Wilson toroid

EMG 25(1) #37054 Electromagnetics , 25:1–19, 2005 Copyright © 2005 Taylor & Francis Inc.ISSN: 0272-6343 print/1532-527X onlineDOI: 10.1080/02726340590522102 Electrostatic Excitation of a Conducting Toroid: Exact Solution and Thin-Wire Approximation ROBERT W. SCHARSTEIN Electrical Engineering Department University of AlabamaTuscaloosa, Alabama, USA HOWARD B. WILSON Aerospace Engineering Department University of AlabamaTuscaloosa, Alabama, USA Laplace’s equation is solved via separation of variables in toroidal coordinates for the electrostatic potential external to a conducting torus placed in a uniform electricfield and excited by an arbitrarily located point charge. The accuracy of the staticthin-wire kernel approximation in an integral equation applied to the circular loopis verified using the exact results in the limit as the toroid shrinks to a ring. Anequivalent lineal charge density from the exact solution agrees remarkably well withthe integral equation solution for the conducting ring. Since the singularity in theHelmholtz Green’s function for the electrodynamic problem is the static singularityconsidered herein, the results confirm the applicability of the thin-wire kernel to thescattering and radiation problems of the circular loop. Keywords toroidal coordinates, Legendre functions, thin-wire kernel, Laplace’s equation Introduction An investigation into the accuracy and applicability of the thin-wire kernel approxima- tion (Wu, 1962; King, 1969) in treating the dynamic electromagnetic wave interactions with a conducting circular loop leads directly to the static analysis of the present paper. Although the exact kernel can be integrated with sinusoidal ring currents to yield sepa-rated series expansions (Werner, 1996) at arbitrary field points, the simple and thereforephysically insightful approximation provided by the thin-wire kernel is still an attractive tool in the numerical and asymptotic analysis of the loop antenna or scatterer. Both thelow-frequency and even the high-frequency behavior of the Helmholtz Green’s functionexp(ikR)/R in a neighborhood of the singular point R=| /vectorr−/vectorr /prime|→ 0 is predominantly that of the static kernel 1 /R. Received 18 May 2004; accepted 8 June 2004. Address correspondence to Robert W. Scharstein, Electrical Engineering Department, Univer- sity of Alabama, Tuscaloosa, AL 35487-0286, USA. E-mail: [email protected] 1 2 R. W. Scharstein and H. B. Wilson (a) (b) Figure 1. Toroid geometry. (a) Axonometric view; (b) cross section of toroidal surface in φ=0,π plane. The next section presents the exact solution in toroidal coordinates for the conducting toroid of Figure 1 excited by a uniform axial electric field, a uniform transverse electricfield, and an arbitrarily located point source. The results of the uniform excitations arerecovered from the general point-source result in the obvious limiting cases of properlypositioned sources at infinity. Induced surface charge distributions (C/m 2) are collapsed to equivalent lineal or ring charge distributions (C/m) in the primary case of interest, the thintoroid. This is used to verify the results of the thin-wire approximation in the integralequation analysis of the third section (“Integral Equation for Ring Charge Density”).Detailed quantitative data is presented in the final section for the case of transverseelectric field excitation, which is of primary relevance to the motivating wave problem. Electrostatic Toroid 3 The boundary value problems for several axisymmetric excitations of the conducting torus are stated, with partial answers, in a collection