jackson_thin wire 1

Charge density on thin straight wire, revisited J. D. Jackson University of California, Berkeley, California 94720 ~Received 12 November 1999; accepted 11 February 2000 ! The question of the equilibrium linear charge density on a charged straight conducting ‘‘wire’’ of finite length as its cross-sectional dimension becomes vanishingly small relative to the length isrevisited in our didactic presentation. We first consider the wire as the limit of a prolate spheroidalconductor with semi-minor axis aand semi-major axis cwhena/c,,1. We then treat an azimuthally symmetric straight conductor of length 2 cand variable radius r(z) whose scale is defined by a parameter a. A procedure is developed to find the linear charge density l(z)a sa n expansion in powers of 1/ L, where L[ln(4c 2/a2), beginning with a uniform line charge density l0. We show, for this rather general wire, that in the limit L..1 the linear charge density becomes essentially uniform, but that the tiny nonuniformity ~of order 1/ L!is sufficient to produce a tangential electric field ~of order L0) that cancels the zeroth-order field that naively seems to belie equilibrium.Wespecializetoarightcircularcylinderandobtainthelinearchargedensityexplicitly,correct to order 1/ L 2inclusive, and also the capacitance of a long isolated charged cylinder, a result anticipated in the published literature 37 years ago. The results for the cylinder are compared withpublished numerical computations. The second-order correction to the charge density is calculatednumerically for a sampling of other shapes to show that the details of the distribution for finite 1/ L vary with the shape, even though density becomes constant in the limit L!‘. We give a second method of finding the charge distribution on the cylinder, one that approximates the charge densityby a finite polynomial in z 2and requires the solution of a coupled set of linear algebraic equations. Perhaps the most striking general observation is that the approach to uniformity as a/c!0i s extremely slow. © 2000 American Association of Physics Teachers. I. INTRODUCTION In recent years, the question, ‘‘Is the linear charge density on a finite length of a charged straight conducting wire ofinfinitesimal cross section constant, or even well defined?’’has been addressed in this journal by Griffiths and Li, 1 Good,2and Andrews.3Although a variety of results are ob- tained for the equivalent of conductors of small transversedimension, the focus was on the limit of vanishing lateralextent. All the authors agree that in that exact limit thecharge density is uniform. There remains, however, a puzzle,stressed to the present author by Bruce D. Winstein. Figure 1 displays a uniform line charge of length 2 cparallel to the z axis and centered about z50. Consider an element of charge dqatz. The sum of the electric fields produced by the un- shaded segments on either side produces no net force on theelementdq, but the shaded segment does. An elementary calculation shows that the force is dF5dqE z5dql0 2pe0z c22z2, ~1! where l0is the uniform linear charge density. Clearly, dq cannot be in equilibrium! How can the equilibrium charge distribution on such a conductor be uniform if there are unbalanced forces? Oneaspect of the problem that might argue for caution is physi-cal. The potential of an isolated finite wire with a fixedamount of charge Qon it grows arbitrarily large as its trans- verse dimensions decrease, increasing as Qln(D/a), where D(a) is a distance of the order of the length ~transverse dimension !of the wire. In the limit a!0, the potential be- comes infinite unless Q!0. Conversely, for the practical situation of a wire placed at fixed potential V, the limit a!0 implies that Qorl 0must vanish. With no linear charge density, there is no force! From this point of view, Griffiths and Li1were right in their doubts about a valid limit—like the cat in Lewis Carroll’s Alice’s Adventures in Wonder- land, in the limit of a!0, nothing is left but the grin. This V fixed,a!0 argument is a red herring, however, as we show below. Physicists never quite go to the limit. Our paper is aimed at clarifying these issues by focusing on the situation for very large D/a, not the limit a50, al- though that limit, properly understood, is smooth. The dis- cussion is frankly pedagogical, without much claim of origi-nality. Some aspects are similar in approach to Andrews, 3 others to the much earlier ~and actually definitive !work of Vainshtein and colleagues4–6and also Waterman.7 We first treat the special case of the long straight ‘‘wire’’ as the limit of a very elongated prolate spheroidalconductor—special because it has a uniform linear charge density, whatever its aspect ratio. In the limit of great elon-gation, the force ~1!applies and is necessary to maintain charge equilibrium, not destroy it! We then examine a‘‘wire’’ consisting of a straight azimuthally symmetric con- ductor of length 2 cand possibly variable radius r(z), which is of the order of a transverse scale parameter a. We consider a/c,,1, or more correctly, L[2 ln(2c/a)..1. In that limit the finishings at the ends of the cylinder and, indeed, whether it is hollow or not, are not significant in determiningthe charge density over the vast majority of the surface. Weshow that there are corrections to the uniform charge density of relative magnitude O(1/L n) times functions of z/c, for n51,2,.... For most shapes, the additions to the charge den- sity peak toward the ends, as expected, and despite being of order 1/ Land smaller, are such that they balance the 789 789 Am. J. Phys. 68~9!, September 2000 http://ojps.aip.org/ajp/ © 2000 American Association of Physics Teachers L-independent force ~1!from the dominant uniform linear density to produce electrostatic equilibrium. The appearanceof a factor of Lin the numerator of the force from these additional charge densities, which are of relative order 1/ L, is what resolves the puzzle of Fig. 1. The tiny nonuniformlinear charge density produces a L-independent force be- cause some of those charges locally get closer and closer to any observation point as a/c!0(L!