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Numerical computation of electromagnetic field for general static and axisymmetric current distribution

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Toshio Fukushima

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4 5 6

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National Astronomical Observatory of Japan / SOKENDAI, 2-21-1, Ohsawa, Mitaka, Tokyo 181-8588, Japan E-mail: [email protected]

Abstract We developed a numerical method to compute the electromagnetic field of arbitrary static and axisymmetric current distribution. The method (i) numerically evaluates a double integral of the electrostatic and magnetostatic potentials of an infinitely thin ring current by the split quadrature method using the double exponential rules, and (ii) derives the electrostatic field and the magnetostatic induction by numerically differentiating the numerically integrated potentials by the central difference formula. A comparison with the exact solution for a poloidal current distribution with an anisotropic Gaussian damping confirmed the 14- and 9-digit accuracy of the potential and the field/induction computed by the new method.

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Keywords: axial symmetry; electrostatics; magnetostatics; numerical

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differentiation; numerical quadrature

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1. Introduction

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The computation of the electromagnetic field for a general axisymmetric three-

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dimensional charge/current distribution is a classic problem in physics and engiPreprint submitted to Journal Computational Physics

July 25, 2017

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neering (Kellog, 1929; MacMillan, 1930). Indeed, its applications are as wide as

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(i) the electron and ion optics (Szil´agyi, 1988), (ii) the charged particle accelera-

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tion (Hamm & Hamm, 2012), (iii) the electron microscopy and spectroscopy (Erni

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et al., 2009), and (iv) the magnetic coil design (Montgomery & Weggel, 1980). Es-

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pecially, it is one of building blocks of the plasma physics and controlled nuclear

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fusion (Kikuchi, 2011; Kikuchi & Azumi, 2015).

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If the spatial distribution of the static electric charge, ρ (x), and of the static

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current vector, J(x), are explicitly known, then the electrostatic scalar potential,

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Φ(x), and the magnetostatic vector potential, A(x), are written as convolutions

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of these distributions with the Newton kernel, 1/|x − x′ |, (Jackson, 1998, equa-

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tions (1.17) and (5.32)) as 1 Φ(x) = 4πε0

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µ0 A(x) = 4π



ρ (x′ ) 3 ′ d x, ′ V |x − x |

(1)



J (x′ ) 3 ′ d x, ′ V |x − x |

(2)

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where the integration is conducted over all the volume occupied by the charge

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and/or the current vector, and ε0 and µ0 are the vacuum permittivity and perme-

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ability, respectively. The associated electrostatic field and the resulting magneto-

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static induction are expressed as

E(x) =

1 4πε0



( ) (x − x′ ) 3 ′ ρ x′ d x, |x − x′ |3

2

(3)

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µ0 B(x) = 4π



( ′ ) (x − x′ ) 3 ′ J x × d x. |x − x′ |3

(4)

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These are nothing but Coulomb’s law and the Biot-Savart law (Jackson, 1998,

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equations (1.5) and (5.14)).

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When the charge/current distribution is finitely bounded, the external electro-

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magnetic field can be expanded in harmonics (Garrett, 1951). However, if the

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evaluation point x is inside the distributions of the charge or current, on the other

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hand, the integral expressions, equations (1)–(4), suffer from the algebraic singu-

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larities. This becomes a serious issue for extended distributions such as encoun-

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tered in the plasma physics.

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Before going further, let us show a practical example. Fig. 1 shows the con-

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tour map on a meridional cross section of a hypothetical charge/current distribu-

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tion. It was designed to resemble the poloidal mode equilibrium solution of the

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plasma current distribution circulating in an ITER-like tokamak (Evangelias &

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Throumoulopoulos, 2016, Fig. 4). Refer to Section 4 later for the detailed model

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description.

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Although the adopted model distribution is infinitely extended in principle, it

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can be practically regarded to be finitely bounded thanks to the Gaussian damping

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around the central ring. Inside this practical boundary, the algebraic singularities

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appear everywhere. Thus, E(x) and/or B(x) are hardly computed by evaluating

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the integral forms by the existing quadrature techniques (Press et al., 2007).

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Therefore, a common practice has been solving Poisson’s equation for Φ(x)

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and A(x) (Jackson, 1998, equations (1.28) and (5.28)), which are nothing but the 3

Cross Section of Charge/Current 0.75 0.5

z

0.25 0 -0.25 -0.5 -0.75 0 0.25 0.5 0.75 1 1.25 1.5 R Figure 1: Cross section of model electric charge/current distribution. Shown is the contour map on the meridional cross section of a hypothetical electric charge/current distribution. The contours are drawn for the levels of the relative magnitude being inverse powers of 2 as ρ /ρ0 = J/J0 = 2−n for n = 1, 2, . . . , 12. Although the distribution is infinitely extended, it is practically bounded in a finite region thanks to the Gaussian damping feature adopted in the model distribution.

