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

Home , Integral transform, Laplace operator, Line integral

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 different 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 |r1 − r2 | electromagnetism. Correspondingly, the electric intensity field of Eq. (4) can be written as 2. The laws of electrostatics   Z −→ −→ ρ(r0 ) Coulomb’s law describes the radial and inverse-square of the  ¯ ¯ 0 −→ E = −ke 5 ¯ −→¯dV = − 5 φ( 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 −→ |−→r − −→r |3 ρ(r0 ) 1 2 −→ ¯ ¯ 0 φ( 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 distribution {ρ,V}. This is the quantity giving rise to the second half −→ XN −→ F −→ −→ −→ −→ = Fi −→ −→ −→ −→ of the title in this manuscript. The reader can appreciate the { q i,ri }→( q , r ) { q i,ri }→( 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 | r − ri | 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 = ρ(r )d V , E · d r = − 5φ · d r = −φ( r ) + φ( r i) (10) −→0 where ρ(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 {ρ,V }→(q, r ) = keq ρ(r ) ¯ ¯3 dV (3) ¯−→ −→0 ¯ V ¯ r − r ¯ closed-line integral vanishes. Also, the curl of Eq. (8) vanishes: 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 ρ(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 position, Eqs. (7) and (8) point out that φ(−→r ) is the potential as the electric intensity field produced by the charge distribu- energy per unit charge. tion {ρ, V }. The reader may also inquire about the equation that φ(−→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 = 4πke ρ(r )dV = 4πkeQ (5) 2 −→ −→ S 5 φ( r ) = −4πkeρ( 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 ) = 4πkeρ( r ) (6) 5 φ( 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 (mϕ) and sin (mϕ) correspond to the alternative even and odd (under ϕ → −ϕ) 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- Z2π 0 tion in spherical and circular cylindrical coordinates, illus- e(in ϕ)e(inϕ) dϕ = δ 0 (24) trating their separation and integration leading to the respec- 2π n ,n tive spherical harmonics and circular cylindrical harmon- 0 0 ics [13,14]. In fact, the Laplace equation in the familiar X∞ e(inϕ)e(inϕ ) spherical (r,ϑ,ϕ) and circular cylindrical (R,ϑ,ϕ) coordinates = δ(ϕ − ϕ0) (25) 2π has the respective forms: n=−∞ ½ · respectively, in terms of the Kronecker-delta symbol and the 1 ∂ ∂ 1 1 ∂ ∂ 2 Dirac-delta function, where the latter is zero for ϕ 6= ϕ0, in- 2 r + 2 sin ϑ r ∂r ∂r r sin ϑ ∂ϑ ∂ϑ finity for ϕ = ϕ0, and its integral is one in the integration ¸¾ 1 ∂2 interval includes the value ϕ = ϕ0. + φ(r, ϑ, ϕ) = 0 (14) sin2 ϑ ∂ϕ2 The polar coordinate 0 ≤ ϑ ≤ π and its associated · ¸ Eq. (19) determine the singularities at ϑ = 0 and π, or 1 ∂ ∂ 1 ∂2 ∂2 R + + φ(R, ϑ, z) = 0 (15) cos (ϑ) = 1 and −1. Eq. (19) is known as the associated R ∂R ∂R R2 ∂ϕ2 ∂z2 Legendre equation; its regular solutions are the associated Legendre polynomials In both cases, the equation is satisfied by separable solu- |m| tions in the following forms: |m| |m| d P` (cos ϑ) θ (ϑ) = P` (cos ϑ) = (sin ϑ) (26) d (cos ϑ)|m| φ(r, ϑ, ϕ) = f(r)θ(ϑ)Φ(ϕ) (16) of degree ` − |m|, where ` = 0, 1, 2, ... and P` (cos ϑ) are φ(R, ϑ, z) = g(R)Φ(ϕ)Z(z) (17) the ordinary Legendre polynomials of degree `. They also have a well-defined parity (−1)`−|m| under ϑ → ϑ − π or The successive factors satisfy the ordinary differential cos ϑ → − cos ϑ. equations The products of the angular functions d2Φ |m| = −m2Φ (18) Ylm(ϑ, ϕ) = NlmP` (cos ϑ) exp (imϕ) (27) dϕ2 are known as the angular spherical harmonics, where Nlm is · 2 ¸ 1 ∂ ∂ m a normalization factor such that sin ϑ − 2 θ(ϑ) = −l(l + 1)θ(ϑ) (19) sin ϑ ∂ϑ ∂ϑ sin ϑ Zπ Z2π · ¸ ? 1 d 2 d l(l + 1) Y`m(ϑ, ϕ)Y`m(ϑ, ϕ) sin ϑdϕdϑ = 1 (28) 2 r − 2 f(r) = 0 (20) r dr dr r 0 0 d2Z It is known that the Fourier basis in Eq. (23) is orthonor- = k2Z (21) dz2 mal. In a similar way, the angular spherical harmonic basis · ¸ of Eq. (27) is orthonormal 1 d d m2 R − g(R) = −k2g(R) (22) R dR dR ρ Zπ Z2π ? Y`0m0 (ϑ, ϕ)Y`m(ϑ, ϕ) sin ϑdϕdϑ = δ`0,`δm0,m (29) 2 2 where -m ,-l(l+1), and k are the respective separation con- 0 0 stants. Notice that Eqs. (18)-(22) have the form of eigen- and complete value equations, involving second-order differential opera- tors which, upon application to the eigenfunction, lead to the X∞ X` ? 0 0 same function multiplied by a constant, called the eigenvalue. Y`0m0 (ϑ , ϕ )Y`m(ϑ, ϕ) The angular coordinate ϕ and the associated Eq. (18) are `=0 m=−` common to both cases. δ(ϕ − ϕ0)δ(ϑ − ϑ0) = (30) The solutions are chosen to those of periodic sin ϑ0 Φ(ϕ + 2π) = Φ(ϕ), restricting the values of the separa- The completeness property implies that any quadratically tion constant to be an integer m = 0, ±1, ±2,. . . integrable function of ϑ and ϕ can be expanded in terms of The set of such solutions, such a basis, X X e(imϕ) F (ϑ, ϕ) = a`mY`m(ϑ, ϕ) (31) Φm(ϕ) = √ (23) 2π ` m

