selvaggi toroidal electrostatics

An application of toroidal functions in electrostatics Jerry Selvaggi,a/H20850Sheppard Salon,b/H20850and M. V. K. Charic/H20850 Electrical, Computer and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 /H20849Received 8 December 2006; accepted 13 April 2007 /H20850 A method employing the use of toroidal functions is introduced for calculating the scalar potential and the electric field from a charged conducting ring. This method is an alternative to thewell-known elliptic integral formulation and is usually easier to formulate than the elliptic integralsolution. © 2007 American Association of Physics Teachers. /H20851DOI: 10.1119/1.2737473 /H20852 I. INTRODUCTION The calculation of the electric scalar potential and the electric field at some arbitrary point in space, not coincidentwith the charged filamentary ring and not on the symmetryaxis, usually involves the use of elliptic integrals. 1–4We show that this use is not necessary and solve this problem byemploying toroidal functions.5–7This approach, unlike the elliptic integral formulation, is not discussed in the physicscurriculum, but is easy to apply. Toroidal functions comefrom the solution of Laplace’s equation in toroidalcoordinates. 8A special class of these functions, called the Legendre functions of the second kind, are uniquely suitedfor handling finite circular cylindrical geometries. The calculation of the scalar potential and the electric field due to a charged ring at an arbitrary point in space not coin-cident with the ring is but one application of the use oftoroidal functions. The toroidal expansion has applications tocoil design used in MRI magnets, 9transformer coils, and circular cylindrical coils with a rectangular cross-section.10 Also, it has been used for calculating the external field frompermanent magnet motors.11–14It has recently been used for determining the gravitational potential.15–17 II. SCALAR POTENTIAL PRODUCED BY A CHARGED RING Figure 1 illustrates the geometry of the problem. To cal- culate the electric scalar potential at an arbitrary point in space, P, not coincident with the charged ring, we start with the differential scalar potential given by d/H9021P=1 4/H9266/H92800/H9261/H20849/H9278/H11032/H20850ad/H9278/H11032 /H20841R−a/H20841, /H208491/H20850 where Ris the position vector of the observation point and a is the position vector of an element of the circular filamen-tary line charge. The distance in circular cylindrical coordi- nates between the source point, /H20849 /H9267/H11032,/H9278/H11032,z/H11032/H20850, and an arbitrary observation point or field point, /H20849/H9267,/H9278,z/H20850, is given by 1 /H20841R−r/H11032/H20841=1 /H20881/H92672+/H9267/H110322+/H20849z−z/H11032/H208502−2/H9267/H9267/H11032cos /H20849/H9278−/H9278/H11032/H20850. /H208492/H20850 In Fig. 1, r/H11032=/H9267/H11032=/H20841a/H20841=aandz/H11032=0. This coordinate system allows us to write Eq. /H208492/H20850as1 /H20841R−a/H20841=1 /H20881/H92672+a2+z2−2/H9267acos /H20849/H9278−/H9278/H11032/H20850. /H208493/H20850 We employ Eqs. /H208493/H20850and /H208491/H20850and write the electric scalar potential as /H9021P=a 4/H9266/H92800/H20885 02/H9266/H9261/H20849/H9278/H11032/H20850d/H9278/H11032 /H20881/H92672+a2+z2−2/H9267acos /H20849/H9278−/H9278/H11032/H20850. /H208494/H20850 Equation /H208494/H20850can be converted into an elliptic integral, which is the approach taken in most advanced textbooks. However,Eq. /H208494/H20850can be used directly without any mathematical ma- nipulations. The Green’s function2,4expansion for Eq. /H208492/H20850is given by18–21 1 /H20841r−r/H11032/H20841=1 /H9266/H20881/H9267/H9267/H11032/H20858 m=0/H11009 /H9255mQm−1/2 /H20849/H9252/H20850cos /H20851m/H20849/H9278−/H9278/H11032/H20850/H20852, /H208495/H20850 where /H9252=/H20849/H92672+/H9267/H110322+/H20849z−z/H11032/H208502/H20850/2/H9267/H9267/H11032/H110221,/H9255mis Neumann’s factor,5which is 1 for m=0 and 2 otherwise. Qm−1/2 /H20849/H9252/H20850is the Legendre function of the second kind and of half-integral degree or a toroidal function of zeroth order.7,21,22These functions are also referred to as Q-functions18and can be represented by the formula12 Qm−1/2 /H20849/H9252/H20850=/H9266 /H208492/H9252/H20850m+1/22m/H20858 n=0/H11009/H208494n+2m−1 /H20850!! 