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differential form of Gauss’s and Ampere’s laws, respectively: ∇2 Φ = −ρ /ε0 ,

(5)

∇2 A = −µ0 J.

(6)

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These equations are elliptic type partial differential equations. They are numeri-

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cally solved by the finite or boundary element methods of various kind (Park et

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al., 1990; Wang & Demerdash, 1990; Brio et al., 1993; Ma et al., 1996; Le-Van et

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al., 2016). Refer to Bellina & Serra (2004) for a concise summary of the numer-

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ical approaches. Nonetheless, the resulting formulation becomes cumbersome in

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general (Demerdash & Wang, 1990) and suffers from the accuracy degrade (Con-

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way, 2001). This is especially true if the boundary conditions are complicated

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(Mitsuoka et al., 2013; Jacques et al., 2016; Mezani et al., 2016).

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Recently, we developed a numerical method to circumvent the difficulties for

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the gravitational field of an axisymmetric mass density distribution (Fukushima,

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2016c). It can be directly applicable to the computation of Φ(x) and E(x) as

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summarized in Appendix A. Therefore, in this article, we adapt the method to the

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computation of A(x) and B(x) for arbitrary axisymmetric distribution of electric

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current. By using the original and adapted methods, we prepared Figs 2 and 3

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showing the bird’s-eye views of E(x) and B(x) of the hypothetical charge/current

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distribution specified in Fig. 1. As will be shown later, these results are of the

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9-digit accuracy, which is far more than necessary.

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Below, we (i) describe the adapted method in Section 2, (ii) examine its com5

Bird’s-Eye View of E E/E0 1 0.5 0.5

0 0 0.5 R

1

z

1.5-0.5

Figure 2: Bird’s-eye view of electrostatic field strength. Displayed is a bird’s-eye view of E ≡ |E(x)|, the magnitude of the electrostatic field caused by the current distribution depicted in Fig. 1.

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Bird’s-Eye View of B B/B0 1 0.5 0.5

0 0 0.5 R

1

z

1.5-0.5

Figure 3: Bird’s-eye view of magnetostatic field strength. Displayed is a bird’s-eye view of B ≡ |B(x)|, the magnitude of the magnetostatic induction caused by the current distribution depicted in Fig. 1.

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putational accuracy and speed in Section 3, and (iii) present its example in Sec-

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tion 4.

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2. Method

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Consider a general static and axisymmetric current distribution. Adopt the cylin-

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drical polar coordinate system, (R, z, ϕ ). In this case, the only non-zero compo-

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nents of J(x) and A(x) are their azimuthal components: J(R, z) ≡ Jϕ (R, z), A(R, z) ≡ Aϕ (R, z).

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(7)

By symmetry, A(x) vanishes on the z-axis as

A(0, z) = 0.

(8)

a(R, z) ≡ A(R, z)/R.

(9)

Therefore, we scale it as

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Denote the lower and upper end point of the radial distribution by RL (≥ 0) and

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RU (≤ +∞), respectively. For simplicity, we assume that J(R, z) vanishes when z ≤

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zL (R) or z ≥ zU (R) where zL (R)(≥ −∞) and zU (R)(≤ +∞) are certain functions of

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R. Then, a(R, z) is expressed as a double integral convolving J(R, z) with Green’s

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function as

∫ RU (∫ zU (R′ )

a(R, z) = RL

zL (R′ )

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) ) ′ F R , z ; R, z dz dR′ , (





(10)

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where we abbreviate the integrand as ( ) ( ) ( ) F R′ , z′ ; R, z ≡ f R′ , z′ ; R, z J R′ , z′ ,

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(11)

while f (R′ , z′ ; R, z) is defined as ( ′ ′ ) ( µ0 ) 16 (R′ )2 S (m (R′ , z′ ; R, z)) f R , z ; R, z = [√ ]3 . 4π 2 2 ′ ′ (R + R) + (z − z)

(12)

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The function f (R′ , z′ ; R, z) is, except for (i) the multiplier R′ caused by the volume

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integration element in the cylindrical coordinate system and (ii) the divisor R in-

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troduced by the scaling, equivalent with the magnetostatic potential evaluated at

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(R, z) of a uniform ring current located at (R′ , z′ ) (Jackson, 1998, equation (5.37)).