‘). The right circular cylinder is treated in detail by two meth- ods~Secs. IV and VI !. Examples are given of the charge densities for shapes other than the spheroid and the cylinder,showing that the details of the distribution for wires of small,but not vanishing, radius depend on the shape. The slowapproach to uniformity with increasing Lis also illustrated ~Sec. V !. II. CHARGED CONDUCTING PROLATE SPHEROID A wire as the limiting case of a conducting prolate spher- oid has been treated by others. We are brief and only stressthe issue of the longitudinal force ~1!as a necessary force, not a puzzle. The spheroidal conductor has semi-major and semi-minor axes, cand~a,b!withb5a. The equilibrium surface charge density is 8 s~r,z!5Q 4pa2c1 Ar2 a41z2 c4, ~2! whereQis the total charge; the spheroid is centered at the origin with the zaxis as the major axis, and r25x21y2.I ti s of interest to find the charge density per unit length dQ/dz. Figure 2 shows the geometry. At the point Pthe element of area around the circumference of width dzisdA 52prdz/cos(a). From the equation of the ellipse ( r2/a2 1z2/c251) and tan a52dr/dz,w efi n ddA52pa2Ar2 a41z2 c4dz. ~3! ButsdA5ldzand because the factor in dAis the recipro- cal of the one in s(r,z), the charge density per unit length in zis l~z!5Q 2c~4! constant and independent of the ratio of minor to major axis! In the limit of a/c!01, the spheroid goes over automati- cally to a uniform line of charge with l05Q/2c. For a conductor the electric field at the surface is normal to it, with magnitude uEu5s/e0. The components of the electric field at the surface may therefore be computed from the surface charge density and the angle a:Ez 5(s/e0)sinaandEr5(s/e0)cosa. We need cosa5Ac22z2 Ac22~12a2/c2!z2, ~5! sina5az cAc22~12a2/c2!z2, where we have used the equation of the ellipse to give ex- pressions as functions of zalone. Note that, to first order in a/c, cos a’1 and sin a’az/c(c22z2)1/2; the normal to the surface ~PN in Fig. 2 !is almost radial, with only a small component in the axial direction. In that limit, however, themagnitude of the electric field is huge. Explicitly, the com-ponents of the electric field at the surface of the spheroid for arbitrary a/care E z5l0 2pe0z @c22~12a2/c2!z2#, ~6! Er5l0c 2pe0aAc22z2 @c22~12a2/c2!z2#. In the limit a2/c2,,1, these expressions reduce to Ez5l0 2pe0z @c22z2#, ~7! Er5l0c 2pe0aAc22z25l0 2pe0r~z!. The axial component is independent of ain the limit a2/c2 ,,1 and agrees with dF/dqfrom Eq. ~1!. It results from the product of an infinite surface charge density ( s}1/a) and a vanishing sin a(}a). The radial component of the electric field in the limit is just the naive result obtained by applying Gauss’s law around the ‘‘wire.’’ It grows without limit asa/c!01, except at the very ends @where ~6!, not ~7!, must be used #. The elongated prolate spheroid specifies one limiting form of a finite ‘‘wire’’ of negligible cross section. It is special inthat its charge density per unit length is constant, indepen- dent of the ratio a/c. For such a uniform charge density, the argument of Sec. I shows that in the limit of a/c!01there must be an axial electric field E zgiven by ~1!or~7!. Con- Fig. 1. Line charge of uniform density. The charge of the shaded portion exerts an unbalanced force to the right on dqatz. Fig. 2. Geometry of the prolate spheroid for computation of the linear charge density. 790 790 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson trary to the discussion there, however, we see here that the axial field is required to guarantee that the charge be in equi-librium on the spheroidal surface. Obviously the spheroidal wire is a very special case, but not the only one. As shown by Griffiths and Li, 1for the general ellipsoidal conductor with arbitrary values of the principal axes (2 a,2b,2c), the linear charge density along a principal axis is constant and given by ~4!, if the third axis is the chosen one. In the limit a,,b,,c, the ellipsoid ap- proximates in a special way a flattened wire. We now turn to a more general ~but cylindrically symmetric !shape. III. GENERAL AZIMUTHALLY SYMMETRIC CONDUCTING CYLINDER A. Linear charge density to order 1 ÕL2inclusive Consider our ‘‘wire’’ to be a long, thin, straight, azimuth- ally symmetric conductor of length 2 cand possibly variable radiusr(z), whose scale is set by the radial parameter a, witha/c,,1. In fact, the criterion for long and thin is more stringent, namely, L[2 ln(2c/a)..1. Specifically, we choose @r~z!#25a2f~z!, ~8! wheref(z) is a smooth positive function, of order unity and bounded above and below for uzu,c, and vanishing at uzu 5c. Later, we specialize to f(z)5(12uz/cu)2andf(z)51 2uz/cun, withna positive even integer. We approach the problem of finding the equilibrium linear charge density on the surface of the conductor through thepotential of a line charge ~zero cylinder radius !evaluated at r5r(z) and beyond, as did Andrews.3We develop a system- atic iterative scheme to find the charge density as a power series in 1/ L, beginning with the lowest order approxima- tion, a line charge of constant density l0extending over the range 2c,z,c. Straightforward integration shows that the electrostatic potential is F~0!~r,z!5l0 4pe0E 2ccdz8 Ar21~z82z!2 5l0 4pe0lnFz1c1Ar21~z1c!2 z2c1Ar21~z2c!2G. ~9! We are interested in the potential near the wire @r’O(a) ,,c,uzu,c#. Introducing the scaled variable z[z/c, and expanding the argument of the logarithm to first order in (r/c)2, we find the potential can be approximated by F~0!~r,z!5l0L 4pe0F111 Lln~12z2!11 Lln~a2/r2! 1~r2/c2!~11z2! 2L~12z2!21flG ~10! provided the observation point is not too close to the ends. The last term can be neglected provided @12uzu#..r/2c,a condition satisfied over most of the wire provided L..1. Using ~8!and omitting the last term in ~10!, we have the approximate but accurate expressionF~0!~r,z!5l0L 4pe0F111 LlnS12z2 f~z!D11 Lln~r2~z!/r2!G. ~11! The integral in ~9!is large, equal to Lin leading order, because of a peaking of the integrand at z85z,but with z/c- andr/r(z)-dependent terms in next order. From ~11!we see that, beyond the leading order, the potential is not constant on the surface of the conductor, r5r(z). We wish to find an addition l1(z) to the constant linear charge density l0such that its potential, when added to ~11!, will remove the zdependence at r5r(z) and give an equi- potential conductor. Evidently, l1/l0will be of order 1/ L. To this end, consider the potential F(1)(r,z) produced by l1(z): F~1!~r,z!51 4pe0E 2ccl1~z8!dz8 Ar21~z82z!2. ~12! We add and subtract l1(z) to the numerator and write F~1!~r,z!5l1~z! 4pe0E 2ccdz8 Ar21~z82z!2 11 4pe0E 2cc@l1~z8!2l1~z!