Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics

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Journal of Electrostatics 64 (2006) 664–672

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Alternative separation of Laplace’s equation in toroidal coordinates andits 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 Laplace’s 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 (ratherthan the toroidal coordinate Z or the azimuthal coordinate cÞ. Applying these solutions to electrostatics leads to solutions, given as

infinite 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 variouscoordinate systems is a classic approach to finding exactsolutions of Laplace’s equation and has been thoroughlystudied [1]. One such set of coordinates is the toroidalsystem, but it will be argued here that some of theusefulness of this coordinate system has been hiddenbecause, while the usual way of separating the variablesis appropriate for some situations, there is another waythat 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 theintersection of a torus, a sphere with its centre on the axisof the torus (the z-axis), and an azimuthal half plane(terminated by the z-axis). The radius and centre of thesphere are determined by the spherical coordinate y,the major and minor radii of the torus are given by thetoroidal coordinate Z, and the particular half plane is

e front matter r 2005 Elsevier B.V. All rights reserved.

stat.2005.11.005

ess: [email protected].

specified by its azimuthal angle c. The scale of thecoordinates is determined by a length a (see Fig. 1). Thedetails of this orthogonal coordinate system are reviewed inSection 2.The traditional method of solving Laplace’s equation by

separation of these variables [2] gives a complete basis ofsolutions of the form ðcosh Z� cos yÞ1=2f ðZÞ YðyÞ CðcÞ,where f ðZÞ is an associated Legendre function P

q

p�1=2ðcosh ZÞ or Q

q

p�1=2ðcosh ZÞ, YðyÞ is sin py or cos py, andCðcÞ is sin qc or cos qc. This basis is particularlyconvenient for axially symmetric situations, for then weset q ¼ 0 and the solution involves the Legendre functionsPp�1=2ðcosh ZÞ or Qp�1=2ðcosh ZÞ (rather than the associatedLegendre functions). This type of solution can be found inseveral textbooks [3–5]. An example from electrostatics isthe potential due to a charged conducting torus; this andseveral other examples are briefly discussed in Appendix B.Here, we show that there is an alternative separation that

gives a basis of the form r�1=2f ðZÞ YðyÞCðcÞ, where r ¼

a sinh Z=ðcosh Z� cos yÞ is the distance from the z-axis, f ðZÞis P

mn�1=2ðcoth ZÞ or Q

mn�1=2ðcoth ZÞ, YðyÞ is sin my or cos my,

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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 ¼ 1 and the sphere has sphere y ¼ p=4. [Then, according to Eq. (1),

r � 1:40a and z � 0:84a.]

r = a

Fig. 2. The reference circle, r ¼ a, z ¼ 0 of the coordinate system is the

intersection of the sphere with the plane z ¼ 0. For any Z it lies inside the

torus.

M. Andrews / Journal of Electrostatics 64 (2006) 664–672 665

and CðcÞ is sin nc or cos nc. This basis is more convenientfor situations where the boundary conditions do notinvolve y, for then we set m ¼ 0 and the solution involvesthe Legendre functions Pn�1=2ðcosh ZÞ or Qn�1=2ðcosh ZÞ.We will see that there are some interesting configurationsin electrostatics where the boundary conditions are ofthis type. The potential due to an arbitrary distributionof charge on a circle can be found in this way, and themethod can be used even when conducting half planes(terminated by the z-axis) are also present. This enableus, for example, to find an expression, as an infinite sumover Legendre functions, for the potential due to a pointcharge between two intersecting conducting planes. In thecase of a point charge outside a single half plane, theinfinite series can be summed to give a closed expression forthe potential. It is also possible to deal with somedistributions of fixed charge inside a portion of aconducting torus when the ends of the portion are closedoff by conducting planes. These matters are discussed inSections 5 and 6.

The singularities of the solutions (of Laplace’s equation)found by this separation can be interpreted, in thecontext of electrostatics, as distributions of charge.Reconstructing the potentials from such a charge distribu-tion (by adding the Coulomb potentials) gives rise to someintegrals involving the Legendre functions, includingHeine’s 1881 representation for Qn�1=2 and an apparentlynew integral that can be expressed in terms of Pa.Approaching these relations from the separation ofLaplace’s equation throws light on the work of Cohlet al. [6], who were mainly interested in the gravitationalapplications of the theory.