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In equation (12), m (R′ , z′ ; R, z) is a function defined as [( ( ′ ′ ) )2 ( ′ )2 ] ′ ′ m R , z ; R, z ≡ 4R R/ R + R + z − z ,

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(13)

while S(m) is a special complete elliptic integral (Fukushima, 2010) defined as S(m) ≡ [(2 − m)K(m) − E(m)] /m2 ,

(14)

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where K(m) and E(m) are the complete elliptic integral of the first and second

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kind with the parameter m ≡ k2 , respectively (Wolfram, 2003). Do not confuse m

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with the modulus k, which has been adopted as the argument of complete elliptic

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integrals in the classic literature (Byrd & Friedman, 1971; Olver et al., 2010). 9

Complete Elliptic Integrals 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

K(m) E(m) S(m) 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

m

Figure 4: Behaviour of complete elliptic integrals. Plotted are the three complete elliptic integrals, K(m), E(m), and S(m) ≡ [(2 − m)K(m) − E(m)]/m2 .

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Refer to Fig. 4 for the behavior of these complete elliptic integrals for the

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standard domain, 0 ≤ m < 1. The precise and fast computation of S(m) is realized

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by the program

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double precision accuracy and runs slightly faster than the exponential function of

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CEIS

(Fukushima, 2016b, Appendix E). Indeed, it has the full

the standard mathematical library.

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The function S(m) has a logarithmic blow-up singularity at m = 1 as clearly

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seen in Fig. 4. As a result, f (R′ , z′ ; R, z) becomes singular when (R′ , z′ ) = (R, z).

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This hinders a proper convergence of the numerical integration of a(R, z) by any

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of the existing numerical quadrature rules (Press et al., 2007, chapter 4). In order

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to resolve this issue, we split the integration intervals at R′ = R and z′ = z as

a(R, z) = a1 (R, z) + a2 (R, z) + a3 (R, z) + a4 (R, z),

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where the split components are written as a1 (R, z) ≡

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a2 (R, z) ≡ 108

a3 (R, z) ≡ 109

a4 (R, z) ≡ 110

(15)

∫ R (∫ z RL

) ) ′ F R , z ; R, z dz dR′ , (

zL (R′ )

∫ RU (∫ z

(

zL (R′ )

R

∫ R (∫ zU (R′ ) RL





)





)

(16)

dR′ ,

(17)

) ( ′ ′ ) ′ F R , z ; R, z dz dR′ ,

(18)

F R , z ; R, z dz

z

∫ RU (∫ zU (R′ ) R



) ) ′ F R , z ; R, z dz dR′ . (





(19)

z

In other words, the domain of the double integral is split into four pieces as illus-

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Split Quadrature 0.75 0.5 3

0.25

4

z

. 0 1

-0.25

2

-0.5 -0.75 0 0.25 0.5 0.75 1 1.25 1.5 R Figure 5: Sketch of double split quadrature. In the numerical integration of the magnetostatic potential of an axisymmetric current distribution at its internal point, we divide the integration domain into four numbered regions radially and vertically separated at the point shown by a bullet.

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trated in Fig. 5. Notice that the split quadrature can be conducted in parallel. This

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will significantly reduce the total computational time if employing an appropriate

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parallel computing tool such as the OpenMP architecture (Fukushima, 2016c).

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Next, we evaluate each piece of integral by the double exponential (DE) quadra-

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ture rule (Takahashi & Mori, 1973). The rule is known as the best available

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method of the numerical quadrature (Bailey et al., 2005). Indeed, the DE rule

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can properly handle the integrable singularities of the integrand such as the blow-

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up logarithmic one of S(m) if the singularities are located at the end points of the

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integration intervals as indicated above. A detailed implementation note of the

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DE rule is available (Fukushima, 2016c).

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At any rate, the split quadrature method is so effective that the integral values

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are very accurately obtained, say at the level of the double precision machine ep-

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silon, in the case of general integrals of the Fermi-Dirac distribution (Fukushima,

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2014). This is also true in the case of the gravitational field for axisymmetric

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three-dimensional objects independently on the shape, size, and finiteness of the

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mass density distribution (Fukushima, 2016c).

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Once a(R, z) is computed, we evaluate B(x) by numerically differentiating a(R, z) as

(

) ∂ a(R, z) BR (R, z) = −R , ∂z R ( ) ∂ a(R, z) . Bz (R, z) = 2a(R, z) + R ∂R z

(20)

(21)

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These expressions in terms of a(R, z) contain no singularities caused by the small-

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ness of R, which appear in the original expressions in terms of A(R, z). Conse13

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quently, their evaluation faces with no numerical difficulties. This is the main

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reason why we prefer a(R, z) to A(R, z) as the basic quantity to be integrated nu-

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merically.

