Editing Poynting vector (section)

==Plane waves==
In a propagating electromagnetic [[plane wave]] in an isotropic lossless medium, the instantaneous Poynting vector always points in the direction of propagation while rapidly oscillating in magnitude. This can be simply seen given that in a plane wave, the magnitude of the magnetic field '''H'''(''r'',''t'') is given by the magnitude of the electric field vector '''E'''(''r'',''t'') divided by ''η'', the [[intrinsic impedance]] of the transmission medium:
<math display="block">|\mathbf{H}| =  \frac {|\mathbf{E}|}{\eta},</math>
where |'''A'''| represents the [[Norm (mathematics)#Euclidean norm|vector norm]] of '''A'''. Since '''E''' and '''H''' are at right angles to each other, the magnitude of their cross product is the product of their magnitudes. Without loss of generality let us take ''X'' to be the direction of the electric field and ''Y'' to be the direction of the magnetic field. The instantaneous Poynting vector, given by the cross product of '''E''' and '''H''' will then be in the positive ''Z'' direction:

<math display="block">\left|\mathsf{S_z}\right| = \left|\mathsf{E_x} \mathsf{H_y}\right| = \frac{\left|\mathsf{E_x}\right|^2}{\eta}.</math>

Finding the time-averaged power in the plane wave then requires averaging over the wave period (the inverse frequency of the wave):
<math display="block">\left\langle\mathsf{S_z}\right\rangle = \frac{\left\langle\left|\mathsf{E_x}\right|^2\right\rangle}{\eta} = \frac{\mathsf{E_\text{rms}^2}}{\eta},</math>
where ''E''<sub>rms</sub> is the [[root mean square]] (RMS) electric field amplitude. In the important case that ''E''(''t'') is sinusoidally varying at some frequency with peak amplitude ''E''<sub>peak</sub>, ''E''<sub>rms</sub> is <math>\mathsf{E_{peak}} / \sqrt{2}</math>, with the average Poynting vector then given by:
<math display="block">\left\langle\mathsf{S_z}\right\rangle = \frac{\mathsf{E_{peak}^2}}{2\eta}.</math>
This is the most common form for the energy flux of a plane wave, since sinusoidal field amplitudes are most often expressed in terms of their peak values, and complicated problems are typically solved considering only one frequency at a time. However, the expression using ''E''<sub>rms</sub> is totally general, applying, for instance, in the case of noise whose RMS amplitude can be measured but where the "peak" amplitude is meaningless. In free space the intrinsic impedance ''η'' is simply given by the [[impedance of free space]] ''η''<sub>0</sub> ≈{{nbsp}}377{{nbsp}}Ω. In non-magnetic dielectrics (such as all transparent materials at optical frequencies) with a specified dielectric constant ''ε''<sub>r</sub>, or in optics with a material whose refractive index <math>\mathsf{n} = \sqrt{\epsilon_r}</math>, the intrinsic impedance is found as:
<math display="block">\eta = \frac{\eta_0}{\sqrt{\epsilon_r}}.</math>

In optics, the value of radiated flux crossing a surface, thus the average Poynting vector component in the direction normal to that surface, is technically known as the [[irradiance]], more often simply referred to as the ''[[intensity (physics)|intensity]]'' (a somewhat ambiguous term).