Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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346 9 Boundary-Value Problems
9.4.4 Example: Charged Ring at Center of Grounded Conducting Sphere
Suppose that a charged ring of radius a is centered within a grounded conducting sphere
of radius b. It is convenient to equation the ˆz axis with the normal to the ring, such that
Ρ
r
q
2Πa
2
∆r a∆
CosΘ
(9.138)
represents the charge density. The electrostatic potential is determined by
Ψ
r
G
r,
r
'
Ρ
r
'
V
'
(9.139)
with
G
r,
r
'
g
r, r
'
Y
ˆr,Y
ˆr
'
(9.140)
g
r, r
'
4Π
2 1
r
<
r
1
>
1
r
>
b
21
(9.141)
Azimuthal symmetry with respect to Φ
'
limits the expansion to terms with m 0, for
which
Y
,0
Θ, Φ
2 1
4Π
1/ 2
P
CosΘ
(9.142)
Therefore, we obtain
Ψ
rq
r
<
r
1
>
1
r
>
b
21
P
0P
CosΘ
(9.143)
where r
<
is the smaller and r
>
is the larger of r and a. Although P
0 can be evaluated in
closed form, the series is not simplified thereby. Nevertheless, the potential can be evalu-
ated numerically using a simple program; however, it is often necessary to include many
terms in order to achieve the desired accuracy. An example is provided in Fig. 9.2 for a/b
and summation up to 50.
9.5 Magnetic Field of Current Loop
The current density for a circular loop of radius a in the xy plane can be expressed as
J J
Φ
ˆ
Φ (9.144)
J
Φ
I∆
CosΘ
∆r a
a
(9.145)
where the azimuthal unit vector can be expressed in either Cartesian or spherical bases as
ˆ
ΦSinΦˆx CosΦˆy
2
Φ
ˆe
Φ
ˆe
(9.146)
9.5 Magnetic Field of Current Loop 347
Figure 9.2. Equipotentials for a charged ring of radius a centered within a grounded sphere of radius
b,wherea/b 0.5.
where
ˆe
ˆe
x
ˆe
y
2
, ˆe
ˆe
x
ˆe
y
2
, ˆe
0
ˆe
z
(9.147)
The multipole expansion of the magnetic vector potential
A
r
1
c
V
'
J
r
'
r
r
'
1
c
0
m
4Π
2 1
Y
,m
Θ, Φ
E
'
r
'
r
'2
J
r
'
r
<
r
1
>
Y
,m
Θ
'
, Φ
'
(9.148)
then becomes
A
r
Ia
c
0
4Π
2 1
r
<
r
1
>
m
Y
,m
Θ, Φ
Φ
'
Y
,m
Π
2
, Φ
'
2
Φ
'
ˆe
Φ
'
ˆe
(9.149)
where r
<
is the smaller and r
>
is the larger of r and a. By writing
Y
,m
Θ, ΦY
,m
Θ, 0
mΦ
ΦY
,m
Θ, Φ
Φ
2Π∆
m,1
Y
,m
Θ, 0 (9.150)
we obtain
A
r2Π
Ia
c
0
4Π
2 1
r
<
r
1
>
2
Y
,1
Θ, 0Y
,1
Π
2
, 0
Φ
ˆe
Y
,1
Θ, 0Y
,1
Π
2
, 0
Φ
ˆe
(9.151)
348 9 Boundary-Value Problems
Figure 9.3. A
Φ
for current loop.
Next we use
Y
,m
m
Y
,m
,Y
,m
Θ, 0Y
,m
Θ, 0 (9.152)
to obtain
A
Φ
r2Π
Ia
c
0
4Π
2 1
r
<
r
1
>
Y
,1
Θ, 0Y
,1
Π
2
, 0
(9.153)
Finally, we use
Y
,1
Π
2
, 0
L
M
M
N
M
M
O
0, 2n
n1
4Π1
21
1/ 2
2n1!!
2n!!
, 2n 1
(9.154)
and
Y
,1
Θ, 0
4Π 1
2 1
1/ 2
P
,1
CosΘ (9.155)
to obtain
A
Φ
r2Π
Ia
c
n0
n
2
n
2n 1!!
n 1!
r
2n1
<
r
2n2
>
P
2n1,1
CosΘ
(9.156)
in terms of associated Legendre polynomials. The appearance of associated Legendre
polynomials in the vector potential, instead of ordinary Legendre polynomials in the scalar
potential for a ring, reflects the different symmetry properties of vector and scalar fields.
Contours of A
Φ
are sketched in Fig. 9.3.