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As for the actual procedure of the numerical differentiation, we once adopted

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Ridders’ method (Ridders, 1982) in computing the gravitational field of two- and

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three-dimensional axisymmetric density distribution (Fukushima, 2016a,c). The

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method obtains the derivative very accurately by employing Richardson’s extrap-

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olation to the zero limit of the test argument difference. However, it is also true

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that the resulting method using Ridders’ method is significantly time-consuming

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in the sense that its CPU time is 2–10 times more than that of single evaluation of

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the potential.

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In order to overcome this difficulty while keeping a similar accuracy, we adopt

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the central difference formula (Olver et al., 2010, equation (3.4.20)) by appropri-

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ately choosing its test argument difference as (

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∂ a(R, z) ∂R

(

)

∂ a(R, z) ∂z

≈ z

) ≈ R

a(R + ∆R, z) − a(R − ∆R, z) , 2∆R

(22)

a(R, z + ∆z) − a(R, z − ∆z) , 2∆z

(23)

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where ∆R and ∆z are test deviations of the arguments. Since the central difference

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formulas are of the second order, we set the deviations as √ √ ∆R = R∗ δ ′ , ∆z = z∗ δ ′ ,

14

(24)

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where δ ′ is the relative error tolerance of the magnetostatic field computation and

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R∗ and z∗ are the typical scale length of the considered distribution. To be consis-

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tent with this setting, we choose δ , the relative error tolerance of the magnetostatic

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potential integration, as

√ δ = δ ′ δ ′.

(25)

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For example, in the IEEE 754 double precision environment, the attainable accu-

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racy of the derivative computation is as limited as

δ ′ = δ 2/3 ≥ ε 2/3 ≈ 2.3 × 10−11 ,

155

(26)

where ε is the double precision machine epsilon expressed as

ε ≡ 2−53 ≈ 1.1 × 10−16 .

(27)

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This limit accuracy is sufficiently high for the practical purposes.

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3. Numerical experiments

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Let us examine the computational accuracy and the computational cost of the new

159

method. First, we measure the accuracy by comparing the computed result with

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a rigorous solution. As such an exact solution, we adopt a pair of the current

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distribution and the magnetostatic vector potential expressed as JG (R, z) ≡ J0 q(ξ , ζ )e(ξ , ζ ),

15

(28)

Bird’s-Eye View of JG JG/J0 1 0.5 0 -0.5 2 1 0

0

1 R

2

-1

3

z

4 -2

Figure 6: Bird’s-eye view of current density distribution of Gaussian toroid. Displayed is a bird’seye view of JG (R, z), a hypothetical current density distribution resulting the magnetostatic potential with an anisotropic Gaussian form damping around a ring.

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Bird’s-Eye View of AG AG/A0 1 0.5 0 -0.5 2 1 0

0

1 R

2

-1

3

z

4 -2

Figure 7: Bird’s-eye view of magnetostatic potential of Gaussian toroid. Displayed is a bird’s-eye view of AG (R, z), the magnetostatic potential of the Gaussian toroid.

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AG (R, z) ≡ ε0 J0 ξ 3 e(ξ , ζ ). 162

Here q(ξ , ζ ) and e(ξ , ζ ) are functions defined as q(ξ , ζ ) = −8ξ − 14ξ0 ξ 2 + q3 (ζ )ξ 3 + 8ξ0 ξ 4 − 4ξ 5 ,

(30)

[ ] e(ξ , ζ ) ≡ exp − (ξ − ξ0 )2 − ζ 2 ,

(31)

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where q3 (ζ ) is an additional function defined as q3 (ζ ) ≡ 16 − 4ξ02 + 2ν 2 − 4ν 2 ζ 2 ,

165

(29)

(32)

and ξ , ξ0 , ζ , and ν are nondimensional quantities defined as

ξ ≡ R/HR , ξ0 ≡ R0 /HR , ζ ≡ z/Hz , ν ≡ HR /Hz ,

(33)

166

while J0 , R0 , HR , and Hz are certain dimensioned constants. Refer to Appendix B

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for the derivation of the exact solution. Indeed, this pair of the magnetostatic po-

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tential and the current density satisfies the vector Poisson’s equation, equation (6),

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and the boundary conditions. For simplicity, we set the parameters of the solution

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as R0 = 2, HR = 1/2, Hz = 1.

(34)

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Figs 6 and 7 provide the bird’s-eye views of JG (R, z) and AG (R, z) for this specific

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choice of parameters.