2. Toroidal coordinates

In the toroidal system, the location of a point is given bythe coordinates Z, y, c where the cartesian coordinates are

ðx; y; zÞ ¼ ða=DÞðsinh Z cosc; sinh Z sinc; sin yÞ (1)

with D:¼ cosh Z� cos y. (The notation A:¼B indicates thatA is defined to be B.) Thus, c is an azimuthal angledenoting a rotation about the z-axis, and the distance fromthis axis is

r ¼ ða=DÞ sinh Z. (2)

The range of the coordinates is ZX0;�poypp; 0pco2p.A little algebra shows that r2 þ z2 þ a2 ¼ 2a2D�1

cosh Z ¼ 2ar coth Z, and the relation

coth Z ¼r2 þ z2 þ a2

2ar(3)

will be often used below. Following from this equation, thesurfaces of constant Z are given by

ðr� a coth ZÞ2 þ z2 ¼ a2=sinh2Z. (4)

For any fixed Z this is the torus generated by rotating aboutthe z-axis a circle C of radius a= sinh Z centred atr ¼ a coth Z; z ¼ 0. As Z!1 this radius becomes smalland the torus collapses to the circle r ¼ a; z ¼ 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 afinite distance of the origin, coincides with the z-axis.In the derivation of Eq. (3), 2a2D�1 cosh Z can also be

written as 2a2 þ 2az cot y, so the surfaces of constant y aregiven by

r2 þ ðz� a cot yÞ2 ¼ a2=sin2 y. (5)

ARTICLE IN PRESSM. Andrews / Journal of Electrostatics 64 (2006) 664–672666

For any fixed y this is a sphere of radius a=j sin yj, centredon z ¼ a cot y; r ¼ 0. This sphere intersects the plane z ¼ 0in the reference circle. (See Fig. 2).

3. Alternative separation of Laplace’s equation

In toroidal coordinates Laplace’s equation r2V ¼ 0becomes [2]

qZðD�1 sinh ZqZV Þ þ qyðD�1 sinh ZqyV Þ

þ ðD sinh ZÞ�1qccV ¼ 0, ð6Þ

which is not immediately separable. But inserting V ¼

U=ffiffirp

into this equation gives

sinh2 Z ðqZZU þ qyyUÞ þ qccU þ 14U ¼ 0,

which does separate giving solutions that are products off ðZÞ, sin my or cos my, and sin nc or cos nc, where

sinh2 Z ðf 00 � m2f Þ � ðn2 � 14Þf ¼ 0.

Changing variable from Z to w:¼ coth Z converts thisequation to

ðw2 � 1Þf 00 þ 2wf 0 �m2

w2 � 1þ n2 �

1

4

� �f ¼ 0, (7)

and the solutions of this are the associated Legendrefunctions P

mn�1=2ðwÞ and Q

mn�1=2ðwÞ. Therefore, the solutions

of Laplace’s equation are products offfiffiffiffiffiffiffia=r

p, P

mn�1=2ðcoth ZÞ

or Qmn�1=2ðcoth ZÞ, sin my or cos my, and sin nc or cos nc.

This is a complete basis; but we will consider only solutionswith m ¼ 0, which still allows arbitrary dependence on Zand c.

Table 1

The asymptotic behaviour of the Legendre functions near the singular

points w ¼ 1 and w ¼ 1

PaðwÞ QaðwÞ

w ¼ 1 1þ 12aðaþ 1Þðw� 1Þ

�g� cðaþ 1Þ �1

2ln

w� 1

2w ¼ 1 Gðaþ 1

2Þffiffiffi

pp

Gðaþ 1Þð2wÞa

ffiffiffipp

Gðaþ 1Þ

Gðaþ 32Þð2wÞ�a�1

Table 2

The asymptotic behaviour offfiffiffiffiffiffiffia=r

pPn�1=2ðwÞ and

ffiffiffiffiffiffiffia=r

pQn�1=2ðwÞ in the three

and (iii) R� a (far from the origin)

