Generalization of the electrostatic potential function for an infinite charge distribution G. Palma,a) R. Oyarzu´n, and U. Raff Department of Physics, University of Santiago, Casilla 307, Correo 2, Santiago, Chile
共Received 30 September 2002; accepted 20 March 2003兲 The asymptotic conditions needed to define the electrostatic potential due to an infinite charge distribution are studied in detail. It is shown that if the charge distribution decreases faster than the square of the distance when 兩r兩 goes to infinity, the convolution integral defining the potential exists, goes to zero as 兩r兩 goes to infinity, and therefore allows the calculation of the electric potential function at any point in space, even if the total charge is infinite. We illustrate the calculation of the electric potential with a simple example of a spherically symmetric infinite charge distribution. © 2003 American Association of Physics Teachers. 关DOI: 10.1119/1.1574039兴
I. INTRODUCTION The concept of the electric potential of a charge distribution plays a fundamental role in classical electromagnetism.1–3 Any undergraduate course teaches us that the difference of the electric potential at two points defines the voltage or potential difference. By using localized charge distributions 共r兲, we also find that the work required to move an electric charge between two points is a measure of the voltage difference. Introductory courses in electricity and magnetism show that if all the sources are localized in some finite region of space, it is possible to define the potential to be zero at infinity 共the usual convention兲.4 –7 Although infinite charge distributions do not exist in the real world, they are the primary choice for studying electrostatics, mostly due to the fact that analytical expressions for the electrostatic potentials 共r兲 can be explicitly calculated either by direct integration or by using special symmetries invoking Gauss’ law. The electrostatic potential of a charge density vanishing outside some large radius becomes (⬁) ⫽0. Nevertheless, the potential due to an infinite charge distribution does not necessarily vanish at infinity, and therefore some asymptotic conditions concerning 共r兲 need to be discussed. In this context, the potential at a finite distance from a charge distribution 共r兲 is equal to the work required to move a unit test charge from infinity to the defined point. With the zero potential rule, and in the absence of delimiting surfaces, the potential of a charge distribution 共r兲 is given by the convolution between the Green’s function of the Poisson equation and the charge distribution 共using cgs units兲:8,9
共 r兲 ⫽
冕
共 r⬘ 兲 d 3 r ⬘ ⫹0 . 兩 r⫺r⬘ 兩
共1兲
The constant 0 is an arbitrary constant chosen so that the scalar potential vanishes at infinity. If the source charges are no longer continuous, the scalar field 共r兲 of Eq. 共1兲 reduces to the well-known expression N
共 r兲 ⫽ 兺
i⫽1
qi ⫹0 , Ri
共2兲
where the q i are N discrete point charges located at distances R i from the observation point r with respect to some coordinate system. However, if the sources are not confined to a 813
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finite region of space, Eq. 共1兲 must be used carefully. For instance, if we consider the classic example of an infinite straight charged wire and try to carry out the integration, we would find that the integral diverges. Although the infinite line charge is discussed as an example in almost every introductory textbook, all the technical aspects involved in the calculation of the electric field using an integration starting from Coulomb’s law are not discussed. Note that for an infinite line charge, the line integral over the electric field vector E evaluated from a finite distance out to infinity diverges logarithmically as one end point goes to infinity. The twodimensional problem of an infinite charged plane leads to a similar situation when the surface charge density is constant. Although mentioned in the literature 共see for instance Refs. 4 and 8兲, this lack of completeness motivates a more general study of a wider class of charge distributions, including non-localized ones, that is, infinite charge distributions. Gibbs2 found two necessary conditions for the asymptotic behavior of a charge distribution such that the integral defining the potential function remains finite: 共i兲 r 3 ⬍K when r increases indefinitely and, 共ii兲 r⬍K ⬘ when r→0, where K and K ⬘ are two arbitrary constants. Note that the first condition is sufficient and stronger than the one we really need, as will be shown below. The correct asymptotic behavior of the charge distribution is the key issue. No restrictive assumption will be made about the electric charge density 共r兲 as a function of r for finite r. An essential contribution was made by Page and Adams,1 which seems to have been overlooked in most more recent monographs about electrodynamics. These authors explicitly discuss the scalar potential as a function of the density which could either vanish beyond some distance or extend to infinity. Their analysis showed that the potential remains finite when the charge density drops to zero as r →⬁ faster than 1/r 2 . Therefore, if we use the result of Ref. 1, we must conclude that the potential due to an infinite line charge distribution does not exist. Nevertheless, due to its simplicity, this problem is usually discussed in the literature solving it by means of Gauss’ law 共see, for instance, Ref. 4兲, yielding, up to an additive constant, the well-known expression (r)⫽⫺2 ln r. This result might suggest an apparent contradiction with the claim of Ref. 1. The explanation however, is as follows: the constant 0 © 2003 American Association of Physics Teachers
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appearing in Eq. 共1兲 cannot be set equal to zero and hence must correspond to the divergent part of the Coulomb integral in our example. This can be explicitly demonstrated by calculating the potential due to a finite line charge q of length 2L placed symmetrically from ⫺L to ⫹L along the z axis, using the Coulomb integral from Eq. 共1兲. A straightforward calculation yields (r)⫽⫺2 ln r ⫹ ln 关 ( 冑r 2 ⫹ (z⫺L) 2 ⫹(L⫺z))( 冑r 2 ⫹(z⫹L) 2 ⫹(z⫹L)) 兴 , where ⫽q/(2L) is the linear charge density and r is the perpendicular distance from the observation point to the z axis. When L approaches infinity, the second term diverges logarithmically with L. When this second term, which is constant in the limit of an infinite length, is interpreted as the constant 0 , we have demonstrated that the well-known result (r)⫽⫺2 ln r is recovered. As shown above, our approach to the problem is somewhat different. We establish the existence of 共r兲 and also find a simple condition for the asymptotic behavior of the charge distribution 共r兲 such that the potential vanishes at infinity. This clarifies the arbitrariness of the potential 0 关Eq. 共2兲兴 which appears in the literature. Moreover, we shall take a different route and discuss an example where the total charge density may be infinite, but nevertheless (⬁)⫽0. Because one has to deal with infinite charge distributions, it is meaningful to establish a clear condition to calculate the potential at some finite point r while being able to discard the arbitrary constant 0 appearing in Eq. 共1兲.