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By employing the new method described in the previous section, we numeri18

Absolute Error of Magnetostatic Potential -13

log10 |∆A|

-14 -15 -16 -17 δ=10-15 -18 0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure 8: Integration error of magnetostatic potential of Gaussian toroid. Shown are the absolute errors of the magnetostatic potential, A(R, z), integrated by the new method with a tiny relative error tolerance, δ = 10−15 . Overlapped are the errors for various values of z as z = 0, 1, 2, 3, and 4 since there are no significant difference among them.

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Absolute Error of Magnetostatic Induction -9

log10 |∆B|

-10 -11 -12 -13 δ=10-15 -14 0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure 9: Computation error of magnetostatic induction vector components of Gaussian toroid. Shown are the absolute errors of the magnetostatic induction vector components, BR (R, z) and Bz (R, z), computed by the new method from the magnetostatic potential integrated with a tiny relative error tolerance, δ = 10−15 . Open and filled circles indicate the errors for BR (R, z) and Bz (R, z), respectively. Overlapped are the errors for various values of z as z = 0, 1, 2, 3, and 4 since there are no significant difference among them.

20

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cally integrated the scaled magnetostatic potential a(R, z) for JG (R, z) with a tiny

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relative error tolerance, δ = 10−15 , and by setting the end points of the integral

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intervals as RL = 0, zL (R′ ) = −∞, RU = zU (R′ ) = +∞.

(35)

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Figs 8 and 9 show the absolute error of the integrated magnetostatic potential,

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A(R, z) ≡ Ra(R, z), and the computed magnetostatic induction vector components, ∆A(R, z) ≡ [A(R, z) − AG (R, z)] /Amax ,

(36)

∆BR (R, z) ≡ [BR (R, z) − BR,G (R, z)] /Bmax ,

(37)

∆Bz (R, z) ≡ [Bz (R, z) − Bz,G (R, z)] /Bmax ,

(38)

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180

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where Amax and Bmax are the maximum values of AG (R, z) and |BG (R, z)|.

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On the other hand, Figs 10 and 11 illustrate the similar absolute errors of the

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integrated electrostatic potential and the computed electrostatic field vector com-

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ponents by means of the original method described in Appendix A. Obviously,

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when the relative error tolerance of the numerical integration is set as tiny as

186

δ = 10−15 , both the original and new methods assure that the integrated poten-

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tials and computed fields are of the 14- and 9-digits accuracy, respectively.

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Next, we move to the computational cost. Figs 12 and 13 show Neval , the

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number of integrand evaluations required to compute a single point value of B

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and E, for the same charge/current distribution. Presented are the results when the

191

relative error tolerance of the potential integration is set as tiny as δ = 10−15 . The 21

Absolute Error of Electrostatic Potential -13

log10 |∆Φ|

-14 -15 -16 -17 δ=10-15 -18 0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure 10: Integration error of electrostatic potential of Gaussian toroid. Shown are the absolute errors of the electrostatic potential, Φ(R, z), integrated by the new method with a tiny relative error tolerance, δ = 10−15 . Overlapped are the errors for various values of z as z = 0, 1, 2, 3, and 4 since there are no significant difference among them.

22

Absolute Error of Electrostatic Field -9

log10 |∆E|

-10 -11 -12 -13 δ=10-15 -14 0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure 11: Computation error of electrostatic field vector components of Gaussian toroid. Shown are the absolute errors of the electrostatic field vector components, ER (R, z) and Ez (R, z), computed by the new method from the electrostatic potential integrated with a tiny relative error tolerance, δ = 10−15 . Open and filled circles indicate the errors for ER (R, z) and Ez (R, z), respectively. Overlapped are the errors for various values of z as z = 0, 1, 2, 3, and 4 since there are no significant difference among them.

23

Neval/106

Computational Cost of Magnetostatic Induction 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6

z=4 z=0

z=2

-15

δ=10 0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure 12: Computational cost of magnetostatic induction of Gaussian toroid. Shown are the number of integrand evaluations, Neval , required in computing BR (R, a) and Bz (R, z). The numbers are plotted as functions of R for some values of z as z = 0, 2, and 4. The integration of the potential is conducted with a tiny relative error tolerance, δ = 10−15 .

24

Computational Cost of Electrostatic Field 1.3

z=4

1.2

Neval/106

1.1

z=0

1 0.9

z=2

0.8 0.7

δ=10-15

0.6 0

0.5

1

1.5

2

2.5

3

3.5

4

R Figure 13: Computational cost of electrostatic field of Gaussian toroid. Same as Fig. 12 but for computing ER (R, a) and Ez (R, z).

