Problem Set 8.
An antenna is constructed from conducting wire wrapped in the shape of a sphere with a
current density given by
J
(
r
,
t
)=[
̂
I
0
a
(
r
−
a
)
sin
e
−
i
ot
]
=[
J
0
(
r
)
e
−
i
ot
]
a) Set up the general expression for the vector and electrostatic potentials,
A
(
r
,
t
)
,
and
(
r
,
t
)
for this localized radiation source in the Coulomb gauge.
b) Find
A
(
r
,
t
)
in the radiation limit.
c) Find the magnetic induction,
B
(
r
,
t
)
, and the electric field,
E
(
r
,
t
)
, in the radiation limit.
d) Find the averaged Poynting vector,
S
(
r
)
ave
, and the power radiated by the antenna.
Solution:
a). First we note that
J
(
r
,
t
)=
0
J
(
r
,
t
)=
J
t
(
r
,
t
)
J
ℓ
(
r
,
t
)=
0
Thus
A
(
r
,
t
)
is given by
A
(
r
,
t
)
=
1
c
∫∫∫ ∫
J
t
(
r
′
,
t
′
)
G
(
r
−
r
′
,
t
−
t
′
)
d
3
r
′
dt
′
=
1
c
∫∫∫ ∫
J
0
(
r
′
)
e
−
i
ot
′
Θ
(
t
−
t
′
)
c

r
−
r
′

(
r
−
r
′
 −
c
(
t
−
t
′
))
d
3
r
′
dt
′
=
exp
(−
i
o
t
)
c
∫∫∫
J
0
(
r
′
)
exp
(
i
o

r
−
r
′
/
c
)

r
−
r
′

d
3
r
′
=[
A
o
(
r
)
exp
(−
i
o
t
)]
with
G
0
(
r
−
r
′
,
o
)=
exp
(
i
o

r
−
r
′
/
c
)

r
−
r
′

=
4
k
∑
l
=
0
∞
∑
m
=−
l
l
j
l
(
kr
<
)[
ij