ARTICLE IN PRESS
Journal of Electrostatics 64 (2006) 664–672
www.elsevier.com/locate/elstat
Alternative separation of Laplace’s 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 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 (rather
than 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 various
coordinate systems is a classic approach to finding exact
solutions of Laplace’s 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, in
particular 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
E-mail address: [email protected].
0304-3886/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.elstat.2005.11.005
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 Laplace’s equation by
separation of these variables [2] gives a complete basis of
solutions of the form ðcosh Z cos yÞ1=2 f ðZÞ YðyÞ CðcÞ,
where f ðZÞ is an associated Legendre function Pqp1=2
ðcosh ZÞ or Qqp1=2 ðcosh ZÞ, YðyÞ is sin py or cos py, and
CðcÞ is sin qc or cos qc. This basis is particularly
convenient for axially symmetric situations, for then we
set q ¼ 0 and the solution involves the Legendre functions
Pp1=2 ðcosh ZÞ or Qp1=2 ðcosh ZÞ (rather than the associated
Legendre functions). This type of solution can be found in
several textbooks [3–5]. 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=2 f ðZÞ YðyÞCðcÞ, where r ¼
a sinh Z=ðcosh Z cos yÞ is the distance from the z-axis, f ðZÞ
is Pmn1=2 ðcoth ZÞ or Qmn1=2 ðcoth ZÞ, YðyÞ is sin my or cos my,
ARTICLE IN PRESS
M. Andrews / Journal of Electrostatics 64 (2006) 664–672
665
2. Toroidal coordinates
r=0
r=a
In the toroidal system, the location of a point is given by
the coordinates Z, y, c where the cartesian coordinates are
r=2a
z=3a
z=2a
ðx; y; zÞ ¼ ða=DÞðsinh Z cos c; sinh Z sin c; sin yÞ
with D:¼ cosh Z cos y. (The notation A:¼B indicates that
A is defined to be B.) Thus, c is an azimuthal angle
denoting a rotation about the z-axis, and the distance from
this axis is
r ¼ ða=DÞ sinh Z.
z=a
z=0
(1)
(2)
The range of the coordinates is ZX0; poypp; 0pco2p.
A little algebra shows that r2 þ z2 þ a2 ¼ 2a2 D1
cosh Z ¼ 2ar coth Z, and the relation
r 2 þ z 2 þ a2
(3)
2ar
will be often used below. Following from this equation, the
surfaces of constant Z are given by
coth Z ¼
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.]
and CðcÞ is sin nc or cos nc. This basis is more convenient
for situations where the boundary conditions do not
involve y, for then we set m ¼ 0 and the solution involves
the Legendre functions Pn1=2 ðcosh ZÞ or Qn1=2 ðcosh ZÞ.
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 distribution
of 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 Laplace’s equation)
found by this separation can be interpreted, in the
context of electrostatics, as distributions of charge.
Reconstructing the potentials from such a charge distribution (by adding the Coulomb potentials) gives rise to some
integrals involving the Legendre functions, including
Heine’s 1881 representation for Qn1=2 and an apparently
new integral that can be expressed in terms of Pa .
Approaching these relations from the separation of
Laplace’s equation throws light on the work of Cohl
et al. [6], who were mainly interested in the gravitational
applications of the theory.
ðr a coth ZÞ2 þ z2 ¼ a2 =sinh2 Z.
(4)
For any fixed Z this is the torus generated by rotating about
the z-axis a circle C of radius a= sinh Z centred at
r ¼ a coth Z; z ¼ 0. As Z ! 1 this radius becomes small
and the torus collapses to the circle r ¼ a; z ¼ 0, which will
be referred to as the reference circle. As Z ! 0 both the
radius, 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), 2a2 D1 cosh Z can also be
written as 2a2 þ 2az cot y, so the surfaces of constant y are
given by
r2 þ ðz a cot yÞ2 ¼ a2 =sin2 y.