25

192

numbers are plotted as functions of R for some values of z as z = 0, 2, and 4. In

193

average, Neval amounts to 0.6–1.6 millions. Since the CPU time to evaluate the

194

complete elliptic integrals is roughly the same as that of the exponential function

195

in the standard mathematical libraries, the expected computational time is around

196

60 ms at an ordinary PC, say that with an Intel Core i7-4600U running at 2.10

197

GHz clock.

198

4. Example

199

In order to show the performance of the new method, we evaluate the magneto-

200

static induction vector of a hypothetical current distribution already depicted in

201

Fig 1. We write its current density as ( J(R, z) ≡ J0

R R0

)2

[ ( ) ( )2 ] R − RT (z) 2 z exp − − , hR (R, z) hz (z)

(39)

202

where (i) J0 is the value of J(R, z) when R = R0 and z = 0, (ii) RT (z) is a function

203

defined as RT (z) ≡ R0 + R1 z + R2 z2 + R3 z3 ,

(40)

204

(iii) hR (R, z) and hz (z) are the scale height of the Gaussian damping in the R- and

205

z-directions defined as hR (R, z) ≡

206

( ) HR 1 + S1 R + S2 R2 1 + P1 D(R, z) + P2 [D(R, z)]2

( ) hz (z) ≡ Hz / 1 + Q1 z + Q2 z2 , 26

,

(41)

(42)

207

where (iv) D(R, z) is a function specifying the normalized radial distance from the

208

peak defined as D(R, z) ≡ [R − RT (z)] /HR ,

209

(43)

while (v) the parameters are chosen as

HR = 0.15, Hz = 0.25, P1 = P2 = Q1 = S1 = 0.2, Q2 = 0.5, 210

R0 = 1, R1 = 0, R2 = −0.3, R3 = −0.1, S2 = −2.

(44)

211

The functional forms and the parameters are experimentally determined so as to

212

mimic the equilibrium solution of the flux function for an ITER-like tokamak

213

(Evangelias & Throumoulopoulos, 2016, Fig. 4). Clearly, the distribution is not

214

plane symmetric and some of its density contours have a kink near the so-called

215

X-point. In any case, the existing methods are hardly applicable to compute its

216

electrostatic field and magnetostatic induction, especially near the peak of the

217

distribution.

218

Using the new method, we evaluated the magnetostatic potential and induction

219

vector of the distribution. Fig. 14 shows the contour maps of the stream line

220

function,

ψ (R, z) ≡ RA(R, z),

(45)

221

which display the magnetic force lines. Meanwhile, the contour maps of the mag-

222

netic induction strength are shown in Fig. 15. Compare them with Figs 16 and 17,

223

the contour maps of the electrostatic potential and field vector of the correspond27

Contour Map of ψ 2

z

1 0 -1 -2 0

1

2

3

4

R Figure 14: Contour map of magnetostatic stream line function. Illustrated is the contour map of ψ (R, z) ≡ RA(R, z), the magnetostatic stream line function of the test current distribution. The contours exhibit the magnetic force lines. The contours are drawn for every 5 per cent level of the peak value.

28

Contour Map of B 2

z

1 0 -1 -2 0

1

2

3

4

R Figure 15: Contour map of magnetostatic field strength. Same as Fig. 14 but for B, the magnetostatic field strength.

29

224

ing charge distribution obtained by the method explained in Appendix A.

225

5. Conclusion

226

By modifying the previous work on the gravitational field computation for an arbi-

227

trary axisymmetric mass density distribution (Fukushima, 2016c), we developed

228

a numerical method to compute the electrostatic and magnetostatic fields of gen-

229

eral static and axisymmetric current density distributions. The method consists of

230

(i) the numerical evaluation of the double integral transform of the charge and/or

231

current of an infinitely thin ring by the split quadrature method employing the

232

double exponential rules, and (ii) the numerical differentiation of the numerically

233

integrated potential by the central difference formula while setting the test argu-

234

ments appropriately. The comparison with the exact analytical solutions confirm

235

the 14- and 9- digit accuracy of the potentials and the fields computed by the new

236

method in the double precision environment. Although the new method requires

237

the quadrature of double integrals, its CPU time is not so large, say 60 ms for a

238

single point magnetostatic induction vector computation executed at an ordinary

239

PC. Thus, it may be an efficient and precise tool to evaluate the electromagnetic

240

field of arbitrary axisymmetric distribution of poloidal charge/current distribution.

241

Acknowledgments

242

243

The author thanks the anonymous referees for their valuable advices to improve the quality of the present article.

30

Contour Map of Φ 2

z

1 0 -1 -2 0

1

2

3

4

R Figure 16: Contour map of electrostatic potential. Same as Fig. 14 but for the electrostatic potential, Φ.