Rev. Mex. F´ıs. E 54 (2) (2008) 153–159 156 L. MEDINA AND E. LEY KOO where the expansion coefficients follow from Eq. (29): In Eqs. (21) and (22) the separation constant could be chosen with the opposite sign, which we express by the an- Zπ Z2π alytical continuation k → iκ. Then, instead of Eqs. (34) ? a`m = Y`m(ϑ, ϕ)F (ϑ, ϕ) sin ϑdϕdϑ (32) and (35), the solutions become 0 0 Z(z) = Ce(iκz) + De(−iκz) (41) The radial Eq. (20) is satisfied by two independent power g(R) = EI (κR) + FK (κR) solutions m m (42) respectively, involving the Fourier basis in longitudinal co- f(r) = Ar` + Br−(`+1) (33) ordinates, and the modified Bessel functions in radial coordinates. The latter are monotonically increasing and decreasing for each given value of `. The positive powers are regular as their arguments change from κR = 0 to κR → ∞. at the origin and diverge as r → ∞, and the inverse powers In conclusion, the general solutions to the Laplace diverge at the origin and tend to zero asymptotically. Eqs. (14) and (15) can be written as a superposition of spher- The products r`Y (ϑ, ϕ) are known as solid spherical `m ical harmonics, from Eqs. (27) and (33), harmonics. The axial coordinate and its associated Eq. (21) X X ³ ´ lead to the exponential solutions ` −(`+1) φ(R, ϑ, z) = a`r + b`r Y`m(ϑ, ϕ) (43) ` m Z(z) = Ce(kz) + De(−kz) (34) or circular cylindrical harmonics, which are regular and singular at z → −∞, respectively, and  Z∞ the other way around at z → ∞. X  φ(R, ϕ, = e(imϕ) [a J (kR) + b N (kR)] The corresponding solutions of Eq. (22) in the circular  mk m mk m radial coordinate are the ordinary Bessel functions m 0 h i o (kz) (−kz) × cke + dke kdk (44) g(R) = EJm(kR) + FNm(kR) (35) from Eqs. (23), (30) and (33), or of the first and second kind, respectively. They are regular  ∞ and singular at the origin R → 0, X Z φ(R, ϕ, z) = e(imϕ) [a I (κR) + b N (κR)] 2  mk m mk m J0(kR) → 1,N0(kR) → ln (kR) (36) m 0 π h i o |m| (iκz) (−iκz) (kR) × cκe + dke κdκ (45) J (kR) → , m 2|m| |m|! using Eqs. (41) and (42). |m| (m − 1)! −|m| The potential in Eq. (43) is expressed as the spherical Nm(kR) → −2 (kR) (37) π harmonic multipole expansion in which the terms with each value of ` are designated as 2` order poles [12]. The potential oscillate for intermediate and larger values of their argument, in Eqs. (44) and (45) are expressed as Fourier series in the and their asymptotic behavior is of the form angular coordinate ϕ. Besides, Eq. (44) involves a Laplace- r ¡ ¡ ¢ ¢ 1 π ordinary Bessel integral transform, and Eq. (45) involves a 2 cos kR − m + 2 2 Jm(kR) = √ , Fourier-modified Bessel integral transform in the longitudi- π kR r ¡ ¡ ¢ ¢ nal and radial coordinates, respectively. The expansion coef- 1 π ficients in Eqs. (43) - (45) are to be determined by the bound- 2 sin kR − m + 2 2 Nm(kR) → √ (38) π kR ary conditions according to each specific application.