wffiffiffiffiffiffiffia=r

pPn�1=2ðwÞ ðn40Þ

r� a z2 þ a2

2ar Cnz2 þ a2

a2

� �n�1=2a

r

� �nd � a

1þd2

2a2

1

R� a R2

2ar CnR2

a2

� �n�1=2a

r

� �n

4. Singularities in the solutions

From the differential equation (7), singularities of theLegendre functions PaðwÞ or QaðwÞ, as a function of thecomplex variable w, can occur only for w ¼ �1 or w ¼ 1.We need consider only the region wX1. Care is needed inaccessing information about these functions, because thesesingularities, if present, are branch points and produce someambiguities. Formulae or numerical values appropriate forapplications that involve wo1 may not be valid here.For example, in Mathematica, the function denoted byLegendreQ(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 Pvariety of Legendre function, either LegendreP(a;w) orLegendreP(a; 0; 3;w) can be used, because these two func-tions are identical for wX1. The asymptotic behaviour [7,8]near w ¼ 1 and w ¼1 is given in Table 1 , except thatP�1=2ðwÞ�p�1

ffiffiffiffiffiffiffiffiffi2=w

plnð8wÞ as w!1. Here, g is Eulers

constant and cðaÞ is the digamma function G0ðaÞ=GðaÞ.Now apply the asymptotic behaviour in Table 1 to find

the behaviour offfiffiffiffiffiffiffia=r

pPn�1=2ðwÞ and


pQn�1=2ðwÞ as

functions of r and z. Singularities can occur only for r ¼ 0

or for w ¼ 1 or w!1. From w ¼ ðr2 þ z2 þ a2Þ=ð2arÞ itfollows that wX1, and w ¼ 1 occurs only on the referencecircle. Also w!1 either for r! 0 or for R!1, where

R:¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2p

is the distance from the origin. The asympto-tic behaviour in these three regions is given in Table 2,

where Cn:¼p�1=2GðnÞ=Gðnþ 12Þ and Dn:¼p1=2Gðnþ 1

2Þ=

Gðnþ 1Þ, and where d:¼½ðr� aÞ2 þ z2�1=2 is the distancefrom the reference circle.Thus, the Q-solutions are bounded as r! 0, have

logarithmic singularities at the reference circle, anddecrease as 1=R or faster as R!1. They thereforecorrespond in electrostatics to some finite distribution ofcharge on the reference circle. The P-solutions diverge toorapidly, both for r! 0 and for R!1, to correspond tofinite distributions of charge. They are not singular on thereference circle. They will be shown in Section 7 to be thepotentials due to distributions of charge along the z-axis;but these distributions are not integrable, so the totalcharge along the z-axis is infinite.

regions (i) r� a (near the z-axis), (ii) d � a (near the reference circle),


pP�1=2ðwÞ


pQn�1=2ðwÞ

2

paffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z2 þ a2p ln

4ðz2 þ a2Þ

ar Dnz2 þ a2

a2

� ��n�1=2r

a

� �n1

�g� c nþ1

2

� �� ln

d

2a

2

pa

Rln4R2

ar DnR2

a2

� ��n�1=2r

a

� �n

ARTICLE IN PRESSM. Andrews / Journal of Electrostatics 64 (2006) 664–672 667

5. Charge distributed around a ring

To consider the potential due to a ring of charge, we usetoroidal coordinates with the ring as reference circle.Continuity of the potential at c ¼ 2p requires that n be aninteger. The simplest case is where n ¼ 0 so that the potentialhas no dependence on the azimuthal angle c. Then

V 0:¼ffiffiffiffiffiffiffia=r

pQ�1=2ðwÞ (8)

satisfies Laplace’s equation except on the reference circleand, from Table 2, approaches pa=R as R!1. It istherefore the potential due to a charge of q ¼ 4p2�0adistributed uniformly around the reference circle. Near thiscircle, from Table 2, V 0� lnð8a=dÞ. Therefore, applyingGauss’s law, the line-charge density on the reference circle is2p�0, which gives the same total charge q.

When n ¼ n, an integer,

V n:¼ffiffiffiffiffiffiffia=r

pQn�1=2ðwÞ 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

Qn�1=2ðwÞ has the same limit as for n ¼ 0; so the line-charge density corresponding to Vn is 2p�0 cos nc. Thetotal charge is zero; so as R!1, Vn falls off faster than1=R. In the case of n ¼ 1, one half of the ring has positivecharge while the other half is negative. Then the ring willhave a dipole moment that can be deduced to be 2

ffiffiffipp

�0a2

from the asymptotic behaviour V1�12

ffiffiffiffiffipp

a2R�3r cosc.Since we have found the potential for any sinusoidal line-

charge density on the reference circle, we can find thepotential due to any distribution of charge on the referencecircle by expressing it as a Fourier series. The only case thatwill be dealt with explicitly here is the delta function; thiswill give the potential due to a point charge. Theappropriate Fourier series, for functions that match bothin magnitude and derivative at c ¼ 0 and c ¼ 2p, is [9]

