Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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336 9 Boundary-Value Problems
9.3.1 Spherical Polar Coordinates
Using the familiar spherical polar coordinates
x r SinΘ CosΦ (9.64)
y r SinΘ SinΦ (9.65)
z r CosΘ (9.66)
the Laplacian takes the form
+
2
Ψ
1
r
2
(
(r
r
2
(Ψ
(r
1
r
2
SinΘ
(
(Θ
SinΘ
(Ψ
(Θ
1
r
2
SinΘ
2
(
2
Ψ
(Φ
2
(9.67)
We propose a separable solution of the form
Ψ
rRr<Θ@Φ (9.68)
for which the Helmholtz equation becomes
1
R
(
(r
r
2
(R
(r
1
SinΘ
1
<
(
(Θ
SinΘ
(<
(Θ
1
SinΘ
2
1
@
(
2
@
(Φ
2
k
2
r
2
0 (9.69)
Recognizing that only one term depends upon Φ, we define a separation constant m using
1
@
2
@
Φ
2
m
2
@
mΦ
(9.70)
leaving
1
R
(
(r
r
2
(R
(r
1
SinΘ
1
<
(
(Θ
SinΘ
(<
(Θ
m
2
SinΘ
2
k
2
r
2
0 (9.71)
Isolating the Θ dependence using a foresightful separation constant
SinΘ
Θ
SinΘ
<
Θ
m
2
< 1 SinΘ
2
< (9.72)
and employing the transformation
x CosΘ SinΘ
<
Θ
1 x
2
<
x
(9.73)
we recognize the associated Legendre equation
1 x
2
2
<
x
2
2x
<
x
1
m
2
1 x
2
<0 <xP
,m
x,Q
,m
x (9.74)
whose solutions are combinations of regular and irregular associated Legendre polynomi-
als. Finally, using
Rr
ur
r
(9.75)
9.3 Separable Coordinate Systems 337
the radial equation reduces to the spherical form of Bessel’s equation
u
''
k
2
1
r
2
u 0
ur
kr
j
kr,n
kr (9.76)
whose solutions for k>0 are composed of regular and irregular spherical Bessel functions.
Therefore, general solutions can be constructed from appropriate linear combinations of
the schematic form
Ψ
r
,m
Ψ
,m
j
kr
n
kr
P
,m
CosΘ
Q
,m
CosΘ
mΦ
mΦ
(9.77)
where Ψ
,m
is a constant coefficient that multiplies terms obtained from the appropriate
choices from each of the three columns. Often the boundary conditions will eliminate
one of the choices from each column, but otherwise we need to superimpose all possible
combinations for the most general conditions. Similarly, depending upon the nature of k it
might be more appropriate to employ modified Bessel functions or
k ! 0 j
kr!r
,n
kr!r
1
(9.78)
Remember, this method provides a useful strategy, but it still must be adapted to the par-
ticular features of the problem at hand.
Often it is much more convenient to combine the angular functions into spherical har-
monics and to use the identity
+
2
ur
r
Y
,m
Θ, Φ
1
r
u
''
r
1
r
2
ur
Y
,m
Θ, Φ (9.79)
The Helmholtz equation then reduces to the radial equation
u
''
r
k
2
1
r
2
ur0 (9.80)
such that Ψ
r can be represented as a multipole expansion of the form
Ψ
r
,m
Ψ
,m
u
kr
kr
Y
,m
Θ, Φ (9.81)
where the Ψ
,m
are expansion coefficients and u
kr is a conveniently normalized solution
to the radial equation and its boundary conditions. Note that we have assumed, for sim-
plicity, that the irregular Legendre functions can be discarded, but generalization should
not be difficult.
