Page 59 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 39

chemical, mechanical, heat, and many other areas of science. By definition, such software
“behemoth” must be extremely complicated and require huge computer resources. Besides, in
this merged product we can expect a lot of logic or coding bugs that could lead to unpredictable
failures and difficult to verify results.

1.7.2 Convection Charges and Currents
In conclusion, consider one more source of electromagnetic fields. Practically any natural
phenomenon might be the source of moving and at rest charges: solar winds and cosmic rays,
volcanic eruptions and tectonic plates drift, charge flow in our brains and nervous system,
massive snowstorms and lightning storms, waterfalls and water evaporation from oceans, earth
surface charges, moving electrically charged clouds, satellites, and their derby flight, etc. In
other words, there is a great deal of natural and human-made sources of the electromagnetic
field whose nature is unknown or has been purely understood, or so complicated that there is
no way to include them into Maxwell’s equations. Moreover, in many such cases, we cannot
validate the continuity equation but can measure some equivalent charge and distributions. Such
other sources, if they required, will be denoted by the same short name with additional subscript
con, for example, instead of and so on. We would like to mention here that the

charge conservation law (1.76) can appear to be broken due to our lack of understanding about
these complex phenomena.

1.8 MAXWELL’s EQUATIONS IN PHASOR FORM

1.8.1 Time and Frequency Domain
The modern communication systems use the wide-ranging diversity of time-varying signals.
They can be periodic or aperiodic, deterministic or random, analog or digital or a mix of all of
them. Perhaps any of communication signals with physical meaning and within broad limits
can be expressed in the frequency domain through the Fourier transform [16] as

∞
() = ∫ () exp(−) (1.77)
−∞
Here () is the signal waveform, () denotes the signal continuous complex spectrum, =
2 is the angular frequency, and is the frequency in Hertz. The inverse Fourier transform

∞
() = ∫ () exp() (1.78)
−∞
recovers the original signal from its complex spectrum. The expression (1.77) reflects the
simple physical fact that the time-varying signal can be represented as an infinite mixture of
more simple monochromatic, i.e. sinusoidal and unlimited in duration, signals in the phasor
form

(, ) = |()| () exp() (1.79)
Here |()| and () is the magnitude and phase, correspondingly. Then the real part ()

of (, ) denoted by the symbol ℜ((, )) and imaginary part () denoted by the symbol

ℑ((, )) are
() = ℜ�|()| () exp()� = |()| cos� + ()� � (1.80)

() = ℑ�|()| () exp()� = |()|sin ( + ())