#dz8 Ar21~z82z!2. The first integral is proportional to F(0)given by ~11!. The second integral has an integrand that is unexceptional near z85zbecause the numerator vanishes there. In fact, in the limit of L..1, we may set r50 in the denominator with the introduction of errors only of the order of r2/c2. We thus have F~1!~r,z!5l1~z! l0F~0!~r,z! 11 4pe0E 2cc@l1~z8!2l1~z!#dz8 uz82zu. ~13! The leading contribution to the first term in ~13!isL@times l1(z)/4pe0#, while the second integral and the remainder of the first term is, as observed by Andrews,3of lower order in L. Focusing on the leading order part of F(1)@because it is already down by one power of Land so of the same order as the second and third terms in ~11!#, we have the sum of F(0) andF(1), evaluated on the cylinder @r5r(z)#as F~0!~r~z!,z!1F~1!~r~z!,z! ’l0L 4pe0F111 LlnS12z2 f~z!D1l1~z! l0G. ~14! To assure constancy of the sum to order 1/ Linclusive, the first-order correction to the charge density must be l1~z! l0521 LlnS12z2 f~z!D. ~15! The complete linear charge density, correct to first order in 1/L, is then l~1!~z!5l0F121 LlnS12z2 f~z!DG. ~16! 791 791 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson This expression is equivalent to the approximation asserted by Andrews3@his Eq. ~3.2!, in my notation # lAndrews ~z!5l0 111 LlnS12z2 f~z!D. ~168! For use below, we note in passing the result for the leading orderpart of F(1)(r,z): F~1!~r,z!’2l0 4pe0lnS12z2 f~z!D1O~1/L!. ~17! With l1(z) known, we can proceed to the next step in our iteration scheme. With ~15!substituted for l1(z)/l0, the potential ~13!, correct to second order in 1/ L, is explicitly F~0!~r,z!1F~1!~r,z! 5l0L 4pe0H121 L2FlnS12z2 f~z!DG2 11 L2I 11 LlnSr2~z! r2DF121 LlnS12z2 f~z!DGJ, ~18! where the integral I, of order unity despite appearances, is I5L l0E 211dx ux2zu@l1~x!2l1~z!# 5E 211dx ux2zuFlnS12z2 f~z!D2lnS12x2 f~x!DG. ~19! The 1/ Lterm at r5r(z), present in F(0), has been elimi- nated by l1, leaving a second-order variation in z.B yt h e same steps from ~12!to~14!with a l2(z) instead of l1,w e can infer that the first line in ~18!is the negative of the charge density l2(z)/l0. The linear charge density, correct to order 1/ L2inclusive, is therefore l~z!5l0H121 LlnS12z2 f~z!D11 L2FFlnS12z2 f~z!DG2 2IGJ. ~20! The scheme of iteration is now clear. The second-order charge density l2(z) can be inserted into ~13!instead of l1(z) to generate F(2)(r,z). The sum of potentials will then be constant up to order 1/ L2, but will have z-dependent terms of order 1/ L3. These can be removed by identifying l3(z), and so on. In higher order, however, the integrals equivalent to Ibecome intractable in terms of known func- tions. Even ~19!needs numerical computation, except for special choices of f(z). Before evaluating ~19!and~20!for specific shapes for f(z), we address the conundrum posed in Sec. I. In Sec. II we saw how the puzzle was not actually a puzzle but a ne- cessity for the spheroidal shape. But what about othershapes? B. General resolution of the issue of the axial electric field and charge equilibrium Because of the shape of the conductor, defined by r(z)i n ~8!, care must be taken to distinguish between the electric field in the zdirection and the electric field tangential to the surface in the r–zplane. If bis the angle between the tan-gent vector and the zaxis in that plane, its tangent is tan b 5dr(z)/dz5r8(z)/c, wherer8(z)5dr/dz. Define the scale of variation of r(z)t ob e O(b), with a,,b. Then r8 5O(ac/b) and tan b5O(a/b),,1. We therefore have sinb’r8/cand cos b’1, with corrections of order ( a2/b2). The components of the zeroth-order electric field in the zand rdirections at the surface are, from ~10!or~11!, Ez~0!52]F~0! c]z5l0 2pe0cSz 12z2D, ~21! Er~0!52]F~0! ]r5l0 2pe0cSc r~z!D. The zeroth-order component of the electric field tangential to the surface is Etan~0!5Ez~0!cosb1Er~0!sinb 5l0 4pe0cF2z 12z212r8 rG 52l0 4pe0cd dzlnS12z2 f~z!D. ~22! The first-order potential is given in leading order by ~17!.I t has only a zcomponent of electric field. With cos b’1, the tangential and zcomponents of the first-order electric field are equal and are Ez~1!5Etan~1!52dF~1! cdz5l0 4pe0cd dzlnS12z2 f~z!D. ~23! Comparison of ~22!and~23!shows that the lowest order and first-order tangential fields cancel, as they must if thesurface is an equipotential. The reader may think this point isa bit of a straw man because the sum of the zeroth- andfirst-order potentials, shown in ~18!, is constant on the sur- face to order 1/ L, and so must give vanishing tangential electric field to that order. More interesting is the sum of thezcomponents of the electric field, E z~0!1Ez~1!52l0 4pe0cd dzln~f~z!!. ~24! Depending on the form of f(z), there is a nonvanishing axial electric field of order L0, but no tangential electric field. Three points are to be made, as follows. ~1!If the surface is the spheroid of Sec. II, f(z)512z2. Then the electric field in the zdirection ~24!is just ~1!. Actually, by looking back at ~15!and~17!, the reader will see that l150 and so F(1)50. In this case there are no corrections to the constant linear charge density, as we al- ready know from Sec. II. ~2!For any shape other than the spheroid, the linear charge density has nonvanishing z-dependent corrections that can be written as an expansion in powers of 1/ L. The ex- ample of the second-order linear charge density for the rightcircular cylinder is considered explicitly in Secs. IV and VI.The second-order charge densities for a family of shapes arediscussed briefly in Sec. V. ~3!For a general shape, the zeroth-order constant charge density l 0generates a potential F(0), which is constant on the conductor to order Lplus corrections of order L0. These order L0terms give rise to the electric field E(0)whose components are given by ~21!. The first-order charge density 792 792 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson l1(z), which is of order 1/ L, generates a potential F(1)of order L0. The electric field E(1)is thus of the same order as E(0)and is such as to combine with it to give a vanishing tangential component of field on the surface. Equilibrium of the surface charge is assured, even in the limit that 1/ L !0, as the charge density approaches uniformity. It is the fact that a nonuniform linear charge density of order Lk causes a potential of order Lk11that resolves the puzzle posed in Sec. I. Physically, the increase by one in the powers ofLis a result of the greater and greater peaking of the integrands in the expressions for the potential or the electric field asa/c!0, the result of contributions from the nonuni- form charge density in the immediate neighborhood of the observation point. IV. RIGHT CIRCULAR CYLINDRICAL CONDUCTOR „1… A. Charge density to order 1 ÕL2inclusive For a right circular cylinder, the radial function f(z)51 foruzu,c. Then the integral I~19!is Icylinder 5E 211dx ux2zu@ln~12z2!2ln~12x2!