l l⫹1 symbols r Ɱ and r Ɑ in Eq. 共5兲 correspond to the radial integration variable when integrating 兩 r⬘ 兩 from 关 0,r 兴 and 关 r,⬁ 兴 , respectively. The integration of the two terms in Eq. 共5兲 is carried out separately. The first term, defined as loc(r), corresponds to the potential produced by a localized charge distribution described by a charge density loc(r), where loc(r) is nonzero inside a sphere of radius R enclosing the localized arbitrary charge distribution. The second term is the asymptotic contribution as(r). If we insert the asymptotic part of the charge distribution from Eq. 共3兲 into Eq. 共5兲 and use the orthogonality relations for the spherical harmonics Y lm ( , ), we obtain:
as共 r兲 ⫽
Because the potential is usually assumed to vanish at infinity, the asymptotic behavior of an infinite charge distribution 共r兲 must be studied with some caution. If there are no delimiting surfaces, the potential is given by Eq. 共1兲. Let the charge distribution 共r兲 be written as:
共 r兲 ⫽ loc共 r兲 ⌰ 共 R⫺ 兩 r兩 兲 ⫹ as共 r兲 ,
共3兲
where ⌰(R⫺ 兩 r兩 ) is the Heaviside 共or step兲 function, loc(r) represents a localized charge distribution, and as stands for the asymptotic part (r→⬁) of the charge distribution 共r兲 for which we shall assume that as(r)⬃ 兩 r兩 ⫺(2⫹ f ) . The constant f is non-negative. In the following, the denominator of Eq. 共1兲 is expanded in terms of spherical harmonics Y lm ( , ), 10 ⬁
l
1 ⫽4 兩 r⫺r⬘ 兩 l⫽0
1
rl
Ɱ * 共 ⬘ , ⬘ 兲 Y lm 共 , 兲 . Y lm 兺 m⫽⫺l 兺 2l⫹1 r Ɑl⫹1
共4兲
If we substitute this expansion into Eq. 共1兲, we obtain the following expression for the potential: ⬁
共 r兲 ⫽4 兺
l
兺
l⫽0 m⫽⫺l
1 Y 共,兲 2l⫹1 lm
冕
d r⬘
l rⱮ
3
l⫹1 rⱭ
* 共 ⬘ , ⬘ 兲关 loc共 r⬘ 兲 ⌰ 共 R⫺ 兩 r⬘ 兩 兲 ⫹ as共 r⬘ 兲兴 . ⫻Y lm
共5兲
The spherical harmonics Y lm ( , ) appearing in Eqs. 共4兲 and 共5兲 are the angular dependent solutions of the wellknown Laplace equation solved in spherical coordinates. The 814
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冕
r
˜R
dr ⬘
1 r⬘
f
⫹4
冕
⬁
r
dr ⬘
1
r⬘
f ⫹1 .