dðc� c0Þ ¼1

2p

X1n¼�1

e{nðc�c0Þ

¼1

2p

X1n¼0

dn cos nðc� c0Þ, ð10Þ

where d0 ¼ 1 and dn ¼ 2 for n ¼ 1; 2; 3; . . . . Since a line-charge density of 2p�0 cos nf produced the potentialffiffiffiffiffiffiffi

a=rp

Qn�1=2ðwÞ cos nf, it follows that a point charge q atangle c0 on the reference circle, which will have line-chargedensity ðq=aÞdðc� c0Þ, will produce the potential

V d ¼q

4p2�0ffiffiffiffiffiarp

X1n¼0

dnQn�1=2ðwÞ cos nðc� c0Þ

¼q


X1n¼�1

Qn�1=2ðwÞe{nðc�c0Þ, ð11Þ

where the latter form uses [6] Qn�1=2ðwÞ ¼ Q�n�1=2ðwÞ. ThisV d must, of course, be just the Coulomb potentialq=ð4p�0jr� r0jÞ, where in cylindrical coordinates r ¼

ðr; z;cÞ and r0 ¼ ða; 0;c0Þ. Therefore, writing r0 instead of

a for greater symmetry, we have the mathematical identity

1

jr� r0j¼

1

pffiffiffiffiffirr0p

X1n¼�1

Qn�1=2ðwÞe{nðc�c0Þ (12)

where w ¼ ðr2 þ r02 þ z2Þ=ð2rr0Þ. Since jr� r0j2 can be writtenas r2 þ r02 þ z2 � 2rr0 cosðc� c0Þ. Eq. (12) can be recast as

X1n¼�1

Qn�1=2ðxÞe{nf ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðx� cosfÞ

p . (13)

As pointed out by Cohl et al. [6] this result, valid for anyxX1 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 morecomplicated infinite sum, but Eq. (12) is claimed [6] to bethe basis for computationally advantageous methods inastrophysics and possibly also in atomic physics. Onereason for this is that when applied to a spatial distributionof charge (or mass) each term in this sum is effectivelydealing with a ring and not just a point. Also the sequenceof Legendre functions can be efficiently calculated becausethey satisfy simple recurrence relations [7].The potential due to a uniform or sinusoidal charge

distribution around a circle can also be found by directintegration. If the line charge density on the circle of radiusa, at angle c0, is 2p�0 cos nc0, then the potential at the pointr ¼ ðr; z;cÞ is

Vn ¼1

2

Z 2p

0

cos nc0

jr� r0jadc0

where r0 ¼ ða; 0;c0Þ, as in the discussion leading to Eq. (12).Using jr� r0j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2arðw� cosðc� c0ÞÞ

p, as before, and

comparing with Eq, 9, gives the integral [10]Z p

0

cos nfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðx� cosfÞ

p df ¼ Qn�1=2ðxÞ. (14)

This integral is equivalent to Heine’s Eq. (13) throughFourier theory.

5.1. Extension to charges inside a torus

These methods can be extended to deal with charge onthe reference circle inside a grounded conducting torus withconstant Z, say Z ¼ Z0. For a uniform distribution ofcharge on the reference circle, the potential is

V ¼ffiffiffiffiffiffiffia=r

p½Q�1=2ðwÞ � P�1=2ðwÞQ�1=2ðw0Þ=P�1=2ðw0Þ�, (15)

where w0:¼ coth Z0. This is the correct potential because itsatisfies Laplace’s equation, becomes zero at Z ¼ Z0, and hasthe same logarithmic singularity on the reference circle as inEq. (8) because


pP�1=2ðwÞ is not singular there. The line

charge density on the circle is therefore 2p�0, as before.Similarly, for a line charge density varying as cos nc the

potential is

V ¼ffiffiffiffiffiffiffia=r

p½Qn�1=2ðwÞ

� Pn�1=2ðwÞQn�1=2ðw0Þ=Pn�1=2ðw0Þ� cos nc. ð16Þ

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

01

2

(a)y

1

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

P. Here P has y ¼ 0, x40 and the charge is one unit from the edge of P

and half a unit from P (so that c ¼ 30�). The diagrams show the potential

on 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.]