#. ~25! We change variables to t5x2z, break up the integration intot,0 andt.0, write out the logs as the sum of four terms and then recombine so that the arguments are linear in t, and finally rescale tto obtain Icylinder 522E 01du uln~12u! 2E 0~11z!/~12z!du uln~11u! 2E 0~12z!/~11z!du uln~11u!. We now break up the second integral into intervals ~0, 1!and (1,(1 1z)/(12z)). Then in the second part we change vari- ablesu!1/uto find Icylinder 52E 01dv vln~12v!1E ~12z!/~11z!1 du ulnu, where v5u2. The first integral can be found in the tables to be2p2/6. The second integral is elementary. The result is I5p2 621 2FlnS11z 12zDG2 . This gives the linear charge density, correct to order 1/ L2 inclusive, l~z!5l0H121 Lln~12z2!11 L2F@ln~12z2!#2 11 2FlnS11z 12zDG2 2p2 6G1O~1/L3!J ~26! and the potential near the wire to the same relative order ~including F(2))i sF~r,z!5l0L 4pe0F111 Ll~z! l0ln~a2/r2!1O~1/L3!G. ~27! On the cylinder ( r5a) the potential is constant to order 1/L2inclusive. Note that, even though l(z) is, strictly speaking, a line charge on the axis ( r50), the potential ~27! forr>acorresponds to a conducting cylinder at r5awith the expected surface charge density ~computed from the ra- dial electric field !equal to l(z)/2pa. It can be shown that the difference is of order a2/c2L215O(e2L/L), with, how- ever, zdependence singular as (1 2z2)2nnear the ends. A referee has pointed out that an alternative expression for l(z), yielding the same result as ~26!to order 1/ L2,i s l~z!5l0 F111 Lln~12z2!GH111 2L2FSlnS11z 12zDD2 2p2 3GJ. ~268! This result combines Andrews’ form of the first-order charge density ~168!with the second-order term 2Icylinder. Empiri- cally, ~268!is a better approximation to the charge density than~26!in that it keeps the potential on the surface r5aan equipotential slightly closer to the ends of the cylinder. The differences disappear rapidly as Lincreases, of course. Our expansion in inverse powers of Lhas advantages in analytic work, in the calculation of the capacitance of the cylinder,for example, see Sec. IVC. B. Practicalities, comparison with numerical calculations of others The range of Lvalues for practical situations can be judged by considering the range of radii for the largest andsmallest diameter copper wires in the American Wire Gauge table:a55.842mm ~AWG No. 0000 !toa53.993 310 22mm~AWG No. 40 !.F o raw i r e1mi n length, the range is L’10!20. Finer or longer wires can be imagined, butL<25 is a likely upper limit for an isolated wire @if indeed anyone was interested in verifying ~26!#. Some numerical calculations exist for the charge density on a cylinder.5,9,10Sakar and Rao9give a few values for the density for L513.82. Their numbers are compared with ~26! in the top half of Fig. 3. Their averaging interval Dz52/9 is sufficiently great that for their largest interval (7/9 ,z,1) we have used ~26!to find a weighted mean position for their point. The agreement is satisfactory. Waterman andPedersen 10are concerned with scattering of electromagnetic waves by conducting cylinders, but in an appendix presentan empirical parametrization of their numerical results forthe charge density for a constant potential along the cylinder. My version of their fit to a range of L’15!185 is l( z) 5l0(12z2)2g, where g51/(L2b) and b’3.025. For L 515, this empirical fit is compared with ~26!in the bottom half of Fig. 3. The agreements for other values of Lare comparably good. Griffiths and Li1give a numerical fit @their Eq. ~3.7!#to the smoothed charge density of their discrete distribution of 200point charges on a line. While not strictly comparable to ~26!, the fit might naively be expected to correspond to c/a ’100–200 or L’10.6–12. In fact, good agreement of ~26! with their formula occurs for L’1761. 793 793 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson Despite the peaking at the ends displayed in Fig. 3, the magnitude of the charge density variation is such that thedeparture from uniformity is rather small. A measure of thenonuniformity is the percentages of charge on the inner half of the cylinder ( uzu,1/2) and the outer half (1/2 ,uzu,1). ForL56(c/a’10), the percentages are 43% ~inner!and 57% ~outer!. For L520(c/a’1.13104), they are 48% ~in- ner!and 52% ~outer!. C. Capacitance of conducting right circular cylinder The capacitance of an isolated long right circular conduct- ing cylinder of radius aand length 2 cis found from ~26!and ~27!. From ~27!we see that the potential of the cylinder is V5l0L/4pe0. The total charge Qis the twice the integral of~26!over the interval, 0 ,z,c. The capacitance is there- fore C54pe02c LE 01l~z! l0dz. ~28! The result of the integration is C54pe02c LH112 L~12ln2! 14 L2F11~12ln2!22p2 12G1O~1/L3!J ~29! or, numerically,C54pe02c LH110.6137056 L11.086766 L2 1O~1/L3!J. The result ~29!is not new. An expression equivalent to it was published by Vainshtein6in 1962. He chooses to define V5L22(12ln2) to suppress formally the term in 1/ Lin ~29!. His form, equivalent to ~29!,i s C54pe02c VH1142p2/3 V21O~1/V3!J. ~30! Obviously, Vainshtein derived the charge density, too, but he does not state it explicitly. The problem of the capacitance of a right circular cylinder has been treated in the literature by numerical methods.11,12 Smythe11used an approximation technique ~described in Ref. 8, pp. 209–211 !to find the capacitance for c/a51,2,4, 8. He gives a convenient interpolation formula, C/4pe0a 52/p10.5539(c/a)0.7587.~The coefficients are from my fit to his numerical values and differ slightly from his. !How does our ~Vainshtein’s !formula for long cylinders compare with Smythe’s? For c/a54(L’4.16) the values are C/4pe0a52.328 @Eq.~29!#, 2.222 ~Smythe !. Forc/a58 (L’5.545), they are C/4pe0a53.307 @Eq.~29!