mark andrews.toroidalcoords

7/25/2019 Mark Andrews.toroidalCoords

1/9

Journal of Electrostatics 64 (2006) 664672

Alternative separation of Laplaces equation in toroidal coordinates and

its application to electrostatics

Mark Andrews

Physics, The Faculties, Australian National University, ACT 0200, Australia

Received 1 November 2004; received in revised form 25 July 2005; accepted 25 November 2005

Available online 28 December 2005

Abstract

The usual method of separation of variables to find a basis of solutions of Laplaces equation in toroidal coordinates is particularly

appropriate for axially symmetric applications; for example, to find the potential outside a charged conducting torus. An alternative

procedure is presented here that is more appropriate where the boundary conditions are independent of the spherical coordinate y(rather

than the toroidal coordinate Z or the azimuthal coordinate c. Applying these solutions to electrostatics leads to solutions, given asinfinite sums over Legendre functions of the second kind, for (i) an arbitrary charge distribution on a circle, (ii) a point charge between

two intersecting conducting planes, (iii) a point charge outside a conducting half plane. In the latter case, a closed expression is obtained

for the potential. Also the potentials for some configurations involving charges inside a conducting torus are found in terms of Legendre

functions. For each solution in the basis found by this separation, reconstructing the potential from the charge distribution

(corresponding to singularities in the solutions) gives rise to integral relations involving Legendre functions.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Laplace equation; Separation of variables; Toroidal coordinates; Legendre polynomials

1. Introduction

The method of separation of variables in various

coordinate systems is a classic approach to finding exact

solutions of Laplaces equation and has been thoroughly

studied [1]. One such set of coordinates is the toroidal

system, but it will be argued here that some of the

usefulness of this coordinate system has been hidden

because, while the usual way of separating the variables

is appropriate for some situations, there is another way

that is more suited to a certain class of problems, inparticular some interesting problems in electrostatics.

The toroidal coordinates[2]of any point are given by the

intersection of a torus, a sphere with its centre on the axis

of the torus (the z-axis), and an azimuthal half plane

(terminated by the z-axis). The radius and centre of the

sphere are determined by the spherical coordinate y,

the major and minor radii of the torus are given by the

toroidal coordinate Z, and the particular half plane is

specified by its azimuthal angle c. The scale of the

coordinates is determined by a length a (see Fig. 1). The

details of this orthogonal coordinate system are reviewed in

Section 2.

The traditional method of solving Laplaces equation by

separation of these variables [2] gives a complete basis of

solutions of the form cosh Z cos y1=2fZ Yy Cc,where fZ is an associated Legendre function Pq

p1=2cosh Z or Qq

p1=2cosh Z, Yy is sinpy or cospy, andCc is sin qc or cos qc. This basis is particularlyconvenient for axially symmetric situations, for then weset q 0 and the solution involves the Legendre functionsPp1=2cosh Z orQp1=2cosh Z (rather than the associatedLegendre functions). This type of solution can be found in

several textbooks [35]. An example from electrostatics is

the potential due to a charged conducting torus; this and

several other examples are briefly discussed in Appendix B.

Here, we show that there is an alternative separation that

gives a basis of the form r1=2fZ YyCc, where r a sinh Z=cosh Z cos y is the distance from thez-axis,fZis P

m

n1=2coth Z or Qmn1=2coth Z, Yy is sin my or cos my,

ARTICLE IN PRESS

www.elsevier.com/locate/elstat

0304-3886/$ - see front matterr 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.elstat.2005.11.005

E-mail address: [email protected].
http://www.elsevier.com/locate/elstathttp://www.elsevier.com/locate/elstat


2/9

andCc is sin ncor cos nc. This basis is more convenientfor situations where the boundary conditions do not

involve y, for then we set m 0 and the solution involvesthe Legendre functions Pn1=2cosh Z or Qn1=2cosh Z.We will see that there are some interesting configurations

in electrostatics where the boundary conditions are of

this type. The potential due to an arbitrary distributionof charge on a circle can be found in this way, and the

method can be used even when conducting half planes

(terminated by the z-axis) are also present. This enable

us, for example, to find an expression, as an infinite sum

over Legendre functions, for the potential due to a point

charge between two intersecting conducting planes. In the

case of a point charge outside a single half plane, the

infinite series can be summed to give a closed expression for

the potential. It is also possible to deal with some

distributions of fixed charge inside a portion of a

conducting torus when the ends of the portion are closed

off by conducting planes. These matters are discussed in

Sections 5 and 6.