of problems (Lebedev, Skalskaya,& Uflyand, 1979). It is not surprising that both the axial and transverse electric fieldexcitations are included in Smythe’s formidable problem set (Smythe, 1968, 1974). Cade(1978a) performs the Kelvin transformation (image theory) for a point source on the axisof a thin toroid and then extends this to accommodate other axisymmetric source distri-butions. Cade comments that the use of toroidal harmonics is a “theory of much elegancebut of questionable value, leading to solutions in terms of slowly converging series offunctions which themselves, individually, are difficult to compute.” The present paperresolves these difficulties. Cade (1978b) also studied general R 3toroids of revolution via a perturbative analysis that starts with the cross-sectional geometry in R2. Exact Solution of Laplace’s Equation in Toroidal Coordinates Toroidal coordinates (ξ,η,φ) are related to cylindrical polar coordinates (ρ,φ,z) via ρ+iz=ccoth/bracketleftbigg1 2(ξ−iη)/bracketrightbigg (1) or ρ=csinhξ coshξ−cosηand z=csinη coshξ−cosη. (2) The observations (ρ−ccothξ)2+z2=(ccschξ)2,ρ2+(z−ccotη)2=(ccscη)2(3) enable the orthogonal circles of constant ξand constant ηto be drawn in the (ρ,z) plane as in Figure 2. Note that the origin of the (ρ,z) plane is mapped to the point (ξ,η) =(0,π) and the point at infinity in the (ρ,z) plane is mapped to the point (ξ,η) =(0,0).A sξ→∞ ,ηis ignorable and (ρ,z) →(c,0). The toroidal surface of Figure 1 is the surface of constant ξ=ξ0,−π≤η≤π, and−π≤φ≤π. With the original dimensions of the toroid a=ccothξ0 and b=ccschξ0, (4) the introduced toroidal parameters are c=/radicalbig a2−b2 and cosh ξ0=a/b. (5) Metric coefficients in the toroidal coordinates are hξ=hη=c coshξ−cosη,h φ=csinhξ coshξ−cosη. (6) Details of the separated differential equations are outlined in several texts (Lebedev, 1972; Moon & Spencer, 1961; Morse & Feshbach, 1953). The reciprocal distance between 4 R. W. Scharstein and H. B. Wilson Figure 2. Circles of constant ξandηin the(ρ,z) plane. the points /vectorrand/vectorr/primein toroidal coordinates is easily written using an addition theorem (Hobson, 1955, §223) and with a recursion relation (Abramowitz & Stegun, 1972, 8.2.5) 1 |/vectorr−/vectorr/prime|=(coshξ−cosη)1/2(coshξ/prime−cosη/prime)1/2 c[2 coshµ−2 cos(η−η/prime)]1/2 =1 cπ(coshξ−cosη)1/2(coshξ/prime−cosη/prime)1/2 ·∞/summationdisplay n=0∞/summationdisplay m=0βepsilonInβepsilonIm(−1)mβGammaI(n−m+1/2) βGammaI(n+m+1/2) ·Qm n−1/2(coshξ>)Pm n−1/2(coshξ<)cosm(φ−φ/prime)cosn(η−η/prime),(7) Electrostatic Toroid 5 where coshµ/definescoshξcoshξ/prime−sinhξsinhξ/primecos(φ−φ/prime)≥1 (8) and with the Neumann number βepsilonIn=2−δ0n. Axial Exciting Electric Field In this case the boundary value problem is axisymmetric and the total harmonic potential is written as the incident plus scattered (or induced) potentials ψ(ρ,z) =−E0z+ψs(ρ,z). (9) The disturbance caused by the introduction of the toroidal obstacle into the uniform field is of no consequence at infinity; hence ψs(∞)=0. The total potential must vanish on the conducting boundary ξ=ξ0. The obvious odd symmetry in ztranslates to odd symmetry in η. An appropriate expansion for the scattered potential in the external region 0≤ξ≤ξ0,−π≤η≤πis ψs(ξ,η) =E0c/radicalbig 2 coshξ−2 cosη∞/summationdisplay n=1anPn−1/2(coshξ) Pn−1/2(coshξ0)sinnη (10) in terms of half-degree toroidal or ring functions of the first kind. The Dirichlet boundary condition is now ψs(ξ0,η)=E0z=E0csinη coshξ0−cosη, (11) so that a Fourier sine series is required in the form sinη [2 coshξ0−2 cosη]3/2=∞/summationdisplay n=1bnsinnη. (12) The