(5)
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.
ARTICLE IN PRESS
M. Andrews / Journal of Electrostatics 64 (2006) 664–672
666
For any fixed y this is a sphere of radius a=j sin yj, centred
on z ¼ a cot y; r ¼ 0. This sphere intersects the plane z ¼ 0
in the reference circle. (See Fig. 2).
3. Alternative separation of Laplace’s equation
In toroidal coordinates Laplace’s equation r2 V ¼ 0
becomes [2]
qZ ðD1 sinh ZqZ V Þ þ qy ðD1 sinh Zqy V Þ
þ ðD sinh ZÞ1 qcc V ¼ 0,
ð6Þ
which
pffiffi is not immediately separable. But inserting V ¼
U= r into this equation gives
sinh2 Z ðqZZ U þ qyy UÞ þ qcc U þ 14U ¼ 0,
which does separate giving solutions that are products of
f ðZÞ, sin my or cos my, and sin nc or cos nc, where
sinh2 Z ðf 00 m2 f Þ ðn2 14Þf ¼ 0.
Changing variable from Z to w:¼ coth Z converts this
equation to
2
m
1
ðw2 1Þf 00 þ 2wf 0 2
þ n2 f ¼ 0,
(7)
4
w 1
and the solutions of this are the associated Legendre
functions Pmn1=2 ðwÞ and Qmn1=2 ðwÞ. Therefore,
pffiffiffiffiffiffiffi the solutions
of Laplace’s equation are products of a=r, Pmn1=2 ðcoth ZÞ
or Qmn1=2 ðcoth ZÞ, sin my or cos my, and sin nc or cos nc.
This is a complete basis; but we will consider only solutions
with m ¼ 0, which still allows arbitrary dependence on Z
and c.
Table 1
The asymptotic behaviour of the Legendre functions near the singular
points w ¼ 1 and w ¼ 1
Pa ðwÞ
Qa ðwÞ
w¼1
1 þ 12 aða þ 1Þðw 1Þ
w¼1
Gða þ 12Þ
pffiffiffi
ð2wÞa
pGða þ 1Þ
1 w1
g cða þ 1Þ ln
2
2
pffiffiffi
pGða þ 1Þ
a1
ð2wÞ
Gða þ 32Þ
4. Singularities in the solutions
From the differential equation (7), singularities of the
Legendre functions Pa ðwÞ or Qa ðwÞ, as a function of the
complex variable w, can occur only for w ¼ 1 or w ¼ 1.
We need consider only the region wX1. Care is needed in
accessing information about these functions, because these
singularities, 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; instead
we must use LegendreQ(a; 0; 3; w), which is their notation
for 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) or
LegendreP(a; 0; 3; w) can be used, because these two functions are identical for wX1. The asymptotic behaviour [7,8]
near w ¼ 1 and
w¼
pffiffiffiffiffiffiffiffi
ffi 1 is given in Table 1 , except that
P1=2 ðwÞp1 2=w lnð8wÞ as w ! 1. Here, g is Eulers
constant and cðaÞ is the digamma function G0 ðaÞ=GðaÞ.
Now apply the asymptotic behaviour in Table 1 to find
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
the behaviour of
a=r Pn1=2 ðwÞ and
a=r Qn1=2 ðwÞ 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 a2 Þ=ð2arÞ it
follows that wX1, and w ¼ 1 occurs only on the reference
circle. Also w ! 1 either for r ! 0 or for R ! 1, where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R:¼ r2 þ z2 is the distance from the origin. The asymptotic behaviour in these three regions is given in Table 2,
where C n :¼p1=2 GðnÞ=Gðn þ 12Þ and Dn :¼p1=2 Gðn þ 12Þ=
Gðn þ 1Þ, and where d:¼½ðr aÞ2 þ z2 1=2 is the distance
from the reference circle.
