Inphysics, thePoynting vector (orUmov–Poynting vector) represents the directionalenergy flux (the energy transfer per unit area, per unit time) orpower flow of anelectromagnetic field. TheSI unit of the Poynting vector is thewatt per square metre (W/m2); kg/s3 inSI base units. It is named after its discovererJohn Henry Poynting who first derived it in 1884.[1]: 132 Nikolay Umov is also credited with formulating the concept.[2]Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding thecurl of an arbitrary vector field to the definition.[3] The Poynting vector is used throughoutelectromagnetics in conjunction withPoynting's theorem, thecontinuity equation expressingconservation of electromagnetic energy, to calculate the power flow in electromagnetic fields.
In Poynting's original paper and in most textbooks, the Poynting vector is defined as thecross product[4][5][6]where bold letters representvectors and
This expression is often called theAbraham form and is the most widely used.[7] The Poynting vector is usually denoted byS orN.
In simple terms, the Poynting vectorS,at a point, gives the magnitude and direction ofsurface power density that are due to electromagnetic fieldsat that point. More rigorously, it is the quantity that must be used to makePoynting'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 localconservation 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 islocally conserved within that region, yielding acontinuity 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.
We can find a relatively simple solution in the case of power transmission through a section ofcoaxial cable analyzed in cylindrical coordinates as depicted in the accompanying diagram. The model's symmetry implies that there is no dependence onθ (circular symmetry) nor onZ (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 onZ).
The coaxial cable is specified as having an innerconductor of radiusR1 and an outer conductor whose inner radius isR2 (its thickness beyondR2 doesn't affect the following analysis). In betweenR1 andR2 the cable contains an idealdielectric material ofrelative permittivityεr and we assume conductors that are non-magnetic (soμ =μ0) and lossless (perfect conductors), all of which are good approximations to real-world coaxial cable in typical situations.
The central conductor is at voltageV and draws a currentI toward the right, so we expect a total power flow ofP =V ·I according to basiclaws 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 (usingGauss's law) 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 voltageV:so that:
The magnetic field, again by symmetry, can be non-zero only in theθ direction, that is, a vector field looping around the center conductor at every radius betweenR1 andR2.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 (+I in the center conductor and −I in the outer conductor), and again the electric field is zero there anyway. UsingAmpère's law in the region fromR1 toR2, which encloses the current +I in the center conductor but with no contribution from the current in the outer conductor, we find at radiusr: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 theZ direction, along the direction of the coaxial cable itself, as we would expect. Again only a function ofr, we can evaluateS(r):whereW is given above in terms of the center conductor voltageV. Thetotal power flowing down the coaxial cable can be computed by integrating over the entire cross sectionA of the cable in between the conductors:
Substituting the earlier solution for the constantW 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 theP =V ·I result can be analytically calculated are: the parallel-plate transmission line,[8] usingCartesian coordinates, and the two-wire transmission line,[9] usingbipolar cylindrical coordinates.
In the "microscopic" version of Maxwell's equations, this definition must be replaced by adefinition in terms of the electric fieldE and themagnetic flux densityB (described later in the article).
It is also possible to combine theelectric displacement fieldD with the magnetic fluxB to get theMinkowski form of the Poynting vector, or useD andH to construct yet another version. The choice has been controversial: Pfeifer et al.[10] summarize and to a certain extent resolve the century-long dispute between proponents of the Abraham and Minkowski forms (seeAbraham–Minkowski controversy).
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., formechanical energy. The Umov–Poynting vector[11] discovered byNikolay Umov in 1874 describes energy flux in liquid and elastic media in a completely generalized view.
The Poynting vector appears inPoynting's theorem (see that article for the derivation), an energy-conservation law:whereJf is thecurrent density offree charges andu is the electromagnetic energy density for linear,nondispersive materials, given bywhere
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 asdissipation, heat, etc. In this definition, bound electrical currents are not included in this term and instead contribute toS andu.
For light in free space, the linear momentum density is
For linear,nondispersive (in which all frequency components travel at the same speed) and isotropic (for simplicity) materials, theconstitutive relations can be written aswhere
Hereε andμ 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.[12]: 262–264
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.[13]
In a propagating electromagneticplane 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 fieldH(r,t) is given by the magnitude of the electric field vectorE(r,t) divided byη, theintrinsic impedance of the transmission medium:where |A| represents thevector norm ofA. SinceE andH 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 takeX to be the direction of the electric field andY to be the direction of the magnetic field. The instantaneous Poynting vector, given by the cross product ofE andH will then be in the positiveZ direction:
Finding the time-averaged power in the plane wave then requires averaging over the wave period (the inverse frequency of the wave):whereErms is theroot mean square (RMS) electric field amplitude. In the important case thatE(t) is sinusoidally varying at some frequency with peak amplitudeEpeak,Erms 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 usingErms 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 impedanceη is simply given by theimpedance of free spaceη0 ≈ 377 Ω. In non-magnetic dielectrics (such as all transparent materials at optical frequencies) with a specified dielectric constantεr, 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 theirradiance, more often simply referred to as theintensity (a somewhat ambiguous term).
The "microscopic" (differential) version of Maxwell's equations admits only the fundamental fieldsE andB, without a built-in model of material media. Only the vacuum permittivity and permeability are used, and there is noD orH. When this model is used, the Poynting vector is defined aswhere
This is actually the general expression of the Poynting vector[dubious –discuss].[14] The corresponding form ofPoynting's theorem iswhereJ is thetotalcurrent density and the energy densityu is given bywhereε0 is thevacuum permittivity. It can be derived directly fromMaxwell's equations in terms oftotal charge and current and theLorentz force law only.