#, 3.320 ~Smythe !. The fractional errors of ~29!compared with Smythe’s values are 4.8% and 0.4% for c/a54 and 8, re- spectively. Vainshtein’s version ~30!is marginally poorer. It appears that ~29!accurately represents the capacitance of a long right circular cylinder for c/a>10. Smythe’s interpola- tion formula suffices for 0 ,c/a,10. V. SECOND-ORDER LINEAR CHARGE DENSITIES FOR OTHER SHAPES The second-order linear charge density for a right circular cylinder was expressible in terms of known functions. In general, a random choice of the radial profile f(z)i n~19!to compute Iwill not result in known functions. However, as remarked earlier, the integrand in ~19!is quite smooth in its behavior at x!z~with opposite limits from above and below because of the denominator !. It is only necessary to break the integral up into two ranges, ( 21,x,z) and ( z,x,1), for numerical integration. Numerical computations of Iwere done for f(z)5(1 2uzu)2, andf(z)512uzun, withnan even integer. The first choice is a ‘‘double cone’’ conductor, shown at the top of Fig. 4 ~a!, along with profiles for the second choice of f(z) withn52, 4, 6, 8, and 10. Note that in the limit n..2 the shape of the ‘‘wire’’ approximates a right circular cylinder. ForL515, the curves of the second-order charge densities for the double cone and n52,4,6,8 are shown, along with that of the right circular cylinder in Fig. 4 ~b!. The trend from n52~spheroid !ton58 shows the progression toward the right circular cylinder. In contrast, the double cone conductor has a linear charge density that seems to tend toward zero atthe ends. The known behavior of the surface charge densitynear the tip of a cone ~Ref. 13, p. 106 !corresponds here to thelinearcharge density l( z)}(12z)1/L. The behavior shown in Fig. 4 ~b!agrees qualitatively with this dependence, but our results cannot be trusted too close to the ends of the conductor. At z50 the double cone has a discontinuity in slope. The behavior at such places can be deduced from two- Fig. 3. Comparison of Eq. ~26!, normalized to unity at z50, with numerical calculations. ~a!Sakar and Rao ~Ref. 9 !,L513.82 ~solid triangles; solid dot is mean position weighted according to the shape of ~26!!.~b!Waterman and Pedersen ~Ref. 10 !,L515~dashed curve is empirical fit to their nu- merical results; see text !. Note the suppressed zero for the ordinates. 794 794 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson dimensional electrostatics ~Ref. 13, p. 78 !. The linear charge density should be singular as uzu22a/pcforuzu,a. For L 515, the exponent has absolute value 7 31024; the singu- larity is too small in magnitude and extent to be discernible in our numerical integration. In Fig. 5 we display the linearcharge densities for the right circular cylinder and for the double cone for L515, 30, and 60 to illustrate the trend toward uniformity with increasing L. The lesson to be learned from Figs. 4 and 5 is that the detailed behavior of thecharge density is shape dependent, even for very thin wires. Only in the limit 1/ L50, unattainable in practice, where it is approached very slowly, do the differences among conduc- tors absolutely disappear. Even then, the shape-dependent z component of electric field ~24!is present! Each cat does have a grin that betrays its identity! Perturbations around a smooth shape can also be treated, but this would take us too far away from our purpose.VI. THIN RIGHT CIRCULAR CONDUCTING CYLINDER „2… An alternative approach to the problem of the conducting cylinder is an approximation scheme similar to the variousnumerical methods employed by others. We wish to deter-mine the linear surface charge density on the actual cylindersuch that the surface is an equipotential. Since the physical situation is symmetric with respect to the plane z50, the charge density l(z) must be even in z. We expand it in a power series of N11 terms in z 2, withN11 initially un- known coefficients. Since the overall scale of the density is arbitrary, there are really only Ncoefficients to be deter- mined. We write l~z!5( k50N Akz2k. ~31! The surface charge density is s(z)5l(z)/2pa. The poten- tial is azimuthally symmetric. We therefore need only the azimuthal average of the Green function 1/ ux2x8u. With both the observation point x5(a,0 ,z5cz) and the source pointx85(acos2Q,asin2u,z85cz8) on the surface, we find K1 ux2x8uL52 pcE 0p/2 du A~2a/c!2sin2u1~z2z8!2.~32! Fig. 4. ~a!Shapes of conductors defined by r(z)/a512uz/cu@double cone # andr(z)2/a2512uz/cun. The transverse ~radial !coordinate has been scaled by a factor of c/6ato make the different shapes visible. ~b!Linear charge densities for the shapes of part ~a!for the double cone and n52,4,6,8 for L515, along with that of a right circular cylinder. Note the suppressed zero for the ordinate. Fig. 5. Behavior of the linear charge density versus z/cwith increasing L ~L515,30,60 !for two different shapes. ~a!Right circular cylinder. ~b! Double cone. 795 795 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson In terms of the linear charge density ~31!and the averaged Green function ~32!, the potential becomes F~a,z!51 4p«02 pE 0p/2 duE 01 dz8l~z8! 3F1 Aa21~z2z8!211 Aa21~z1z8!2G.~33! Here we have introduced the quantity a52(a/c)sinuand exploited the symmetry of l(z8) to restrict the z8range to ~0, 1!. When Fis expanded in a Taylor series in zonly even powers occur: F~a,z!5F~0!11 2!z2]2F ]z2~0!11 4!z4]4F ]z4~0!1fl. Here and below we suppress the explicit dependence on a. The longitudinal electric field ~timesc!is evidently cEz52]F ]z52Fz]2F ]z2~0!11 3!z3]4F ]z4~0!1flG. ~34! If we now substitute the power series ~31!forl(z8) into ~33! and require that the first Nterms in the expansion ~34!vanish ~in order to approximate an equipotential on the surface !,w e obtainNcoupled linear equations for the Nunknowns, yk 5Ak/A0. Implicit in the discussion so far is the idea that the ends of the cylinder are unimportant. We must address that issuebefore proceeding further. If the cylinder is capped by hemi-spheres of radius a, or a similarly smooth joining, there will be no singularity in the surface charge density at z56c. Even with flat plates or a hollow tube, the charge density, though singular ~Ref. 13, p. 78 !at r5a,z56c, is inte- grable. The charges within a distance of order aof each end aredQ5O@2pa2s(0)#5O@al(0)#.14The axial electric field in the central region from the ends is thus estimated to be cuEz~ends!u54dQz 4p«0c~12z2!25OSa cl~0! 