Editing Poynting vector (section)

==Time-averaged Poynting vector==

The above form for the Poynting vector represents the ''instantaneous'' power flow due to ''instantaneous'' electric and magnetic fields. More commonly, problems in electromagnetics are solved in terms of [[sinusoidal]]ly varying fields at a specified frequency. The results can then be applied more generally, for instance, by representing incoherent radiation as a superposition of such waves at different frequencies and with fluctuating amplitudes.

We would thus not be considering the instantaneous {{math|'''E'''(''t'')}} and {{math|'''H'''(''t'')}} used above, but rather a complex (vector) amplitude for each which describes a coherent wave's phase (as well as amplitude) using [[phasor]] notation. These complex amplitude vectors are ''not'' functions of time, as they are understood to refer to oscillations over all time. A phasor such as {{math|'''E'''<sub>m</sub>}} is understood to signify a sinusoidally varying field whose instantaneous amplitude {{math|'''E'''(''t'')}} follows the real part of {{math|'''E'''<sub>m</sub>&thinsp;''e<sup>jωt</sup>''}} where {{mvar|ω}} is the (radian) frequency of the sinusoidal wave being considered.

In the time domain, it will be seen that the instantaneous power flow will be fluctuating at a frequency of 2''ω''. But what is normally of interest is the ''average'' power flow in which those fluctuations are not considered. In the math below, this is accomplished by integrating over a full cycle {{math|1=''T'' = 2''π'' / ''ω''}}. The following quantity, still referred to as a "Poynting vector", is expressed directly in terms of the phasors as:

<math display="block">\mathbf{S}_\mathrm{m} = \tfrac{1}{2} \mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^* ,</math>

where <sup>∗</sup> denotes the complex conjugate. The time-averaged power flow (according to the instantaneous  Poynting vector averaged over a full cycle, for instance) is then given by the ''real part'' of {{math|'''S'''<sub>m</sub>}}. The imaginary part is usually ignored, however, it signifies "reactive power" such as the interference due to a [[standing wave]] or the [[Electromagnetic radiation#Near and far fields|near field]] of an antenna. In a single electromagnetic [[plane wave]] (rather than a standing wave which can be described as two such waves travelling in opposite directions), {{math|'''E'''}} and {{math|'''H'''}} are exactly in phase, so {{math|'''S'''<sub>m</sub>}} is simply a real number according to the above definition.

The equivalence of {{math|Re('''S'''<sub>m</sub>)}} to the time-average of the ''instantaneous'' Poynting vector {{math|'''S'''}} can be shown as follows.

<math display="block">\begin{align}\mathbf{S}(t) &= \mathbf{E}(t) \times \mathbf{H}(t)\\
&= \operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} e^{j\omega t}\right) \times \operatorname{Re}\!\left(\mathbf{H}_\mathrm{m} e^{j\omega t}\right)\\
&= \tfrac{1}{2}\! \left(\mathbf{E}_\mathrm{m} e^{j\omega t} + \mathbf{E}_\mathrm{m}^* e^{-j\omega t}\right) \times \tfrac{1}{2}\! \left(\mathbf{H}_\mathrm{m} e^{j\omega t} + \mathbf{H}_\mathrm{m}^* e^{-j\omega t}\right)\\
&= \tfrac{1}{4}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^* + \mathbf{E}_\mathrm{m}^* \times \mathbf{H}_\mathrm{m} + \mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m} e^{2j\omega t} + \mathbf{E}_\mathrm{m}^* \times \mathbf{H}_\mathrm{m}^* e^{-2j\omega t}\right)\\
&= \tfrac{1}{2} \operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^*\right) + \tfrac{1}{2}\operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m} e^{2j\omega t}\right)\! .
\end{align}</math>

The average of  the instantaneous Poynting vector '''S'''  over time is given by:
<math display="block">\langle\mathbf{S}\rangle = \frac{1}{T} \int_0^T \mathbf{S}(t)\, dt = \frac{1}{T} \int_0^T\! \left[\tfrac{1}{2} \operatorname{Re}\! \left(\mathbf{E}_\mathrm{m} \times \mathbf{H}_\mathrm{m}^*\right) + \tfrac{1}{2} \operatorname{Re}\! \left({\mathbf{E}_\mathrm{m}} \times {\mathbf{H}_\mathrm{m}} e^{2j\omega t}\right)\right]dt.</math>

The second term is the double-frequency component having an average value of zero, so we find:
<math display="block">\langle \mathbf{S}\rangle = \tfrac{1}{2} \operatorname{Re}\! \left({\mathbf{E}_\mathrm{m}} \times \mathbf{H}_\mathrm{m}^*\right) =  \operatorname{Re}\! \left(\mathbf{S}_\mathrm{m}\right) </math>

According to some conventions, the factor of 1/2 in the above definition may be left out. Multiplication by 1/2 is required to properly describe the power flow since the magnitudes of {{math|'''E'''<sub>m</sub>}} and {{math|'''H'''<sub>m</sub>}} refer to the ''peak'' fields of the oscillating quantities. If rather the fields are described in terms of their [[root mean square]] (RMS) values (which are each smaller by the factor <math>\sqrt{2}/2</math>), then the correct average power flow is obtained without multiplication by 1/2.