If the field at a given
point is given by the Poynting vector, ,
is expressed as ,
W/m2.
Here, ,
which is the direction of propagation. The average
value of in
time is ,
W/m2.
It shows the density of
the net (or average) electromagnetic power transfer
and the direction of its flow.
Using phasor notation,

or

where is the
complex Poynting vector.
The physical meaning of
the complex Poynting vector is similar to that of complex
power in electrical circuits. Firstly, its real part, , shows how
much power density leaves the point—it is delivered
to the surrounding medium in a specific direction.
Secondly, its imaginary part, , shows the
magnitude of the reactive power density, which fluctuates
to and fro with double frequency, without any contribution
to the net power flow in space.
Figure
1 is an illustration of the time-domain and
the complex Poynting vector. It plots the value of
the time-domain Poynting vector when and :
.
This can also be expressed
as .
The total power density (red
line), is the sum of: (1) the term (blue
curve), and (2) the term (green
curve), which is a sinusoidal function with double
frequency.
The first term is always
a positive number (it assumes values between 0 and ), thus, indicating
that its respective power density has a steady flow
without a change in the direction of propagation. Its
average is exactly (the
real part of the complex Poynting vector).
On the other hand, the
second term changes its sign, thus, indicating that
it corresponds to a power flow along for quarter
of a period, and then along for the
next quarter-period. This term has a zero net power
transfer. Its magnitude is exactly (the
imaginary part of the complex Poynting vector).
Notice that the overall
power density becomes
negative for short periods of time, for example, for . This is when
the second (reactive) term is negative and larger than
the first (propagating or active) term. |