thumb|300x300px|An electric dipole (oscillating here along the axis) results in [[dipole radiation, whose electric field strength (colored) and Poynting vector (arrows) are shown for its plane.]]

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/m<sup>2</sup>); kg/s<sup>3</sup> 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 <math>\mathbf{S}</math> is defined as the cross product

<math display=block>\mathbf{S} = \mathbf{E} \times \mathbf{H},</math>

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:

<math display="block">\nabla\cdot \mathbf{S} = -\frac{\partial u}{\partial t}</math>

where <math>u</math> 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 R<sub>1</sub> and an outer conductor whose inner radius is R<sub>2</sub> (its thickness beyond R<sub>2</sub> doesn't affect the following analysis). In between R<sub>1</sub> and R<sub>2</sub> the cable contains an ideal dielectric material of relative permittivity ε<sub>r</sub> and we assume conductors that are non-magnetic (so μ = μ<sub>0</sub>) and lossless (perfect conductors), all of which are good approximations to real-world coaxial cable in typical situations.

center|600px|thumb|Illustration of electromagnetic power flow inside a [[coaxial cable according to the <span style="color:green">Poynting vector S</span>, calculated using the <span style="color:red">electric field E</span> (due to the voltage V) and the <span style="color:blue">magnetic field H</span> (due to current I).]]

thumb|The electric field in a transmission line complying with Snell's law.

thumb|right|350px|[[Direct current|DC power transmission through a coaxial cable showing relative strength of electric (<math>E_r</math>) and magnetic (<math>H_\theta</math>) fields and resulting Poynting vector (<math>S_z = E_r \cdot H_\theta</math>) 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 R<sub>1</sub> and R<sub>2</sub>.]]

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 (<math>R_1 < r < R_2</math>), 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:

<math display=block>E_r(r) = \frac{W}{r}</math>

W can be evaluated by integrating the electric field from <math>r = R_2</math> to <math>R_1</math> which must be the negative of the voltage V:

<math display=block>-V = \int_{R_2}^{R_1} \frac{W}{r} dr = -W \ln \left(\frac{R_2}{R_1}\right)</math>

so that:

<math display=block>W = \frac{V}{\ln(R_2/R_1)}</math>

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 R<sub>1</sub> and R<sub>2</sub>. 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. Using Ampère's law in the region from R<sub>1</sub> to R<sub>2</sub>, 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:

<math display=block>\begin{align}

I = \oint_C \mathbf{H} \cdot ds &= 2 \pi r H_\theta(r) \\

H_\theta(r) &= \frac {I}{2 \pi r}

\end{align}</math>

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(r):

<math display=block>S_z(r) = E_r(r) H_\theta(r) = \frac{W}{r} \frac {I}{2 \pi r} = \frac{W \, I} {2 \pi r^2}</math>

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:

<math display=block>\begin{align}

P_\text{tot}

&= \iint_\mathbf{A} S_z (r, \theta)\, dA = \int_{R_1}^{R_2} 2 \pi r dr S_z(r) \\

&= \int_{R_1}^{R_2} \frac{W\, I}{r} dr = W\, I\, \ln \left(\frac{R_2}{R_1}\right).

\end{align}</math>

Substituting the earlier solution for the constant W we find:

<math display=block>P_\mathrm{tot} = I \ln \left(\frac{R_2}{R_1}\right) \frac{V}{\ln(R_2/R_1)} = V \, I</math>

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 (described later in the article).

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 (see Abraham–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., 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 (see that article for the derivation), an energy-conservation law:

<math display="block">\frac{\partial u}{\partial t} = -\mathbf{\nabla} \cdot \mathbf{S} - \mathbf{J_\mathrm{f \cdot \mathbf{E},</math>

where J<sub>f</sub> is the current density of free charges and u is the electromagnetic energy density for linear, nondispersive materials, given by

<math display="block">u = \frac{1}{2}\! \left(\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H}\right)\! ,</math>

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

<math> \frac{\langle S \rangle}{c^2} . </math>

For linear, nondispersive (in which all frequency components travel at the same speed) and isotropic (for simplicity) materials, the constitutive relations can be written as