Thus, the Q-solutions are bounded as r ! 0, have
logarithmic singularities at the reference circle, and
decrease as 1=R or faster as R ! 1. They therefore
correspond 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 to
finite 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.
Table 2
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
The asymptotic behaviour of a=r Pn1=2 ðwÞ and a=r Qn1=2 ðwÞ 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)
ra
da
Ra
w
pffiffiffiffiffiffiffi
a=r Pn1=2 ðwÞ ðn40Þ
z2 þ a2
2ar
Cn
1þ
R2
2ar
d2
2a2
z2 þ a2
a2
n1=2 a n
r
1
pffiffiffiffiffiffiffi
a=r Qn1=2 ðwÞ
2
a
4ðz2 þ a2 Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln
p z2 þ a2
ar
Dn
1
Cn
pffiffiffiffiffiffiffi
a=r P1=2 ðwÞ
R2
a2
n1=2 a n
r
2 a 4R2
ln
pR
ar
2
n1=2 z þ a2
r n
2
a
a
1
d
ln
g c n þ
2
2a
2 n1=2 R
r n
Dn 2
a
a
ARTICLE IN PRESS
M. Andrews / Journal of Electrostatics 64 (2006) 664–672
a for greater symmetry, we have the mathematical identity
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 ¼ 2p requires that n be an
integer. The simplest case is where n ¼ 0 so that the potential
has no dependence on the azimuthal angle c. Then
pffiffiffiffiffiffiffi
V 0 :¼ a=r Q1=2 ðwÞ
(8)
satisfies Laplace’s equation except on the reference circle
and, from Table 2, approaches pa=R as R ! 1. It is
therefore the potential due to a charge of q ¼ 4p2 0 a
distributed uniformly around the reference circle. Near this
circle, from Table 2, V 0 lnð8a=dÞ. Therefore, applying
Gauss’s law, the line-charge density on the reference circle is
2p0 , which gives the same total charge q.
When n ¼ n, an integer,
pffiffiffiffiffiffiffi
V n :¼ a=r Qn1=2 ðwÞ cos nc
(9)
can be analysed in a similar way, and of course sin nc
would
pffiffiffiffiffiffiffi do just as well as cos nc. Near the reference circle
a=r Qn1=2 ðwÞ has the same limit as for n ¼ 0; so the linecharge density corresponding to V n is 2p0 cos nc. The
total charge is zero; so as R ! 1, V n falls off faster than
1=R. In the case of n ¼ 1, one half of the ring has positive
charge while the other half is negative. Then the ring
pffiffiffi will2
have a dipole moment that can be deduced
to
be
2
p 0 a
pffiffiffiffiffi
from the asymptotic behaviour V 1 12 p a2 R3 r cos c.
Since we have found the potential for any sinusoidal linecharge 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 that
will 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 ¼ 0 and c ¼ 2p, is [9]
1
0
1 X
dðc c0 Þ ¼
e{nðcc Þ
2p n¼1
1
1 X
¼
dn cos nðc c0 Þ,
2p n¼0
ð10Þ
1
X
q
p
ffiffiffiffi
ffi
dn Qn1=2 ðwÞ cos nðc c0 Þ
Vd ¼
4p2 0 ar n¼0
1
X
0
q
p
ffiffiffiffi
ffi
Qn1=2 ðwÞe{nðcc Þ ,
2
4p 0 ar n¼1
1
X
0
1
1
p
ffiffiffiffiffi
¼
Qn1=2 ðwÞe{nðcc Þ
0
0
jr r j p rr n¼1
(12)
where w ¼ ðr2 þ r0 2 þ z2 Þ=ð2rr0 Þ. Since jr r0 j2 can be written
as r2 þ r0 2 þ z2 2rr0 cosðc c0 Þ. Eq. (12) can be recast as
1
X
n¼1
p
Qn1=2 ðxÞe{nf ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
2ðx cos fÞ
(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 sequence
of 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 angle c0 , is 2p0 cos nc0 , then the potential at the point
r ¼ ðr; z; cÞ is
Z
1 2p cos nc0
a dc0
Vn ¼
2 0 jr r0 j
0
where r0 ¼ ða; 0; cp
Þ, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
as in the discussion ffileading to Eq. (12).
