ENSE NANZA REVISTA MEXICANA DE FISICA E 52 (1) 9094 JUNIO 2006

Electrostatic and magnetostatic fields for Bessel-Fourierdistributed sources on infinite planes

M.A. RosalesDepartamento de Fsica y Matematicas, Universidad de las Americas, Puebla,

Apartado postal 408, Exhacienda Sta. Catarina Martir, 72820 Cholula, Pue., Mexico.

E. Ley-KooInstituto de Fsica, Universidad Nacional Autonoma de Mexico,

Apartado Postal 20-364, 01000 Mexico, D. F., Mexico.

Recibido el 28 de septiembre de 2005; aceptado el 6 de diciembre de 2005

The fields produced by Bessel-Fourier distributions of static electric charges on an infinite plane are constructed by integrating the differentialequations for the electrostatic potential and the electric intensity field, subject to the boundary conditions at the source plane. Similarly, thefields produced by Bessel-Fourier distributions of stationary electric currents on an infinite plane are constructed via the integration of thedifferential equations for the potential and the magnetic induction field, subject to the corresponding boundary conditions.Keywords: Bessel-Fourier distributed sources; static electric charges and intensity fields; stationary electric currents and magnetic inductionfields.

Los campos producidos por distribuciones de Bessel-Fourier de cargas electricas en un plano infinito se construyen integrando las ecuacionesdiferenciales para el potencial electrostatico y el campo de intensidad electrica, sujetos a las condiciones de frontera en el plano de la fuente.Similarmente, se construyen los campos producidos por corrientes electricas estacionarias en un plano infinito, por medio de la integracionde las ecuaciones diferenciales para el potencial magnetostatico y el campo de induccion magnetica, sujetos a las condiciones de fronteracorrespondientes.

Descriptores: Distribuciones de Bessel-Fourier de fuentes; cargas estaticas y campo de intensidad electricas; corrientes electricas estacionar-ias y campos de induccion magnetica.

PACS: 41.20.Cv; 41.20.Gz

1. IntroductionSome electrostatic and magnetostatic configurations, basedon simple geometries and source distributions, are studiedin introductory courses, illustrating the concepts and laws ofelectrostatics and magnetostatics, respectively, and some oftheir applications [1,2]. The simplest and starting situation inelectrostatics involves a point charge, with its Coulomb radialand inverse-distance electrostatic potential field. Throughthe superposition principle, other geometries with uniformcharge distributions and their respective fields are identified.For a uniformly charged infinite straight line: radial andinverse-distance electric intensity field, and logarithm-of-the-distance electrostatic potential. For a uniformly charged in-finite plane: normal and uniform electric intensity field andlinear-in-the-distance electrostatic potential. The respectiveequipotential surfaces are concentric spheres, coaxial cylin-ders, and parallel planes. For a uniformly charged sphericalsurface, the electric intensity field is Coulombic outside andvanishes inside. Similarly, for a uniformly charged cylindri-cal surface, the electric intensity field vanishes inside, and isradial and inverse-distance outside. The superposition of thefields of two concentric spheres, two coaxial cylinders, andtwo parallel planes with equal charges of opposite signs leadsto the vanishing of the fields outside, and their confinementinside the respective capacitors, for which the ratio of chargeto potential difference, or capacitance, can be calculated.

In magnetostatics, the magnetic point charge situationdoes not exist. The simplest situation corresponds to an infi-

nite straight wire carrying a stationary electric current, whichwas used by Oersted in his discovery that the electric cur-rent reorients a magnetic compass needle. Ampe`re formu-lated his right-hand rule to define the relative directions ofthe current and the needle: if the extended thumb representsthe current direction, the fingers curling around the thumb in-dicate the direction in which the needle is oriented at eachposition. Ampe`re also formulated his circulation law: thecirculation of the magnetic induction field around a curve Cis proportional to the electric current intensity crossing thesurface S of which C is the perimeter. The magnetic induc-tion field for the straight wire geometry is inversely propor-tional to the distance, and its lines are coaxial circles. Thereaders attention is called to recognizing the difference in di-rections of the field lines, and the same distance dependenceof the field magnitudes, for the same straight line geometryin the electrostatic and magnetostatic cases. Again, by theapplication of the superposition principle, other geometrieswith uniform electric current distributions and their respec-tive magnetic induction fields can be established. For an in-finite straight circular cylinder with a stationary uniformlydistributed current moving along its straight generatrices, themagnetic induction field vanishes inside, and coincides out-side with that of the same total current moving along its axis.For the same infinite cylinder with a solenoidal winding, astationary, uniformly-distributed current produces a magneticinduction field, which is uniform and along the axial direc-tion inside, and vanishes outside. For an infinite plane with a

ELECTROSTATIC AND MAGNETOSTATIC FIELDS FOR BESSEL-FOURIER DISTRIBUTED SOURCES ON INFINITE PLANES 91

uniformly distributed current, the magnetic induction field isuniform, parallel to the plane, and perpendicular to the direc-tion of the current. The electrostatic and magnetostatic fieldsfor the planes with uniform sources share the uniformity, anddiffer in directions.