Fourier coefficients bnare evaluated by starting with an integral representation (Lebedev, 1972, 7.10.10) and performing an integration by parts: Qn−1/2(coshξ)=/integraldisplayπ 0cosnx [2 coshξ−2 cosx]1/2dx =1 n/integraldisplayπ 0sinnxsinx [2 coshξ−2 cosx]3/2dx,(13) such that the total potential is written entirely in toroidal coordinates as ψ(ξ,η) =4E0c π/radicalbig 2 coshξ−2 cosη ·∞/summationdisplay n=1nQn−1/2(coshξ0)Pn−1/2(coshξ)−Qn−1/2(coshξ)Pn−1/2(coshξ0) Pn−1/2(coshξ0) ·sinnη. (14) 6 R. W. Scharstein and H. B. Wilson This result can be transformed to Smythe’s (1968, 1974) answers using Whipple’s identity (Cohl et al., 2000). Charge density on the surface ξ=ξ0is σ(η)=−ε0∂ ∂nψ(ξ 0,η)=ε0 hξ∂ψ(ξ 0,η) ∂ξ=ε0 c[coshξ0−cosη]∂ψ(ξ 0,η) ∂ξ. (15) Differentiation of the total potential gives, with the Wronskian (Lebedev, 1972, 7.7.2), σ(η)=2ε0E0 π[2 coshξ0−2 cosη]3/2 sinhξ0∞/summationdisplay n=1n Pn−1/2(coshξ0)sinnη. (16) Asb/a→0,ξ0→∞ and the n=1 term dominates, so that the surface charge density on a thin toroid is σ(η)−−−−−→ b/a→02ε0E0sinη. (17) This is the classical solution for the charge density on an infinitely long circular cylinder in the corresponding R2potential problem. Equipotentials from (14) are drawn in Figure 3 for a reasonably fat (b/a=0.8) toroid. Normalized surface charge density of Figure 4 shows that plus and minus chargestend to accumulate on opposing inner surfaces of the fat toroid ( b/a=0.8), while the thin toroid ( b/a=0.1) displays the limiting behavior of (17). Figure 3. Equipotentials in the axial plane for axial excitation ˆzE0. Case: a=1,b=0.8. Electrostatic Toroid 7 Figure 4. Normalized surface charge density for axial excitation. Solid curve: b/a=0.8, dashed curve:b/a=0.1. Transverse Exciting Electric Field With the source electric field polarized in the xdirection, i.e., /vectorE=ˆxE0, the incident po- tential is −E0x=−E0ρcosφand an appropriate expansion for the scattered potential is ψs(ξ,η,φ) =E0ccosφ/radicalbig 2 coshξ−2 cosη∞/summationdisplay n=0cnP1 n−1/2(coshξ) P1 n−1/2(coshξ0)cosnη. (18) Vanishing of the total potential on the toroid surface ξ=ξ0now requires a Fourier series in the form [2 coshξ0−2 cosη]−3/2=∞/summationdisplay n=0βepsilonIndncosnη, (19) that is, dn=1 π/integraldisplayπ 0cosnη (2 coshξ0−2 cosη)3/2dη=−1 πQ/prime n−1/2(coshξ0) (20) by (13) above. The total potential is ψ(ξ,η,φ) =−2E0c πcosφ/radicalbig 2 coshξ−2 cosη∞/summationdisplay n=0βepsilonIncosnη (21) ·sinhξ0Q/prime n−1/2(coshξ0)P1 n−1/2(coshξ)−sinhξQ/prime n−1/2(coshξ)P1 n−1/2(coshξ0) P1 n−1/2(coshξ0), 8 R. W. Scharstein and H. B. Wilson in agreement with published results (Smythe, 1968, 1974) upon enlisting the Whipple and several recursion formulae. The Wronskian relationship Q/prime ν(ζ)P1/prime ν(ζ)−Q/prime/prime ν(ζ)P1 ν(ζ)=W/bracketleftBig Q1 ν(ζ),P1 ν(ζ)/bracketrightBig +ζ (ζ2−1)3/2Q1 ν(ζ)P1 ν(ζ) =−ν(ν+1) (ζ2−1)3/2+ζ (ζ2−1)3/2Q1 ν(ζ)P1 ν(ζ)(22) permits the reduction ∂ ∂ξ/bracketleftBig sinhξ0Q/prime n−1/2(coshξ0)P1 n−1/2(coshξ)−sinhξQ/prime n−1/2(coshξ)P1 n−1/2(coshξ0)/bracketrightBig ξ=ξ0 =−(n−1/2)(n+1/2) sinhξ0. (23) The induced surface charge density, as in (15), is hence σ(η,φ) =ε0E0 πcosφ[2 coshξ0−2 cosη]3/2 sinhξ0∞/summationdisplay n=0βepsilonIn(n−1/2)(n+1/2) P1 n−1/2(coshξ0)cosnη. (24) Only the n=0,1 terms are significant for a very thin toroid (cosh ξ0=a/b→∞ ), resulting in σ(η,φ) −−−−−→ b/a→0ε0E0cosφ/braceleftbiggcoshξ0 ξ0−2(1−ln 2)+3 cosη/bracketleftbigg 1−1 2ξ0−4(1−ln 2)/bracketrightbigg/bracerightbigg . (25) An equivalent ring (lineal) charge density (C/m) is σβlscript(φ)=2πb lim b/a→0σ(η,φ) =2πε0E0a ξ0−2(1−ln 2)cosφ. (26) Equipotentials (21) in