9.6 Inhomogeneous Wave Equation 349
The magnetic field is obtained using
A A
Φ
ˆ
ΦB
r
1
r SinΘ
(
(Θ
SinΘA
Φ
,B
Θ
1
r
(
(r
rA
Φ
,B
Φ
0 (9.157)
The angular derivative can also be expressed as
1
SinΘ
(
(Θ
(
(x
1
SinΘ
(
(Θ
SinΘA
Φ
(
(x
1 x
2
1/ 2
A
Φ
x
(9.158)
where x CosΘ. Expressing the associated Legendre polynomial as
P
,1
x
1 x
2
1/ 2
P
'
x
(
(x
1 x
2
1/ 2
P
,1
x
1 x
2
P
''
x2xP
'
x
(9.159)
and using the Legendre differential equation, we find
1
SinΘ
(
(Θ
SinΘP
,1
CosΘ
1P
CosΘ
(9.160)
and obtain
B
r
2Π
Ia
cr
n0
n
2
n
2n 1!!
n!
r
2n1
<
r
2n2
>
P
2n1
CosΘ
(9.161)
The B
Θ
component is easily evaluated but must be separated into inner and outer regions,
whereby
r<a B
Θ
4Π
Ia
cr
n0
n
2
n
2n 1!!
n!
r
2n1
a
2n2
P
2n1,1
CosΘ
(9.162)
r>a B
Θ
2Π
Ia
cr
n0
n
2
n
2n 1!!
n 1!
a
2n1
r
2n2
P
2n1,1
CosΘ
(9.163)
9.6 Inhomogeneous Wave Equation
9.6.1 Spatial Representation of Time-Independent Green Function
Consider an inhomogeneous scalar wave equation of the form
1
c
2
(
2
(t
2
+
2
Ψ
r,t4ΠΡ
r,t (9.164)
where Ψ represents the wave amplitude, c is the phase velocity, and Ρ represents a source
that we assume is localized in both space and time. It is useful to perform a Fourier analysis
of the time dependence
Ρ
r,t
Ω
2Π
Ωt
Ρ
Ω
r (9.165)
Ψ
r,t
Ω
2Π
Ωt
Ψ
k
r (9.166)
350 9 Boundary-Value Problems
such that the spatial dependence is described by the inhomogeneous Helmholtz equation
+
2
k
2
Ψ
k
r4ΠΡ
Ω
r (9.167)
where k Ω/c is the wave number. One could, of course, perform a Fourier analysis of
the spatial dependence also, but we prefer to begin with a more traditional partial-wave
expansion of the Green function, which satisfies an equation of the form
+
2
G k
2
G 4Π∆
r
r
'
(9.168)
with boundary conditions to be specified later. Once we have the Green function, the solu-
tion to the original wave equation becomes
Ψ
k
rΦ
k
r
G
k
r,
r
'
Ρ
Ω
r
'
V
'
(9.169)
where Φ
k
is a solution to the homogeneous equation
+
2
k
2
Φ
k
r0 (9.170)
that is determined by matching the boundary conditions.
For example, in the time-independent formalism for scattering by a localized distribu-
tion, one specifies that the asymptotic wave function takes the form of an incident plane
wave plus an outgoing spherical wave of the form
r D r
'
Ψ
k
r Exp
k ,
r f
k
'
,
k
Expk
'
r
r
(9.171)
where the scattering amplitude f is the amplitude of the spherical wave. The incident wave
vector
k
Ω
c
ˆ
k specifies both the frequency Ω and direction
ˆ
k for the incident wave while
the direction of the outgoing wave is specified by
ˆ
k
'
ˆr. The magnitude of the scattered
and incident wave vectors are equal for elastic scattering, such that k
'
k Ω/c,but
differ for inelastic scattering. If we choose the ˆz axis along the incident direction, we may
write the asymptotic wave function for elastic scattering as
r D r
'
Ψ
k
r Expkz f
k
Θ, Φ
Expkr
r
(9.172)
Including the time dependence, the planes of constant phase in the incident component
Φ
r,tExpkz Ωt (9.173)
are clearly seen to travel in the direction of increasing z with phase velocity c. Similarly,
spheres of constant phase in the scattered wave
Ψ
sc
r,t-
Expkr Ωt
r
(9.174)
move radially outward with the same phase velocity. Thus, outgoing boundary conditions
require
r D r
'
G
k
r,
r
'
Expkr
r
(9.175)
However, a more rigorous treatment would represent the incoming wave by a localized
wave packet of finite extent, rather than a plane wave.