The singularities of the solutions (of Laplaces equation)

found by this separation can be interpreted, in the

context of electrostatics, as distributions of charge.

Reconstructing the potentials from such a charge distribu-

tion (by adding the Coulomb potentials) gives rise to some

integrals involving the Legendre functions, including

Heines 1881 representation for Qn1=2 and an apparentlynew integral that can be expressed in terms of Pa.

Approaching these relations from the separation of

Laplaces equation throws light on the work of Cohl

et al. [6], who were mainly interested in the gravitational

applications of the theory.

2. Toroidal coordinates

In the toroidal system, the location of a point is given by

the coordinates Z, y, c where the cartesian coordinates are

x;y; z a=Dsinh Z cos c; sinh Z sin c; sin y (1)

with D: cosh Z cos y. (The notation A:B indicates thatA is defined to be B.) Thus, c is an azimuthal angledenoting a rotation about the z-axis, and the distance from

this axis is

r a=D sinh Z. (2)The range of the coordinates isZX0;poypp; 0pco2p.A little algebra shows that r2 z2 a2 2a2D1cosh Z 2ar coth Z, and the relation

coth Z r2 z2 a2

2ar (3)

will be often used below. Following from this equation, the

surfaces of constant Z are given byr a coth Z2 z2 a2=sinh2Z. (4)For any fixedZthis is the torus generated by rotating about

the z-axis a circle C of radius a= sinh Z centred atr a coth Z; z 0. As Z ! 1 this radius becomes smalland the torus collapses to the circle r a; z 0, which willbe referred to as the reference circle. As Z ! 0 both theradius, and the distance to the centre, of the circle C

become large; then that part of the torus that is within a

finite distance of the origin, coincides with the z-axis.

In the derivation of Eq. (3), 2a2D1 cosh Z can also bewritten as 2a2

2az cot y, so the surfaces of constant y are

given by

r2 z a cot y2 a2=sin2 y. (5)

ARTICLE IN PRESS

z=0

z=a

z=2a

z=3a

r=0

r=a

r=2a

Fig. 1. The toroidal coordinates of any point are given by the intersection

of a sphere, a torus, and an azimuthal half plane. The torus shown here

has Z 1 and the sphere has sphere y p=4. [Then, according to Eq. (1),r 1:40a an d z 0:84a.]

r = a

Fig. 2. Thereference circle, r a, z 0 of the coordinate system is theintersection of the sphere with the plane z

0. For any Z it lies inside the

torus.

M. Andrews / Journal of Electrostatics 64 (2006) 664672 665


3/9

For any fixed y this is a sphere of radius a=j sin yj, centredon z a cot y; r 0. This sphere intersects the plane z 0in the reference circle. (SeeFig. 2).

3. Alternative separation of Laplaces equation

In toroidal coordinates Laplaces equation r2V 0becomes[2]

qZD1 sinh ZqZV qyD1 sinh ZqyV D sinh Z1qccV 0, 6

which is not immediately separable. But inserting VU=

ffiffir

p into this equation gives

sinh2 Z qZZU qyyU qccU 14U 0,which does separate giving solutions that are products of

fZ, sin my or cos my, and sin nc or cos nc, wheresinh2 Z f00 m2f n2 1

4f 0.

Changing variable from Z to w: coth Z converts thisequation to

w2 1f00 2wf0 m2

w2 1 n2 1

4

f 0, (7)

and the solutions of this are the associated Legendre

functions Pm

n1=2w and Qmn1=2w. Therefore, the solutionsof Laplaces equation are products of

ffiffiffiffiffiffiffia=r

p ,P

m

n1=2coth Zor Q

m

n

1=2

coth Z

, sin my or cos my, and sin nc or cos nc.

This is a complete basis; but we will consider only solutionswith m 0, which still allows arbitrary dependence on Zand c.

4. Singularities in the solutions

From the differential equation (7), singularities of the

Legendre functions Paw or Qaw, as a function of thecomplex variable w, can occur only for w 1 or w 1.We need consider only the region wX1. Care is needed in

accessing information about these functions, because thesesingularities, if present, are branch points and produce some

ambiguities. Formulae or numerical values appropriate for

applications that involve wo1 may not be valid here.