31

Contour Map of E 2

z

1 0 -1 -2 0

1

2

3

4

R Figure 17: Contour map of electrostatic field strength. Same as Fig. 16 but for the magnitude of the electrostatic field, E ≡ |E|.

32

244

Appendix A. Electrostatic field computation

245

Let us summarize the new method to compute the electrostatic field for a general

246

static and axisymmetric charge distribution. It is a simplification of the method

247

for the gravitational field computation (Fukushima, 2016c). First, the electrostatic

248

potential, Φ(R, z), is computed by the split quadrature as Φ(R, z) = Φ1 (R, z) + Φ2 (R, z) + Φ3 (R, z) + Φ4 (R, z),

249

where each piece is written as Φ1 (R, z) ≡

250

Φ2 (R, z) ≡ 251

Φ3 (R, z) ≡ 252

Φ4 (R, z) ≡ 253

(A.1)

∫ R (∫ z RL

zL (R′ )

) ( ′ ′ ) ′ G R , z ; R, z dz dR′ ,

(A.2)

) ) ′ G R , z ; R, z dz dR′ ,

(A.3)

) ( ′ ′ ) ′ G R , z ; R, z dz dR′ ,

(A.4)

) ( ′ ′ ) ′ G R , z ; R, z dz dR′ .

(A.5)

∫ RU (∫ z

(

zL (R′ )

R

∫ R (∫ zU (R′ ) RL



z

∫ RU (∫ zU (R′ ) R



z

The integrand is expressed as ( ) ( ) ( ) G R′ , z′ ; R, z ≡ g R′ , z′ ; R, z ρ R′ , z′ ,

33

(A.6)

254

where (i) ρ (R, z) is the charge distribution function and (ii) g (R′ , z′ ; R, z) is another

255

kernel function defined as (

) g R , z ; R, z = ′



(

1 4πε0

)

4R′ K (m (R′ , z′ ; R, z)) √ , 2 2 ′ ′ (R + R) + (z − z)

(A.7)

256

where K(m) is the complete elliptic integral of the first kind (Wolfram, 2003). This

257

kernel function g (R′ , z′ ; R, z) is, except for the multiplication factor R′ , equivalent

258

with the electrostatic potential evaluated at (R, z) of a uniform ring charge located

259

at (R′ , z′ ). Refer to Fukushima (2010) for the gravitational case.

260

On the other hand, the axisymmetric electrostatic field vector components in

261

the cylindrical coordinate system are evaluated by numerical differentiations of

262

Φ(R, z) by means of the central difference formula as ER (R, z) ≈

Φ (R + ∆R, z) − Φ (R − ∆R, z) , 2∆R

(A.8)

Ez (R, z) ≈

Φ (R, z + ∆z) − Φ (R, z − ∆z) , 2∆z

(A.9)

263

264

where ∆R and ∆z are the test deviations of the arguments as described in the main

265

text.

266

Appendix B. Reference magnetostatic solution

267

Let us find an exact solution of the pair of the static toroidal current distribution,

268

J(R, z), and the magnetostatic potential function, A(R, z). They must satisfy the

34

269

azimuthal component of Poisson’s equation for the magnetic vector potential as

∂ 2A 1 + ∂ R2 R 270

(

) ∂A A ∂ 2A − 2 + 2 = −µ0 J, ∂R R ∂z

(B.1)

where µ0 is the vacuum permeability.

271

In general, it is difficult to solve the equation for A(R, z) when J(R, z) is given.

272

However, the reverse process is straightforward as long as the boundary conditions

273

are satisfied. In fact, once A(R, z) is specified, J(R, z) is automatically obtained by

274

the partial differentiation. If both of them satisfy the proper boundary conditions,

275

then they become the solution pair.

276

For simplicity, we assume that A(R, z) and J(R, z) are expressed as A(R, z) ≡ µ0 J0 p(ξ , ζ )e(ξ , ζ ),

(B.2)

J(R, z) ≡ J0 q(ξ , ζ )e(ξ , ζ ),

(B.3)

277

278

where (i) e(ξ , ζ ) is already defined in equation (31) as [ ] e(ξ , ζ ) ≡ exp − (ξ − ξ0 )2 − ζ 2 ,

(B.4)

279

(ii) p(ξ , ζ ) and q(ξ , ζ ) are unknown functions to be determined to satisfy equa-

280

tion (B.1) and the boundary conditions, (iii) ξ , ξ0 , and ζ are nondimensional

281

quantities already defined as

ξ ≡ R/HR , ξ0 ≡ R0 /HR , ζ ≡ z/Hz , 35

(B.5)

282

and (iv) R0 , HR , and Hz are certain dimensioned constants. In the present case, the

283

proper boundary conditions are

lim A(R, z) = lim A(R, z) = A(0, z) = 0,

(B.6)

lim J(R, z) = lim J(R, z) = J(0, z) = 0.

