4 Time harmonic sources and
retarded
poten
tials
•
The solution of forced wave equation
∇
2
V

μ
o
±
o
∂
2
V
∂t
2
=

ρ
±
o
for scalar potential
V
is most conveniently obtained in the frequency
domain:
Consider a timeharmonic forcing function
ρ
and a timeharmonic re
sponse
V
expressed as
ρ
(
r
,t
)=
Re
{
˜
ρ
(
r
)
e
jωt
}
and
V
(
r
,t
)=
Re
{
˜
V
(
r
)
e
jωt
}
in terms of phasors
˜
ρ
(
r
)
and
˜
V
(
r
)
.
Then, the above wave equation transforms — upon replacing
∂
∂t
by
jω
—intophasorformas
∇
2
˜
V
+
μ
o
±
o
ω
2
˜
V
=

˜
ρ
±
o
.
•
For
ω
=0
the above equation reduces to Poisson’s equation, which we
know has, with an impulse forcing
˜
ρ
(
r
)=
δ
(
r
)
,
an impulse response solution
˜
V
(
r
)=
1
4
π±
o

r

≡
1
4
π±
o
r
1
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View Full Documentwhere
r
≡
r

denotes the distance of the observing point
r
from the
impulse point located at the origin.
–
Note that this impulse response
∝
1
/r
is symmetric with respect
to the origin just like the impulse input
δ
(
r
)
.
Note that by substituting
the source function
δ
(
r
)
and
response function
1
4
π±
o
r
back
into the Poisson’s equation
we obtain an equality
∇
2
±
1

r

²
=

4
πδ
(
r
)
,
which is a useful vector iden
tity.
We now
postulate
and subsequently prove that for
ω
≥
0
,theimpu
l
sere

sponse solution of the forced wave equation — i.e., with forcing function
˜
ρ
(
r
)=
δ
(
r
)
—is
˜
V
(
r
)=
e

jk

r

4
π±
o

r

with
k
≡
ω
√
μ
o
±
o
=
ω
c
.
Proof:
For
˜
ρ
(
r
)=
δ
(
r
)
the source of the forced wave equation (for an
arbitrary
ω
)is
symmetric with respect to the origin
,imp
ly
ingthatthecorre
sponding solution
˜
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 Fall '08
 Staff
 Electromagnet, Frequency, potential solution, Fundamental physics concepts, wave equation

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