the meridian plane are drawn in Figure 5 for b/a=0.01. The extremely thin toroid allows a noticeable potential variation in the “interior” region ρ<a . The curves in Figure 6 of normalized surface charge density (24) at φ=0 illustrate the importance of the constant (independent of η) term as b/a changes from the selected values 0.8 to 0.1. Point Charge Excitation No loss in generality is suffered by the Green’s function in the presence of the axi-symmetric toroid by placing the point charge qat the azimuthal coordinate φ=0. The spherical coordinates of the source point (r s,θs,0)transform to the toroidal coor- dinates cothξs=r2 s+c2 2crssinθs, cotηs=r2 s−c2 2crscosθs. (27) Electrostatic Toroid 9 Figure 5. Equipotentials in the meridian ( z=0) plane for transverse excitation ˆxE0. Case: a=1, b=0.01. Figure 6. Normalized surface charge density for transverse excitation. Solid curve: b/a=0.8, dashed curve: b/a=0.1. 10 R. W. Scharstein and H. B. Wilson The free-field or source potential due to a point charge qat(ξs,ηs,0), in the absence of the toroid, is from (7), ψi(ξ,η,φ) =q 4π2ε0c(coshξ−cosη)1/2(coshξs−cosηs)1/2∞/summationdisplay n=0∞/summationdisplay m=0βepsilonInβepsilonIm(−1)m ·βGammaI(n−m+1/2) βGammaI(n+m+1/2)Qm n−1/2(coshξ>)Pm n−1/2(coshξ<)cosmφcosn(η−ηs). (28) The total potential, that is, the sum of this exciting field plus an induced or scattered potential, can now be written by inspection in these natural toroidal coordinates. As inall the cases of this paper, the scattered potential must be bounded external to the toruswhere 0 ≤ξ≤ξ 0, and the total potential vanishes on the conducting surface ξ=ξ0. Thus, the total potential is ψ(ξ,η,φ) =q 4π2ε0c(coshξ−cosη)1/2(coshξs−cosηs)1/2 ·∞/summationdisplay n=0∞/summationdisplay m=0βepsilonInβepsilonIm(−1)mβGammaI(n−m+1/2) βGammaI(n+m+1/2)cosmφcosn(η−ηs) Pm n−1/2(coshξ0) ·/bracketleftBig Pm n−1/2(coshξ0)Qm n−1/2(coshξ>)Pm n−1/2(coshξ<) −Qm n−1/2(coshξ0)Pm n−1/2(coshξs)Pm n−1/2(coshξ)/bracketrightBig .(29) It is advantageous to compute the point-source potential directly, as in (7), and separately evaluate the double series contribution for the scattered potential. The surface chargedensity, as in (15), simplifies nicely because the gamma functions in the applicableWronskian cancel those in the series such that σ(η,φ) =−q 4π2c2sinhξ0(coshξs−cosηs)1/2(coshξ0−cosη)3/2 ·∞/summationdisplay n=0∞/summationdisplay m=0βepsilonInβepsilonImPm n−1/2(coshξs) Pm n−1/2(coshξ0)cosmφcosn(η−ηs).(30) Asb/a→0 the appropriate asymptotic forms of the denominator functions are P−1/2(coshξ0)−−−−→ ξ0→∞2 π(ξ0+2l n2)e−ξ0/2, Pm −1/2(coshξ0)−−−−→ ξ0→∞2√πξ0−ln 2 βGammaI(1/2−m)e−ξ0/2(m=1,2,...), Pm n−1/2(coshξ0)−−−−→ ξ0→∞βGammaI(n)√πβGammaI(n −m+1/2)eξ0(n−1/2)/parenleftbiggm=0,1,... n=1,2,.../parenrightbigg .(31) Electrostatic Toroid 11 Obviously, then, only the terms with n=0 matter in the case of an extremely thin toroid, and the approximate lineal charge density, independent of ηas expected, is σβlscript(φ)=2πb lim b/a→0σ(η,φ) =−q 2πa/radicalBigg sinhξ0 sinhξs(coshξs−cosηs)1/2∞/summationdisplay m=0βepsilonImQm−1/2(cothξs) Qm−1/2(cothξ0)cosmφ,(32) where b(coshξ0−cosη)3/2 c2sinhξ0−−−−−→ b/a→01 a, (33) and Whipple’s relation (Cohl et al., 2000) has been used to transform the ratio of first- kindPm −1/2functions of varying order to a ratio of second-kind Qm−1/2functions of varying degree and order zero. In this limiting case, the argument of the denominatorfunction is cothξ 0=/bracketleftBig 1−(b/a)2/bracketrightBig−1/2 ≈1+1 2(b/a)2=1+2α2, (34) where the small parameter α=b/2aarises naturally in the thin-wire approximation of the next section. Using