For example, in Mathematica, the function denoted by

LegendreQ(a; w) is not satisfactory for our purposes; insteadwe must use LegendreQ(a; 0; 3; w), which is their notationfor the Legendre function of the second kind of type 3.

For this function the w-plane is not cut for wX1. For the P

variety of Legendre function, either LegendreP(a; w) orLegendreP(a; 0; 3; w) can be used, because these two func-tions are identical forwX1. The asymptotic behaviour[7,8]

near w

1 and w 1

is given in Table 1 , except that

P1=2wp1 ffiffiffiffiffiffiffiffiffi

2=wp

ln8w as w ! 1. Here, g is Eulersconstant and ca is the digamma function G0a=Ga.

Now apply the asymptotic behaviour in Table 1to find

the behaviour offfiffiffiffiffiffiffi

a=rp

Pn1=2w andffiffiffiffiffiffiffi

a=rp

Qn1=2w asfunctions ofr and z. Singularities can occur only for r 0or for w 1 or w ! 1. From w r2 z2 a2=2ar itfollows that wX1, and w 1 occurs only on the referencecircle. Also w ! 1 either for r ! 0 or for R ! 1, whereR:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 z2

p is the distance from the origin. The asympto-

tic behaviour in these three regions is given in Table 2,

where Cn:p1=2Gn=Gn 12 and Dn:p1=2Gn 12=Gn 1, and where d:r a

2

z2

1=2

is the distancefrom the reference circle.

Thus, the Q-solutions are bounded as r ! 0, havelogarithmic singularities at the reference circle, and

decrease as 1=R or faster as R ! 1. They thereforecorrespond in electrostatics to some finite distribution of

charge on the reference circle. The P-solutions diverge too

rapidly, both for r ! 0 and for R ! 1, to correspond tofinite distributions of charge. They are not singular on the

reference circle. They will be shown in Section 7 to be the

potentials due to distributions of charge along the z-axis;

but these distributions are not integrable, so the total

charge along the z-axis is infinite.

ARTICLE IN PRESS

Table 1

The asymptotic behaviour of the Legendre functions near the singular

points w 1 and w 1

Paw Qaw

w 1 1 12

aa 1w 1 g ca 1 12

lnw 1

2w 1 Ga 1

2

ffiffiffipp

Ga 1 2wa

ffiffiffip

p Ga 1

Ga 32 2w

a1

Table 2

The asymptotic behaviour offfiffiffiffiffiffiffi

a=rp

Pn1=2w andffiffiffiffiffiffiffi

a=rp

Qn1=2w in the three regions (i) r a (near the z-axis), (ii) d a (near the reference circle),and (iii) R a (far from the origin)

w ffiffiffiffiffiffiffi

a=rp

Pn1=2w n40ffiffiffiffiffiffiffi

a=rp

P1=2wffiffiffiffiffiffiffi

a=rp

Qn1=2w

r a z2 a22ar Cn

z2 a2a2

n1=2a

r

n 2p

affiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiz2 a2

p ln 4z2 a2ar Dn

z2 a2a2

n1=2r

a

nd a

1 d2

2a2

1 1 g c n 12

ln d

2a

R a R22ar Cn

R2

a2 n1=2

a

r n 2

p

a

Rln

4R2

ar DnR2

a2 n1=2

r

a n

M. Andrews / Journal of Electrostatics 64 (2006) 664672666


4/9

5. Charge distributed around a ring

To consider the potential due to a ring of charge, we use

toroidal coordinates with the ring as reference circle.

Continuity of the potential at c 2p requires that n be aninteger. The simplest case is wheren 0 so that the potentialhas no dependence on the azimuthal angle c. Then

V0:ffiffiffiffiffiffiffi

a=rp

Q1=2w (8)satisfies Laplaces equation except on the reference circle

and, from Table 2, approaches pa=R as R ! 1. It istherefore the potential due to a charge of q 4p20adistributed uniformly around the reference circle. Near this

circle, from Table 2, V0 ln8a=d. Therefore, applyingGausss law, the line-charge density on the reference circle is

