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SOLUTION OF BASIC EQUATIONS OF ELECTRODYNAMICS 177
′ ′
⁄
1 ( ,−|− | )
(, ) = ∫ ′ ′
′
4 0 |− | � (4.46)
′ ′
⁄
0 ( ,−|− | ) ′
(, ) = ∫ ′
′
4 |− |
It is the common practice to present (4.46) introducing Green’s function as
′ ′
⁄
δ( −[−�− � ])
′
′
(, ; , ) = ⎫
′
|− |
⎪
1
′
′
′
′
(, ) = ∫ (, ; , ) ( , ) ′ (4.47)
′
4 0 ⎬
0 ′ ′ ′ ′ ′ ⎪
(, ) = ∫ (, ; , ) ( , ) ⎭
′
4
Green’s function identifies the fields created by a point-source of infinite density, located at
point and emitting the radiation at moment of time . The way we have developed (4.33) is
′
′
intuitive and not rigorous, but it is physically sound and help immensely to get straight to the
right solution. We leave as an exercise for the reader to verify (4.46) or (4.47) by putting them
into (4.22) or (4.23).
4.1.10 Potentials and Green’s Function in Space-Frequency Domain
As usual, let us start from the one-dimensional version of (4.29) in free-of-sources unbounded
medium and consider the scalar potential in spherical coordinate system (see Appendix I for
2
Laplacian operator ∇ ). Assuming that the magnitude (i.e. the time factor is omitted) of this
potential depends on radial component only we have
2
1 ()
2
� 2
� 2
∇ + = + = 0 (4.48)
2
Evidently, this equation can be rewritten as
2
() + () = 0 (4.49)
� 2
2
The solutions of (4.49) are well-known [1 - 4] and describe pure harmonic motions like sin��
�
and cos��. Then using phasor notifications we can obtain
�
− � �
(, ) = + (4.50)
Here and are some arbitrary constants and according to (4.14), (4.15), = − where
�
1
2
and are both positive numbers. Then reinstating the omitted time factor we finally have
1 2
�− � � �+ � �
(, , , ) = (, ) = + (4.51)
Let us review the behavior of each terms in (4.51) as a function of radius. Evidently,
�− � � = − 2 (− 1 ) � (4.52)
�+ � � = 2 (+ 1 )
The bottom equity in (4.52) tell us something strange that the amplitude of outgoing EM wave
exponentially increases as it goes away from the origin without any infusion of energy from the