It is also useful to commit to memory some of the standard Green functions for open
boundary conditions. Namely, for Poisson’s equation we find
+
2
G
r,
r
'
4Π∆
r,
r
'
G
r,
r
'
1
r
r
'
4Π
2 1
r
<
r
1
>
Y
ˆr,Y
ˆr
'
(9.82)
338 9 Boundary-Value Problems
where we remind the reader of the convenient notation
Y
ˆr,Y
ˆr
'
m
Y
,m
Θ, ΦY
,m
Θ
'
, Φ
'
m
Y
,m
Θ, ΦY
,m
Θ
'
, Φ
'
m
m
Y
,m
Θ, ΦY
,m
Θ
'
, Φ
'
(9.83)
Similarly, for the Helmholtz equation one finds
+
2
k
2
G
r,
r
'
4Π∆
r,
r
'
G
r,
r
'
Expk
r
r
'
r
r
'
4Πk
j
kr
<
h
kr
>
Y
ˆr,Y
ˆr
'
(9.84)
where h
j
n
are spherical Hankel functions for outgoing () or incoming ()
boundary conditions. The radial factors are obtained by interface matching with the aid of
the Wronskian for the radial differential equation; we leave the algebra as an exercise for
the reader.
9.3.2 Cylindrical Coordinates
Using the cylindrical coordinates
x ΞCosΦ (9.85)
y ΞSinΦ (9.86)
z z (9.87)
the Laplacian takes the form
+
2
Ψ
1
Ξ
(
(Ξ
Ξ
(Ψ
(Ξ
1
Ξ
2
(
2
Ψ
(Φ
2
(
2
Ψ
(z
2
(9.88)
We propose a separable solution of the form
Ψ
r
RΞ@ΦZz (9.89)
for which the Helmholtz equation becomes
1
R
1
Ξ
(
(Ξ
Ξ
(R
(Ξ
1
@
1
Ξ
2
(
2
@
(Φ
2
1
Z
(
2
Z
(z
2
k
2
0 (9.90)
Separating first the Φ dependence using trigonometric functions with periodic boundary
conditions and then the z dependence using hyperbolic functions assuming finite range,
9.4 Spherical Expansion of Dirichlet Green Function for Poisson’s Equation 339
we obtain
1
@
(
2
@
(Φ
2
m
2
@z
mΦ
,
mΦ
(9.91)
1
Z
(
2
Z
(z
2
Α
2
n
ZzSinhΑ
n
z, CoshΑ
n
z (9.92)
1
Ξ
(
(Ξ
Ξ
(R
(Ξ
k
2
Α
2
m
2
Ξ
2
R 0 RΞ J
m
Β
n
Ξ,N
m
Β
n
Ξ (9.93)
with Β
2
n
k
2
Α
2
n
(9.94)
where we have assumed, somewhat arbitrarily, that the separation constants m, Α, Β are
real; if not, we replace a trigonometric function by an exponential function or a Bessel
function by a modified Bessel function. The schematic notation f f
1
,f
2
indicates a
suitable choice or linear combination of solutions that satisfies the appropriate boundary
conditions. Therefore, complete solutions can be constructed from linear superpositions of
the schematic form
Ψ
r
m,n
Ψ
m,n
J
m
Β
n
Ξ
N
m
Β
n
Ξ
SinhΑ
n
z
CoshΑ
n
z
mΦ
mΦ
(9.95)
Depending upon the boundary conditions, it might be more convenient to employ
trigonometric functions for Z and then modified Bessel functions for R. Alternatively,
sometimes it is more convenient to separate the R dependence first and then match across
an interface with respect to Z. If the range of the angular variable is restricted, different
boundary conditions and different angular functions could be needed. It takes experience
to anticipate the optimum choices for a particular problem, and some of that experience
can be acquired by solving the problems at the end of the chapter! In the remainder of this
chapter we will concentrate upon spherical geometries, but many problems with cylindri-
cal geometries are provided. Rather than attempt to apply the results derived here directly,
it is usually better to perform the separation of variables for each problem anew in order
to make informed decisions regarding the order of separation and the nature of separation
constants.