4p«04z ~12z2!2D. ~35! The end contribution is O(a/c) compared to ~1!or~21!, the magnitudes of axial electric field involved in establishing equilibrium on the long cylinder. For L..1, we clearly havea/c52e2L/2,,1. For example, L510 corresponds to a/c’0.013; L520, toa/c’1024. We may safely neglect end effects for L..1 provided the observation point is not too close to either end. We saw the same sort of restriction in Sec. III. Now we implement our approximation scheme. We de- mand ]2jF ]z2j~0!50 forj51,2,3,...,N. ~36! In~33!we need ] ]zF1 Aa21~z6z8!2G56] ]z8F1 Aa21~z6z8!2G, so that]2j ]z2jF1 Aa21~z6z8!2G z505]2j ]z82jF1 Aa21z82G. ~37! TheNconstraints ~36!are (j51,2,3,...,N) 054 pE 0p/2 du( k50N AkE 01 dzz2k]2j ]z2jF1 Aa21z2G. ~38! We define the coefficients in the set of algebraic equations as bjk54 p~2j!!E 0p/2 duE 01 dzz2k]2j ]z2jF1 Aa21z2G. ~39! Withyk5Ak/A0, theNequations of ~38!can be written ( k51N bjkyk52bj0,j51,2,3,...,N. ~40! The coefficients bjkare evaluated in the Appendix. The re- sults for small a/care bjk51 k2jforjÞk, ~41a! bjj5L21( p512j1 p. ~41b! First we look at the large Llimit. To leading order in L, the coupled equations ~40!reduce to the uncoupled set, Lyj51 j,j51,2,3,...,N. We may consider the limit of N!‘and find the charge density, l~z!5A0F111 L( j51‘z2j jG5A0F121 Lln~12z2!G,~42! in agreement with the first-order correction in ~26!found above. To go beyond the asymptotic limit, we may solve the coupled equations by matrix methods. For small Nthe alge- bra is manageable by hand. For example, the N52 results for the coefficients are y152~3L214! ~6L225!~L23!16, ~43! y253~L21! ~6L225!~L23!16. In Fig. 6 this N52 charge density, correct to order z4inclu- sive, is compared with ~26!, normalized to unity at z50, for L520. We see that the agreement is good for uzu,0.7, but the quartic fails to be singular enough for larger uzu. Also shown are the charge densities for N55 andN515, found with the Pascal linear equation tools ‘‘ludcmp’’ and ‘‘lubksb’’ ~Ref. 15, pp. 683, 684 !on a Macintosh. Because the results for N515 and ~26!are virtually indistinguishable, we plot points for the analytic form to show the agreement. As expected intuitively, the higher the degree Nof the poly- nomial, the better the representation at uzuvalues near unity. 796 796 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson VII. SUMMARY AND CONCLUSIONS We have revisited the problem of the charge density on a ‘‘wire,’’ defined in Sec. II as a very elongated prolate con-ducting spheroid and more generally in Sec. III, where theissue of the ‘‘unbalanced’’ force is resolved. For a conductor of length 2 cwith variable radius r(z) of order a, we give an iteration scheme to generate the linear charge density as a power series in 1/ L, where L[ln(4c 2/a2). The linear charge density to second order in 1/ Linclusive is shown to be l~z!5l0H121 LlnS12z2 f~z!D11 L2FFlnS12z2 f~z!DG2 2IGJ. ~20! whereIis given by ~19!and@r(z)/a#25f(z).@See~8!.#For the special case of the right circular cylinder ( f(z)51) the integralIcan be evaluated analytically. The linear charge density is then l~z!5l0H121 Lln~12z2!11 L2F@ln~12z2!#2 11 2FlnS11z 12zDG2 2p2 6G1O~1/L3!J. ~26! We show that the first-order correction in l(z), though very small for large L, produces an electric field tangent to the surfaceof the conductor that just cancels the tangential com- ponent of the zeroth-order ‘‘unbalanced’’ field ~1!to give equilibrium to the charge on the surface. Sections IV and VI treat the practical situation of a con- ducting right circular cylinder. We compare the result ~26! with numerical results in the published literature in Fig. 3.Calculation of the total charge leads to an expression for thecapacitance of an isolated long conducting right circular cyl-inder. A numerical comparison with the results of Smythe 11 for modest L(c/a54, 8!shows that our result ~or equiva- lently, Vainshtein’s6!forCis an excellent representation ~better than 1% !forc/a>10 or L>6. In Sec. V we present results obtained by numerical inte- gration for the second-order charge density on long conduc-tors of different shapes. Figure 4 illustrates how the charge density along the conductor varies with shape for finite L, while Fig. 5 shows the approach to uniformity with increas-ingLfor two different shapes. Even in the limit of infini- tesimala/c, when the charge density is essentially uniform inzand the conductor is virtually a line of charge, the shape leaves its signature in the longitudinal electric field ~24!. In Sec. VI a different approach to the right circular cylin- der, involving a polynomial approximation in zto the charge density, is presented. With Nindependent coefficients in the expansion of l( z) to order z2N, we demand that the first N terms in a Taylor series in zof the tangential electric field at r5avanish. The vanishing of the corresponding derivatives of the potential leads to a set of Ninhomogeneous coupled linear algebraic equations for the coefficients in the chargedensity. To leading order in L, the equations become trivial; the limitN!‘can be taken; the series can be summed; the first-order analytic result in ~26!is obtained. For arbitrary L, matrix inversion methods can be used, by hand for small N or by computer for larger N. The results of the polynomial approximation approach are compared with the analytic form~26!in Fig. 6. It is worth stressing two striking things that emerge here.~1!The specific behavior of the linear charge density and the fields on the wire depend on its ‘‘primordial’’ shape be- fore the limit a/c,,1 is taken. The prolate spheroid has a uniform linear charge density l(z) for all aspect ratios and a longitudinal electric field that in the limit approaches ~1!. For the general azimuthally symmetric conductor, the linearcharge density can be expressed as a series in inverse powersofL, with the tangential electric field vanishing and the asymptotic axial electric field depending on the shape, asgiven by ~24!~which, of course, includes the prolate spher- oid as a special case !. The right circular cylinder has a non- uniform linear charge density, but no axial electric field be-cause its surface is parallel to the zaxis and the surface charge is in equilibrium there. ~2!The second striking thing is the extreme slowness of the approach to uniformity of the linear charge density on the cylinder as a function of c/a. Forc/a510, 1/ L50.1669; for c/a.10 4,1 /L50.050, only a factor of 3.3 smaller. For 1/L50.01, we need c/ag2.631021! In passing we note that the technique of Sec. III ~and also Sec. IV !can be applied to the long-wavelength limit of the radar cross-section problem,4,5,6,7,10for which one requires the charge density ~for calculation of the electric dipole mo- ment!for a potential varying linearly along the length of the cylinder. The relevant charge density is odd in z, but other- wise the approaches are identical. ACKNOWLEDGMENTS I thank David J. Griffiths for first bringing the problem to my attention in early 1997 and Bruce D. Winstein for re-igniting my interest in Spring 1999. George H. Trilling was amost helpful competitor/collaborator in May 1999. As pre-liminary and incomplete fragments of this paper appeared atthat time, he and David Griffiths engaged me in valuabledialogues. I also thank George for a critical reading of thismanuscript. Finally, let me thank both referees—one for ameticulous reading of the manuscript and the suggestion ofmany small but significant improvements in clarity, the otherfor the observation of other ways of generating numericalapproximations to the equilibrium charge density. Fig. 6. Comparison of second-order analytic charge density ~26!for the right circular cylinder, normalized to unity at z50,~solid points !withN 52, 5, and 15 polynomial approximations ~continuous curves !forL520. Note the suppressed zero for the ordinate. 