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SOLUTION OF BASIC EQUATIONS OF ELECTRODYNAMICS 181

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= = = [s] (4.63)
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4. The equity (, ) = − defines the phase, i.e. space-time phase shift or delay of the
expending EM wave of single frequency. In general, the phase distribution at any fixed
moment of time is characterized by its phase wavefront meaning the position of the

surface on which the phase of a wave is constant or

− = . (4.64)

Drop a pebble in a calm pond and you get a two-dimensional circular wave on the surface
of the pond running outward and forming the pattern likewise in Figure 4.2.3.

5. According to (4.61) and (4.64) the surfaces of constant field intensity and phase coincides
at any fixed moment of time. Figure 4.2.3 illustrating (4.64) at an arbitrary but fixed
moment of time is, evidently, the dissection of the spheres shown sketchy in Figure 4.2.4a
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and is the resemblance of the Chinese ivory Puzzle Ball displayed in Figure 4.2.4b. It is
clear that the shape of each phase wavefront is an ideal sphere. That is why the waves are
radiated by the infinitesimal radiator can be called spherical.

a) b)
Figure 4.2.4. a) Several consecutive images of phase wave front, b) Chinese ivory Puzzle Ball

6. The remarkable fact is that the far-away EM waves emitted by any type of finite-size
radiators or finite set of such radiators are obligated to be very close to spherical. To prove
it assume that the observation points ≫ . As such, | − | ≅ (1 − ∘ ) (see
′
′
′
0
Figure 4.1.1) for any point inside . Physically it means that we measure the fields
′
′
created by an arbitrary antenna in so called far-field zone. For modern antennas such as
large and ultra large radio telescopes, radars phased arrays and submarine radio
communication antenna ground systems this zone can be tens or even hundreds kilometers
far-off. As a matter of fact, Green’s function in (4.55) is independent of the variable .
′
′
�− � |− |� �− � �
′ � 0 ∘ ′ (4.65)
(, , ) = ≅
′
|− |
Thus this factor can be pulled outside of the integral in (4.56)
5 Public Domain Image, source: https://www.pinterest.com/pin/503066220849382214/