The reader may ask if the simple and familiar situationsjust described can be modified in a simple way in order togenerate other easy-to-understand and interesting situationsof electrostatics and magnetostatics. The answer is yes, andis illustrated in the textbooks at the higher levels [3-7], specif-ically via the multipole expansions of the respective potentialfields. However, most of the presentations in the books arelimited to the fields outside a region where the sources arelocated, ignoring the fact that the fields inside are alreadypresent in the same expansion, and that there are geometricaland physical connections between the inner and outer fields.This was the motivation for writing the article Multipole Ex-pansions Inside and Outside [8]. Other harmonic expansionsin two dimensions in circular, elliptic, and parabolic cylin-drical coordinates [9-12], and in three dimensions in circu-lar cylindrical, parabolic, and spheroidal coordinates [13,14],in toroidal coordinates [15-18], and in bispherical coordi-nates [19,20] have been studied by our group in connectionwith the analysis of different electric, magnetic, and elec-tronic devices.

In a recent work, harmonic static charge and stationarycurrent distributions in planes were assumed to be of the cir-cular and hyperbolic cosine types in one of the Cartesian co-ordinates, and the corresponding electrostatic and magneto-static fields were constructed [21]. In this contribution, theharmonic source distributions on the planes are chosen tobe of the Bessel-Fourier type, in circular cylindrical coor-dinates. The changes from [21] to this work in the type ofdistribution and in the coordinates translate also into changesin the dimensionality of the associated fields from two tothree. There is also a change at the didactic and mathemati-cal levels, requiring the use of Bessel radial functions insteadof the (co)sine functions in the Cartesian coordinate. Sec-tion 2 presents the construction of the electrostatic potentialand electric intensity fields from the assumed Bessel-Fourierstatic charge distributions in a plane. Section 3 illustratesthe corresponding construction of the magnetostatic potentialand magnetic induction fields for the Bessel-Fourier distribu-tions of the stationary electric current in a plane. Section 4contains a description of the results of the previous sections,including a comparison of the respective fields, and also theirconnections with other systems.

2. Construction of the electrostatic poten-tial and electric intensity fields for Bessel-Fourier distributed static charges on an in-finite plane

The electric intensity field ~E(~r) is connected with its electro-static charge sources via Gausss law in differential equation

and boundary condition forms:

~E = 4pi (1)( ~E2 ~E1) n = 4pi, (2)

where is the volume charge density, is the surface chargedensity, and n is a unit vector normal to the boundary surface.The conservative nature of the electrostatic field is expressedby the corresponding equations:

~E = 0 (3)( ~E2 ~E1) n = 0. (4)

Integration of Eq. (3) is accomplished by introducing theelectrostatic potential(~r):

~E = . (5)Substitution of Eq. (5) into Eq. (1) leads to the Poisson

equation for the potential,

2 = 4pi. (6)For the situation of charge distributed in an infinite plane,

the volume density vanishes and Eqs. (1) and (6) become ~E = 0 (7)2 = 0. (8)

Consequently, the electric intensity field must besolenoidal, Eq. (7), and irrotational, Eq. (3); while the poten-tial must be harmonic, Eq. (8).

The separability and integrability of Eq. (8) in circularcylindrical coordinates lead to the harmonic solutions

(, , z) = [AmJm() +BmNm()]

[Cm cosm+Dm sinm][Emez + Fmez] (9)in terms of radial Bessel functions, angular (co)sine func-tions, and longitudinal exponential functions, with separationconstants and m = 0, 1, 2, 3, . . . [22].

The Bessel-Fourier surface charge distributions in thez = 0 plane

C = 0Jm() cosm (10)S = 0Jm() sinm (11)

determine the values of the parameters and coefficients inEq. (9).

Specifically, for the charge distribution of Eq. (10), thepotential becomes

C(, , z 0) = 0Jm() cosmez (12)C(, , z 0) = 0Jm() cosmez (13)

exhibiting its harmonic nature inherited from the source, itscontinuity at z = 0, and its vanishing for |z| .

Rev. Mex. Fs. E 52 (1) (2006) 9094

92 M.A. ROSALES AND E. LEY-KOO

Then the electric intensity field follows directly from thesubstitution of Eqs. (12) and (13) into Eq. (5):

~EC(, , z 0) = 0 [J m() cosm

mJm() sinm kJm() cosm

]ez (14)

~EC(, , z 0) = 0 [J m() cosm

mJm() sinm+ kJm() cosm

]ez (15)

Notice that the tangential components of the field at thesource plane, z = 0, are continuous, so that Eq. (4) is satis-fied, while the normal components in the k direction exhibita discontinuity which, via Eqs. (2) and (10), determines thevalue of 0:

20 = 4pi0. (16)

For the charge distribution of Eq. (13), the final resultsare directly written, for the potential:

S(, , z 0) = 2pi0

Jm() sinm ez (17)

S(, , z 0) = 2pi0

Jm() sinm ez (18)

and for the electric intensity field:

~ES(, , z 0) = 2pi0 [J m() sinm

mJm()

cosm+ kJm() sinm]ez (19)

~ES(, , z 0) = 2pi0 [J m() sinm

mJm()

cosm kJm() sinm]ez (20)

The familiar case of the uniformly charged plane fol-lows from Eq. (10) for the limit situation when 0 andm = 0, taking into account the fact that J0( 0) 1,J 0( 0) 0 and ez 1 z. Then, the potentialin Eqs. (12) and (13) becomes linear in the distance from thesource plane, and the field in Eqs. (14) and (15) becomes uni-form and normal to the source plane. with the discontinuity4pi0 associated with Gausss law, Eq. (2).