797 797 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson APPENDIX The coefficients bjkin Sec. VI are defined as bjk54 p~2j!!E 0p/2 duE 01 dzz2k]2j ]z2jF1 Aa21z2G,~A1! where a5(2a/c)sinuandj51,2,...,Nwhilek50,1,2,...,N. In anticipation of choosing a/c,,1, we consider the de- rivative in ~A1!in the limit a!0. Then lim a!0]2j ]z2jF1 Aa21z2G5]2j ]z2jS1 zD5~2j!! z2j11. ~A2! If we insert this limiting result into ~A1!, we see that if k .jthezintegrand is nonsingular. Thus, with the neglect of terms of order O(a2/c2) and for k.j, we have bjk54 pE 0p/2 duE 01 dzz2~k2j!2151 k2j~k.j!.~A3! Next consider k5j. We have bjj54 p~2j!!E 0p/2 duE 01 dzz2j]2j ]z2jF1 Aa21z2G.~A4! We integrate by parts twice, recognizing that the odd deriva- tives of 1/( a21z2)1/2vanish at z50 and the even deriva- tives are finite ~foraÞ0), while the qth derivative at z51i s (21)q(q)! plusO(a2/c2). Neglecting the corrections we then have bjj522 2j22 2j211bj21,j21. ~A5! Repeated use of this recursion relation gives us bjj522( p532j1 p1b11. ~A6! We now must evaluate b11: b1154 pE 0p/2 duE 01 dzz2]2 ]z2F1 Aa21z2G. ~A7! Two integrations by parts yields b1152314 pE 0p/2 duE 01 dz1 Aa21z2. ~A8! The zintegration gives ln @(11A11a2)/a#’ln(2/a) 5ln@c/(asinu)#. Here we have neglected, as usual, correc- tions ofO(a2/c2). The final integral is b1152314 pE 0p/2 du@ln~c/a!2ln~sinu!# 52312@ln~c/a!1ln2#5L22~111 2!. ~A9! We substitute into ~A6!to obtain the general result for the diagonal element, bjj5L22( p512j1 p. ~A10!We now turn to the case of k,j. We put j5k1n, with n51,2,.... Starting with ~A1!instead of ~A4!, we follow the same procedure of two integrations by parts to obtain the general recursion relation, bk1n,k52~4k12n21! ~k1n!~2k12n21! 1k~2k21! ~k1n!~2k12n21!bk1n21,k21. ~A11! Now consider the special case k50. The integral ~A1!re- duces to @neglecting O(a2/c2)# bn054 p~2n!!E 0p/2 duE 01 dz]2n ]z2nF1 Aa21z2G 54 p~2n!!E 0p/2 duF]2n21 ]z2n21F1 Aa21z2GG 01 54 p~2n!!p 2@2~2n21!!#521 n. ~A12! We now use the recursion relation ~A11!to evaluate bn11,1: bn11,1521 ~n11!~2n11!F2n1311 nG521 n5bn,0. ~A13! This result suggests that perhaps bk1n,kis independent of k. To test this idea we add and subtract 1/ ntobk1n21,k21on the right-hand side of ~A11!. The result is Sbk1n,k11 nD5k~2k21! ~k1n!~2k12n21!Sbk1n21,k2111 nD. ~A14! Beginning at k51 with ~A12!, we can iterate ~A14!succes- sively. The left-hand side is equal to zero for any k. We thus have established that for j.k, bjk51 k2j2~k,j!. ~A15! In summary, ~A3!and~A15!combine to give, for kÞj, bjk51 k2j, ~A16a ! while for k5j,~A10!is bjj5L22( p512j1 p. ~A16b ! These expressions for the coefficients have corrections of orderO(a2/c2), appropriately neglected for L..1. 1D. J. Griffiths and Y. Li, ‘‘Charge density on a conducting needle,’’ Am. J. Phys.64, 706–714 ~1996!. 2R. H. Good, ‘‘Comment on ‘Charge density on a conducting needle,’’’ Am. J. Phys. 65, 155–156 ~1997!. 3Mark Andrews, ‘‘Equilibrium charge density on a conducting needle,’’ Am. J. Phys. 65, 846–850 ~1997!. 4P. L. Kapitsa, V. A. Fock, and L. A. Vainshtein, ‘‘Static boundary prob- lems for a hollow cylinder of finite length,’’ Zh. Tekh. Fiz. 29, 1177–1187 ~1959!@Sov. Phys. Tech. Phys. 4, 1077–1087 ~1960!#. 5L. A. Vainshtein, ‘‘Static boundary problems for a hollow cylinder of 798 798 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson finite length. II. Numerical results,’’ Zh. Tekh. Fiz. 32, 1157–1164 ~1962! @Sov. Phys. Tech. Phys. 7, 855–860 ~1963!#. 6L. A. Vainshtein, ‘‘Static boundary problems for a hollow cylinder of finite length. III. Approximate formulas,’’ Zh. Tekh. Fiz. 32, 1165–1173 ~1962!@Sov. Phys. Tech. Phys. 7, 861–866 ~1963!#. 7P. C. Waterman, ‘‘Matrix methods in potential theory and electromagnetic scattering,’’ J. Appl. Phys. 50, 4550–4566 ~1979!. 8W. R. Smythe, Static and Dynamic Electricity ~McGraw–Hill, New York, 1968!, 3rd ed., Sec. 5.02, pp. 123–124. 9T. K. Sakar and S. M. Rao, ‘‘An iterative method of solving electrostatic problems,’’ IEEE Trans. Antennas Propag. AP-30 ~4!, 611–616 ~1982!. 10P. C. Waterman and J. C. Pedersen, ‘‘Scattering by finite wires,’’ J. Appl. Phys.72, 349–359 ~1992!.11W. R. Smythe, ‘‘Charged right circular cylinder,’’ J. Appl. Phys. 27, 917– 920~1956!;33, 2966–2967 ~1962!. 12P. K. Wang, ‘‘Calculation of electrostatic fields surrounding finite circular cylinder conductors,’’ IEEE Trans. Antennas Propag. 32, 956–962 ~1984!. 13J. D. Jackson, Classical Electrodynamics ~Wiley, New York, 1998 !, 3rd ed. 14A. R. Djordjevic ´, ‘‘Comments on ‘Calculation of electrostatic fields sur- rounding finite circular cylinder conductors,’’’ IEEE Trans. Antennas Propag.33, 683–684 ~1985!. 15W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, The Art of Scientific Computing ~Cambridge U.P., Cambridge, 1986 !. VERSATILE THEORISTS A hasty investigation of the angular distribution of the neutron beam, performed as soon as the accelerator started functioning, indicated a forward distribution, as expected, but with a minimumat deflection zero. This was reported in a colloquium attended by Oppenheimer, and he immedi-ately gave a learned theoretical explanation of the phenomenon. I listened to it and then said thatit was better to check if by chance there was not a lead brick just in front of the target, projectinga shadow. Immediately after the colloquium somebody rushed to check my hypothesis. It wascorrect. Emilio Segre `,A Mind Always in Motion—The Autobiography of Emilio Segre `~University of California Press, Berkeley, 1993!,p .2 2 9 . 