Laplace and Poisson equations; spherical and circular cylindrical Harmonic functions. Este art´ıculo ilustra una manera practica´ de conectar y coordinar la ensenanza˜ y aprendizaje de la f´ısica y las matematicas.´ El punto de partida es el potencial electrostatico´ que se obtiene en el curso introductorio de electromagnetismo a partir del potencial de Coulomb y del principio de superposicion´ para cualquier distribucion´ de carga. Tambien´ se reconoce la necesidad de construir soluciones de las ecuaciones diferenciales de Laplace y de Poisson, identificando al potencial de Coulomb como una funcion´ generadora de funciones armonicas.´ Cor- respondientemente, tambien´ se reconoce la conveniencia de expresar al potencial electrostatico´ en terminos´ de su desarrollo multipolar en coordenadas esfericas,´ o de transformadas integrales basadas en funciones armonicas´ en diferentes sistemas de coordenadas. Estas conex- iones proporcionan una motivacion´ para maestros y alumnos para adquirir las matematicas´ necesarias como una herramienta basica´ en el estudio de la teor´ıa electromagnetica,´ la optica´ y mecanica´ cuantica.´ Descriptores: Electrostatica;´ ecuaciones diferenciales de Laplace y de Poisson; funciones armonicas´ esfericas´ y cil´ındricas circulares. PACS: 41.20.Cv1 1. Introduction This article addresses the general problem of connecting and coordinating the study of physics and mathematics in dif- ferent areas and on different levels. Specific facets of the Mathematics and physics have always been closely inter- problem have been explored in [5] and in a series of dialogues woven in a two-way process. The former is not only the under the title of ”Mathematics motivated by physics” [6]. language of the latter; in addition, it often determines to a Emphasis was placed on the construction of mathematical large extent the content and meaning of physical concepts bridges to make the transition from introductory courses and theories themselves. Consequently, progress in the study in mechanics, fluids, thermodynamics, electromagnetism of fundamental physics increasingly depends on the avail- and quantum mechanics to their junior/senior/graduate level ability of new mathematical tools. It is a well-known fact counterparts. The second half of the title of this manuscript that there has been a close interrelationship between mathe- blends the physical and mathematical elements to guide col- matics and physics throughout their historical development. leagues and students in their respective tasks of teaching and Modern mathematics and physics were born in the 17th cen- learning electrostatics, identifying and constructing the ap- tury through Newton’s formulation of the laws of mechanics propriate mathematical tools. and the invention of the infinitesimal calculus [1-2]. New- ton changed the face of scientific research by placing the full The starting points are the physical laws of electrostatics force of mathematics at the service of physical enquiry, be- expressed in their integral and differential equation forms, re- coming a unique example of coordination in invention and viewed in Sec. 2. Section 3 is devoted to the solutions to discovery by a single individual. In contrast, Einstein had the Laplace equation in some illustrative coordinate systems, to learn Riemannian geometry in order to formulate the the- corresponding to the so-called harmonic function bases. In ory of general relativity [3], while Born recognized the ma- Sec. 4, the harmonic function expansions of the Coulomb trix mathematics behind Heisenberg’s formulation of quan- potential in spherical and circular cylindrical coordinates are tum mechanics [4]. comparatively analyzed, contrasting their discrete and con- 154 L. MEDINA AND E. LEY KOO tinuous natures, respectively. The physical and mathemat- The Coulomb force of Eq. (1) is conservative and can ical elements identified in Secs. 3 and 4 are the basis for be written as the negative of the gradient of the so-called representing the electrostatic potential as harmonic function Coulomb potential energy, expansions in Sec. 5, characterizing them according to the µ ¶ specific coordinates involved. Section 6 contains discussions ¡! 