9.4 Spherical Expansion of Dirichlet Green Function for Poisson’s
Equation
We seek to construct a Green function for Poisson’s equation that vanishes on the concen-
tric spheres r a and r b and satisfies
+
2
G
r,
r
'
4Π∆
r
r
'
,G 0onr a, b (9.96)
340 9 Boundary-Value Problems
in the enclosed volume. It is natural to employ spherical polar coordinates and to make a
multipole expansion of the form
G
r,
r
'
g
r, r
'
Y
ˆr,Y
ˆr
'
(9.97)
∆
r
r
'
∆r r
'
r
2
Y
ˆr,Y
ˆr
'
(9.98)
where g
satisfies the radial equation
1
r
2
rg
r
2
1
r
2
g
r, r
'
4Π
r
2
∆r r
'
(9.99)
Solutions to the homogeneous equation that satisfy either the inner or the outer boundary
condition take the form
r<r
'
g
A
r
a
r
a
1
(9.100)
r>r
'
g
B
r
b
r
b
1
(9.101)
and are subject to matching conditions
g
>
g
<
B
r
'
b
r
'
b
1
A
r
'
a
r
'
a
1
(9.102)
$g
'
4Π
r
'
2
B
r
'
r
'
b
1
r
'
b
1
A
r
'
r
'
a
1
r
'
a
1
4Π
r
'
2
(9.103)
across the interface. It is convenient to rewrite these equations as
B1 ΒA1 Α (9.104)
B Β 1 A Α 1 4Πr
'
1
(9.105)
where
Α
a
r
'
21
Β
b
r
'
21
(9.106)
It is now easy, albeit tedious, to solve these equations
sol Solve
B1 ΒA1 Α,B Β 1 A Α 14Π
r
!
1
,
A, B
1/ / Simpli fy
A
4Π1 Β
r
#
1
1 2Α Β
,B
4Π1 Α
r
#
1
1 2Α Β
9.4 Spherical Expansion of Dirichlet Green Function for Poisson’s Equation 341
and to construct the two pieces
A
r
a
r
a
1
,B
r
b
r
b
1
/ .sol / .
Α
a
r
!
21
, Β
b
r
!
21
/ / Simpli fy
4Π
r
a
1
r
a
1
b
r
#
12
r
#
1
1 2
a
r
#
12
b
r
#
12
,
4Π
r
b
1
r
b
1
a
r
#
12
r
#
1
1 2
a
r
#
12
b
r
#
12
Therefore, combining the two pieces, we obtain the Dirichlet Green function for the vol-
ume between concentric grounded spheres as
g
r, r
'
4Π
2 1
1
r
<
r
>
1
b
21
r
21
>
r
21
<
a
21
b
21
a
21
(9.107)
where r
<
is the smaller and r
>
is the larger of r and r
'
. Notice that this function is simply
the product of the inner and outer solutions with a normalization based upon their Wron-
skian; that general form should be familiar from our work on Sturm–Liouville systems and
appears often in various guises.
Several important special cases
a ! 0 g
r, r
'
4Π
2 1
r
<
r
1
>
1
r
>
b
21
(9.108)
b !g
r, r
'
4Π
2 1
r
<
r
1
>
3
4
4
4
4
5
1
a
r
<
21
6
7
7
7
7
8
(9.109)
a ! 0,b!g
r, r
'
4Π
2 1
r
<
r
1
>
(9.110)
can be obtained from the limiting behaviors when one or both of the radii assume extreme
values. The first case listed above describes the interior of a sphere of radius b, the second
describes the exterior of a sphere of radius a, while the third describes an open geometry
without bounding surfaces. The result for open geometry should already be familiar from
the multipole expansion
1
r
r
'
4Π
2 1
r
<
r
1
>
Y
ˆr
, Y
ˆr
'
(9.111)
for the potential of a unit point charge in empty space.
Given an arbitrary charge distribution Ρ
r
'
in the volume between concentric spheres
with specified potentials Ψ
a
Ψa, Θ, Φ and Ψ
b
Ψb, Θ, Φ, we can now compute the
electrostatic potential anywhere within the enclosed volume by integrating
Ψ
r
G
r,
r
'
Ρ
r
'
V
'
1
4Π
Ψ
r
'
+
'
G
r,
r
'
,
S
'
(9.112)
342 9 Boundary-Value Problems
taking care to interpret the directed element of surface area as outward with respect to the
enclosed volume such that
r
'
a
S
'
a
2
E
'
ˆr
'
(9.113)
r
'
b
S
'
b
2
E
'
ˆr
'
(9.114)
where ˆr
'
is the radial unit vector. Thus, the second contribution to the potential produced
by a point charge in the presence of a grounded sphere must represent the potential due to
charges on the surface of the conductor. Several examples of the use of this Green func-
tion are given below. Some of these examples are simple enough to evaluate symbolically,
but generally numerical integration would be required for nontrivial distributions. Never-
theless, a mathematician would consider the problem solved because numerical integration
can be performed by computer in a straightforward manner. A physicist, on the other hand,
would probably require a more practical demonstration that the numerical method works!