799 799 Am. J. Phys., Vol. 68, No. 9, September 2000 J. D. Jackson muonic system in which the muon spends considerable time within the nucleus. The energy value is adjusted upward un-til the right-hand boundary condition is satisfied ~see Fig. 1 !. The large difference between the seed value and the final energy value ~218.916 vs 210.413 MeV for the 1 sstate! demonstrates the significance of the finite size of the nucleus for muonic atoms. The fine structure corrections outlined by Tiburzi and Hol- stein @their Eqs. ~16!–~18!#can be easily added to the MATHCAD worksheet ~omitted here for the sake of brevity !,using traditional numerical algorithms, so that a thorough comparison of theory and experiment can be made. a!Electronic mail: [email protected] 1B. C. Tiburzi and B. R. Holstein, ‘‘Bound states of a uniform spherical charge distribution-revisited!,’’ Am. J. Phys. 68, 640–648 ~2000!. 2J. Zablotney, ‘‘Energy levels of a charged particle in the field of spheri- cally symmetric uniform charge distribution,’’ Am. J. Phys. 43, 168–172 ~1975!. 3F. Rioux, ‘‘Direct numerical integration of the radial equation,’’ Am. J. Phys.59, 474–475 ~1991!. Comment on ‘‘Charge density on a thin straight wire, revisited,’’ by J. D. Jackson †Am. J. Phys. 68 „9…, 789–799 „2000…‡ O. F. de Alcantara Bonfim and David Griffithsa) Reed College, Portland, Oregon 97202 ~Received 5 September 2000; accepted 12 September 2000 ! @DOI: 10.1119/1.1326076 # Jackson’s paper1supports an emerging consensus2that the linear charge density on a conducting wire is uniform, in thezero radius limit. This is easily proved for the special case ofan ellipsoid, but Jackson demonstrates that it holds regard-less of shape. This conclusion is so counterintuitive that wedecided to reexamine the original numerical studies, 3based on discrete charge distributions, that appeared to confirm themore plausible hypothesis that the charge accumulates pref-erentially near the ends. We place Ncharges at equal spacing on the interval 0 ,x<1: q 1atx151/N, q2atx252/N, fl ~1!qnatxn5n/N, flq NatxN51 ~and equal charges at the corresponding points on 21<x ,0!, together with a single charge q0atx050. We then adjust the charges so that the Coulomb force on each of them exceptqN~which is subject to an extra confining force !is zero: ( j51Nqj ~n1j!21q0 n21( j51n21qj ~n2j!22( j5n11Nqj ~j2n!2 50~n51,2,...,N21!, ~2! subject to the constraint q012( n51N qn51 ~3! ~the scaled total charge on the wire !. This does not determine the charge at the center—the force on q0is automaticallyzero, by symmetry. To ensure continuity we choose q0 5q1. What remains is a set of Nlinear equations for the N unknown charges. Griffiths and Li2solved this system numerically for Nup to 100, and persuaded themselves that the linear charge den-sity was approaching a nontrivial limiting form—fairly flat in the center, but with spikes at the ends ( x561). They were seduced by extraordinarily slow convergence as N Fig. 1. Linear charge density on a needle, as a function of position. The calculation was done using 2 N11 point charges equally spaced on the interval from 21t o 11, and requiring that the net force on each charge ~except the end two !vanish. The total charge on the needle is 1. ~a!Solid line:N516384; dashed line: N532.~b!Expanded view of the right end; this time the dashed line is N51024. 515 515 Am. J. Phys. 69~4!, April 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers !‘, as we can see from Fig. 1, which extends the calcula- tion out to N516384: As Nincreases, the charge density approaches 1/2, except at the very ends, which occupy a decreasing portion of the length and contain a diminishingfraction of the total charge. In Fig. 2 we plot the charge density at the center ( l(0) 5Nq 0) as a function of N, to demonstrate the ~painfully slow ! approach to 0.5. Jackson shows that the natural expansion parameter is L21, where L[ln(4c2/a2), with 2cthe length of the wire and aits characteristic ‘‘radius,’’ and he suggests that for the discrete model this translates to L;2l nN.I n Fig. 2 the solid line is a best fit of the form l~0!5P11P2 lnN1P3 ~lnN!2; ~4! for our data ~withNranging from 32 to 16384 !P150.500, P2520.152, and P3520.123. In Fig. 3 we plot the charge density at the ends of the wire (l(1)5NqN), as a function of N. It seems clear that this quantity increases without limit—in fact, our data are well represented by the functional forml~1!5FQ11Q2 lnN1Q3 ~lnN!2GlnN, ~5! withQ150.0719,Q250.912, and Q3520.874 ~solid line !. Nevertheless, these ‘‘rabbit ears’’ in l(x) are of decreasing significance as N!‘, in the sense that they occupy a dimin- ishing portion of the total length and contain a smaller and smaller fraction of the total charge. ACKNOWLEDGMENT We thank Nicholas Wheeler for illuminating discussions of this problem. a!Electronic mail: Griffi[email protected] 1J. D. Jackson, ‘‘Charge density on a thin straight wire, revisited,’’ Am. J. Phys.68, 789–799 ~2000!. 2R. H. Good, ‘‘Comment on ‘Charge density on a conducting needle,’’’ Am. J. Phys. 65, 155–156 ~1997!; Mark Andrews, ‘‘Equilibrium charge density on a conducting needle,’’ ibid.65, 846–850 ~1997!; Nicholas Wheeler, ‘‘Construction and applications of the fractional calculus’’ ~un- published !. 3D. J. Griffiths and Ye Li, ‘‘Charge density on a conducting needle,’’ Am. J. Phys.64, 706–714 ~1996!. PREPARATION? Gibbs began his lectures on thermodynamics with the Carnot cycle, which he always got wrong. After getting thoroughly mixed up he concluded the first lecture with an apology, and inthe second lecture he gave it letter perfect. It was in this way he introduced entropy, rather than inthe formal way in the ‘‘Heterogeneous Substances’’. E. B. Wilson, a student of J. Willard Gibbs, as quoted by Clifford Truesdell in J. Serrin ~editor !,New Perspectives in Thermodynamics ~Springer, New York, 1986 !, p. 107. Fig. 2. Charge density at the center of the needle, as a function of N. Dots represent the numerical results. The solid line is the best fit of the form l(0)5P11P2/lnN1P3/(lnN)2~forNranging from 32 to 16384 !, which occurs for P150.500,P2520.152, and P3520.123. Evidently l~0!ap- proaches the uniform density value of 0.5, as Nincreases. Fig. 3. Charge density at the ends of the needle ( x561), as a function of N. Dots represent the numerical results. The solid line is the best fit of the form l(1)5@Q11Q2/lnN1Q3/(lnN)2#lnN, which occurs for Q1 50.0719,Q250.912, and Q3520.874. Evidently l~61!diverges as N increases. 516 516 Am. J. Phys., Vol. 69, No. 4, April 2001 Notes and Discussions