2p0, which gives the same total charge q.When n n, an integer,

Vn:

ffiffiffiffiffiffiffia=rp Qn1=2w cos nc (9)can be analysed in a similar way, and of course sin ncwould do just as well as cos nc. Near the reference circleffiffiffiffiffiffiffi

a=rp

Qn1=2w has the same limit as for n 0; so the line-charge density corresponding to Vn is 2p0cos nc. Thetotal charge is zero; so as R ! 1, Vn falls off faster than1=R. In the case ofn 1, one half of the ring has positivecharge while the other half is negative. Then the ring will

have a dipole moment that can be deduced to be 2ffiffiffi

pp

0a2

from the asymptotic behaviour V112ffiffiffiffiffi

pp

a2R3r cos c.Since we have found the potential for any sinusoidal line-

charge density on the reference circle, we can find the

potential due to any distribution of charge on the reference

circle by expressing it as a Fourier series. The only case thatwill be dealt with explicitly here is the delta function; this

will give the potential due to a point charge. The

appropriate Fourier series, for functions that match both

in magnitude and derivative at c 0 and c 2p, is [9]

dc c0 12p

X1n1

e{ncc0

12p

X1n0

dncos nc c0, 10

where d0 1 and dn 2 for n 1; 2; 3;. . . . Since a line-charge density of 2p0cos nf produced the potentialffiffiffiffiffiffiffi

a=rp

Qn1=2w cos nf, it follows that a point charge q atanglec0 on the reference circle, which will have line-chargedensityq=adc c0, will produce the potential

Vd q4p20

ffiffiffiffiffiar

pX1n0

dnQn1=2w cos nc c0

q4p20

ffiffiffiffiffiar

pX1

n1Qn1=2we{ncc

0, 11

where the latter form uses[6] Qn1=2w Qn1=2w. ThisVd must, of course, be just the Coulomb potential

q=

4p0

jr

r0

j, where in cylindrical coordinates r

r; z; c and r0 a; 0; c0. Therefore, writing r0 instead of

afor greater symmetry, we have the mathematical identity

1

jr r0j 1

pffiffiffiffiffi

rr0p

X1n1

Qn1=2we{ncc0 (12)

wherew r2 r02 z2=2rr0. Since jr r0j2 can be writtenasr2

r02

z2

2rr0cos

c

c0

. Eq. (12) can be recast as

X1n1

Qn1=2xe{nf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2x cos f

p . (13)As pointed out by Cohl et al. [6] this result, valid for any

xX1 and any angle f, was proved by Heine in 1881.

It may seem that little is gained by writing the inverse

distance between two points as this apparently more

complicated infinite sum, but Eq. (12) is claimed [6] to be

the basis for computationally advantageous methods in

astrophysics and possibly also in atomic physics. One

reason for this is that when applied to a spatial distribution

of charge (or mass) each term in this sum is effectively

dealing with a ring and not just a point. Also the sequenceof Legendre functions can be efficiently calculated because

they satisfy simple recurrence relations[7].

The potential due to a uniform or sinusoidal charge

distribution around a circle can also be found by direct

integration. If the line charge density on the circle of radius

a, at anglec0, is 2p0cos nc0, then the potential at the point

r r; z; c is

Vn1

2

Z 2p0

cos nc0

jr r0j a dc0

wherer0 a; 0; c0, as in the discussion leading to Eq. (12).Using

jr

r0

j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2arw cosc c0

p , as before, andcomparing with Eq, 9, gives the integral[10]Z p0

cos nfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2x cos f

p df Qn1=2x. (14)This integral is equivalent to Heines Eq. (13) through

Fourier theory.

5.1. Extension to charges inside a torus

These methods can be extended to deal with charge on

the reference circle inside a grounded conducting torus with

constant Z, say Z

Z0. For a uniform distribution of

charge on the reference circle, the potential is

Vffiffiffiffiffiffiffi

a=rp

Q1=2w P1=2wQ1=2w0=P1=2w0, (15)where w0: coth Z0. This is the correct potential because itsatisfies Laplaces equation, becomes zero at Z Z0, and hasthe same logarithmic singularity on the reference circle as in

Eq. (8) becauseffiffiffiffiffiffiffi

a=rp

P1=2w is not singular there. The linecharge density on the circle is therefore 2p0, as before.

