Poynting vector

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The Poynting vector describes the energy flux (J·m⁻²·s⁻¹) of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside independently co-discovered the Poynting vector.

It points in the direction of energy flow and its magnitude is the power per unit area crossing a surface that is normal to it. 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:

$\mathbf{S} = \mathbf{E} \times \mathbf{H} = \frac{1}{\mu} \mathbf{E} \times \mathbf{B},$

where E is the electric field, H and B are the magnetic field and magnetic flux density respectively, and $μ$ is the permeability of the surrounding medium. For an electromagnetic wave propagating in free space $μ$ becomes $μ 0$ , the permeability of free space.

For example, the Poynting vector near an ideally conducting wire is parallel to the wire axis - so electric energy is flowing in space outside of the wire. The Poynting vector becomes tilted toward wire for a resistive wire, indicating that energy flows from the e/m field into the wire, producing resistive Joule heating in the wire.

Since the electric and magnetic fields of an electromagnetic wave oscillate, the magnitude of the Poynting vector also oscillates. The average of the magnitude over one period of the wave is called the irradiance or intensity, I:

$I = |\left \langle S \right \rangle_T|.$

For time-harmonic fields with some fixed frequency ω and time-dependence $\!\, e^{i \omega t}$ , the Poynting vector varies with the cosine-squared function, at twice the frequency of the electromagnetic wave. Since the average value of this function over one cycle is equal to half its peak value, the time-averaged power density is given by $I = \begin{matrix} \frac{1}{2} \end{matrix} \mathrm{Re}(\mathbf{E} \times \mathbf{H}^*)$ , where * denotes the complex conjugate.

S divided by the square of the speed of light in a vacuum is the density of the linear momentum of the electromagnetic field.

The time-averaged intensity <S> divided by the speed of light in a vacuum is the radiation pressure exerted by an electromagnetic wave on the surface of a target.