- 1) so that the potential only dies off as 1/r rather
than the quasi-static l/r2 . Using the relationships Q= I/jw
(12) could have been obtained immediately
and c = l/vj,
from (6) and (7) with the Lorentz gauge condition of Eq. (13) in
Section 9-1-1:
=- ---
A
]
- -
- jo \r ' ar
(r2-A,)
r sin 0 00
(
sin 0)
2
-:
p.idlc
4io= •+ jkr)
7 e cos 0
Qdl
=v 2(1 +j)e
9-2-3
COS 0
(13)
The Electric and Magnetic Fields
Using (6), the fields are directly found from (8) as
i = vxx^
ar
= -i*
-r
idl
k sin I
41
0
ýikr
+I
1
(jikr)
e-
(14)
(14)
671
Radiation from PointDipoles
VxH
•
iE
(
1
jw(flS)i,
0
a0
sin
r
jWE
laI
- r ar(rHa)io
1
idlk 2
K
41
[2
i
+io sin
sin
2
I1
-+
Sjkr
( jkr)2
cos
2
(jkr)
+
I3
(jkr)
)3)
(jkr)
((15)
e
Note that even this simple source generates a fairly
complicated electromagnetic field. The magnetic field in (14)
points purely in the k direction as expected by the right-hand
2
rule for a z-directed current. The term that varies as 1/r
is
called the induction field or near field for it predominates at
distances close to the dipole and exists even at zero frequency.
The new term, which varies as 1/r, is called the radiation field
since it dominates at distances far from the dipole and will be
shown to be responsible for time-average power flow away
from the source. The near field term does not contribute to
power flow but is due to the stored energy in the magnetic field
and thus results in reactive power.
3
The 1/r
terms in (15) are just the electric dipole field terms
present even at zero frequency and so are often called the
electrostatic solution. They predominate at distances close to
the dipole and thus are the near fields. The electric field also
has an intermediate field that varies as l/r 2 , but more
important is the radiation field term in the i 0 component,
which varies as I/r. At large distances (kr>> ) this term
dominates.
In the far field limit (kr >> 1), the electric and magnetic fields
are related to each other in the same way as for plane waves:
limr E =
r>>1
=HEsin Oe E
jkr
k,
Eo=0
Id~2
4r
E
(16)
The electric and magnetic fields are perpendicular and their
ratio is equal to the wave impedance 71= V/-LIE. This is because
in the far field limit the spherical wavefronts approximate a
plane.
9-2-4
Electric Field Lines
Outside the dipole the volume charge density is zero, which
allows us to define an electric vector potential C:
V-E=O
E=VxC
672
Radiation
Because the electric field in (15) only has r and 0 components,
C must only have a 4 component, Co(r, 0):
E= Vx C=
1
I-(sin
a
OC
)i -
,
r sin 0 8
1a
-(rCO)ie
(18)
r ar
We follow the same procedure developed in Section 4-4-3b,
where the electric field lines are given by
dr
E,
rdB
E,
(
-(sin
80
OC )
(19)
a
sin 0•(rCO)
which can be rewritten as an exact differential,
a(r sin
Cs) dr+
ar
80
(r sin BC,) dO = 0
d(r sin OC,)= 0
(20)
so that the field lines are just lines of constant stream-function
r sin OC,. C, is found by equating each vector component in
(18) to the solution in (15):
1
a
rsin 0
sin
Idl k2
=
r
[2-2
cos
1
+
jkr
1
(rC
=
4E=-
sin
+
+
e
(21)
which integrates to
, Idl•
4
e
sin 0
j
r
(kr)
r
Then assuming I is real, the instantaneous value of C, is
"
C, = Re (C, eiw)
dl
-dI
sin
_
,sin 6 cos (at- kr)+
sin (_.t - kr)\
41 rE
kr
(23)
kr
so that, omitting the constant amplitude factor in (23), the
field lines are
rC6 sin 0 = const• sin 2 0(cos (ot - kr) + sint
-L1-·-·ILL-·----~---·I~-·-----·-----·--
kr
kr)
const
9-2
t =O0
(a)
dipole field solution
The electric field lines for a point electric dipole at wt = 0 and ot = 7r/2.
674
Radiation
These field lines are plotted in Figure 9-2 at two values of
time. We can check our result with the static field lines for a
dipole given in Section 3-1-1. Remembering that k = o/c, at
low frequencies,
Scos (wt - kr)
lim
w~o{ sin (ot - kr)
kr
1
(t-r/c)
r/c
(25)
t
r/c
so that, in the low-frequency limit at a fixed time, (24)
approaches the result of Eq. (6) of Section 3-1-1:
lim sin 2 0
= const
(26)
Note that the field lines near the dipole are those of a static
dipole field, as drawn in Figure 3-2. In the far field limit
lim sin 2 0 cos (wt - kr) = const
(27)
kr >>l
the field lines repeat with period A = 2ir/k.
9-2-5
Radiation Resistance
Using the electric and magnetic fields of Section 9-2-3, the
time-average power density is
Electromagnetic Field Theory: A Problem Solving Approach Part 70
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