the above toroidal-to-spherical coordinate transformations, thisequivalent lineal charge density becomes, since c→aandξ 0→∞ asb/a→0, σβlscript(φ)=−q 2π√arssinθs∞/summationdisplay m=0βepsilonImQm−1/2(cothξs) Qm−1/2(1+2α2)cosmφ ( −π≤φ≤π). (35) This expression is in exact agreement with the independent solution (62) of the thin-wire integral equation of the subsection on “Point Charge Excitation.” Equipotentials are drawn in the source plane ( y=0o rφ=0,π) in Figure 7 for a representative toroid excited by a point source at (ρs,φs,zs)=(4,0,4). The scaled lineal charge density (32) of a thin toroid with b/a=0.001 is graphed as a function of φin Figure 8 when the exciting point charge lies in the plane of the ring ( θs=π/2). Three cases are presented: the point charge half a radius inside ( rs=0.5), half a radius outside ( rs=1.5), and close to the outside ( rs=1.1) of the ring of unit radius a=1. The uniform exciting fields of this section are recovered as special cases of this more general point-source excitation by considering a pair of equal and opposite point charges(Jackson, 1999). The uniform axial field derives from placing +q/2 a distance r son the positive zaxis, i.e., at the spherical coordinates (rs,0,0)accompanied by −q/2 the same distance rsalong the negative zaxis, i.e., at the spherical coordinates (rs,π,0). Then, as rs→∞ : ξs→0, cotη± s→±rs 2c, sinη± s→±2c rs, cosηs→1−2(c/rs)2, (coshξs−cosηs)1/2→√ 2c rs,Pm n−1/2(coshξs)→δm,0, cosn(η−ηs)−cosn(η+ηs)=2 sinnηsinnηs→4cn rssinnη.(36) 12 R. W. Scharstein and H. B. Wilson Figure 7. Equipotentials in the y=0 plane for point-source excitation. Case: a=1,b=0.8; source coordinates (xs,ys,zs)=(4,0,4). Figure 8. Lineal charge density for point-source excitation. θs=π/2,φs=0,a=1,b=0.001, rs=0.5,1.1,1.5. Electrostatic Toroid 13 This superposition of the two appropriately adjusted potentials is ψs→−qc π2ε0r2s(2 coshξ−2 cosη)1/2∞/summationdisplay n=1nQn−1/2(coshξ0) Pn−1/2(coshξ0)Pn−1/2(coshξ)sinnη, (37) exactly equal to the scattered component of (14) in the previous subsection, where E0=−q 4πε0r2s(38) is the magnitude of the z-directed uniform electric field. Thex-directed uniform exciting field is likewise obtained by placing +q/2a t (rs,π/2,0)(along the +xaxis) and −q/2 at the diametrically opposite point (rs,π/2,π). The required limiting geometrical forms in this case are, as rs→∞ and with ηs=0: cothξs→rs 2c, coshξs→1+2(c/rs)2, sinhξs→2c rs, coshξs−cosηs→2(c/rs)2.(39) The combination cos mφ−cosm(φ−π)furnishes the sum of contributions ψs→q 2π2ε0rs(2 coshξ−2 cosη)1/2∞/summationdisplay n=0βepsilonIn∞/summationdisplay m=1,3,...βGammaI(n−m+1/2) βGammaI(n+m+1/2)cosmφcosnη ·Qm n−1/2(coshξ0)Pm n−1/2(coshξs)Pm n−1/2(coshξ) Pm n−1/2(coshξ0). (40) The leading term in an adaptation of an asymptotic form (Olver, 1997) Pm n−1/2(coshξs)−−−−→rs→∞βGammaI(n+m+1/2) βGammaI(m+1)βGammaI(n−m+1/2)/parenleftbiggc rs/parenrightbiggm (41) shows that only the m=1 term is of any consequence: ψs→qc 2π2ε0r2scosφ(2 coshξ−2 cosη)1/2 ·∞/summationdisplay n=0βepsilonInQ1 n−1/2(coshξ0)P1 n−1/2(coshξ) P1 n−1/2(coshξ0)cosnη.(42) This result is exactly equal to the scattered term of (21) with Q1 n−1/2(coshξ0)=sinhξ0Q/prime n−1/2(coshξ0) (43) and with the x-directed incident electric field E0also given by (38). 