(B.7)

R→+∞

z→±∞

284

R→+∞

z→±∞

285

The conditions at infinity are satisfied by the Gaussian form of damping unless

286

p(ξ , ζ ) and/or q(ξ , ζ ) increases more rapidly. In order to satisfy the last condition

287

for A(R, z) simply, we assume that p(ξ , ζ ) is a cubic monomial of ξ as p(ξ , ζ ) ≡ ξ 3 .

(B.8)

288

One may think that the degree could be lower if only required is the zero value

289

condition, p(0, ζ ) = 0. Nevertheless, if p(ξ , ζ ) contains lower degree monomials

290

like ξ or ξ 2 , then q(ξ , ζ ) will not satisfy the corresponding boundary condition

291

of the current density.

292

At any rate, q(ξ , ζ ) is obtained by partial differentiations as q(ξ , ζ ) = −8ξ − 14ξ0 ξ 2 + q3 (ζ )ξ 3 + 8ξ0 ξ 4 − 4ξ 5 ,

293

(B.9)

where q3 (ζ ) is the function already defined in equation (32) as q3 (ζ ) ≡ 16 − 4ξ02 + 2ν 2 − 4ν 2 ζ 2 ,

36

(B.10)

294

while ν is the ratio of scale heights in the R- and z-directions as

ν ≡ HR /Hz .

(B.11)

295

Obviously, this solution satisfies the condition q(0, ζ ) = 0, and therefore guaran-

296

tees the boundary condition, J(0, z) = 0.

297

Notice that, although p(ξ ) is positive definite, q(ξ , ζ ) is not so. As a result,

298

depending on the location, R and z, the sign of J(R, z) will be alternative even if

299

that of A(R, z) remains to be the same.

300

301

302

Since the magnetostatic potential is analytically obtained, the associated magnetostatic induction vector is automatically calculated as ] [ BR (R, z) = 2µ0 J0 ξ 3 ζ e(ξ , ζ ) /Hz ,

(B.12)

[ ] ( ) Bz (R, z) = 2µ0 J0 ξ 2 2 + ξ0 ξ − ξ 2 e(ξ , ζ ) /HR .

(B.13)

303

Thus, the solution is completed.

304

Appendix C. Reference electrostatic solution

305

Following the magnetostatic case in Appendix B, we will construct an exact solu-

306

tion of the pair of the static toroidal charge distribution, ρ (R, z), and the electro-

307

static potential, Φ(R, z), satisfying Poisson’s equation as

∂ 2Φ 1 + ∂ R2 R

(

) ∂Φ ∂ 2 Φ −ρ , + 2 = ∂R ∂z ε0 37

(C.1)

308

where ε0 is the vacuum permittivity. By a similar procedure, we obtained an exact

309

solution in the Gaussian form as Φ(R, z) = (ρ0 /ε0 ) ξ 3 e(ξ , ζ ),

(C.2)

ρ (R, z) = ρ0 r(ξ , ζ )e(ξ , ζ ).

(C.3)

310

311

Here (i) r(ξ , ζ ) is a function defined as r(ξ , ζ ) ≡ −9ξ − 14ξ0 ξ 2 + q3 (ζ )ξ 3 + 8ξ0 ξ 4 − 4ξ 5 ,

(C.4)

312

while q3 (ζ ) is the function defined in equation (32), (ii) ξ , ξ0 , ζ , and ν are nondi-

313

mensional quantities already introduced in Appendix B, and (iii) ρ0 , R0 , HR , and

314

Hz are certain dimensioned constants. The obtained solutions satisfy the boundary

315

conditions: lim Φ(R, z) = lim Φ(R, z) = Φ(0, z) = 0,

(C.5)

lim ρ (R, z) = lim ρ (R, z) = ρ (0, z) = 0.

(C.6)

R→+∞

z→±∞

316

R→+∞

z→±∞

317

Once the electrostatic potential is determined, the associated electrostatic field

318

vector is computed as

319

] [ ( ) ER (R, z) = ρ0 ξ 2 3 + 2ξ0 ξ − 2ξ 2 e(ξ , ζ ) / (ε0 HR ) ,

(C.7)

[ ] Ez (R, z) = −2ρ0 ξ 3 ζ e(ξ , ζ ) / (ε0 Hz ) .

(C.8)

38

320

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