0
Using jr r j ¼ 2arðw cosðc c0 ÞÞ, as before, and
comparing with Eq, 9, gives the integral [10]
Z p
cos nf
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi df ¼ Qn1=2 ðxÞ.
(14)
2ðx cos fÞ
0
This integral is equivalent to Heine’s Eq. (13) through
Fourier theory.
5.1. Extension to charges inside a torus
where d0 ¼ 1 and dn ¼ 2 for n ¼ 1; 2; 3; . . . . Since a linecharge
p
ffiffiffiffiffiffiffi density of 2p0 cos nf produced the potential
a=r Qn1=2 ðwÞ cos nf, it follows that a point charge q at
angle c0 on the reference circle, which will have line-charge
density ðq=aÞdðc c0 Þ, will produce the potential
¼
667
ð11Þ
where the latter form uses [6] Qn1=2 ðwÞ ¼ Qn1=2 ðwÞ. This
V d must, of course, be just the Coulomb potential
q=ð4p0 jr r0 jÞ, where in cylindrical coordinates r ¼
ðr; z; cÞ and r0 ¼ ða; 0; c0 Þ. Therefore, writing r0 instead of
These methods can be extended to deal with charge on
the reference circle inside a grounded conducting torus with
constant Z, say Z ¼ Z0 . For a uniform distribution of
charge on the reference circle, the potential is
pffiffiffiffiffiffiffi
V ¼ a=r ½Q1=2 ðwÞ P1=2 ðwÞQ1=2 ðw0 Þ=P1=2 ðw0 Þ, (15)
where w0 :¼ coth Z0 . This is the correct potential because it
satisfies Laplace’s equation, becomes zero at Z ¼ Z0 , and has
the same logarithmic
pffiffiffiffiffiffiffisingularity on the reference circle as in
Eq. (8) because a=r P1=2 ðwÞ is not singular there. The line
charge density on the circle is therefore 2p0 , as before.
Similarly, for a line charge density varying as cos nc the
potential is
pffiffiffiffiffiffiffi
V ¼ a=r ½Qn1=2 ðwÞ
Pn1=2 ðwÞQn1=2 ðw0 Þ=Pn1=2 ðw0 Þ cos nc.
ð16Þ
ARTICLE IN PRESS
M. Andrews / Journal of Electrostatics 64 (2006) 664–672
668
And for an arbitrary distribution of charge
reference circle, one can find the potential as
these terms using the Fourier series of the
density. In particular, for a point charge q at
the reference circle, the potential is
V¼
around the
a sum over
line charge
angle c0 on
1
X
q
p
ffiffiffiffi
ffi
dn ½Qn1=2 ðwÞ
4p2 0 ar n¼0
Pn1=2 ðwÞQn1=2 ðw0 Þ=Pn1=2 ðw0 Þ cos nc,
ð17Þ
following the analysis leading to Eq. (11). The part
involving Qn1=2 ðwÞ is just the Coulomb potential due to
the charge q while the part involving Pn1=2 ðwÞ is the
potential 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 be
the origin of the azimuthal angle, c ¼ 0. If b is the angle
between the planes, then the second plane is at c ¼ b. The
solutions of Laplace’s equation (except onpthe
ffiffiffiffiffiffiffi reference
circle) that become zero on both planes are a=r Qnp=b1=2
ðwÞ sinðnpc=bÞ, where n ¼ 1; 2; 3; . . . . These correspond to
a line-charge density on the reference circle of 2p0 sin
ðnpc=bÞ. Again one could construct the potential for
an 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 ¼ 0 and at c ¼ b, and
that is [11]
dðc c0 Þ ¼
1
2X
npc
npc0
sin
.