14 R. W. Scharstein and H. B. Wilson Figure 9. Toroid geometry in bastardized cylindrical polar coordinates. Integral Equation for Ring Charge Density Treatment of the conducting ring of finite thickness is facilitated by considering the exaggerated geometry of Figure 9, where the position of an arbitrary point on the ringsurface is /vectorr=(a+bcosβ)ˆρ+bsinβˆz (44) in terms of an angle βthat describes the location around the circumference of the reasonably thin ring. Note that βis not, in general, equal to ηof the toroidal coordinates, but it is in the limit of a vanishingly thin ring. With the field point /vectorron the surface of the thin toroid (44) and with the source point right on the axis /vectorr /prime=aˆρ/primeaccording to this equivalent line charge model, the subject distance function is |/vectorr−/vectorr/prime|2=4a2sin2/bracketleftbigg1 2(φ−φ/prime)/bracketrightbigg +4absin2/bracketleftbigg1 2(φ−φ/prime)/bracketrightbigg cosβ+b2. (45) The average of this over βeliminates the cos βterm, which is evidently one source of the static form of the thin-wire kernel approximation (Wu, 1962) |/vectorr−/vectorr/prime|≈2a/braceleftbigg sin2/bracketleftbigg1 2(φ−φ/prime)/bracketrightbigg +(b/2a)2/bracerightbigg1/2 (46) that was also adopted for subsequent loop antenna and scatterer solutions (King, 1969). Vanishing of the total (incident plus scattered) potential on the surface of the thin ring gives the integral equation 1 4πε0/integraldisplay Cσβlscript(/vectorr/prime) |/vectorr−/vectorr/prime|dβlscript/prime=−ψi(/vectorr) ( /vectorr∈C) (47) for the induced lineal charge density, where dβlscript/prime=adφ/prime. Electrostatic Toroid 15 Transverse Uniform Electric Field Excitation In this case all the fields vary like cos φ, so the single unknown is the magnitude σβlscript0of the induced lineal charge density σβlscript(/vectorr/prime)=σβlscript0cosφ/prime. (48) The integral equation is therefore /integraldisplayπ −πcosφ/prime /braceleftbigg sin2/bracketleftbigg1 2(φ/prime−φ)/bracketrightbigg +(b/2a)2/bracerightbigg1/2dφ/prime=8πε0E0a σβlscript0cosφ( −π≤φ≤π). (49) A change of integration variable θ=φ/prime−φ, withα=b/2a, produces I(α)=/integraldisplayπ 0cosθ /bracketleftbigg sin21 2θ+α2/bracketrightbigg1/2dθ=4πε0E0a σβlscript0. (50) This integral is our familiar toroidal function of degree 1/2, which can alternately be expressed in terms of the complete elliptic integral of the first and second kinds toexploit tabulated asymptotic behavior I(α)=2/integraldisplay π 0cosθ /bracketleftbig 2(1+2α2)−2 cosθ/bracketrightbig1/2dθ =2Q1/2(1+2α2) =4α2+2√ α2+1K/bracketleftBig (α2+1)−1/2/bracketrightBig −4/radicalbig α2+1E/bracketleftBig (α2+1)−1/2/bracketrightBig =2/bracketleftbig ln(4/α)−2/bracketrightbig +1 2/bracketleftbig 3l n(4/α)−1/bracketrightbig α2+1 32/bracketleftbig 15 ln(α/4)+23/2/bracketrightbig α4+O(α6). (51) The leading term is most useful in the present case of a thin toroid, whereupon the desired coefficient of the lineal charge density is σβlscript0≈2πε0E0a ln(4/α)−2≈2πε0E0a ξ0−2(1−ln 2)(52) in agreement with (35), since for b/a→0, ξ0=cosh−1a b≈ln2a b=− lnα. (53) 16 R. W. Scharstein and H. B. Wilson Point Charge Excitation The forcing term in the integral equation (47) is now the incident potential ψi(/vectorr)=q 4πε0|/vectorr−/vectorrs|, (54) where the source charge qis located at /vectorrs=(rs,θs,0)=(ρs,0,zs)=(ξs,ηs,0) (55) in spherical, cylindrical polar, and toroidal coordinates, respectively. A convenient repre- sentation (Morse & Feshbach, 1953) for the reciprocal distance of the exciting potential is 1 |/vectorr−/vectorrs|=1 π√ρρs∞/summationdisplay m=0βepsilonImQm−1/2(coshχ)cosmφ, (56) where coshχ/definesρ2+ρ2 s+(z−zs)2 2ρρs. (57) The integral equation, with the thin-wire kernel approximation, is therefore /integraldisplayπ −πσβlscript(φ/prime) /braceleftbigg sin2/bracketleftbigg1 2(φ−φ/prime)/bracketrightbigg +(b/2a)2/bracerightbigg1/2dφ/prime =−2q π√arssinθs∞/summationdisplay m=0βepsilonImQm−1/2/parenleftbigga2+r2 s 2arssinθs/parenrightbigg cosmφ ( −π≤φ≤π).