The two alternative definitions of the Poyntingvector are equal in vacuum or in non-magnetic materials, whereB =μ0H. In all other cases, they differ in thatS = (1/μ0)E ×B and the correspondingu are purely radiative, since the dissipation term−J ⋅E covers the total current, while theE ×H definition has contributions from bound currents which are then excluded from the dissipation term.[15]
Since only the microscopic fieldsE andB occur in the derivation ofS = (1/μ0)E ×B and the energy density, assumptions about any material present are avoided. The Poynting vector and theorem and expression for energy density are universally valid in vacuum and all materials.[15]
The above form for the Poynting vector represents theinstantaneous power flow due toinstantaneous electric and magnetic fields. More commonly, problems in electromagnetics are solved in terms ofsinusoidally varying fields at a specified frequency. The results can then be applied more generally, for instance, by representing incoherent radiation as a superposition of such waves at different frequencies and with fluctuating amplitudes.
We would thus not be considering the instantaneousE(t) andH(t) used above, but rather a complex (vector) amplitude for each which describes a coherent wave's phase (as well as amplitude) usingphasor notation. These complex amplitude vectors arenot functions of time, as they are understood to refer to oscillations over all time. A phasor such asEm is understood to signify a sinusoidally varying field whose instantaneous amplitudeE(t) follows the real part ofEm ejωt whereω is the (radian) frequency of the sinusoidal wave being considered.
In the time domain, it will be seen that the instantaneous power flow will be fluctuating at a frequency of 2ω. But what is normally of interest is theaverage power flow in which those fluctuations are not considered. In the math below, this is accomplished by integrating over a full cycleT = 2π /ω. The following quantity, still referred to as a "Poynting vector", is expressed directly in terms of the phasors as:where∗ denotes thecomplex conjugate. The time-averaged power flow (according to the instantaneous Poynting vector averaged over a full cycle, for instance) is then given by thereal part ofSm. The imaginary part is usually ignored[12][clarification needed], however, it signifies "reactive power" such as the interference due to astanding wave or thenear field of an antenna. In a single electromagneticplane wave (rather than a standing wave which can be described as two such waves travelling in opposite directions),E andH are exactly in phase, soSm is simply a real number according to the above definition.
The equivalence ofRe(Sm) to the time-average of theinstantaneous Poynting vectorS can be shown as follows.
The average of the instantaneous Poynting vectorS over time is given by:
The second term is the double-frequency component having an average value of zero, so we find:
According to some conventions, the factor of 1/2 in the above definition may be left out. Multiplication by 1/2 is required to properly describe the power flow since the magnitudes ofEm andHm refer to thepeak fields of the oscillating quantities. If rather the fields are described in terms of theirroot mean square (RMS) values (which are each smaller by the factor), then the correct average power flow is obtained without multiplication by 1/2.
If a conductor has significant resistance, then, near the surface of that conductor, the Poynting vector would be tilted toward and impinge upon the conductor.[9]: figs.7,8 Once the Poynting vector enters the conductor, it is bent to a direction that is almost perpendicular to the surface.[16]: 61 This is a consequence ofSnell's law and the very slow speed of light inside a conductor. The definition and computation of the speed of light in a conductor can be given.[17]: 402 Inside the conductor, the Poynting vector represents energy flow from theelectromagnetic field into the wire, producing resistiveJoule heating in the wire. For a derivation that starts with Snell's law see Reitz page 454.[18]: 454
The density of the linear momentum of the electromagnetic field isS/c2 whereS is the magnitude of the Poynting vector andc is the speed of light in free space. Theradiation pressure exerted by an electromagnetic wave on the surface of a target is given by
The Poynting vector occurs in Poynting's theorem only through itsdivergence∇ ⋅S, that is, it is only required that thesurface integral of the Poynting vector around a closed surface describe the net flow of electromagnetic energy into or out of the enclosed volume. This means that adding asolenoidal vector field (one with zero divergence) toS will result in another field that satisfies this required property of a Poynting vector field according to Poynting's theorem. Since thedivergence of any curl is zero, one can add thecurl of any vector field to the Poynting vector and the resulting vector fieldS′ will still satisfy Poynting's theorem.
However even though the Poynting vector was originally formulated only for the sake of Poynting's theorem in which only its divergence appears, it turns out that the above choice of its formis unique.[12]: 258–260, 605–612 The following section gives an example which illustrates why it isnot acceptable to add an arbitrary solenoidal field toE ×H.
The consideration of the Poynting vector instatic fields shows the relativistic nature of the Maxwell equations and allows a better understanding of the magnetic component of theLorentz force,q(v ×B). To illustrate, the accompanying picture is considered, which describes the Poynting vector in a cylindricalcapacitor, which is located in anH field (pointing into the page) generated by a permanent magnet. Although there are only static electric and magnetic fields, the calculation of the Poynting vector produces a clockwise circular flow of electromagnetic energy, with no beginning or end.
While the circulating energy flow may seem unphysical, its existence is necessary to maintainconservation of angular momentum. The momentum of an electromagnetic wave in free space is equal to its power divided byc, the speed of light. Therefore, the circular flow of electromagnetic energy implies anangular momentum.[19] If one were to connect a wire between the two plates of the charged capacitor, then there would be a Lorentz force on that wire while the capacitor is discharging due to the discharge current and the crossed magnetic field; that force would be circumferential to the central axis and thus add angular momentum to the system. That angular momentum would match the "hidden" angular momentum, revealed by the Poynting vector, circulating before the capacitor was discharged.
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