sin
b n¼1
b
b
ðwÞ cos 12nf. In the Appendix it is shown that
Sðw; fÞ
1
X
1
1
:¼
Q1=2n1=2 ðwÞ cos nf þ Q1=2 ðwÞ
2
2
n¼1
"
!#
2 cos 12f
1
1
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p þ arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ð21Þ
2ðw cos fÞ
2ðw cos fÞ 2
Therefore, the potential V ðrÞ at position r with cylindrical
coordinates ðr; z; cÞ due to a point charge q at ðr0 ; z0 ; c0 Þ and
a conducting half plane at c ¼ 0 is
1
q
pffiffiffiffiffi ½Sðw; c c0 Þ Sðw; c þ c0 Þ,
V ðrÞ ¼
(22)
4p0 p rr0
where w ¼ ½r2 þ r0 2 þ ðz z0 Þ2 =ð2rr0 Þ. An expression
equivalent 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.
0
x 1
(a)
(18)
1
2
5
(19)
This problem of a point charge between two intersecting
conducting planes appears in Batygin’s collection [12].
There cylindrical coordinates were used to give a completely different expression for the potential.
An interesting special case is where b ¼ 2p. This is the
case of a point charge outside a single semi-infinite
conducting plane with a straight boundary (a half plane).
The potential is
1
1
2q X
1
1
pffiffiffiffiffi
V¼
Q1=2n1=2 ðwÞ sin nc sin nc0 .
4p0 p ar n¼1
2
2
1
y
1
1
4q X
npc
npc0
pffiffiffiffiffi
sin
.
Qnp=b1=2 ðwÞ sin
4p0 b ar n¼1
b
b
0
2
2
For a point charge q at c0 we require a line charge density
ðq=aÞdðc c0 Þ and therefore the potential is
V¼
2
1
y
(20)
cos 12nðc þ c0 Þ
Inserting 2 sin 12nc sin 12nc0 ¼ cos 12nðc c0 Þ P
1
shows that we require sums of the form
n¼1 Q1=2n1=2
0
0.5
0.2
0.1
(b)
0
0.05
0.02
x
1
0.01
2
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.]
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669
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
Z 1
1
lðz0 Þ dz0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V P0 ¼
4p0 1
r2 þ ðz0 zÞ2
Z 1
a
dz0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
¼
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
p 1
a2 þ z0 2 r2 þ ðz0 zÞ2
The correctness of the relation
Z
a 1
dz0
q
ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 þ z0 2 r2 þ ðz0 zÞ2
rffiffiffi
2
a
a þ r2 þ z2
¼
P1=2
r
2ar
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 of
the 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.
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 ¼ Z0 ) closed off by
conducting planar ends at c ¼ 0 and c ¼ b. Thus, the
potential when there is a point charge q at c ¼ c0 is
V¼
1
1
4q X
pffiffiffiffiffi
½Q
ðwÞ Pnp=b1=2 ðwÞ
4p0 b rr0 n¼1 np=b1=2
Qnp=b1=2 ðw0 Þ=Pnp=b1=2 ðw0 Þ sin
npc
npc0
sin
.
b
b
ð23Þ
An example is shown in Fig. 4.
7. Reconstructing the P-solutions from their singularities
pffiffiffiffiffiffiffi
The solutions V Pn :¼ a=rPn1=2 ðcoth ZÞ cos nc of Laplace’s 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,
pffiffiffiffiffiffifficonsider the case where n ¼ 0, so that
V P0 :¼ a=rP1=2 ðcoth ZÞ. From Table 2, V P0 ð2a=pÞða2 þ z2 Þ1=2 ln r close to the z-axis. This corresponds
to a line charge density of lðzÞ ¼ 4a0 ða2 þ z2 Þ1=2 on the
ð24Þ
confirms that the solution V P0 is just the potential due to
the charge density lðzÞ 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 ¼ 0) of Eq. (27) proved below. The relation
between P1=2 and the complete elliptic integral [7] is well
known, and hence
Z 1
dz0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
a2 þ z0 2 r2 þ ðz0 zÞ2
4
ðr aÞ2 þ z2
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K
.