9.6 Inhomogeneous Wave Equation 351
Recognizing that k ! 0 G !
r
r
'
1
, we seek a solution of the form
G
gR
R
(9.176)
where R
r
r
'
. It is convenient to temporarily shift the coordinate system to make
r
'
!0
and to employ
+
2
fg f +
2
g g+
2
f 2+ f ,+g +
2
g
r
1
r
+
2
g
2
r
2
(g
(r
4Π∆
r (9.177)
Using the spherical symmetry of gr to write
+
2
gr
1
r
(
2
(r
2
rgr rg
''
r2g
'
r (9.178)
the Helmholtz equation becomes
g
''
rk
2
gr4Π1 gr∆r (9.179)
Therefore, by requiring g01 we immediately obtain g a Expkrb Expkr
with a b 1. Outgoing boundary conditions require G -
kR
/Rfor R !, such that
G
k
r,
r
'
Expk
r
r
'
r
r
'
(9.180)
represents an outgoing spherical wave produced by a point source at
r
'
. Similarly, the
Green function for incoming boundary conditions becomes
G
k
r,
r
'
Expk
r
r
'
r
r
'
G
k
r,
r
'
(9.181)
It is also useful to express these functions in terms of spherical Hankel functions
G
k
r,
r
'
Expk
r
r
'
r
r
'
kh
0
k
r
r
'
(9.182)
where only 0 is needed because the Green function is spherically symmetric with
respect to the distance from the source.
Suppose that Ρ
Ω
represents a localized source, such as an antenna, that radiates out-
going waves and that there is no incident wave. We can then drop the Φ
k
contribution and
employ the Green function for outgoing boundary conditions, such that the solution to the
inhomogeneous Helmholtz equation takes the form
Ψ
k
r
Expk
r
r
'
r
r
'
Ρ
Ω
r
'
V
'
(9.183)
In the far field, where r is much greater than any r
'
for which Ρ
Ω
has appreciable strength,
we may use
r D r
'
r
r
'
*r
1
ˆr ,
r
'
r
Expk
r
r
'
r
r
'
*
Expkr
r
Exp
k,
r
'
(9.184)
where
k kˆr to approximate the Green function. Note that it is important to include the
dependence of the phase of the exponential upon r
'
because cancellations between various
352 9 Boundary-Value Problems
parts of the source are crucial to the amplitude and angular distribution of the scattered
wave, but the variation of the denominator is much less important and can be neglected for
a localized source. Thus, in the far field we obtain an asymptotic approximation
r D r
'
Ψ
k
r f
k
Expkr
r
(9.185)
that has the form of an outgoing spherical wave whose angular distribution
f
k
Exp
k ,
r
'
Ρ
Ω
r
'
V
'
(9.186)
is governed by the Fourier transform of the source. The quantity f
k is known as the
form factor. Similar form factors appear throughout theories of radiation or scattering. The
interpretation of the form factor should now be clear. When the field point is sufficiently
far from the source to treat the rays received from various parts of the source as parallel,
the total amplitude is given by the sum of amplitudes from each part of the source with a
phase, relative to the center, given by
k ,
r
'
.
It is often useful to expand the form factor in terms of multipoles of the source. Expand-
ing the plane wave in spherical harmonics
Exp
k ,
r
'
4Π
j
kr
'
Y
ˆ
k,Y
ˆr
'
(9.187)
the angular distribution can be expressed in the form
f
k
C
, Y
ˆ
k f Θ, Φ
,m
C
,m
kY
,m
Θ, Φ (9.188)
where the coefficients
C
,m
k
j
krY
,m
ˆrΡ
Ω
rV (9.189)
are Fourier–Bessel multipole amplitudes for the source.
9.6.2 Partial-Wave Expansion
It will often be useful to have a partial-wave expansion of the Helmholtz Green function.
Let
G
k
r,
r
'
g
r, r
'
Y
ˆr,Y
ˆr
'
(9.190)
∆
r
r
'
∆r r
'
rr
'
Y
ˆr,Y
ˆr
'
(9.191)
where the boundary conditions will be specified later and where the k dependence in g
is
left implicit. Using
+
2
g
rY
,m
Θ, Φ
1
r
(
2
rg
r
(r
2
1
r
2
g
r
Y
,m
Θ, Φ (9.192)
9.6 Inhomogeneous Wave Equation 353
the radial equation becomes
(
2
g
r, r
'
(r
2
2
r
(g
r, r
'
(r
k
2
1
r
2
g
r, r
'
4Π
∆r r
'
rr
'
(9.193)
The homogeneous equation is recognized as the spherical Bessel equation. If we require
g
to be regular for r<r
'
and to obey outgoing boundary conditions for r>r
'
, then we
expect g
to take the form
g
r, r
'
Aj
kr
<
h
kr
>
(9.194)
where A is a constant, j
is the regular spherical Bessel function, and h
is the outgoing
spherical Hankel function. The constant is determined by the discontinuity in slope
$
(g
r, r
'
(r
rr
'
4Π
r
2
A
3
4
4
4
4
4
5
j
kr
(h
kr
(r
h
kr
( j
kr
(r
6
7
7
7
7
7
8
4Π
r
2
(9.195)
where the term in the final parentheses is the Wronskian between the solutions in the inner
and outer regions. Using
h
x j
xn
x j
x
(h
x
(x
h
x
( j
x
(x
j
xn
'
xn
x j
'
x (9.196)
and the Wronskian
j
xn
'
xn
x j
'
x
x
2
A 4Πk (9.197)
we finally obtain the partial-wave expansion
G
k
r,
r
'
Expk
r
r
'
r
r
'
4Πk
j
kr
<
h
kr
>
Y
ˆr,Y
ˆr
'
(9.198)
The asymptotic form of the wave function for r D r
'
, where r
'
is limited by the source Ρ,
can now be written as
r D r
'
Ψ
k
r Φ
k
rk
,m
h
krY
,m
ˆr
j
kr
'
Y
,m
ˆr
'
Ρ
Ω
r
'
V
'
(9.199)
The angular dependence of the scattered wave is coupled to the angular properties of its
source. Using the asymptotic behavior of the Hankel function
h
kr
Expkr
1
2
Π
kr
(9.200)
we find
Ψ
k
r Φ
k
r
kr
r
,m
C
,m
kY
,m
ˆr (9.201)
C
,m
j
kr
'
Y
,m
ˆr
'
Ρ
Ω
r
'
V
'
(9.202)
Thus, the scattered wave is expanded in outgoing waves with definite orbital angular
momentum whose amplitudes are determined by a multipole expansion of the Fourier–
Bessel transform of the source.
