Page 155 - Maxwell House

POYNTING's THEOREM 135

x means that the energy flux (3.51) is responsible for power delivery from external space
∗
to the volume V increasing the energy stored in this volume. Such sort of power can be
considered as the source of electromagnetic fields inside V similar to some source currents
inside.

The real part of Poynting’s vector. The real or in-phase component Poynting’s vector carries
not only information about the direction of power flow. If the integrated real part of complex
Poynting’s vector (3.51), for example, is positive the power leaving the volume V forever in the
form of radiation.

The imaginary part of Poynting’s vector. The imaginary part or quadrature part of Poynting’s
vector carries information about evanescent (i.e. reactive, see Chapter 1) fields meaning that
some part of energy or power stored in EM fields crossing back and forth the closed surface A.
Evidently, the time domain average of such bouncing Poynting’s vector flux is equal to zero in
no-loss systems and can serve as one of indicator of possible resonances within the volume V.
This kind of information is very useful in many situations like the development of numerical
computer model.

The structure of all other integrals in (3.49) is analogous to those in the time-space domain. So
their physical interpretation is evident and can be omitted. Taking the real and imaginary part
of (3.49), we can find the magnitude of the in-phase and quadrature component of the power
balance

′′
′′

ℜ� ()� = ∫ [ ()|| + () ]/2 + ℜ� ()� � (3.52)
Σ
0
0

ℑ� ()� = 2( () − ()) + ℑ� ()�
Σ
Here the magnitude of the reactive electric () and magnetic () energy accumulated

within the volume V
0 ′
() = ∫ ()|| ⁄
2

2 � (3.53)
0

′
2
() = ∫ ()|| ⁄

2
Eventually, the first in-phase term on the right side of (3.52) corresponds to the energy loss,
electric and magnetic, within the volume V while the last one describes either the power leaving
the volume through external radiation or coming inside from the sources located outside. It is
customary to call the in-phase component as the active or time irreversible power and the
quadrature component as the reactive/stored or time reversible power [1, 2].
Pointing’s theorem formulated above is not only the principal instrument to verify the accuracy
of Maxwell's equations solutions, but it is a power tool for the interpretation of many significant
physical phenomena. Let us start from so-called Uniqueness Theorem.