ð25Þ
ðr þ aÞ2 þ z2
ðr þ aÞ2 þ z2
For n ¼ 1; 2; 3; . . . Table 2 shows that, for r a,
pffiffiffiffiffiffiffi
V Pn :¼ a=rPn1=2 ðcoth ZÞ cos ncMðzÞrn cos nc,
(26)
potential
where MðzÞ:¼C n anþ1 ðz2 þ a2 Þn1=2 . The
rn cos nc corresponds to what one might call a cylindrical
n-pole. It satisfies Laplace’s equation except at r ¼ 0 and
corresponds 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
V Pn is produced by an n-pole of strength MðzÞ. The radial
component E r of the electric field due to the potential
rn cos nc is E r ¼ nrn1 cos nc. To generate V Pn consider a
cylinder of radius b a, with surface charge density
sðz; cÞ ¼ 0 E r MðzÞ ¼ 0 nMðzÞbn1 cos nc. To calculate
the potential due to this cylinder of charge, take a
slice of height dz0 at z ¼ z0 . It will be a circle of radius b
with line charge density 2sðz0 ; cÞ dz0 . (The extra factor
of 2 is required because only half of the charge on
the cylinder contributes to the outward field.) We already know [Eq. (9)] that the potential at ðr; z; cÞ due
to a circle ofpradius
b at z0 with line charge density
ffiffiffiffiffiffiffi
2p0 cos nc is b=rQn1=2 ðr2 þ ðz z0 Þ2 þ b2 Þ=ð2brÞ cos nc,
and from Table 2, for R b this potential will become Dn ðb2 =R2 Þnþ1=2 ðr=bÞn where R2 ¼ r2 þ ðz z0 Þ2 .
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M. Andrews / Journal of Electrostatics 64 (2006) 664–672
670
Therefore, combining the contributions from all these
slices, and using C n Dn ¼ 1=n,
rffiffiffi
2
Z 1
rn
ða2 þ z0 2 Þn1=2 dz0
a
a þ r 2 þ z2
.
¼
P
n1=2
r
pan1 1 ðr2 þ ðz z0 Þ2 Þnþ1=2
2ar
(27)
V Pn
The correctness of this relation shows that
is solely due
to the n-pole distribution along the z-axis.
To verify Eq. (27) put a ¼ 1 for simplicity
and substitute
R1
u ¼ ðaz0 þ 1Þ=ðz0 aÞ into I:¼ 1 ðz0 2 þ 1Þn ½ðz0 zÞ2 þ
r2 n1 dz0 , with R a a1 ¼ z1 ðr2 þ z2 1Þ, to give
1
I ¼ 2ð1 z=aÞn1 0 ðu2 þ 1Þn ðu2 þ t2 Þn1 du, where t2 ¼
ð1þazÞ=ð1z=aÞ. But ð1þazÞð1 z=aÞ ¼ r2 so t ¼ ð1 þ azÞ=r
and t1 ¼ ð1 z=aÞ=r. Substituting u ¼ x2 in I puts it
into a standard hypergeometric form [13] giving I ¼
pðt rÞn1 F ðn þ 1; 12; 1; 1 t2 Þ, which can be expressed as
[14] I ¼ prn1 Pn ð12 ½t þ t1 Þ, and t þ t1 ¼ ðr2 þ z2 þ 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 we
have shown that the relation is valid for any n, even though
the context here requires n to be an integer for continuity in c.