354 9 Boundary-Value Problems
9.6.3 Momentum Representation of Time-Independent Green Function
Another useful representation of the Helmholtz Green function is obtained by expanding
G
r,
r
'
3
k
'
2Π
3
Β
k
'
,
r
'
Φ
k
'
,
r (9.203)
where Φ
k,
r is an eigenfunction of the homogeneous equation
+
2
k
2
Φ
k,
r0 (9.204)
and where
+
2
k
2
G
r,
r
'
4Π∆
r
r
'
(9.205)
with the appropriate boundary conditions. Using Φ
k,
rExp
k ,
r and applying the
differential operator +
2
k
2
we obtain
+
2
k
2
G
r,
r
'
3
k
'
2Π
3
k
2
k
'2
Β
k
'
,
r
'
Exp
k
'
,
r
4Π
3
k
'
2Π
3
Exp
k
'
,
r
r
'
(9.206)
where in the last step we employ the Fourier representation of the delta function. Using
the orthogonality of plane waves, we deduce
Β
k
'
,
r
'
4Π
Exp
k
'
,
r
'
k
'2
k
2
G
r,
r
'
4Π
3
k
'
2Π
3
Exp
k
'
,
r
r
'
k
'2
k
2
(9.207)
The angular integral can be done by expanding the plane wave, such that
G
r,
r
'
2
Π
0
k
'
k
'2
j
0
k
'
R
k
'2
k
2
2
ΠR
0
k
'
k
'
Sink
'
R
k
'2
k
2
1
ΠR
k
'
k
'
Expk
'
R
k
'2
k
2
(9.208)
where R
r
r
'
. Ordinarily we would exploit the symmetry of the integrand to replace
the sine function by an exponential and to extend the range of integration with respect
to k
'
to so that we can use a great semicircular contour closed in the upper half-
plane, but the integrand has singularities at k
'
k that are on the contour. Fortunately,
we still have some unused information, namely the boundary conditions, which can be
used to determine the proper handling of these singularities. Imposing outgoing boundary
conditions in the far field, R !, suggests that we should include the positive pole and
exclude the negative pole. With this choice we obtain the same outgoing Green function
G
k
r,
r
'
Expk
r
r
'
r
r
'
(9.209)
as before. Alternatively, incoming boundary conditions are satisfied by including the neg-
ative and excluding the positive pole. The appropriate contours for these conditions are
sketched in Fig. 9.4.
9.6 Inhomogeneous Wave Equation 355
Re k
Imk
k
k
Imk
k
k
Re k
Figure 9.4. left: outgoing bc, right: incoming bc.
Re k
Imk
k
k
Re k
Imk
k
k
Figure 9.5. left: outgoing bc, right: incoming bc.
Another method for achieving the same result is to modify the Helmholtz equation by
adding an infinitesimal ∆
k ! k
∆
2
k
'
k
∆
2
(9.210)
such that the poles at k
'
are shifted off the real axis asymmetrically, as sketched in Fig. 9.5.
If ∆!0
, we obtain outgoing boundary conditions while if ∆!0
we obtain incoming
boundary conditions. Therefore, the momentum representation of the Green function can
be expressed as
˜
G
k, k
'
4Π
k
2
k
'2
∆
(9.211)
where ∆ is a positive infinitesimal and where the spatial representation is obtained using
the Fourier transform
G
k
r,
r
'
4Π
3
k
'
2Π
3
Exp
k
'
,
r
r
'
k
2
k
'2
∆
(9.212)