9.4.1 Example: Multipole Expansion for Localized Charge Distribution
Suppose that we seek the electrostatic potential
Ψ
r
G
r,
r
'
Ρ
r
'
V
'
(9.115)
for distances r D R where R represents the radius containing all of the charge, such that
G
r,
r
'
g
r, r
'
Y
ˆr,Y
ˆr
'
(9.116)
g
r, r
'
4Π
2 1
r
<
r
1
>
B
4Π
2 1
r
'
r
1
(9.117)
The potential then takes the form
Ψ
r
,m
q
,m
r
1
Y
,m
ˆr (9.118)
where the multipole amplitudes are
q
,m
4Π
2 1
r
Y
,m
Θ, ΦΡ
rV (9.119)
The lowest nonvanishing multipoles tend to dominate for r D R.
9.4.2 Example: Point Charge Near Grounded Conducting Sphere
The potential due to a point charge near a grounded conducting sphere has two contribu-
tions, one from the point charge itself and another from the distribution of surface charge
needed to maintain the conductor at constant potential. One can show that the contribu-
tion of the induced surface charge density is equivalent to the potential produced by an
9.4 Spherical Expansion of Dirichlet Green Function for Poisson’s Equation 343
image charge. Suppose first that the point charge is inside the conducting sphere, such that
r, r
'
<b. By writing
r
<
r
1
>
r
>
b
21
1
b
r
<
r
>
b
2
1
b
rr
'
b
2
r
b
b
r
r
1
b
(9.120)
we obtain
4Π
2 1
r
<
r
1
>
r
>
b
21
Y
ˆr,Y
ˆr
'
r
b
b
1
r
r
b
(9.121)
where
r
b
r
'
b/r
'
2
is the location of an image charge outside a sphere of radius b induced
by a point charge inside the sphere at
r
'
. Similarly, by writing
r
<
r
1
>
a
r
<
21
1
a
a
2
r
<
r
>
1
1
a
a
2
rr
'
r
a
a
r
a
r
1
(9.122)
we obtain
4Π
2 1
r
<
r
1
>
a
r
<
21
Y
ˆr,Y
ˆr
'
r
a
a
1
r
r
a
(9.123)
where
r
a
r
'
a/r
'
2
is the location of an image charge inside a sphere of radius a induced
by a point charge outside the sphere at
r
'
. Therefore, the potential for a point charge at
r
'
in the presence of a ground conducting sphere of radius R becomes
Ψ
r
q
r
r
'
q
i
r
r
i
with q
i
q
R
r
'
,
r
i
R
r
'
2
r
'
(9.124)
where q
i
is the magnitude and
r
i
is the location of the image charge. The same expression
is obtained whether the charge is inside or outside the sphere but, of course, the potential
vanishes in the region on the opposite side of the conducting surface from the point charge.
The actual surface charge density is determined by the normal derivative of the elec-
trostatic potential according to
Σ
1
4Π
(Ψ
(r
rR
(9.125)
which, after simple but tedious algebra, reduces to
Σ
q
4ΠR
r
'2
R
2
R
2
2r
'
R CosΘ r
'2
3/ 2
(9.126)
where CosΘ ˆr , ˆr
'
is the polar angle on the sphere relative to the line between its center
and the point charge. A simple calculation shows that the total induced charge
2ΠR
2
1
1
ΣCosΘ
L
M
M
N
M
M
O
q, r
'
<R
qR/ r
'
,r
'
>R
(9.127)
is not necessarily equal to the image charge. If the point charge is inside the grounded
sphere the induced charge is equal to q so that the net charge and the potential outside
344 9 Boundary-Value Problems
0 0.5 1 1.5 2 2.5 3
T
0.5
1
1.5
2
2.5
3
V 4SR
2
sq
r
4R
r
3R
r
2R
0 0.5 1 1.5 2 2.5 3
T
5
10
15
20
25
30
r
0.75R
r
0.50R
r
0.25R
Figure 9.1. Surface charge density induced upon a grounded sphere of radius R by a point charge at
r
'
.
the sphere vanish. On the other hand, if the point charge is outside the sphere the total
induced charge is inversely proportional to the distance from the center of the sphere.
