Page 198 - Maxwell House

178 Chapter 4

sources. Evidently, we have to discard this part of solution as a physically meaningless and
irrelevant one. Then following in much the same way as before we can define the EM potential
′
created in point P (see Figure 4.1.1) by an infinitesimal source ( , ) at frequency as

′
( ,) �− � |− |�
′

′
′
(, , , ) = (4.53)
′
4 0 |− |
Plugging this into the general solution (4.32) yields
′
�− � |− |�
1 ′ ′
(, , ) = ∫ ( ) (4.54)
′
′
4 0 |− |
Therefore, Green’s function for monochromatic fields can be written as
′
�− � |− |�

′
(, , , ) = (4.55)
′
|− |
This function as before describes the potential at the observation point that is emitted by a
unit point-source. If so, in the presence of sources the general harmonic solution of (4.29) yields
1
′
′
(, , ) = ∫ (, , , ) ( ) ′
′

4 0 � (4.56)
0 ′ ′ ′
(, , ) = ∫ (, , , ) ( )
4 ′
The integrals in (4.56) looks fantastic creating the illusion that any electrodynamics problem
might be solve at once as soon as EM waves sources are given. Unfortunately, there are several
major difficulties to evaluate (4.56) numerically. The obvious ones are the singularity as | −
′
′
| → 0 and rapid oscillating factor �− � |− |� in (4.55) that makes numerical approach
problematic and in some cases prohibitive for large-scale problem.
Let us come back to the Lorentz’s gauge (4.21). We know (see Chapter 1) that for harmonic
processes the operation of differentiation in time can be replaced by the factor . Bearing it in
mind, we have from (4.21) after some transformations
1
∘ + + = ∘ + +
0
0
2
2
= ∘ + = 0 (4.57)
0 0
Here is the relative complex dielectric constant defined by (2.12) in Chapter 2 when the

polarization loss in dielectric medium is negligible low. Therefore,
2

= − ∘ (4.58)

It means that the scalar potential can be excluded from (4.19)
2

= ∙ ( ∘ ) −
� (4.59)
1
= ×
0
Finally, we have gotten all information required for the practical engineering analysis.