Poynting vector

In physics, the Poynting vector represents the directional energy flux or power flow of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre ; 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 is defined as the cross product
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. 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, then electromagnetic energy is locally conserved within that region, yielding a continuity equation as a special case of Poynting's theorem:
where 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 nor on Z. The model 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, and over a sufficiently short segment of cable.
The coaxial cable is specified as having an inner conductor of radius R₁ and an outer conductor whose inner radius is R₂. In between R₁ and R₂ the cable contains an ideal dielectric material of relative permittivity ε_r and we assume conductors that are non-magnetic and lossless, all of which are good approximations to real-world coaxial cable in typical situations.
File:Coax-poynting.png|thumb|right|350px|DC power transmission through a coaxial cable showing relative strength of electric and magnetic fields and resulting Poynting vector at a radius r from the center of the coaxial cable. The broken magenta line shows the cumulative power transmission within radius r, half of which flows inside the geometric mean of R₁ and R₂.
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, symmetry dictates that it is in the radial direction and it can be shown that they must obey the following form:
W can be evaluated by integrating the electric field from to which must be the negative of the voltage V:
so that:
The magnetic field, again by symmetry, can be non-zero only in the θ direction, that is, a vector field looping around the center conductor at every radius between R₁ and R₂. Inside the conductors themselves the magnetic field may or may not be zero, but this is of no concern since the Poynting vector in these regions is zero due to the electric field being zero. Outside the entire coaxial cable, the magnetic field is identically zero since paths in this region enclose a net current of zero, and again the electric field is zero there anyway. Using Ampère's law in the region from R₁ to R₂, which encloses the current +I in the center conductor but with no contribution from the current in the outer conductor, we find at radius r:
Now, from an electric field in the radial direction, and a tangential magnetic field, the Poynting vector, given by the cross-product of these, is only non-zero in the Z direction, along the direction of the coaxial cable itself, as we would expect. Again only a function of r, we can evaluate S:
where W is given above in terms of the center conductor voltage V. The total power flowing down the coaxial cable can be computed by integrating over the entire cross section A of the cable in between the conductors:
Substituting the earlier solution for the constant W we find:
that is, the power given by integrating the Poynting vector over a cross section of the coaxial cable is exactly equal to the product of voltage and current as one would have computed for the power delivered using basic laws of electricity.
Other similar examples in which the P = V · I result can be analytically calculated are: the parallel-plate transmission line, using Cartesian coordinates, and the two-wire transmission line, using bipolar cylindrical coordinates.

Other forms

In the "microscopic" version of Maxwell's equations, this definition must be replaced by a definition in terms of the electric field E and the magnetic flux density B.
It is also possible to combine the electric displacement field D with the magnetic flux B to get the Minkowski form of the Poynting vector, or use D and H to construct yet another version. The choice has been controversial: Pfeifer et al. summarize and to a certain extent resolve the century-long dispute between proponents of the Abraham and Minkowski forms.
The Poynting vector represents the particular case of an energy flux vector for electromagnetic energy. However, any type of energy has its direction of movement in space, as well as its density, so energy flux vectors can be defined for other types of energy as well, e.g., for mechanical energy. The Umov–Poynting vector discovered by Nikolay Umov in 1874 describes energy flux in liquid and elastic media in a completely generalized view.

Interpretation

The Poynting vector appears in Poynting's theorem, an energy-conservation law:
where J_f is the current density of free charges and u is the electromagnetic energy density for linear, nondispersive materials, given by
where

E is the electric field;
D is the electric displacement field;
B is the magnetic flux density;
H is the magnetizing field.

The first term in the right-hand side represents the electromagnetic energy flow into a small volume, while the second term subtracts the work done by the field on free electrical currents, which thereby exits from electromagnetic energy as dissipation, heat, etc. In this definition, bound electrical currents are not included in this term and instead contribute to S and u.
For light in free space, the linear momentum density is
For linear, nondispersive and isotropic materials, the constitutive relations can be written as
where

ε is the permittivity of the material;
μ is the permeability of the material.

Here ε and μ are scalar, real-valued constants independent of position, direction, and frequency.
In principle, this limits Poynting's theorem in this form to fields in vacuum and nondispersive linear materials. A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms.
One consequence of the Poynting formula is that for the electromagnetic field to do work, both magnetic and electric fields must be present. The magnetic field alone or the electric field alone cannot do any work.

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 is given by the magnitude of the electric field vector E divided by η, the intrinsic impedance of the transmission medium:
where represents the 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:
Finding the time-averaged power in the plane wave then requires averaging over the wave period :
where E_rms is the root mean square electric field amplitude. In the important case that E is sinusoidally varying at some frequency with peak amplitude E_peak, E_rms is, with the average Poynting vector then given by:
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_rms 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 η₀ ≈377Ω. In non-magnetic dielectrics with a specified dielectric constant ε_r, or in optics with a material whose refractive index, the intrinsic impedance is found as:
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.