In either case, the induced charge is concentrated nearby when the point charge is near
the surface and is spread more uniformly as the distance between the point charge and
the surface increases. Figure 9.1 show the angular distribution of induced surface charge
density for several values of r
'
/R.
Although it is probably easier to obtain the Green function for a single grounded sphere
using the method of images than the expansion in spherical harmonics, the problem of two
concentric spheres would be rather difficult to solve using images because an infinite set of
images is required. The expansion in spherical harmonics, by contrast, is no more difficult
for a spherical shell than for a sphere and contains the latter as a special case. The Green
function method is much more versatile than the method of images.
9.4.3 Example: Specified Potential on Surface of Empty Sphere
Suppose that Ψ f Θ, Φ is specified on the surface of an empty sphere of radius R.The
interior potential is then determined by
Ψ
r
R
2
4Π
f Θ, Φ
(G
r,
r
'
(r
'
r
'
R
E
'
(9.128)
where
S
'
ˆr
'
R
2
E
'
is the directed differential area and
G
r,
r
'
g
r, r
'
Y
ˆr,Y
ˆr
'
(9.129)
g
r, r
'
4Π
2 1
r
<
r
1
>
1
r
>
R
21
(9.130)
is the Green function for the interior of a sphere. The radial derivative is simply
(g
r, r
'
(r
'
r
'
R
4Π
R
2
r
R
(9.131)
9.4 Spherical Expansion of Dirichlet Green Function for Poisson’s Equation 345
such that
r<RΨr, Θ, Φ
,m
r
R
Y
,m
Θ, Φ
Y
,m
Θ
'
, Φ
'
f
Θ
'
, Φ
'
E
'
(9.132)
can be evaluated numerically for any surface potential. Similarly, the exterior potential is
determined by
g
r, r
'
4Π
2 1
r
<
r
1
>
3
4
4
4
4
5
1
R
r
<
21
6
7
7
7
7
8
(g
r, r
'
(r
'
r
'
R
4Π
R
2
R
r
1
(9.133)
Then, using
S
'
ˆr
'
R
2
E
'
as the outward normal to the boundary of the exterior region,
we obtain
r>RΨr, Θ, Φ
,m
R
r
1
Y
,m
Θ, Φ
Y
,m
Θ
'
, Φ
'
f
Θ
'
, Φ
'
E
'
(9.134)
These results can now be combined in the form
Ψr, Θ, Φ
,m
r
<
r
1
>
Ψ
,m
Y
,m
Θ, Φ (9.135)
where now r
<
is interpreted as the smaller and r
>
as the larger of r and R and where the
multipole amplitudes are given by
Ψ
,m
Y
,m
Θ, Φ f Θ, ΦE (9.136)
The same results could have been obtained by writing the eigenfunction expansions for Ψ
in the interior and exterior regions and matching the boundary condition at the surface
without using the Green function. (Try it!)
Suppose that a point charge is found near a sphere of specified potential. According
to the superposition principle, the electrostatic potential can be constructed by adding the
potential for a point charge near a grounded sphere to the contribution by a sphere with
specified surface potential in otherwise empty space. More generally, if there is a charge
distribution near a sphere with specified potential, one adds the contribution for each ele-
ment of charge, q
'
Ρ
r
'
V
'
for a grounded sphere to the contribution of the surface
potential, as represented by the formula
Ψ
r
G
r,
r
'
Ρ
r
'
V
'
1
4Π
Ψ
r
'
+
'
G
r,
r
'
,
S
'
(9.137)
where the appropriate Green function is used in the interior and exterior regions and where
S
'
is radially outward for the interior or inward for the exterior regions. The surface
contribution can be represented by a multipole expansion while the net contribution of the
charge density and its corresponding induced surface charges can be represented either by
multipoles or by images.