1 F 1!2 = ¡ 5 ke q1 q2 ¡! ¡! = ¡ 5 (U12) (7) of the extensions to other coordinates, and to other areas of jr1 ¡ r2 j electromagnetism. Correspondingly, the electric intensity field of Eq. (4) can be written as 2. The laws of electrostatics 0 1 Z ¡! ¡! ½(r0 ) Coulomb’s law describes the radial and inverse-square of the @ ¯ ¯ 0A ¡! E = ¡ke 5 ¯ ¡!¯dV = ¡ 5 Á( r ) (8) ¯¡!r ¡ r0 ¯ distance force between two electrically charged point parti- V cles [7-12]: ¡! ¡! where ¡! (r1 ¡ r2 ) F 1!2 = ke q1 q2 (1) Z ¡! j¡!r ¡ ¡!r j3 ½(r0 ) 1 2 ¡! ¯ ¯ 0 Á( r ) = ke ¯ ¡!¯dV (9) The superposition principle applies in electrostatics, and ¯¡!r ¡ r0 ¯ V states that the force of a collection of charges on a test charge is the vector sum of Coulomb forces: is the electrostatic potential produced by the charge distribu- tion f½,Vg. This is the quantity giving rise to the second half ¡! XN ¡! F ¡! ¡! ¡! ¡! = Fi ¡! ¡! ¡! ¡! of the title in this manuscript. The reader can appreciate the f q i;ri g!( q ; r ) f q i;ri g!( q ; r ) i=1 physical and mathematical elements in it. In Sec. 5, sev- eral alternative mathematical representations are introduced XN (¡!r ¡ ¡!r ) = k q q i (2) explicitly. e i ¡! ¡! 3 i=1 j r ¡ ri j If the line integral of electric intensity field is evaluated using Eq. (8), In the case in which the collection of charges is contin- uously distributed in a volume V, so that the charge element Zr Zr 0 ¡!0 3 0 ¡! ¡! ¡! ¡! ¡! associated with a differential volume is dq = ½(r )d V , E ¢ d r = ¡ 5Á ¢ d r = ¡Á( r ) + Á( r i) (10) ¡!0 where ½(r ) is the charge volume density, the sum in Eq. (2) ri ri becomes an integral: the result depends only on the initial and final points of the in- ³ ¡!´ Z ¡!r ¡ r0 tegration path, and is independent of the path chosen. When ¡! ¡!0 0 ¡! the path is closed, and therefore both points coincide, the F f½;V g!(q; r ) = keq ½(r ) ¯ ¯3 dV (3) ¯¡! ¡!0 ¯ V ¯ r ¡ r ¯ closed-line integral vanishes. Also, the curl of Eq. (8) van- ishes: Since the forces in Eqs. (2) and (3), are proportional to ¡! the magnitude of the charge q, it is possible to identify 5 £ E (¡!r ) = 0 (11) ³ ¡!´ ¡! Z ¡! 0 ¡! F ¡! r ¡ r Equations (7)-(11) are different forms of expressing the E (¡!r ) = = k ½(r0 ) dV 0 (4) e ¯ ¡!¯3 conservative character of the electrostatic field. While Eq. (4) q ¯¡! 0 ¯ ¡! V ¯ r ¡ r ¯ indicates that E (¡!r ) is the force per unit charge at a given po- sition, Eqs. (7) and (8) point out that Á(¡!r ) is the potential as the electric intensity field produced by the charge distribu- energy per unit charge. tion f½; V g. The reader may also inquire about the equation that Á(¡!r ) Gauss’ law of electrostatics follows from the evaluation must satisfy. The answer follows from the substitution of of the flux integral of Eq. (4), becoming I Z Eq. (8) into Eq. (6) with the result: ¡! ¡! ¡! 0 0 E ¢ d a = 4¼ke ½(r )dV = 4¼keQ (5) 2 ¡! ¡! S 5 Á( r ) = ¡4¼ke½( r ); (12) V where S is the closed surface bounding the volume V, and Q the so-called Poisson equation, involving the Lapacian or 2 is the electric charge contained inside that volume. The dif- Laplace operator 5 . ferential equation form of Gauss’ law is obtained by applying At the points where there is no charge, Eq. (12) reduces Gauss’ flux or divergence theorem to Eq. (5): to the so-called Laplace equation, ¡! ¡! ¡! 2 ¡! 5 ¢ E ( r ) = 4¼ke½( r ) (6) 5 Á( r ) = 0 (13) Rev. Mex. F´ıs. E 54 (2) (2008) 153–159 MATHEMATICS MOTIVATED BY PHYSICS: THE ELECTROSTATIC POTENTIAL IS THE COULOMB INTEGRAL. 155 3. Harmonic functions in spherical and circu- is the well-known Fourier basis. The real and imaginary parts lar cylindrical coordinates cos (m') and sin (m') correspond to the alternative even and odd (under ' !¡') Fourier bases. The orthonormality The solutions to the Laplace equation are called harmonic and completeness properties of the basis are expressed, by functions. Here we review the explicit forms of the equa- Z2¼ 0 tion in spherical and circular cylindrical coordinates, illus- e(in ')e(in') d' = ± 0 (24) trating their separation and integration leading to the respec- 2¼ n ;n tive spherical harmonics and circular cylindrical harmon- 0 0 ics [13,14].
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