The ordinary Bessel functions are quadratically inte- 4. Harmonic expansions of the Coulomb po- grable and form an orthonormal and complete set of functions: tential Z∞ In order to solve the Poisson differential equation, Eq. (12), 0 −→ 0 δ(k − k ) for any source distribution with electric charge density ρ( r ), Jm(k R)Jm(kR)RdR = (39) k 0 it is sufﬁcient to use the Green function technique. The Green 0 function G(−→r , −→r 0) satisﬁes the Poisson equation for a unit ∞ −→ Z 0 electric point charge located at the position r 0: 0 δ(R − R ) Jm(kR)Jm(kR )kdk = 0 (40) R 2 −→ −→0 −→ −→0 0 ∇ G( r , r ) = −4πδ( r − r ) (46)

Rev. Mex. F´ıs. E 54 (2) (2008) 153–159 MATHEMATICS MOTIVATED BY PHYSICS: THE ELECTROSTATIC POTENTIAL IS THE COULOMB INTEGRAL. . . 157 where the Dirac-delta function represents appropriately the Then the reader can identify Eq. (50) with the Coulomb electric charge density of the chosen source. Its volume inte- integral transform of the charge density already discussed in gral that contains the point where the charge is located is connection with Eq. (9), in Sec. 2. Z δ(−→r − −→r 0)dV 0 = 1 (47) The Coulomb potential, or inverse of the distance between the source point and the ﬁeld point, is known to be The solution to the Poisson Eq. (12) is given by the in- the generating function of the spherical harmonics [13, 14] tegral of the product of the electric charge density and the Green function: ∞ Z 1 1 X r` < ˆ0 −→ −→0 −→ −→0 0 −→ −→ = q P (ˆr · r ) φ( r ) = ρ( r )G( r , r )dV (48) | r − r 0] r`+1 r2 + r02 − 2rr0(ˆr · rˆ0) `=0 > If the Laplace operator is applied to both sides of the last X∞ ` X` equation, the result is 4π r< ? 0 0 Z = `+1 Y`m(ϑ , ϕ )Y`m(ϑ, ϕ) (51) 2` + 1 r> ∇2φ(−→r ) = ρ(−→r 0)∇2G(−→r , −→r 0)dV 0 `=0 m=−` Z −→0 −→ −→0 0 −→ where r< and r> represent the smaller and larger of r and = −4π ρ( r )δ( r − r )dV = −4πρ( r ) (49) 0 r . Expansion of the -1/2 power of powers (r) of the where use is made of Eq. (46) and of the integration with the trinomial generates the Legendre polynomials with argument Dirac-delta function, showing that φ(−→r ) is indeed a solution to Eq. (12). rˆ · rˆ0 = sin ϑ sin ϑ0 cos(ϕ − ϕ0) + cos ϑ cos ϑ0 (52) On the other hand, the potential of the unit point charge is simply the Coulomb potential, with which the Green function can be identiﬁed: and the latter in turn generate the spherical harmonics, via the −→ −→0 1 so-called addition theorem involving the sum over m. G( r , r ) = −→ −→ (50) | r − r 0] The inverse of the distance in Eq. (9) is also the generating function of the circular cylindrical harmonics [15]:

∞ Z∞ 1 1 X 0 im(ϕ−ϕ ) 0 −k(z>−z<) p = e Jm(kR)Jm(kR )e dk (53) R2 + R02 − 2RR0 cos(ϕ − ϕ0) + (z − z0)2 π m=−∞ 0 ∞ X∞ Z 1 1 im(ϕ−ϕ0) −iκ(z−z0) p = e Im(κR<)Km(κR>)e dκ (54) R2 + R02 − 2RR0 cos(ϕ − ϕ0) + (z − z0)2 π m=−∞ 0 Notice that the spherical and circular cylindrical harmonic expansions of Eqs. (51), (53) and (54), are particular cases of the general solutions to the Laplace Equation described by Eqs. (43), (44) and (45) in Sec. 3. Notice also, that the Poisson Eq. (12) reduces to the Laplace equation for all points r 6= r0, making possible the harmonic expansions of the Coulomb potential.

