Chapter 12: The Poynting Vector

Illustration 1

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.

 
Figures (click to enlarge)
Figure 1

Figure 1: The time-dependent Poynting vector.