Similarly, for a line charge density varying as cos nc the

potential is

V ffiffiffiffiffiffiffia=rp Qn1=2w Pn1=2wQn1=2w0=Pn1=2w0 cos nc. 16

ARTICLE IN PRESS



5/9

And for an arbitrary distribution of charge around the

reference circle, one can find the potential as a sum over

these terms using the Fourier series of the line charge

density. In particular, for a point charge q at angle c0 onthe reference circle, the potential is

V q4p20

ffiffiffiffiffiar

p X1n0

dn Qn1=2w

Pn1=2wQn1=2w0=Pn1=2w0 cos nc, 17

following the analysis leading to Eq. (11). The part

involving Qn1=2w is just the Coulomb potential due tothe charge q while the part involving Pn1=2w is thepotential due to the induced charge on the torus.

6. Charges between intersecting conducting planes

Take the rotational axis of the toroidal system to lie on

the intersection of the planes and let one of the planes bethe origin of the azimuthal angle, c 0. If b is the anglebetween the planes, then the second plane is at c b. Thesolutions of Laplaces equation (except on the reference

circle) that become zero on both planes areffiffiffiffiffiffiffi

a=rp

Qnp=b1=2w sinnpc=b, where n 1; 2; 3;. . . . These correspond toa line-charge density on the reference circle of 2p0sinnpc=b. Again one could construct the potential foran arbitrary charge along the portion of the reference

circle between the planes using its Fourier series. For a

point charge we need the delta-function appropriate

for functions that are zero at c 0 and at c b, andthat is [11]

dc c0 2b

X1n1

sinnpc

b sin

npc0

b . (18)

For a point charge q at c 0 we require a line charge densityq=adc c0 and therefore the potential is

V 14p0

4q

bffiffiffiffiffi

arp

X1n1

Qnp=b1=2w sinnpc

b sin

npc0

b . (19)

This problem of a point charge between two intersecting

conducting planes appears in Batygins collection [12].

There cylindrical coordinates were used to give a com-

pletely different expression for the potential.

An interesting special case is where b 2p. This is thecase of a point charge outside a single semi-infinite

conducting plane with a straight boundary (a half plane).

The potential is

V 14p0

2q

pffiffiffiffiffi

arp

X1n1

Q1=2n1=2w sin1

2nc sin

1

2nc0. (20)

Inserting 2 sin 12nc sin 1

2nc0

cos 1

2n

c

c0

cos 1

2n

c

c0

shows that we require sums of the form P1n1Q1=2n1=2

w cos 12nf. In the Appendix it is shown that

Sw; f

:X1n1

Q1=2n1=2w cos1

2nf 1

2Q1=2w

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w cos f

p 12 p arctan

2 cos 12fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2w cos fp !" #

. 21

Therefore, the potentialVr at position r with cylindricalcoordinates r; z; c due to a point chargeq at r0; z0; c0 anda conducting half plane at c 0 is

Vr 14p0

q

pffiffiffiffiffi

rr0p Sw; c c0 Sw; c c0, (22)

where w r2 r02 z z02=2rr0. An expressionequivalent to Eq. (22) has been found using a different

method [15]. Fig. 3 shows the potential (on a plane of

constant z) for an example of this system.

ARTICLE IN PRESS

0 1 2

0

1

2

x

y

(b)

0.010.020.050.1

0.2

0.5

1 2 5

0

2

x

0

1

2

(a) y

1

Fig. 3. The potential due to a point charge outside a conducting half plane

P. Here P has y 0, x40 and the charge is one unit from the edge of Pand half a unit from P (so that c 30). The diagrams show the potentialon the plane perpendicular to the edge of P and 0.3 units from the charge.

(a) General features of the potential. (b) Some equipotential curves; but

note that the values of the potential are relatively small behind P.

[All parameters are the same in the two diagrams.]



6/9

The methods of this section can be combined with those

in Section 5 to deal with charges on the reference circle

inside a portion of a torus (with Z Z0) closed off byconducting planar ends at c 0 and c b. Thus, thepotential when there is a point charge q at c c0 is

V 14p0

4q

bffiffiffiffiffi

rr0p

X1n1

Qnp=b1=2w Pnp=b1=2w

Qnp=b1=2w0=Pnp=b1=2w0 sinnpc

b sin

npc0

b . 23

An example is shown inFig. 4.

7. Reconstructing the P-solutions from their singularities

The solutions VPn:ffiffiffiffiffiffiffi

a=rp

Pn1=2coth Z cos nc of La-places equation correspond to a charge distribution along

the z-axis. Here, the potential will be reconstructed from

that charge distribution by adding the contributions from

each small part of the z-axis.