<math display="block">\mathbf{D} = \varepsilon \mathbf{E},\quad \mathbf{B} = \mu\mathbf{H},</math>

where

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

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(r, t) is given by the magnitude of the electric field vector E(r, t) divided by η, the intrinsic impedance of the transmission medium:

<math display="block">|\mathbf{H}| = \frac {|\mathbf{E}|}{\eta},</math>

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:

<math display="block">\left|\mathsf{S_z}\right| = \left|\mathsf{E_x} \mathsf{H_y}\right| = \frac{\left|\mathsf{E_x}\right|^2}{\eta}.</math>

Finding the time-averaged power in the plane wave then requires averaging over the wave period (the inverse frequency of the wave):

<math display="block">\left\langle\mathsf{S_z}\right\rangle = \frac{\left\langle\left|\mathsf{E_x}\right|^2\right\rangle}{\eta} = \frac{\mathsf{E_\text{rms}^2{\eta},</math>

where E<sub>rms</sub> is the root mean square (RMS) electric field amplitude. In the important case that E(t) is sinusoidally varying at some frequency with peak amplitude E<sub>peak</sub>, E<sub>rms</sub> is <math>\mathsf{E_{peak / \sqrt{2}</math>, with the average Poynting vector then given by:

<math display="block">\left\langle\mathsf{S_z}\right\rangle = \frac{\mathsf{E_{peak}^2{2\eta}.</math>

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<sub>rms</sub> 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 η<sub>0</sub> ≈377Ω. In non-magnetic dielectrics (such as all transparent materials at optical frequencies) with a specified dielectric constant ε<sub>r</sub>, or in optics with a material whose refractive index <math>\mathsf{n} = \sqrt{\epsilon_r}</math>, the intrinsic impedance is found as:

<math display="block">\eta = \frac{\eta_0}{\sqrt{\epsilon_r.</math>

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 (a somewhat ambiguous term).

Formulation in terms of microscopic fields

The "microscopic" (differential) version of Maxwell's equations admits only the fundamental fields E and B, without a built-in model of material media. Only the vacuum permittivity and permeability are used, and there is no D or H. When this model is used, the Poynting vector is defined as

<math display="block">\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B},</math>

where

  • μ<sub>0</sub> is the vacuum permeability;
  • E is the electric field vector;
  • B is the magnetic flux.

This is actually the general expression of the Poynting vector. The corresponding form of Poynting's theorem is

<math display="block">\frac{\partial u}{\partial t} = - \nabla \cdot \mathbf{S} -\mathbf{J} \cdot \mathbf{E},</math>

where J is the total current density and the energy density u is given by

<math display="block">u = \frac{1}{2}\! \left(\varepsilon_0 |\mathbf{E}|^2 + \frac{1}{\mu_0} |\mathbf{B}|^2\right)\! ,</math>

where ε<sub>0</sub> is the vacuum permittivity. It can be derived directly from Maxwell's equations in terms of total charge and current and the Lorentz force law only.

The two alternative definitions of the Poynting vector are equal in vacuum or in non-magnetic materials, where . In all other cases, they differ in that and the corresponding u are purely radiative, since the dissipation term covers the total current, while the E × H definition has contributions from bound currents which are then excluded from the dissipation term.

Since only the microscopic fields E and B occur in the derivation of 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. This is a consequence of Snell'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. Inside the conductor, the Poynting vector represents energy flow from the electromagnetic field into the wire, producing resistive Joule heating in the wire. For a derivation that starts with Snell's law see Reitz page 454.

Radiation pressure

The density of the linear momentum of the electromagnetic field is S/c<sup>2</sup> where S is the magnitude of the Poynting vector and c is the speed of light in free space. The radiation pressure exerted by an electromagnetic wave on the surface of a target is given by

<math display="block">P_\mathrm{rad} = \frac{\langle S\rangle}{\mathrm{c.</math>

Uniqueness of the Poynting vector

The Poynting vector occurs in Poynting's theorem only through its divergence , that is, it is only required that the surface 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 a solenoidal vector field (one with zero divergence) to S will result in another field that satisfies this required property of a Poynting vector field according to Poynting's theorem. Since the divergence of any curl is zero, one can add the curl of any vector field to the Poynting vector and the resulting vector field S′ 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 form is unique. 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.

See also

  • Wave vector

References

Further reading