M. Andrews / Journal of Electrostatics 64 (2006) 664–672668

And for an arbitrary distribution of charge around thereference circle, one can find the potential as a sum overthese terms using the Fourier series of the line chargedensity. In particular, for a point charge q at angle c0 onthe reference circle, the potential is

V ¼q


X1n¼0

dn ½Qn�1=2ðwÞ

� Pn�1=2ðwÞQn�1=2ðw0Þ=Pn�1=2ðw0Þ� cos nc, ð17Þ

following the analysis leading to Eq. (11). The partinvolving Qn�1=2ðwÞ is just the Coulomb potential due tothe charge q while the part involving Pn�1=2ðwÞ 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 onthe intersection of the planes and let one of the planes bethe origin of the azimuthal angle, c ¼ 0. If b is the anglebetween the planes, then the second plane is at c ¼ b. Thesolutions of Laplace’s equation (except on the referencecircle) that become zero on both planes are


pQnp=b�1=2

ðwÞ sinðnpc=bÞ, where n ¼ 1; 2; 3; . . . . These correspond toa line-charge density on the reference circle of 2p�0 sinðnpc=bÞ. Again one could construct the potential foran arbitrary charge along the portion of the referencecircle between the planes using its Fourier series. For apoint charge we need the delta-function appropriatefor functions that are zero at c ¼ 0 and at c ¼ b, andthat is [11]

dðc� c0Þ ¼2

b

X1n¼1

sinnpcb

sinnpc0

b. (18)

For a point charge q at c0 we require a line charge densityðq=aÞdðc� c0Þ and therefore the potential is

V ¼1

4p�0

4q

bffiffiffiffiffiarp

X1n¼1

Qnp=b�1=2ðwÞ sinnpcb

sinnpc0

b. (19)

This problem of a point charge between two intersectingconducting planes appears in Batygin’s collection [12].There cylindrical coordinates were used to give a com-pletely different expression for the potential.

An interesting special case is where b ¼ 2p. This is thecase of a point charge outside a single semi-infiniteconducting plane with a straight boundary (a half plane).The potential is

V ¼1

4p�0

2q

pffiffiffiffiffiarp

X1n¼1

Q1=2n�1=2ðwÞ sin1

2nc sin

1

2nc0. (20)

Inserting 2 sin 12nc sin 1

2nc0 ¼ cos 1

2nðc� c0Þ � cos 1

2nðcþ c0Þ

shows that we require sums of the formP1

n¼1Q1=2n�1=2

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

Sðw;fÞ

:¼X1n¼1

Q1=2n�1=2ðwÞ cos1

2nfþ

1

2Q�1=2ðwÞ

¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðw� cosfÞp 1

2pþ arctan

2 cos 12fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðw� cosfÞp !" #

. ð21Þ

Therefore, the potential V ðrÞ at position r with cylindricalcoordinates ðr; z;cÞ due to a point charge q at ðr0; z0;c0Þ anda conducting half plane at c ¼ 0 is

V ðrÞ ¼1

4p�0

q

pffiffiffiffiffirr0p ½Sðw;c� c0Þ � Sðw;cþ c0Þ�, (22)

where w ¼ ½r2 þ r02 þ ðz� z0Þ2�=ð2rr0Þ. An expressionequivalent to Eq. (22) has been found using a differentmethod [15]. Fig. 3 shows the potential (on a plane ofconstant z) for an example of this system.

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 ¼ 0 and c ¼ b. In this example,

Z ¼ 2, b ¼ 45� and the charge is at c ¼ 10�. The potential is shown for the

plane z ¼ 0:2a. [For Z ¼ 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 ¼ 0 plane.

M. Andrews / Journal of Electrostatics 64 (2006) 664–672 669

The methods of this section can be combined with thosein Section 5 to deal with charges on the reference circleinside a portion of a torus (with Z ¼ Z0) closed off byconducting planar ends at c ¼ 0 and c ¼ b. Thus, thepotential when there is a point charge q at c ¼ c0 is

V ¼1

4p�0

4q

bffiffiffiffiffirr0p

X1n¼1

½Qnp=b�1=2ðwÞ � Pnp=b�1=2ðwÞ

Qnp=b�1=2ðw0Þ=Pnp=b�1=2ðw0Þ� sinnpcb

sinnpc0

b. ð23Þ

An example is shown in Fig. 4.