First, consider the case where n 0, so thatVP0:


p P1=2coth Z. From Table 2, VP0

2a=p

a2

z2

1=2 ln rclose to thez-axis. This corresponds

to a line charge density of lz 4a0a2 z21=2 on the

z-axis. This is not integrable to a finite amount of charge.

Integrating the Coulomb potential from each small

segment of the z-axis gives

VP0 1

4p0

Z 11

lz0 dz0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 z0 z2q a

p

Z 11

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 z02

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 z0 z2

q .The correctness of the relation

a

p

Z 11



q

ffiffiffia

r

r P1=2

a2 r2 z22ar

24

confirms that the solution VP0 is just the potential due to

the charge density l

z

along the z-axis. I have not found

the integral in Eq. (24) in any of the standard collections,and Mathematica and Maple fail on it although in

principle it can be treated as an elliptic integral. It is a

special case (n 0) of Eq. (27) proved below. The relationbetween P1=2 and the complete elliptic integral [7] is wellknown, and henceZ 1

1



q 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

a

2

z2q

K r a2 z2r a2 z2

. 25

For n 1; 2; 3;. . . Table 2shows that, for r a,VPn:


p Pn1=2coth Z cos ncMzrn cos nc, (26)

where Mz:Cnan1z2 a2n1=2. The potentialrn cos nc corresponds to what one might call a cylindricaln-pole. It satisfies Laplaces equation except at r 0 andcorresponds to 2n lines of charge, all parallel to the z-axis,

alternatively positive and negative, arranged to make a

cylinder coaxial with the z-axis, in the limit where the

radius of the cylinder tends to zero. So Eq. (26) implies that

VPn is produced by an n-pole of strength Mz. The radialcomponent E

r of the electric field due to the potential

rn cos ncis Er nrn1 cos nc. To generateVPn consider acylinder of radius b a, with surface charge densitysz; c 0ErMz 0nMzbn1 cos nc. To calculatethe potential due to this cylinder of charge, take a

slice of height dz0 at z z0. It will be a circle of radius bwith line charge density 2sz0; c dz0. (The extra factorof 2 is required because only half of the charge on

the cylinder contributes to the outward field.) We al-

ready know [Eq. (9)] that the potential at r; z; c dueto a circle of radius b at z0 with line charge density2p0cos nc is

ffiffiffiffiffiffiffib=r

p Qn1=2r2 z z02 b2=2br cos nc,

and from Table 2, for R

b this potential will be-

come Dnb2=R2n1=2r=bn where R2 r2 z z02.

ARTICLE IN PRESS

Fig. 4. The potential due to a point charge inside a portion of a

conducting torus and on the reference circle of the torus. The portion is

closed by conducting planar ends at c 0 and c b. In this example,Z 2,b 45 and the charge is at c 10. The potential is shown for theplanez 0:2a. [ForZ 2 the inner radius of the torus is 0:2757::a:] Part ofthe reference circle is shown and the position of the charge is indicated by

a heavy dot. Also shown are the two part-circles where the torus intersects

the z 0 plane.



7/9

Therefore, combining the contributions from all these

slices, and using CnDn 1=n,rn

pan1

Z 11

a2 z02n1=2dz0r2 z z02n1=2

ffiffiffi

a

r

r Pn1=2

a2 r2 z22ar

.

(27)

The correctness of this relation shows that VPn is solely dueto the n-pole distribution along the z-axis.

To verify Eq. (27) put a 1 for simplicity and substituteu az0 1=z0 a into I: R11z02 1nz0 z2r2n1 dz0, with a a1 z1r2 z2 1, to giveI 21 z=an1 R1

0 u2 1nu2 t2n1 du, where t2

1az=1z=a. But 1az1z=a r2 sot 1 az=rand t1 1 z=a=r. Substituting u x2 in I puts itinto a standard hypergeometric form [13] giving Ipt rn1Fn 1;12; 1; 1 t2, which can be expressed as[14] I prn1Pn12 t t1, and t t1 r2 z2 1=r.Now restoring a gives Eq. (27).

It is remarkable that the integral in Eq. (27) can be sosimply expressed in terms of the Legendre function, and we

have shown that the relation is valid for any n, even though

the context here requiresnto be an integer for continuity inc.

