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Chapter 3
Simple Radiating Systems
A simple radiating system consists of a localized charge density
ρ
(
r
,t
)
and a localized current density
J
(
r
)
.Th
e
system is considered to be localized if its dimensions are small compared to the wavelength of the radiation. We will consider
radiating sources in a vacuum and begin with the equations for the vector and scalar potentials in the Lorentz gauge and
Gaussian units:
∇
2
φ
(
r
)
−
1
c
2
∂
2
∂t
2
φ
(
r
,t
)=
−
4
πρ
(
r
);
∇
2
A
(
r
)
−
1
c
2
∂
2
2
A
(
r
,t
−
4
π
c
J
(
r
)
.
(3.334)
To analyze a radiating system we use the retarded Green’s function for the basic wave equation (Eq. 68) :
G
(
r
−
r
0
−
t
0
Θ
(
t
−
t
0
)
c

r
−
r
0

δ
(

r
−
r
0

−
c
(
t
−
t
0
))
.
(3.335)
The vector and scalar potentials are given by:
A
(
r
A
h
(
r
)+
ZZZ Z
J
μ
r
0
−

r
−
r
0

c
¶
Θ
(
t
−
t
0
)
c

r
−
r
0

δ
(

r
−
r
0

−
c
(
t
−
t
0
))
d
3
r
0
dt
0
&
φ
(
r
φ
h
(
r
ρ
μ
r
0
−

r
−
r
0

c
¶
Θ
(
t
−
t
0
)
c

r
−
r
0

δ
(

r
−
r
0

−
c
(
t
−
t
0
))
d
3
r
0
dt
0
.
Using
δ
(

r
−
r
0

−
c
(
t
−
t
0
)) =
1
c
δ
(
t
0
−
t
+

r
−
r
0

/c
))
to do the
t
0
integration one
f
nds
A
(
r
A
h
(
r
1
c
ZZZ
J
μ
r
0
−

r
−
r
0

c
¶
1

r
−
r
0

d
3
r
0
and
(3.336)
φ
(
r
φ
h
(
r
1
c
ρ
μ
r
0
−

r
−
r
0

c
¶
1

r
−
r
0

d
3
r
0
,
(3.337)
where
A
h
and
φ
h
are solutions to the homogenous wave equations (no sources)& they will be taken to be zero.
3.1
Localized harmonically oscillating source
We consider
f
rst sources which oscillate at a
f
xed
frequency,
ω
:
ρ
(
r
ρ
0
(
r
)exp(
−
iωt
)
(3.338)
J
(
r
J
0
(
r
−
iωt
)
.
(3.339)
The real part of these source functions and of the potentials will be the physical parameters. The vector potential generated
by this current source is
A
(
r
exp (
−
iωt
)
c
J
0
(
r
0
)
exp (
iω

r
−
r
0

/c
)

r
−
r
0

d
3
r
0
.
(3.340)
The spatial variation of the vector potential is contained in
A
0
(
r
1
c
J
0
(
r
0
)
exp (
iω

r
−
r
0

/c
)

r
−
r
0

d
3
r
0
.
(3.341)
73
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View Full DocumentThe spatial dependence of the magnetic induction is given by
B
0
(
r
)=
∇
×
A
0
(
r
)
(3.342)
and (since at the observation point
J
(
r
,
t
)=0
) the electric
f
eld is given by
∇
×
B
(
r
,t
1
c
∂
∂t
E
(
r
,
t
−
i
ω
c
E
(
r
,
t
)
,
(3.343)
E
(
r
ic
ω
∇
×
B
(
r
)
.
(3.344)
The spatial dependence of the electric
f
eld is
E
0
(
r
ic
ω
∇
×
B
0
(
r
)
.
We have already restricted our sources to be localized with a characteristic dimension,
d
, satisfying
d
¿
c/ω
=
λ/
2
π.
The exponential term in the integrand for the vector potential suggests that the potential will have a different spatial
dependence depending on the range of
r
:
d
¿
r
¿
c/ω
(the near
f
eld or static zone)
r
À
c/ω
(the far
f
eld or radiation zone)
To evaluate
A
0
(
r
)
one needs an appropriate expansion for the following Green’s function:
G
0
(
r
−
r
0
,ω
exp (
iω

r
−
r
0

/c
)

r
−
r
0

where
[
∇
2
+
k
2
]
G
0
(
r
−
r
0
−
4
πδ
(
r
−
r
0
)
and
k
=
ω/c
.
This Green’s function for the Helmholtz equation (the wave equation with
∂
2
G
c
2
2
=
ω
2
c
2
2
G
=
k
2
G
) can be written in spherical
coordinates as follows:
G
(
r
−
r
0
∞
X
l
=0
l
X
m
=
−
l
g
l
(
r, r
0
)
Y
lm
(
θ,φ
)
Y
lm
¡
θ
0
,φ
0
¢
∗
where
(3.345)
where
g
l
(
r, r
0
)
is a solution of
d
dr
μ
r
2
d
dr
g
l
(
r, r
0
)
¶
+
¡
kr
2
−
l
(
l
+1)
¢
g
l
(
0
−
4
(
r
−
r
0
)
,k
=
ω
c
.
(3.346)
The
g
l
(
0
)
can be expanded in solutions to the spherical Bessel equation:
d
dr
μ
r
2
d
dr
f
C
(
)
¶
+
¡
k
2
r
2
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 Spring '10
 G.
 Charge, Current

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