jackson_thin wire 2

Charge density on a thin straight wire: The first visit J. D. Jacksona) University of California, Berkeley, California 94720 ~Received 24 August 2001; accepted 9 November 2001 ! The problem of the linear charge density on a long straight conducting wire was considered and solved using a variational technique over 123 years ago. We describe the who and where andsummarize his results. We also eat humble pie. © 2002 American Association of Physics Teachers. @DOI: 10.1119/1.1432973 # The question of the linear charge density on a long straight conducting wire and its limiting form as the ratio of radius tolength vanishes has been treated in at least six papers bothbefore and after the publication of Griffiths and Li 1in 1996. My own paper2in 2000 aspired to be the last. Griffiths and Li, in their second paragraph, say, ‘‘The question soundssimple enough, and it must surely have been considered longago by Sommerfeld, Smythe, Stratton, or maybe even Max- well himself $emphasis added %. However, we have found no reference to it in the literature.’’ They may perhaps be for-given for overlooking the Russian publications in the 1960s~Refs. 5–7 in Ref. 2 !because the Russians were concerned mostly with the total charge and capacitance of a long cylin-der. But we are all guilty of a most egregious failure insearching the prior literature, as I now describe. Recently, in looking for something else, I was astounded and mortified to find, near the end of the second volume ofThe Scientific Papers of James Clerk Maxwell , 3a paper en- titled ‘‘On the electrical capacity of a long narrow cylinder, and of a disk of sensible thickness.’’4The relevant volume of the Proceedings of the London Mathematical Society re-veals that the paper was communicated on behalf of Maxwellby a Mr. Tucker at the meeting of the society of March 14,1878, with Lord Rayleigh, President, in the chair. Maxwelldied 20 months later on November 5, 1879. This paper ap-pears to be his last published research in electricity and mag-netism and almost his last on any subject. It is a concise,masterful treatment of the topic. Let me make clear that, in my admiration of Maxwell’s work and my chagrin at not being aware of it, I do not wishto imply that the recent papers are mere rehashes. They ap-proach the problem differently, 1or in more diversity and depth,2and add to our understanding of the physics involved. Nevertheless, we can only guess how knowledge of Max-well’s paper would have changed the number and content ofour recent publications—surely considerably. Maxwell’s opening words are, ‘‘The distribution of elec- tricity in equilibrium on a straight line without breadth is auniform one. We may expect, therefore, that the distributionon a cylinder will approximate uniformity as the radius ofthe cylinder diminishes.’’ He proceeds to discuss Cavend-ish’s upper and lower limits for the capacity before launchinginto a variational calculation that is the equivalent of Prob-lem 1.18 of my textbook. 5He chooses to expand the linear charge density l(z) in a finite sum of even order Legendre polynomials, assumes that the length is very long compared to the radius, and ignores the contribution to the potentialfrom the end surfaces. His approach is closely similar to thesecond method in my paper. 2He considers first a two-term approximation ( L50,2) and then three ( L50,2,4). He gives lower limits for the capacity6and then gives the charge den-sityl(z) from his three-term approximation in the limit of very large ratio of length to radius. In the notation of my paper, Maxwell’s expression is l~z!5Q 2cF111 LGM~z!G, ~1! whereQis the total charge, 2 cis the length of the cylinder, ais its radius, z5z/cwith range ~21, 1!, and L52 ln(2c/a). The function GM(z) of Maxwell is GM~z!57 16~9z422z2217/15 !55 3P2~z!19 10P4~z!.~2! For comparison, the recent first-order ~in 1/L!expression has2,7 G~z!52l n2 222ln~12z2!. ~3! We plot the two functions G(z) in Fig. 1 for comparison. The heavier, slightly oscillatory curve is Maxwell’s approxi- mation. More terms in the expansion in Legendre polynomi-als would smooth out the oscillations, but the three-term ap- proximation is remarkably good. In fact, if we expand G( z) @Eq.~3!#in Legendre polynomials, we find the first two terms in the series are just Maxwell’s GM(z)@Eq.~2!#. Below his equivalent of Eqs. ~1!and~2!, Maxwell contin- ues, ‘‘which shows that, as the ratio of the length to thediameter increases, the density becomes more nearly uni-form, and the deviation from uniformity becomes more con-fined to the parts near the ends of the cylinder.’’ Could theconclusion of the investigation be said more succinctly? Fig. 1. Comparison of the correction functions GM(z) andG(z). Maxwell’s is the solid curve. 409 409 Am. J. Phys. 70~4!, April 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers Of course, as indicated earlier, we as workers in this same vineyard need not be ashamed of having written our papers.Among other things, the linear charge density is now knownto higher order in 1/ L; the results have been generalized to shapes other than the right circular cylinder; the puzzle ofhow the charges can be in equilibrium on the cylinder with a constant linear density as a/c!0 has been answered. Still, I draw the following lessons from this story: ~a!There isalmostnothing new under the sun, especially in a mature subject such as electrostatics. 8 ~b!To avoid embarrassment and/or an editor’s wrath, au- thors must search the previous literature assiduouslyand search again. ~c!The giants in any field are far in advance of the mere mortals ~in this case, over 123 years! !. Mea culpa atque culpa ceteri ACKNOWLEDGMENT I thank Mark W. Jackson for helpful discussions. a!Electronic mail: [email protected] 1D. J. Griffiths andY. Li, ‘‘Charge density on a conducting needle,’’Am. J. Phys.64,7 0 6– 7 1 4 ~1996!. 2J. D. Jackson, ‘‘Charge density on a thin straight wire, revisited,’’Am. J.Phys.68, 789–799 ~2000!.This paper contains references to other relevant papers not cited here. 3J. C. Maxwell, The Scientific Papers of James Clerk Maxwell , edited by W. D. Nivens, 2 vols. bound as one ~Dover reprint, NewYork !, Vol. 2, pp. 672–680. 4J. C. Maxwell, ‘‘On the electrical capacity of a long narrow cylinder, andof a disk of sensible thickness,’’ Proc. London Math. Soc. IX,9 4 – 1 0 1 ~1878!. 5J. D. Jackson, Classical Electrodynamics ~Wiley, New York, 1998 !, 3rd ed., p. 53. 6Maxwell reproduces his paper ~Ref. 4 !, mutatis mutandis, in The Electrical Researchers of Henry Cavendish , edited by J. C. Maxwell ~Cambridge U.P., Cambridge, 1879 !, Note 12, pp. 393–400, and compares his theoret- ical capacitance to Cavendish’s measurements of 1771. 7The form of G(z)~to compare with Maxwell’s !results from taking the first-order terms in 1/ Lfrom my Eqs. ~26!and~29!of Ref. 2, and express- ingl~z!in terms of the total charge Qrather than the linear charge density parameter l0. 8A notable exception is the remarkable theorem discovered in 1956 by Thompson and Lampard @A. M. Thompson, and D. G. Lampard, ‘‘A new theorem in electrostatics and its application to calculable standards of ca-pacitance,’’ Nature ~London !177,8 8 8 ~1956!#. The practical realization and relevance for electrical standards are discussed briefly by Zimmerman@N. M. Zimmerman, ‘‘Aprimer on electrical units in the Syste `me Interna- tional,’’ Am. J. Phys. 66, 324–331 ~1998!#. I have given a pedagogical proof of the theorem with examples @J. D. Jackson, ‘‘Acurious and useful theorem in two-dimensional electrostatics,’’ Am. J. Phys. 67, 107–115 ~1999!#. 410 410 Am. J. Phys., Vol. 70, No. 4, April 2002 J. D. Jackson