8. Conclusion

An alternative method of separating variables in toroidal

coordinates provides a simple route to a basis of solutions

of Laplaces equation appropriate for boundary conditions

that are independent of the spherical coordinate y. This

gives the potential for a class of problems in electrostatics,

and reconstructing the potential from the charge distribu-tions (corresponding to singularities in the solutions) gives

rise to some relations involving Legendre functions.

Appendix A. Sum of series over Legendre-Q functions

We require sums of the form

X1n1

Q1=2n1=2w cos1

2nf

X1

n0Qnw cos n

1

2

f

X1

n1Qn1=2w cos nf. A:1

The second of these sums can be found from HeinesEq. (13)

X1n1

Qn1=2w cos nf 1

2Q1=2w

12

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w cos f

p . (A.2)From the generating relation (see a few lines below) we can

deduce the first

X1n0

Qnw cos n 1

2

f

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w cos fp arctan

2cos 12

fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w cos fp !. A:3

Adding these gives

Sw; f

:X1n1

Q1=2n1=2w cos1

2nf 1

2Q1=2w

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w cos f

p 12 p arctan

2cos 1

2

fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w cos f

p !" #.A:4

A.1. Derivation of Eq. (A.3)

The generating relation[16,17]for Qnw isX1n0

hnQnw

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2wh h2p ln w h ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2wh h2pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w2 1p !

. A:5

If h e{f, then 1 2wh h2 2e{fw cos f. Werequire w41 and therefore, with u:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w cos f

p ,

X1n0

Qnwe{n1=2f

{u

ln w e{f {e1=2{fuffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w2 1p

{u

lnu 2sin 12 f

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2 1

p u 2{ cos 12

f

,

and the real and imaginary parts of this giveX1n0

Qnw cos n 1

2

f 1

uarctan

2cos 12

f

u

(A.6)

X1n0

Qnw sin n 1

2

f

1u

lnu 2sin 1

2fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2w 1p

! 1

uarcsinh

2sin 12

fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w 1

p !

.

A:7

Appendix B. Comparison with the traditional separation

The method of separation usually found in textbooks

[1,4] inserts Vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cosh Z cos yp

U into Laplaces (6)

instead of Vffiffiffiffiffiffiffi

1=rp

U as in Section 3. This is not

essentially different, because cosh Z cos y a=r sin y,but the separation is slightly different and leads to

associated Legendre functions of cosh Z (instead of

coth Z). The result is solutions of Laplaces equation that

are products offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cosh Z cos yp

, Pm

n1=2cosh Z orQ

m

n1=2cosh Z, sin ny or cos ny, and sin mc or cos mc. Notethat now the lower index is associated with theydependence

(while in Section 3 it was the upper index). This means that

ARTICLE IN PRESS



8/9

in this traditional approach the simpler Legendre functions

will suffice for axially symmetric situations. (Another way to

see the equivalence of the two approaches to separation of

the variables is to note[18]that bothffiffiffiffiffiffiffiffiffiffiffiffi

sinh Zp

Pm

n1=2cosh Zand

ffiffiffiffiffiffiffiffiffiffiffiffisinh Z

p Q

m

n1=2cosh Z can be expressed as linearcombinations of Pnm1=2coth Z and Qnm1=2coth Z.Therefore, a term (in the traditional expansion) of theform

ffiffiffiffiD

p P

m

n1=2cosh Z sin ny sin mc can be written, usingEq. (2), as a linear combination of two terms of the form

r1=2 Pnm1=2coth Z sin ny sin mc and r1=2 Qnm1=2coth Zsin ny sin mc.)

Thus for axially symmetric systems, the potential can be

written as a sum of terms that are products of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh Z cos y

p , Pn1=2cosh Z or Qn1=2cosh Z, and

sin ny or cos ny. (The continuity in y requires that

n 0; 1; 2;. . . .)

B.1. Example: The potential outside a charged conducting

torus

The Qn1=2cosh Z are too divergent at Z 0 (whichcorresponds to the z-axis). The boundary condition that

V V0 (a constant) for Z Z0 (specifying the conductingtoroidal surface), being even in y, excludes terms in sin y.

Thus, the potential outside the torus must have the form

Vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cosh Z cos yp X1

n0anPn1=2cosh Z cos ny. (B.1)

Imposing the condition that V V0 for Z Z0 is easilydone by comparing this equation with Heines Eq. (13) in

the form

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2x cos yp X1

n0dnQn1=2x cos ny, (B.2)

where d0 1 and dn 2 for n40. Thus, the potentialoutside the conducting torusZ Z0 at potential V V0 is

V V0p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cosh Z cos y

p X1n0

dn

Qn1=2cosh Z0Pn1=2cosh Z0

Pn1=2cosh Z cos ny. B:3

B.2. Further examples

These deal with some cases where there are charges

inside a torus.