Adding these gives
Sðw; fÞ
1
X
1
1
Q1=2n1=2 ðwÞ cos nf þ Q1=2 ðwÞ
:¼
2
2
n¼1
"
!#
2 cos 12 f
1
1
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p þ arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
2ðw cos fÞ
2ðw cos fÞ 2
ðA:4Þ
A.1. Derivation of Eq. (A.3)
The generating relation [16,17] for Qn ðwÞ is
1
X
n¼0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
1
w h þ 1 2wh þ h2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln
.
w2 1
1 2wh þ h2
ðA:5Þ
{f
ðw cos fÞ.
We
If h ¼ e{f , then 1 2wh þ h2 ¼ 2e
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
require w41 and therefore, with u:¼ 2ðw cos fÞ,
1
X
8. Conclusion
hn Qn ðwÞ
Qn ðwÞe{ðnþ1=2Þf
n¼0
An alternative method of separating variables in toroidal
coordinates provides a simple route to a basis of solutions
of Laplace’s 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 distributions (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
1
X
1
Q1=2n1=2 ðwÞ cos nf
2
n¼1
1
1
X
X
1
¼
Qn ðwÞ cos n þ f þ
Qn1=2 ðwÞ cos nf. ðA:1Þ
2
n¼0
n¼1
The second of these sums can be found from Heine’s
Eq. (13)
1
X
n¼1
1
1
2p
Qn1=2 ðwÞ cos nf þ Q1=2 ðwÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
.
2
2ðw cos fÞ
(A.2)
From the generating relation (see a few lines below) we can
deduce the first
1
X
1
Qn ðwÞ cos n þ f
2
n¼0
!
2 cos 12 f
1
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
ðA:3Þ
2ðw cos fÞ
2ðw cos fÞ
{
w e{f þ {e1=2{f u
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ ln
u
w2 1
u 2 sin 12 f
{
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þ 2{ cos f ,
¼ ln
u
2
2 w2 1
and the real and imaginary parts of this give
1
X
2 cos 12 f
1
1
Qn ðwÞ cos n þ f ¼ arctan
u
2
u
n¼0
(A.6)
1
Qn ðwÞ sin n þ f
2
n¼0
1
X
!
!
u 2 sin 12 f
2 sin 12 f
1
1
ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ arcsinh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
¼
u
u
2ðw 1Þ
2ðw 1Þ
ðA:7Þ
Appendix B. Comparison with the traditional separation
The method of separation usually found in textbooks
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
[1,4] inserts V ¼ cosh Z cos y U into Laplace’s (6)
pffiffiffiffiffiffiffi
instead of V ¼ 1=r U 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 to
associated Legendre functions of cosh Z (instead of
coth Z). The result is p
solutions
of Laplace’s
equation that
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
are products of
cosh Z cos y, Pmn1=2 ðcosh ZÞ or
Qmn1=2 ðcosh 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
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M. Andrews / Journal of Electrostatics 64 (2006) 664–672
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
pffiffiffiffiffiffiffiffiffiffiffiffi
the variables
is
to
note
[18]
that
both
sinh
Z Pmn1=2 ðcosh ZÞ
pffiffiffiffiffiffiffiffiffiffiffiffi m
and
sinh Z Qn1=2 ðcosh ZÞ can be expressed as linear
combinations of Pnm1=2 ðcoth ZÞ and Qnm1=2 ðcoth ZÞ.
Therefore,
pffiffiffiffi a term (in the traditional expansion) of the
form D Pmn1=2 ðcosh ZÞ sin ny sin mc can be written, using
Eq. (2), as a linear combination of two terms of the form
r1=2 Pnm1=2 ðcoth ZÞ sin ny sin mc and r1=2 Qnm1=2 ðcoth ZÞ
sin ny sin mc.)
Thus for axially symmetric systems, the potential can be
written
as a ffi sum of terms that are products of
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cosh Z cos y, Pn1=2 ðcosh ZÞ or Qn1=2 ðcosh ZÞ, and
sin ny or cos ny. (The continuity in y requires that
n ¼ 0; 1; 2; . . . .)