The source distributions in Eqs. (19) and (11) for a cho-sen value of m = 1, 2, 3, . . ., share the same form and differonly in their relative orientations. The same properties arenaturally inherited by the potentials and the electric intensityfields, as follows from the comparisons of Eqs. (12)-(13) vs(17)-(18) and (14)-(15) vs (19)-(20).

3. Construction of the magnetic potential andmagnetic induction fields for Bessel-Fourierdistributed stationary currents on an infi-nite plane

The magnetic induction field

B(~r) is connected with its sta-tionary electric current sources by Ampe`res law in its differ-ential equation and boundary condition forms:

~B = 4pic~J (21)

( ~B2 ~B1) n = 4pic~K (22)

where ~J is the current per unit area and ~K is the current perunit length. The non-existence of magnetic monopoles is ex-pressed by the magnetic Gauss law in its differential equationand boundary condition forms:

~B = 0 (23)( ~B2 ~B1) n = 0. (24)

The solenoidal character of the magnetic induction field,Eq. (23), allows its expression as the curl of the magneto-static vector potential ~A(~r):

~B = ~A(~r). (25)Substitution of this expression into Eq. (21) leads to the

equation satisfied by the potential

( ~A) = 4pic~J

( ~A)2 ~A = 4pic~J. (26)

By using the transverse gauge,

~A = 0,equation (26) reduces to Poissons equation

2 ~A = 4pic~J. (27)

For the current sources distributed on the plane z = 0,the current density outside the plane vanishes, ~J = 0. Then,the magnetic induction field becomes irrotational, and the po-tential satisfies Laplaces equation, since Eqs. (21) and (27)reduce to

~B = 0 (28)2 ~A = 0. (29)

The stationary electric currents on the source plane arechosen as the linear density distributions of the Bessel-Fourier types:~KCm = I0 [kJm() cosm]

= I0

[m

Jm() sinm J m() cosm

](30)

Rev. Mex. Fs. E 52 (1) (2006) 9094

ELECTROSTATIC AND MAGNETOSTATIC FIELDS FOR BESSEL-FOURIER DISTRIBUTED SOURCES ON INFINITE PLANES 93

~KCm = I0 [kJm() sinm]

= I0

[m

Jm() cosm J m() sinm

]. (31)

The harmonic magnetostatic potential, solutions ofEq. (29), share the Bessel-Fourier character of the sources.In the case of Eq. (30) we consider the potential above andbelow the source plane:

~ACm(, , z 0) = ACm0

[m

Jm() sinmJ m() cosm

]ez (32)

~ACm(, , z 0) = ACm0

[m

Jm() sinmJ m() cosm

]ez. (33)

which is solenoidal and continuous at z = 0.The magnetic induction field is obtained when the poten-

tial from these equations are substituted into Eq. (25):~BCm(, , z 0) = ACm0 [J m() cosm

+m

Jm() sinm+ kJm() cosm

]ez (34)

~BCm(, , z 0) = ACm0 [J m() cosm

mJm() sinm+ kJm() cosm

]ez (35)

Notice now that the normal components of the magneticinduction fields at z = 0 are continuous, Eq. (24) for n = k;while its tangential components are discontinuous and pro-portional to the linear current density, Eq. (30), consistentwith Eq. (22) and determining the value of AC0m:

2AC0m =4picI0. (36)

For the current distribution of Eq. (31), the correspond-ing magnetic potential and magnetic induction fields take thefinal forms:

~ASm(, , z>

94 M.A. ROSALES AND E. LEY-KOO

potentials were constructed first borrowing the harmonicityof the sources, then the solenoidal and irrotational characterof the force fields is guaranteed, and the boundary conditionsare applied to determining the magnitude parameter for thepotential, Eq. (36).

In Ref. 21 the electromagnetic fields associated withtime-harmonically-varying currents distributed harmonicallyon an infinite plane were constructed. The correspondingproblem, for currents with Bessel-Fourier, Mathieu, and We-

ber distributions on an infinite plane, has also been investi-gated using circular, elliptic, and parabolic cylindrical coor-dinates, respectively. It is recognized that the correspondingelectromagnetic radiation fields correspond to the respectivePropagation Invariant Optical Fields [23]. Their static limitscorrespond to the Bessel-Fourier cases studied here, and tothe respective harmonic sources, potentials, and force fieldsin elliptic and parabolic geometries.

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Rev. Mex. Fs. E 52 (1) (2006) 9094