5. Harmonic expansions of the electrostatic potential

Substitution of the harmonic expansions of the Coulomb potential Eqs. (51), (53) and (54) into Eq. (9) leads to the respective harmonic expansions of the electrostatic potential:

∞ π 2π X∞ X` Z Z Z ` 4π 0 0 0 0 0 0 0 r< 02 ? 0 0 φ(r, ϑ, ϕ) = dr sin ϑ dϑ dϕ ρ(r , ϑ , ϕ ) `+1 r Y`m(ϑ , ϕ )Y`m(ϑ, ϕ) (55) 2` + 1 r> `=0 m=−` 0 0 0 Z∞ Z∞ Z2π Z∞ 1 X∞ φ(R, ϑ, z)= (2−δ ) dkJ (kR) dR0J (kR0)R0 dϕ0 cos m(ϕ−ϕ0) dz0ρ(R0, ϕ0, z0)e−k(z>−z<) (56) π m,0 m m m=0 0 0 0 −∞ Z∞ Z∞ Z2π Z∞ 1 X∞ φ(R, ϑ, z)= (2−δ ) dκ dR0I (κR )K (κR )R0 dϕ0 cos m(ϕ−ϕ0) dz0ρ(R0, ϕ0, z0) cos κ(z−z0) (57) π m,0 m < m > m=0 0 0 0 −∞

Rev. Mex. F´ıs. E 54 (2) (2008) 153–159 158 L. MEDINA AND E. LEY KOO

Equation (55) gives the familiar spherical multipole ex- sake of space and illustration, the analysis in this paper has pansion of the electrostatic potential, described in detail in been limited to spherical and circular cylindrical coordinates. Ref. 14 for the regions inside and outside the region where The interested reader may consult [15] for the harmonic the sources are located. It must be pointed out that in most expansions of the Coulomb potential harmonics in prolate books, the study of this type of expansion is limited only to spheroidal, oblate spheroidal, and paraboloidal harmonics. the region far from the sources. The expansions in circular Also the relationship between the electrostatic potential and cylindrical harmonics of Eqs. (56) and (57) involve Fourier the electric charge density has its counterpart in the relation- series in angular coordinates, and integral transforms of the ship between the magnetostatic potential and the electric cur- Laplace-ordinary Bessel and of the Fourier-modified Bessel rent density, being connected by a Coulomb integral trans- types in longitudinal and radial coordinates, respectively. form and by the Poisson equation [14]. The same ideas are Integrations of the charge density with harmonic func- extended to the case of electromagnetic fields. Instead of tions over the primed coordinates provide the coefficients in the Laplace and Poisson equations, the sourceless and source −→ the complementary harmonic expansion of ρ( r ). The steps Helmholtz equations must be used. The Coulomb potential as followed from Eq. (9) of Sec. 2 passing through Secs. 3 a Green function is replaced by the outgoing spherical wave −→ −→0 and 4 and ending in Eqs. (55) - (57) of the present section eik| r − r |/−→r −−→r 0. The complete electromagnetic multipole provide the bridge for arriving at the mathematical integral expansion is also available in Ref. 16. transforms motivated by the physical integral transform of It must also be pointed out that the mathematics of elec- the electrostatic potential. tromagnetism is also useful in quantum mechanics, where the superposition principle is also valid. For instance, the 6. Discussion spherical harmonics are the eigenfunctions of angular mo- ˆ2 ˆ 2 mentum ` and `z, with eigenvalues ~ `(`+1) and m~ for the The contents of this article, as anticipated in the Introduction, square of its magnitude and its components along the z-axis. take the review of the laws of electrostatics, in particular the Plane waves and spherical waves are the mathematical tools connection between the electric source density and the elec- for describing the scattering of electromagnetic and quantum trostatic potential, Eqs. (9) and (12), as the motivation for waves. In conclusion, the interested reader is invited to iden- learning about the solutions to the Laplace equation in Sec. 3, tify and develop the appropriate mathematics for the field of the harmonic expansions of the Coulomb potential in Sec. 4, physics of his/her choice. and the harmonic expansions of the electrostatic potential in Sec. 5. The outline presented here emphasizes the special place that harmonic functions play in the study of electro- Acknowledgement statics and it is the aim of the authors to motivate the reader, whether teacher or student, to learn about them in detail. Al- One of the authors (E.L.K.) on sabbatical leave at the Uni- though the center of the attention has been the electrostatic versity of Wisconsin-Milwaukee with support from DGAPA- potential, once it is available, Eq. (9) serves to construct the UNAM wishes to thank Professor Hans Volkmer of the electric intensity field and its harmonic expansions. For the School of Mathematical Sciences for his hospitality.

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Rev. Mex. F´ıs. E 54 (2) (2008) 153–159