7. Reconstructing the P-solutions from their singularities

The solutions VPn :¼


pPn�1=2ðcoth ZÞ cos nc of La-

place’s equation correspond to a charge distribution alongthe z-axis. Here, the potential will be reconstructed fromthat charge distribution by adding the contributions fromeach small part of the z-axis.

First, consider the case where n ¼ 0, so thatV P

0 :¼ffiffiffiffiffiffiffia=r

pP�1=2ðcoth ZÞ. From Table 2, VP

0��

ð2a=pÞða2 þ z2Þ�1=2 ln r close to the z-axis. This correspondsto a line charge density of lðzÞ ¼ 4a�0ða2 þ z2Þ�1=2 on the

z-axis. This is not integrable to a finite amount of charge.Integrating the Coulomb potential from each smallsegment of the z-axis gives

V P0 ¼

1

4p�0

Z 1�1

lðz0Þdz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðz0 � zÞ2

q¼

a

p

Z 1�1

dz0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ z02

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðz0 � zÞ2

q .

The correctness of the relation

a

p

Z 1�1



q¼

ffiffiffia

r

rP�1=2

a2 þ r2 þ z2

2ar

� �ð24Þ

confirms that the solution VP0 is just the potential due to

the charge density lðzÞ along the z-axis. I have not foundthe integral in Eq. (24) in any of the standard collections,and Mathematica and Maple fail on it although inprinciple it can be treated as an elliptic integral. It is aspecial case (n ¼ 0) of Eq. (27) proved below. The relationbetween P�1=2 and the complete elliptic integral [7] is wellknown, and henceZ 1�1



q

¼4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðrþ aÞ2 þ z2q K

ðr� aÞ2 þ z2

ðrþ aÞ2 þ z2

� �. ð25Þ

For n ¼ 1; 2; 3; . . . Table 2 shows that, for r� a,

VPn :¼


pPn�1=2ðcoth ZÞ cos nc�MðzÞr�n cos nc, (26)

where MðzÞ:¼Cn a�nþ1ðz2 þ a2Þn�1=2. The potential

r�n cos nc corresponds to what one might call a cylindricaln-pole. It satisfies Laplace’s equation except at r ¼ 0 andcorresponds to 2n lines of charge, all parallel to the z-axis,alternatively positive and negative, arranged to make acylinder coaxial with the z-axis, in the limit where theradius of the cylinder tends to zero. So Eq. (26) implies thatVP

n is produced by an n-pole of strength MðzÞ. The radialcomponent Er of the electric field due to the potentialr�n cos nc is Er ¼ nr�n�1 cos nc. To generate V P

n consider acylinder of radius b� a, with surface charge densitysðz;cÞ ¼ �0ErMðzÞ ¼ �0nMðzÞb�n�1 cos nc. To calculatethe potential due to this cylinder of charge, take aslice of height dz0 at z ¼ z0. It will be a circle of radius b

with line charge density 2sðz0;cÞdz0. (The extra factorof 2 is required because only half of the charge onthe cylinder contributes to the outward field.) We al-ready know [Eq. (9)] that the potential at ðr; z;cÞ dueto a circle of radius b at z0 with line charge density2p�0 cos nc is

ffiffiffiffiffiffiffib=r

pQn�1=2ðr

2 þ ðz� z0Þ2 þ b2Þ=ð2brÞ cos nc,

and from Table 2, for R� b this potential will be-come Dnðb

2=R2Þnþ1=2ðr=bÞn where R2 ¼ r2 þ ðz� z0Þ2.


Therefore, combining the contributions from all theseslices, and using CnDn ¼ 1=n,

rn

pan�1

Z 1�1

ða2 þ z02Þn�1=2dz0

ðr2 þ ðz� z0Þ2Þnþ1=2¼

ffiffiffia

r

rPn�1=2

a2 þ r2 þ z2

2ar

� �.