共6兲
We see that the first integral exists whenever f ⬎0 and goes to zero when r→⬁, even when f ⫽1. The lower integration limit of the first integral, labeled as ˜R , denotes some large radius for which the asymptotic form of the charge density holds. The second integral also vanishes when r→⬁. Therefore, we can conclude that the potential of a charge distribution 共even if its support is not finite兲 vanishes at infinity only if the charge density (r)→0 is faster than the inverse of the square of the distance. Note that the total charge given by Q⫽
II. ASYMPTOTIC CONDITION FOR THE CHARGE DENSITY
4 r
冕
共 r⬘ 兲 d 3 r ⬘ ,
共7兲
need not be finite. In fact, the evaluation of the integral in Eq. 共7兲 yields: ⬁ 1 Q⫽ loc共 r⬘ 兲 d 3 r ⬘ ⫹4 dr ⬘ f . 共8兲 ˜R r⬘ r ⬘ ⭐R
冕
冕
The first term corresponds to the total charge associated with the localized charge density within a radius R, and the second integral corresponds to the contribution obtained from the asymptotic part of the total charge distribution and diverges for f ⭐1. In other words, the total electric charge is finite only if f ⬎1. III. A SPHERICAL SYMMETRIC INFINITE CHARGE DISTRIBUTION In the following, the conclusion of Sec. II is illustrated using a spherically symmetric infinite charge distribution. Assume the charge is distributed over the whole space according to 1 ˜ 兲. 共 r兲 ⫽ loc共 r兲 ⌰ 共 R⫺ 兩 r兩 兲 ⫹ 0 3 ⌰ 共 兩 r兩 ⫺R 共9兲 兩 r兩 The first term corresponds to the localized charge distribution within a spherical region of radius R. Its contribution to the potential is a well-known expression that is discussed in many references, for example, Refs. 9 and 11. However, as mentioned, the asymptotic part of the spherical charge distribution 关the second term of Eq. 共9兲兴 must be analyzed carefully. A straightforward calculation yields the asymptotic contribution to the scalar potential as(r),
as共 r兲 ⫽
4 ˜ 兲 ⫹1 兴 , 关 ln共 r/R r as Palma, Oyarzu´n, and Raff
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which goes to zero as 兩 r兩 →⬁. The total charge associated with this part of the charge distribution diverges logarithmically for large distances. According to our conclusion in Sec. II, we see immediately that, due to the asymptotic behavior of the linear charge density of an infinite wire, it is impossible to obtain (⬁)⫽0. The problem of finding the charge density induced by an electric charge q at some finite distance a from an infinite conducting plate is worth mentioning as a wellknown classical problem.11 A straightforward calculation of yields ⬃1/r 3 , where r⫽ 冑(x⫺a) 2 ⫹y 2 , and therefore we conclude that (⬁)⫽0 in this case. Indeed the total charge induced is equal to ⫺q, which reduces the problem to a localized charge distribution. IV. CONCLUSIONS Infinite electric charge distributions offer intriguing and challenging theoretical problems that are ubiquitous in graduate and undergraduate courses of electromagnetism. We have shown that, if the charge density goes to zero faster than the inverse of the squared distance as r→⬁, the potential of a charge distribution goes to zero at infinity, even if the total charge is not bounded. Under these circumstances it is possible to define the potential at the point r, which is equivalent to the work required to move a unit test charge from infinity to r.
ACKNOWLEDGMENTS The authors want to thank E. Matute for helpful comments. G.P. was supported in part through Project Nos. FONDECYT #1020010 and DICYT. U.R. is grateful for partial support by DICYT, University of Santiago. a兲
Electronic mail:
[email protected] L. Page and N. I. Adams, Electrodynamics 共Van Nostrand, New York, 1940兲, pp. 48 –51. 2 W. Gibbs, Vector Analysis 共Dover, New York, 1960兲, pp. 204 –210. 3 O. Heaviside, Electromagnetic Theory 共Chelsea, New York, 1971兲, Vol. I, pp. 202–206. 4 E. M. Purcell, Electricity and Magnetism 共McGraw–Hill, New York, 1965兲, pp. 35– 43. 5 R. K. Wangsness, Electromagnetic Fields 共Wiley, New York, 1986兲, 2nd ed., pp. 68 –70. 6 H. A. Haus and J. R. Melcher, Electromagnetic Fields and Energy 共Prentice–Hall International, 1989兲, pp. 90–92. 7 K. R. Demarest, Engineering Electromagnetics 共Prentice–Hall, Upper Saddle River, NJ, 1998兲, pp. 107–113. 8 J. R. Reitz, Foundation of Electromagnetic Theory 共Addison–Wesley, Reading, MA, 1972兲, 2nd ed., pp. 39– 41. 9 J. D. Jackson, Classical Electrodynamics 共Wiley, New York, 1975兲, 2nd ed., pp. 34 – 45, 136 –138. 10 G. Arfken, Mathematical Methods for Physicists 共Academic, New York, 1970兲, 2nd ed., pp. 569–584. 11 W. Greiner, Klassische Elektrodynamik 共Verlag Harri Deutsch, Frankfurt am Main, Theoretische Physik, Band 3, 1991兲, pp. 60– 62. 1
PHYSICISTS TURNED ELDER STATESMEN Loomis followed his passion for science to Washington, and then into war, but political influence was something that neither interested him nor held any allure. He did not care to join the ranks of physicists-turned-elder statesmen who were trotted out at conventions and government seminars, to be ‘‘exhibited as lions at Washington tea parties,’’ as the distinguished physicist Samuel K. Allison described ‘‘the awe and gratitude of the scientifically illiterate lay world.’’ Independence was a luxury he could afford, and it enabled him to remain detached, and slightly above, the postwar scramble for position and power that consumed so many of his colleagues. Jennett Conant, Tuxedo Park 共Simon & Schuster, New York, NY, 2002兲, p. 286.
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