B.1. Example: The potential outside a charged conducting
torus
The Qn1=2 ðcosh ZÞ are too divergent at Z ¼ 0 (which
corresponds to the z-axis). The boundary condition that
V ¼ V 0 (a constant) for Z ¼ Z0 (specifying the conducting
toroidal surface), being even in y, excludes terms in sin y.
Thus, the potential outside the torus must have the form
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X
V ¼ cosh Z cos y
an Pn1=2 ðcosh ZÞ cos ny.
(B.1)
n¼0
Imposing the condition that V ¼ V 0 for Z ¼ Z0 is easily
done by comparing this equation with Heine’s Eq. (13) in
the form
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X
p ¼ 2ðx cos yÞ
dn Qn1=2 ðxÞ cos ny,
(B.2)
n¼0
where d0 ¼ 1 and dn ¼ 2 for n40. Thus, the potential
outside the conducting torus Z ¼ Z0 at potential V ¼ V 0 is
1
V 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X
2ðcosh Z cos yÞ
dn
V¼
p
n¼0
Qn1=2 ðcosh Z0 Þ
Pn1=2 ðcosh ZÞ cos ny.
Pn1=2 ðcosh Z0 Þ
ðB:3Þ
B.2. Further examples
These deal with some cases where there are charges
inside a torus.
1. The potential inside the torus Z ¼ Z0 , when there is a
uniformly charged ring on the reference circle (r ¼ a,
z ¼ 0), has the form
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V ¼ cosh Z cos y½P1=2 ðcosh ZÞ
Q1=2 ðcosh ZÞ P1=2 ðcosh Z0 Þ=Q1=2 ðcosh Z0 Þ. ðB:4Þ
The line-charge density can be deduced from the
logarithmic singularity in P1=2 ðcosh ZÞ at the ring, as for
Eq. (8). This case was also considered in Section 5; it can be
671
treated by either approach because the boundary conditions do not depend on y or c. The two different looking
results, Eq. (B.4) and Eq. (15), are equivalent because [19]
sffiffiffiffiffiffiffiffiffiffiffiffi
1
2
Q
ðcoth ZÞ,
(B.5)
P1=2 ðcosh ZÞ ¼
p sinh Z 1=2
sffiffiffiffiffiffiffiffiffiffiffiffi
2
P1=2 ðcoth ZÞ.
Q1=2 ðcosh ZÞ ¼ p
sinh Z
(B.6)
2. Similarly,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V ¼ cosh Z cos y ½P1=2 ðcosh ZÞ
Q1=2 ðcosh ZÞ P1=2 ðcosh Z0 Þ=Q1=2 ðcosh Z0 Þ cos y ðB:7Þ
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 y can be replaced by
cosðy y0 Þ.
3. The potential between two tori (with the same
reference circle r ¼ a, z ¼ 0) held at different potentials
can be expressed as a sum of the form
V¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X
cosh Z cos y
½an Pn1=2 ðcosh ZÞ
n¼0
þ bn Qn1=2 ðcosh ZÞ cos ny
ðB:8Þ
and the coefficients an and bn can be found by solving the
two linear equations that come from V ¼ V 0 at Z ¼ Z0 and
V ¼ V 1 at Z ¼ Z1 , and comparing with Eq. (B.2) in each
case.
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, PrenticeHall, 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, McGrawHill, London, 1953, p. 115, Eq. (5).
ARTICLE IN PRESS
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M. Andrews / Journal of Electrostatics 64 (2006) 664–672
[14] A. Erdlyi (Ed.), Higher Transcendental Functions, vol. I, McGrawHill, 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.
[18] H.S. Cohl, J.E. Tohline, A.R.P. Rau, H.M. Srivastava, Astron.
Nachr. 321 (5/6) (2000) 363–372 (Eq. 31, 32).
[19] H.S. Cohl, J.E. Tohline, A.R.P. Rau, H.M. Srivastava, Astron.
Nachr. 321 (5/6) (2000) 363–372 (Eq. 33, 34).