(58) Expanding the unknown charge density in a Fourier series σβlscript(φ/prime)=−2q π√arssinθs∞/summationdisplay n=0βepsilonInAnQn−1/2/parenleftbigga2+r2 s 2arssinθs/parenrightbigg cosnφ/prime(−π≤φ/prime≤π) (59) requires evaluation of I(n,α) =/integraldisplayπ −πcosnφ/prime /radicalBigg sin2/bracketleftbigg1 2(φ/prime−φ)/bracketrightbigg +α2dφ/prime, (60) whereα=b/2ais always the important parameter. Letting θ=φ/prime−φand noting the periodicity and symmetry in the integrand shows I(n,α) =2 cosnφ/integraldisplayπ 0cosnθ/radicalbigg sin21 2θ+α2dθ =4 cosnφ/integraldisplayπ 0cosnθ /bracketleftbig 2(1+2α2)−2 cosθ/bracketrightbig1/2dθ =4 cosnφ Q n−1/2(1+2α2).(61) Electrostatic Toroid 17 Equality of the Fourier series in φon the left- and right-hand sides of the integral equation (58) now reveals the complete solution for the ring charge density σβlscript(φ)=−q 2π√arssinθs∞/summationdisplay n=0βepsilonInQn−1/2/parenleftbigga2+r2 s 2arssinθs/parenrightbigg Qn−1/2(1+2α2)cosnφ ( −π≤φ≤π). (62) Sinceα/lessmuch1, the interpretation of some limiting cases is assisted by the approximations Q−1/2(1+2α2)−−−−−→ b/a→0ln2a b≈ξ0 (63) and Q1/2(1+2α2)−−−−−→ b/a→02l n2−2+ln2a b≈ξ0−2(1−ln 2). (64) Ifθs→0, then coth ξs→∞ and only the n=0 term is significant and σβlscript(φ)→−q 2ξ0/radicalbig r2s+a2. (65) This term ∝1/ξ0≈0 to the order of approximation here. In this case the incident electric field is axial to the ring and the dominant term in the surface charge varies assinη, known from the exact solution (17). Accordingly, the thin-wire approximation of (46) in (47) is trivial in such a case where the incident electric field has no componenttangential to the infinitesimally thin ring. Ifθ s=π/2 andrs→∞ , then coth ξs→rs/2aand σβlscript(φ)−−−−→rs→∞−q 2rs/bracketleftbigg1 Q−1/2(1+2α2)+a/rs Q1/2(1+2α2)cosφ/bracketrightbigg −−−−→rs→∞−q 2rsξ0−qa 2r2s[ξ0−2(1−ln 2)]cosφ =−q 2rsξ0+2πε0E0a ξ0−2(1−ln 2)cosφ(66) where the familiar (38) is the magnitude of the incident electric field, which is essentially uniform and polarized in the xdirection, transverse to the ring. The cos φterm agrees with (26), as required. The first term that is constant in φis not present in the analysis in the subsection on the “Transverse Exciting Electric Field,” where the φ-dependence of all the fields is cos φat the outset. Following again the symmetric limiting proce- dure (Jackson, 1999), the uniform transverse excitation is correctly recovered from thepoint-source solution by considering a pair of point sources, +q/2a t(r s,π/2,0)and −q/2a t(rs,π/2,π) asrs→∞ . The constant terms from each of these contributions then cancel. 18 R. W. Scharstein and H. B. Wilson Summary of Main Implication for the Dynamic Problem One important implication of the electrostatic analysis of this work is its support of the thin-wire kernel approximation (Wu, 1962). The case of a transverse electric fieldexhibits the most interaction with a thin ring and displays the dominant, static featuresof the motivating electrodynamic problem. Therefore, the differences between the exactand approximate solutions for the induced charge density in this case merit quantifi-cation. The known cos φvariation is disregarded in the normalization of the thin-wire result (52) σ βlscript(φ) ε0E0acosφ=2π ln(8a/b)−2=B0, (67) where the constant B0is a convenient label. A comparison with the exact surface charge density of (24) is effected through the construction 2πbσ(η,φ) ε0E0acosφ=2b a(2 coshξ0−2 cosη)3/2 sinhξ0∞/summationdisplay n=0βepsilonIn(n−1/2)(n+1/2) P1 n−1/2(coshξ0)cosnη =C0(η)+C1(η)cosη+C2(η)cos 