In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area, per unit time) or power flow of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m2); kg/s3 in SI base units. It is named after its discoverer John Henry Poynting who first derived it in 1884. Nikolay Umov is also credited with formulating the concept. Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition. The Poynting vector is used throughout electromagnetics in conjunction with Poynting's theorem, the continuity equation expressing conservation of electromagnetic energy, to calculate the power flow in electromagnetic fields.
Definition In Poynting's original paper and in most textbooks, the Poynting vector S {\displaystyle \mathbf {S} } is defined as the cross product
S = E × H , {\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} ,}
where bold letters represent vectors and
E is the electric field vector; H is the magnetic field's auxiliary field vector or magnetizing field. This expression is often called the Abraham form and is the most widely used. The Poynting vector is usually denoted by S or N. In simple terms, the Poynting vector S, at a point, gives the magnitude and direction of surface power density that are due to electromagnetic fields at that point. More rigorously, it is the quantity that must be used to make Poynting's theorem valid. Poynting's theorem essentially says that the difference between the electromagnetic energy entering a region and the electromagnetic energy leaving a region must equal the energy converted or dissipated in that region, that is, turned into a different form of energy (often heat). Poynting's theorem is simply a statement of local conservation of energy. If electromagnetic energy is not gained from or lost to other forms of energy within some region (e.g., mechanical energy or heating), then electromagnetic energy is locally conserved within that region, yielding a continuity equation as a special case of Poynting's theorem:
∇ ⋅ S = − ∂ u ∂ t {\displaystyle \nabla \cdot \mathbf {S} =-{\frac {\partial u}{\partial t}}}
where u {\displaystyle u} is the energy density of the electromagnetic field. This frequent condition holds in the following simple example in which the Poynting vector is calculated and seen to be consistent with the usual computation of power in an electric circuit.
Example: Power flow in a coaxial cable We can find a relatively simple solution in the case of power transmission through a section of coaxial cable analyzed in cylindrical coordinates as depicted in the accompanying diagram. The model's symmetry implies that there is no dependence on θ (circular symmetry) nor on Z (position along the cable). The model (and solution) can be considered simply as a DC circuit with no time dependence, but the following solution applies equally well to the transmission of radio frequency power, as long as we are considering an instant of time (during which the voltage and current don't change), and over a sufficiently short segment of cable (much smaller than a wavelength, so that these quantities are not dependent on Z). The coaxial cable is specified as having an inner conductor of radius R1 and an outer conductor whose inner radius is R2 (its thickness beyond R2 doesn't affect the following analysis). In between R1 and R2 the cable contains an ideal dielectric material of relative permittivity εr and we assume conductors that are non-magnetic (so μ = μ0) and lossless (perfect conductors), all of which are good approximations to real-world coaxial cable in typical situations.
The central conductor is at voltage V and draws a current I toward the right, so we expect a total power flow of P = V · I according to basic laws of electricity. By evaluating the Poynting vector, however, we are able to identify the profile of power flow in terms of the electric and magnetic fields inside the coaxial cable. The electric field is zero inside of each conductor, but between the conductors ( R 1 < r < R 2 {\displaystyle R_{1}<r<R_{2}} ), symmetry dictates that it is in the radial direction and it can be shown (using Gauss's law) that they must obey the following form:
E r ( r ) = W r {\displaystyle E_{r}(r)={\frac {W}{r}}}
W can be evaluated by integrating the electric field from r = R 2 {\displaystyle r=R_{2}} to R 1 {\displaystyle R_{1}} which must be the negative of the voltage V:
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