22n/H20849n+m/H20850!n!1 /H208492/H9252/H208502n./H208496/H20850 Snow22has given the most complete account of the math- ematical properties of the toroidal functions. This in-depthreference should be consulted by anyone who wants to betterunderstand toroidal functions and their properties. However, to apply the Q-function successfully, Eqs. /H208495/H20850and /H208496/H20850are all that are necessary. Equation /H208495/H20850represents a Fourier cosine series expansion of the inverse distance in cylindrical coordinates whoseweighting coefficients are the toroidal functions given by Eq./H208496/H20850. This expansion is reminiscent of the expansion of an arbitrary periodic function in terms of a Fourier series whoseweighting coefficients are found from certain orthogonalityconditions. By employing Eqs. /H208495/H20850and /H208496/H20850, we can write Eq. /H208494/H20850as /H9021 P=a 4/H92662/H928001 /H20881/H9267a/H20858 m=0/H11009 /H9255mQm−1/2 /H20849/H9252/H20850/H20885 02/H9266 /H9261/H20849/H9278/H11032/H20850cos /H20851m/H20849/H9278 −/H9278/H11032/H20850/H20852d/H9278/H11032, /H208497/H20850 where /H9252=/H20849/H92672+a2+z2/H20850/2/H9267a/H110221. 724 724 Am. J. Phys. 75 /H208498/H20850, August 2007 http://aapt.org/ajp © 2007 American Association of Physics Teachers For a uniformly charged ring, /H9261/H20849/H9278/H11032/H20850=/H92610=q/2/H9266aand the m=0 term in Eq. /H208497/H20850is the only term that survives the inte- gration. As a result, we obtain a simple-looking expressionfor the electric scalar potential at an arbitrary point in spacenot coincident with the charged ring, /H9021 P=q 4/H92662/H92800/H20881/H9267aQ−1/2 /H20849/H9252/H20850. /H208498/H20850 From Eq. /H208496/H20850,Q−1/2 /H20849/H9252/H20850is given by Q−1/2 /H20849/H9252/H20850=/H9266/H20858 n=0/H11009/H208494n−1 /H20850!! 22n/H20849n!/H208502/H20873/H9267a /H92672+a2+z2/H208742n+1/2 . /H208499/H20850 If we employ Eq. /H208499/H20850, we can write Eq. /H208498/H20850as /H9021P=q 4/H9266/H92800/H20858 n=0/H11009/H208494n−1 /H20850!! 22n/H20849n!/H208502/H20849/H9267a/H208502n /H20849/H92672+a2+z2/H208502n+1/2. /H2084910/H20850 Equation /H2084910/H20850is the infinite series solution for the scalar potential of a uniformly charged ring of radius aand is valid at an arbitrary point that is not coincident with the chargedring. Likewise, for a harmonic charge density given by /H9261/H20849 /H9278/H11032/H20850=/H92610cos /H20849p/H9278/H11032/H20850, /H2084911/H20850 where p=1,2,3,..., the integral in Eq. /H208497/H20850is easily evalu- ated. A few three-dimensional surface plots have been cre-ated to visualize the potential and electric field components. The total charge, q, used for all the plots is 1 /H9262C and the radius of the charged ring a=1 m. The observation cylinder includes both the cylindrical hull and the circular caps. Thecylindrical hull, which surrounds the source, has a length of 2 m and has a radius /H9267=2 m. The circular caps are located at z= ±1 m and the radial coordinate of the caps varies from /H9267=1 20mt o/H9267=2 m. The cylindrical surface plots were cre- ated using Tecplot23by importing a data file that includes the Cartesian coordinates and its associated scalar potential orelectric field component. The data file was created byMaple, 24but other packages of this type will work.For a uniform charge density the total charge on the source is nonzero and, therefore, the total scalar potential is domi- nated in the far field by its monopole contribution. If /H9261/H20849/H9278/H11032/H20850 =/H92610cos /H20849p/H9278/H11032/H20850with integer p/H333561, the total charge on the cir- cular ring is zero. If p=1, then the total scalar potential is dominated in the far field by its dipole contribution. The reason is that if p=1, the m=1 term is the only term that survives the integration in Eq. /H208497/H20850and the scalar potential is /H9021P=qQ1/2/H20849/H9252/H20850 4/H92662/H92800/H20881/H9267acos /H20849/H9278/H20850. /H2084912/H20850 The