1. The potential inside the torus Z Z0, when there is auniformly charged ring on the reference circle (r a,z 0), has the formV

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh Z cos y

p P1=2cosh Z

Q1=2cosh Z P1=2cosh Z0=Q1=2cosh Z0. B:4The line-charge density can be deduced from the

logarithmic singularity in P

1=2

cosh Z

at the ring, as for

Eq. (8). This case was also considered in Section 5; it can be

treated by either approach because the boundary condi-

tions do not depend on y or c. The two different looking

results, Eq. (B.4) and Eq. (15), are equivalent because [19]

P1=2cosh Z 1

p

ffiffiffiffiffiffiffiffiffiffiffiffi2

sinh Z

s Q1=2coth Z, (B.5)

Q1=2cosh Z pffiffiffiffiffiffiffiffiffiffiffiffi

2

sinh Z

s P1=2coth Z. (B.6)

2. Similarly,


cosh Z cos yp

P1=2cosh Z Q1=2cosh Z P1=2cosh Z0=Q1=2cosh Z0 cos yB:7

corresponds to a uniform line dipole around the reference

circle inside the torus. The orientation of the dipole can be

arbitrarily changed since the latter cos ycan be replaced by

cosy

y0

.

3. The potential between two tori (with the same

reference circle r a, z 0) held at different potentialscan be expressed as a sum of the form


cosh Z cos yp X1

n0anPn1=2cosh Z

bnQn1=2cosh Z cos ny B:8and the coefficients an and bn can be found by solving the

two linear equations that come fromV V0 at Z Z0 andV V1 at Z Z1, and comparing with Eq. (B.2) in eachcase.

References

[1] P. Moon, D.E. Spencer, Field Theory Handbook, Springer, Berlin,

1961.

[2] P. Moon, D.E. Spencer, Field Theory Handbook, Springer, Berlin,

pp. 112115.

[3] J. Vanderlinde, Classical Electromagnetic Theory, Wiley, New York,

1993, pp. 356360.

[4] W.R. Smythe, Static and Dynamic Electricity, McGraw-Hill,

London, 1939, p. 60.

[5] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, London, 1941,

p. 218.

[6] H.S. Cohl, J.E. Tohline, A.R.P. Rau, H.M. Srivastava, Astron.

Nachr. 321 (5/6) (2000) 363372.[7] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions,

National Bureau of Standards, 1964 (Chapter 8).

[8] The asymptotic behavior of the Legendre functions can be gleaned

from Ref. [7], but an easier source to use is at the web address:

ofunctions.wolfram.com/HypergeometricFunctions4.

[9] G. Barton, Elements of Greens Functions and Propagation, Oxford

University Press, Oxford, 1989 (Eq. 1.3.11).

[10] N.N. Lebedev, Special Functions and their Applications, Prentice-

Hall, Englewood Cliffs, NJ, 1965, p. 188.

[11] G. Barton, Elements of Greens Functions and Propagation, Oxford

University Press, Oxford, 1989 (Eq. 1.3.9).

[12] V.V. Batygin, I.N. Toptygin, Problems in Electrodynamics, Academic

Press, New York, 1962, pp. 4546.

[13] A. Erdlyi (Ed.), Higher Transcendental Functions, vol. I, McGraw-

Hill, London, 1953, p. 115, Eq. (5).

ARTICLE IN PRESS



9/9

[14] A. Erdlyi (Ed.), Higher Transcendental Functions, vol. I, McGraw-

Hill, London, 1953, p. 173, Eq. (5).

[15] K.I. Nikoskinen, I.V. Lindell, IEEE Trans. Antennas Propagation 43

(2) (1995) 179187.

[16] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, fourth

ed., Cambridge University Press, Cambridge, 1962, p. 321.

[17] E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics,

Cambridge University Press, Cambridge, 1939, p. 69.


Nachr. 321 (5/6) (2000) 363372 (Eq. 31, 32).


Nachr. 321 (5/6) (2000) 363372 (Eq. 33, 34).

ARTICLE IN PRESS


mark andrews.toroidalcoords

Documents