2η+···,(68) which is nota Fourier series in ηowing to the factor before the summation. However, a meaningful understanding of the relative importance of the terms is achieved by notingthat the coefficient functions called C 0(η)andC2(η)are most important when η= π/2, for example. Similarly, the coefficient C1(η)has its biggest effect when η=0. The variation of these coefficients C0(π/2),C2(π/2),C1(0)withb/a is graphed in Figure 10, along with the thin-wire result, presently B0in this normalization. The striking feature of this result is the accuracy of the thin-wire result B0compared to the exact result embodied in C0, even for toroids that are definitely fat. Of course, the constant (independent of ηorβ) term of the thin-wire approximation is insufficient to completely Figure 10. Coefficients of normalized charge density for transverse excitation. Solid curves: exact solution; Dashed curve: B0from thin-wire kernel approximation. Electrostatic Toroid 19 Table 1 Relative error between B0andC0(π/2) b/a ξ 0 Error ( −%) 10−614.5 6 .8×10−12 10−512.2 6 .8×10−10 10−49.9 6 .9×10−8 10−37.6 7 .1×10−6 10−25.3 7 .6×10−4 10−13.0 8 .9×10−2 0.2 2.3 4 .0×10−1 0.5 1.3 3 .7×100 characterize a toroid of nonnegligible thickness, where the terms involving C1andC2 become significant. Numerical values of the percentage difference or relative error =B0−C0(π/2) C0(π/2)×100% (69) are given in Table 1. References Abramowitz, M., and I. A. Stegun. 1972. Handbook of mathematical functions with formulas, graphs, and mathematical tables . Washington, DC: NBS. Cade, R. 1978a. The Kelvin transformation for a torus. J. Appl. Phys . 49:5722–5727. Cade, R. 1978b. A perturbation method for solving torus problems in electrostatics. J. Inst. Maths Applics . 21:265–284. Cohl, H. S., J. E. Tohline, A. R. P. Rau, and H. M. Srivastava. 2000. Developments in determining the gravitational potential using toroidal functions. Astron. Nachr . 321:363–372. Gil, A., J. Segura, and N. M. Temme. 2000. Computing toroidal functions for wide ranges of the parameters . CWI Technical Report MAS-R0014, Amsterdam. Hobson, E. W. 1955. The theory of spherical and ellipsoidal harmonics . New York: Chelsea. Jackson, J. D. 1999. Classical electrodynamics , 3rd ed. New York: Wiley, p. 63. King, R. W. P. 1969. The loop antenna for transmission and reception. In Antenna theory Part 1 , ed. R. E. Collin and F. J. Zucker (pp. 458–482). New York: McGraw-Hill. Lebedev, N. N. 1972. Special functions and their applications . New York: Dover. Lebedev, N. N., I. P. Skalskaya, and Y . S. Uflyand. 1979. Worked problems in applied mathematics . New York: Dover, pp. 236–237. Moon, P., and D. E. Spencer. 1961. Field theory handbook . Berlin: Springer-Verlag, pp. 112–115. Morse, P. M., and H. Feshbach. 1953. Methods of theoretical physics. Part II . New York: McGraw- Hill, pp. 1301–1304. Olver, F. J. 1997. Asymptotics and special functions . Wellesley, MA : A K Peters, pp. 174, 178. Smythe, W. R. 1968. Static and dynamic electricity , 3rd ed. New York: McGraw-Hill, p. 239. Smythe, W. R. 1974. Solutions to problems in static and dynamic electricity . Pasadena: California Institute of Technology. NTIS PB232797, p. 74. Werner, D. H. 1996. An exact integration procedure for vector potentials of thin circular loop antennas. IEEE Trans. Antennas Propagat . 44:157–165. Wu, T. T. 1962. Theory of the thin circular loop antenna. J. Math. Phys . 3:1301–1304. Copyright ofElectromagnetics istheproperty ofTaylor &Francis Ltdand itscontent maynotbecopied oremailed tomultiple sitesorposted toalistserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.