Q1/2/H20849/H9252/H20850term includes a spherical dipole contribution, which is the major contribution to the scalar potential. Figure 2, which is a plot of Eq. /H2084912/H20850, illustrates the dipole pattern. If/H9261/H20849/H9278/H11032/H20850=/H92610cos /H208492/H9278/H11032/H20850, the total charge on the circular ring is zero and the total scalar potential is dominated by its quad- rupole contribution. This is true because only the m=2 term survives the integration in Eq. /H208497/H20850. The scalar potential for this case is /H9021P=qQ3/2/H20849/H9252/H20850 4/H92662/H92800/H20881/H9267a/H208512 cos2/H20849/H9278/H20850−1 /H20852. /H2084913/H20850 Figure 3 shows a plot of the scalar potential represented by Fig. 3. Scalar potential if /H9261/H20849/H9278/H11032/H20850=/H92610cos /H208492/H9278/H11032/H20850. Fig. 1. Charged ring. Fig. 2. Scalar potential if /H9261/H20849/H9278/H11032/H20850=/H92610cos/H9278/H11032. 725 725 Am. J. Phys., Vol. 75, No. 8, August 2007 Selvaggi, Salon, and Chari Eq. /H2084913/H20850and illustrates the quadrupole pattern, which domi- nates the scalar potential. Note that if the linear charge density of a charged ring is harmonic in cos /H20849p/H9278/H20850, then Eq. /H208497/H20850acts like a filter, filtering out only those specific harmonic components where m=p. III. ELECTRIC FIELD PRODUCED BY A CHARGED RING The electric field components due to a ring of charge with a harmonic charge density, /H9261/H20849/H9278/H11032/H20850=/H92610cos /H20849/H9278/H11032/H20850, calculated from E=−/H11633/H9021P, are E/H9267=q/H20853g1Q−1/2 /H20849/H9252/H20850+g2Q1/2/H20849/H9252/H20850/H20854 4/H92662/H92800/H9267/H20881/H9267aDcos /H20849/H9278/H20850, /H2084914a /H20850 Ez=q/H20853g3Q−1/2 /H20849/H9252/H20850+g4Q1/2/H20849/H9252/H20850/H20854 4/H92662/H92800/H20881/H9267aDcos /H20849/H9278/H20850, /H2084914b /H20850E/H9278=qQ1/2/H20849/H9252/H20850 4/H92662/H92800/H9267/H20881/H9267asin/H20849/H9278/H20850, /H2084914c /H20850 where g1=/H9267a/H20849/H92672−a2−z2/H20850,g2=/H20849−/H92672a2+/H92672z2+a4+2z2a2+z4/H20850, g3=2/H9267za,g4=−z/H20849/H92672+a2+z2/H20850, and D=/H20849/H92672+a2−2/H9267a+z2/H20850 /H11003/H20849/H92672+a2+2/H9267a+z2/H20850. Equation /H2084914/H20850is easily computed using Maple, in which it is easily shown that dQ1/2/H20849/H9261/H20850 d/H9261=/H9261Q1/2/H20849/H9261/H20850−Q−1/2 /H20849/H9261/H20850 2/H20849/H92612−1 /H20850. /H2084915/H20850 The three-dimensional surface plot of the radial component given by Eq. /H2084914a /H20850is shown in Fig. 4. We can continue this process for any value of pin Eq. /H2084911/H20850to produce higher multipole patterns. IV. DISCUSSION Another method of attack, which is widely employed for calculating the electric field from a charged ring but not dis-cussed in this paper, is spherical harmonic analysis. This method requires two solutions; one valid for R/H11022aand one valid for R/H11021a. A spherical harmonic solution suffers from slow convergence as the observation point moves closer tothe source and, therefore, the series solution developed fromthis method is not very useful for accurately calculating near-field solutions. With a toroidal expansion, only one seriessolution is needed, which is valid for observation points in-side or outside the ring’s radius. Equations /H208498/H20850and /H2084910/H20850or /H2084912/H20850–/H2084914/H20850are valid at an arbitrary point in space not coinci- dent with the source It is useful to know what the toroidal functions and their derivatives look like. Figures 5 and 6 illustrate a few of the Q-functions and their corresponding derivatives. We see from Figs. 5 and 6 that the Q-functions, unlike the Legendre polynomials, are monotonic functions and highlyconvergent. 25 a/H20850Electronic address: [email protected] b/H20850Electronic address: [email protected] c/H20850Electronic address: [email protected] 1F. R. Zypman, “Off-axis electric field of a charged ring,” Am. J. Phys. 74, 4, 295–300 /H208492006 /H20850. 2J. D. Jackson, Classical Electrodynamics /H20849Wiley, New York, 1999 /H20850,3 r d ed., pp. 117–119, 181–183. Fig. 4. 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