Fundamentals of Electromagnetics - RF/Microwave Interaction with Biological Tissues - page 25

energy is interpreted as a field concentration, stored in space, expressed in
joules (J). The total electromagnetic energy stored in a given volume is
obtained by integrating the energy over the volume. The time derivative of
energy yields watts, hence power.
We should now have a better understanding of the second sentence of this
section: In the time domain, Poynting's theorem expresses an equality between
the spatial variation of EM power and the time variation of EM energy, the
sum of the electric and magnetic energies. It is often said that Poynting's
theorem expresses the conservation of energy. What it precisely expresses is
that, for a given volume, if there is a net flow of EM power penetrating into
the volume, then the EM energy increases in the volume, a possible difference
between the two quantities being the power dissipation within the volume
because of the medium conductivity. In the time domain, Poynting's theorem
can be expressed in either differential or integral form.
Expressed in the frequency domain, the real part of the complex Poynting
vector at a point is equal to the average value of the real power flux, physi-
cally measurable, at that point. When integrated over the surface limiting a
given volume, it is equal to the real power dissipated in the considered volume
due to whole of the electric, magnetic, and conductive losses. Contrary to the
time-domain theorem, the frequency-domain theorem shows that the imagi-
nary part of the Poynting vector is not related to the total frequency-domain
EM energy: It is related to the difference of the magnetic and electric
energies. Hence, it vanishes when the two energies are equal. This situation is
called resonance , where the power flux is entirely real. In the frequency
domain also, Poynting's theorem can be expressed in either differential or
integral form.
Poynting's theorem can be used in establishing a general expression for the
impedance of an EM structure, for instance an antenna [3]. The structure is
placed inside a virtual closed surface and the expression relates the energy
stored and the power dissipated in the bounded volume.
Poynting's theorem expresses the equality between the space variation of
the EM power and the time variation of the EM energy. This form of the
theorem is sufficient in most cases, at least in media where the current is a con-
duction current. In some cases, however, a generalized form may be necessary,
for instance when the current is a convection current, due to moving charges,
in vacuum or other media. This may be the case in plasmas, magnetohydro-
dynamics, and microwave tubes. Tonks has established such a generalized form
of Poynting's theorem [12], obtaining equality for the conservation of energy,
where the power is the sum of EM term and a kinetic term, while the energy
is the sum of EM energy and kinetic energy. On both sides of the equations,
the EM and kinetic terms cannot be separated. This expresses the possi-
bility of transforming one term into the other, for instance EM energy into
kinetic power, as in a particle accelerator, as well as kinetic energy into EM
power, as in solar eruptions. In most cases, however, and in biological
applications in particular, the usual form of Poynting's theorem is quite
satisfactory.