(27)

The correctness of this relation shows that VPn is solely due

to the n-pole distribution along the z-axis.To verify Eq. (27) put a ¼ 1 for simplicity and substitute

u ¼ ðaz0 þ 1Þ=ðz0 � aÞ into I :¼R1�1ðz02 þ 1Þn½ðz0 � zÞ2þ

r2��n�1 dz0, with a� a�1 ¼ z�1ðr2 þ z2 � 1Þ, to giveI ¼ 2ð1� z=aÞ�n�1

R10 ðu

2 þ 1Þnðu2 þ t2Þ�n�1 du, where t2 ¼

ð1þazÞ=ð1�z=aÞ. But ð1þazÞð1�z=aÞ ¼ r2 so t ¼ ð1þ azÞ=r

and t�1 ¼ ð1� z=aÞ=r. Substituting u ¼ x2 in I puts itinto a standard hypergeometric form [13] giving I ¼

pðt rÞ�n�1F ðnþ 1; 12; 1; 1� t�2Þ, which can be expressed as[14] I ¼ pr�n�1Pnð

12½tþ t�1�Þ, and tþ t�1 ¼ ðr2 þ z2þ 1Þ=r.

Now restoring a gives Eq. (27).It is remarkable that the integral in Eq. (27) can be so

simply expressed in terms of the Legendre function, and wehave shown that the relation is valid for any n, even thoughthe context here requires n to be an integer for continuity in c.

8. Conclusion

An alternative method of separating variables in toroidalcoordinates provides a simple route to a basis of solutionsof Laplace’s equation appropriate for boundary conditionsthat are independent of the spherical coordinate y. Thisgives the potential for a class of problems in electrostatics,and reconstructing the potential from the charge distribu-tions (corresponding to singularities in the solutions) givesrise to some relations involving Legendre functions.

Appendix A. Sum of series over Legendre-Q functions

We require sums of the form

X1n¼1


2nf

¼X1n¼0

QnðwÞ cos nþ1

2

� �fþ

X1n¼1

Qn�1=2ðwÞ cos nf. ðA:1Þ

The second of these sums can be found from Heine’sEq. (13)

X1n¼1

Qn�1=2ðwÞ cos nfþ1

2Q�1=2ðwÞ ¼

12pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðw� cosfÞp . (A.2)

From the generating relation (see a few lines below) we candeduce the first

X1n¼0

QnðwÞ cos nþ1

2

� �f


2ðw� cosfÞp arctan

2 cos 12fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðw� cosfÞ

p !

. ðA:3Þ

Adding these gives

Sðw;fÞ

:¼X1n¼1


2nfþ

1

2Q�1=2ðwÞ


2ðw� cosfÞp 1

2pþ arctan

2 cos 12fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðw� cosfÞp !" #

.

ðA:4Þ

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

The generating relation [16,17] for QnðwÞ isX1n¼0

hnQnðwÞ

¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 2whþ h2p ln

w� hþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2whþ h2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2 � 1p

!. ðA:5Þ

If h ¼ e{f, then 1� 2whþ h2¼ �2e{fðw� cosfÞ. We

require w41 and therefore, with u:¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðw� cosfÞ

p,

X1n¼0

QnðwÞe{ðnþ1=2Þf

¼{

uln

w� e{f þ {e1=2{fuffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2 � 1p

� �

¼{

uln

u� 2 sin 12f

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiw2 � 1p uþ 2{ cos

1

2f

� �� ,

and the real and imaginary parts of this give

X1n¼0

QnðwÞ cos nþ1

2

� �f ¼

1

uarctan

2 cos 12f

u

� �(A.6)

X1n¼0

QnðwÞ sin nþ1

2

� �f

¼�1

uln

u� 2 sin 12fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðw� 1Þp

!¼

1

uarcsinh

2 sin 12fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðw� 1Þp

!.

ðA:7Þ

Appendix B. Comparison with the traditional separation

The method of separation usually found in textbooks

[1,4] inserts V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh Z� cos y

pU into Laplace’s (6)

instead of V ¼ffiffiffiffiffiffiffi1=r

pU as in Section 3. This is not

essentially different, because cosh Z� cos y ¼ ða=rÞ sin y,but the separation is slightly different and leads toassociated Legendre functions of cosh Z (instead ofcoth Z). The result is solutions of Laplace’s equation thatare products of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh Z� cos y

p, P

mn�1=2ðcosh ZÞ or

Qmn�1=2ðcosh ZÞ, sin ny or cos ny, and sin mc or cos mc. Note

that now the lower index is associated with the y dependence(while in Section 3 it was the upper index). This means that

