Poynting vector

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The Poynting vector is the cross product of the electric field and the magnetic field, named after its inventor John Henry Poynting. Oliver Heaviside independently co-discovered the Poynting vector.

For an electromagnetic wave it points in the direction of energy flow and its magnitude is the power per unit area crossing a surface which is normal to it. (The fact that it points perhaps contributes to the frequency with which its name is often misspelled.) In this sense it can be thought of as the energy flux (J·m^-2·s^-1) of an electromagnetic wave. It is derived by considering the conservation of energy and taking into account that the magnetic field can do no work. It is given the symbol S (in bold because it is a vector) and, in SI units, is given by:

<math>\mathbf{S} = \mathbf{E} \times \mathbf{H} = \frac{1}{\mu} \mathbf{E} \times \mathbf{B}</math>,

where E is the electric field, H and B are the magnetic field and magnetic flux density respectively, and <math>\mu</math> is the permeability of the surrounding medium. For an electromagnetic wave propagating in free space <math>\mu</math> becomes <math>\mu_0</math>, the permeability of free space.

Since the electric and magnetic fields of an electromagnetic wave oscillate, the magnitude of the Poynting vector changes with time. The average of the magnitude over a long time T (longer than the period of the wave) is called the irradiance or intensity, I:

<math>I = |\left \langle S \right \rangle_T|</math>.

For time-harmonic fields with some fixed frequency ω and time-dependence <math>\!\, e^{i \omega t}</math>, this time-average power density is given by <math>I = \begin{matrix} \frac{1}{2} \end{matrix} \mathrm{Re}(\mathbf{E} \times \mathbf{H}^*) </math>.

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