Wave Power in a Lossless Medium

Consider the following uniform plane wave, described in terms of the phasor representation of its electric field intensity:

Here

is a complex-valued constant associated with the source of the wave, and

is the positive real-valued propagation constant. Therefore, the wave is propagating in the

direction in lossless media.

The solution is not to seek total power, but rather power per unit area. This quantity is known as the spatial power density, or simply “power density,” and has units of W/m

.¹ Then, if we are interested in total power passing through some finite area, then we may simply integrate the power density over this area. Let’s skip to the answer, and then consider where this answer comes from. It turns out that the instantaneous power density of a uniform plane wave is the magnitude of the Poynting vector

Note that this equation is dimensionally correct; i.e. the units of

(V/m) times the units of

(A/m) yield the units of spatial power density (V⋅A/m

, which is W/m

). Also, the direction of

is in the direction of propagation (reminder: Section 9.5), which is the direction in which the power is expected to flow. Thus, we have some compelling evidence that

is the power density we seek. However, this is not proof – for that, we require the Poynting Theorem, which is a bit outside the scope of the present section, but is addressed in the “Additional Reading” at the end of this section.

A bit later we’re going to need to know

for a uniform plane wave, so let’s work that out now. From the plane wave relationships (Section 9.5) we find that the magnetic field intensity associated with the electric field in Equation 9.7.1 is

where

is the real-valued impedance of the medium. Let

be the phase of

; i.e.,

. Then

and

Now applying Equation 9.7.2,

As noted earlier,

is only the instantaneous power density, which is still not quite what we are looking for. What we are actually looking for is the time-average power density

– that is, the average value of

over one period

of the wave. This may be calculated as follows:

Since

, the definite integral equals

. We obtain

It is useful to check units again at this point. Note (V/m)

divided by

is W/m

, as expected.

Equation 9.7.5 is the time-average power density (units of W/m

) associated with a sinusoidally-varying uniform plane wave in lossless media.

Note that Equation 9.7.5 is analogous to a well-known result from electric circuit theory. Recall the time-average power PavePave (units of W) associated with a voltage phasor

across resistance

is

which closely resembles Equation 9.7.5. The result is also analogous to the result for a voltage wave on a transmission line (Section 3.20), for which:

where

is a complex-valued constant representing the magnitude and phase of the voltage wave, and

is the characteristic impedance of the transmission line.

Here is a good point at which to identify a common pitfall.

and

are the peak magnitudes of the associated real-valued physical quantities. However, these quantities are also routinely given as root mean square (“rms”) quantities. Peak magnitudes are greater by a factor of

, so Equation 9.7.5 expressed in terms of the rms quantity lacks the factor of

.

Considering the prevalence of phasor representation, it is useful to have an alternative form of the Poynting vector which yields time-average power by operating directly on field quantities in phasor form. This is

, defined as:

(Note that the magnetic field intensity phasor is conjugated.) The above expression gives the expected result for a uniform plane wave. Using Equations 9.7.1 and 9.7.3, we find

which yields

as expected.

• 1

  Be careful: The quantities power spectral density (W/Hz) and power flux density (W/(m2·Hz)) are also sometimes referred to as “power density.” In this section, we will limit the scope to spatial power density (W/m2).

• “Poynting vector” on Wikipedia.
• “Poynting's theorem“ on Wikipedia