ARTICLE IN PRESSM. Andrews / Journal of Electrostatics 64 (2006) 664–672 671

in this traditional approach the simpler Legendre functionswill suffice for axially symmetric situations. (Another way tosee the equivalence of the two approaches to separation ofthe variables is to note [18] that both

ffiffiffiffiffiffiffiffiffiffiffiffisinh Z

pPmn�1=2ðcosh ZÞ

andffiffiffiffiffiffiffiffiffiffiffiffisinh Z

pQ

mn�1=2ðcosh ZÞ can be expressed as linear

combinations of Pnm�1=2ðcoth ZÞ and Qn

m�1=2ðcoth ZÞ.Therefore, a term (in the traditional expansion) of theform

ffiffiffiffiDp

Pmn�1=2ðcosh ZÞ sin ny sin mc can be written, using

Eq. (2), as a linear combination of two terms of the formr�1=2 Pn

m�1=2ðcoth ZÞ sin ny sin mc and r�1=2 Qnm�1=2ðcoth ZÞ

sin ny sin mc.)Thus for axially symmetric systems, the potential can be

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

p, Pn�1=2ðcosh ZÞ or Qn�1=2ðcosh ZÞ, and

sin ny or cos ny. (The continuity in y requires thatn ¼ 0; 1; 2; . . . .)

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

torus

The Qn�1=2ðcosh ZÞ are too divergent at Z ¼ 0 (whichcorresponds to the z-axis). The boundary condition thatV ¼ V0 (a constant) for Z ¼ Z0 (specifying the conductingtoroidal surface), being even in y, excludes terms in sin y.Thus, the potential outside the torus must have the form

V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh Z� cos y

p X1n¼0

anPn�1=2ðcosh ZÞ cos ny. (B.1)

Imposing the condition that V ¼ V 0 for Z ¼ Z0 is easilydone by comparing this equation with Heine’s Eq. (13) inthe form

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðx� cos yÞ

p X1n¼0

dnQn�1=2ðxÞ cos ny, (B.2)

where d0 ¼ 1 and dn ¼ 2 for n40. Thus, the potentialoutside the conducting torus Z ¼ Z0 at potential V ¼ V 0 is

V ¼V 0

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðcosh Z� cos yÞ

p X1n¼0

dn

Qn�1=2ðcosh Z0Þ

Pn�1=2ðcosh Z0ÞPn�1=2ðcosh ZÞ cos ny. ðB:3Þ

B.2. Further examples

These deal with some cases where there are chargesinside a torus.

1. The potential inside the torus Z ¼ Z0, when there is auniformly charged ring on the reference circle (r ¼ a,z ¼ 0), has the form


p½P�1=2ðcosh ZÞ

�Q�1=2ðcosh ZÞP�1=2ðcosh Z0Þ=Q�1=2ðcosh Z0Þ�. ðB:4Þ

The line-charge density can be deduced from thelogarithmic singularity in P�1=2ðcosh ZÞ at the ring, as forEq. (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 lookingresults, Eq. (B.4) and Eq. (15), are equivalent because [19]

P�1=2ðcosh ZÞ ¼1

p

ffiffiffiffiffiffiffiffiffiffiffiffi2

sinh Z

sQ�1=2ðcoth ZÞ, (B.5)

Q�1=2ðcosh ZÞ ¼ p

ffiffiffiffiffiffiffiffiffiffiffiffi2

sinh Z

sP�1=2ðcoth ZÞ. (B.6)

2. Similarly,


p½P1=2ðcosh ZÞ

�Q1=2ðcosh ZÞP1=2ðcosh Z0Þ=Q1=2ðcosh Z0Þ� cos y ðB:7Þ

corresponds to a uniform line dipole around the referencecircle inside the torus. The orientation of the dipole can bearbitrarily changed since the latter cos y can be replaced bycosðy� y0Þ.3. The potential between two tori (with the same

reference circle r ¼ a, z ¼ 0) held at different potentialscan be expressed as a sum of the form


p X1n¼0

½anPn�1=2ðcosh ZÞ

þ bnQn�1=2ðcosh ZÞ� cos ny ðB:8Þ

and the coefficients an and bn can be found by solving thetwo linear equations that come from V ¼ V0 at Z ¼ Z0 andV ¼ V 1 at Z ¼ 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. 112–115.

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

1993, pp. 356–360.

[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) 363–372.

[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. 45–46.

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

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


[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) 179–187.

[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) 363–372 (Eq. 31, 32).


Nachr. 321 (5/6) (2000) 363–372 (Eq. 33, 34).

Documents

Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics