Poynting vector

File:DipoleRadiation.gif

Dipole radiation of a dipole vertically in the page showing electric field strength (colour) and Poynting vector (arrows) in the plane of the page.

In physics, the Poynting vector represents the directional energy flux density (the rate of energy transfer per unit area) of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m²). It is named after its inventor John Henry Poynting who first derived it in 1884. Oliver Heaviside and Nikolay Umov also independently discovered the Poynting vector.

Definition

In Poynting's original paper and in many textbooks, the Poynting vector is defined as

$\mathbf{S} = \mathbf{E} \times \mathbf{H},$

where bold letters represent vectors and

• E is the electric field vector;
• H is the magnetic field vector.

This expression is often called the Abraham form. The Poynting vector is usually denoted by S or N.

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 described later in the article).

It is also possible to combine the electric displacement field D with the magnetic flux density 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

File:Poynting vectors of DC circuit.svg

A DC circuit consisting of a battery (V) and resistor (R), showing the direction of the Poynting vector (S, blue) in the space surrounding it, along with the fields it is derived from; the electric field (E, red) and the magnetic field (H, green). In the region around the battery the Poynting vector is directed outward, indicating power flowing out of the battery into the fields; in the region around the resistor the vector is directed inward, indicating field power flowing into the resistor. Across any plane P between the battery and resistor, the Poynting flux is in the direction of the resistor.

The Poynting vector appears in Poynting's theorem (see that article for the derivation), an energy-conservation law:

$\frac{\partial u}{\partial t} = -\mathbf{\nabla} \cdot \mathbf{S} - \mathbf{J_\mathrm{f}} \cdot \mathbf{E},$

where J_f is the current density of free charges and u is the electromagnetic energy density for linear, nondispersive materials, given by

$u = \frac{1}{2}\! \left(\mathbf{E} \cdot \mathbf{D} + \mathbf{B} \cdot \mathbf{H}\right)\! ,$

where

• E is the electric field;
• D is the electric displacement field;
• B is the magnetic flux density;
• H is the magnetic 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 linear, nondispersive and isotropic(for simplicity) materials, the constitutive relations can be written as

$\mathbf{D} = \varepsilon \mathbf{E},\quad \mathbf{H} = \frac{1}{\mu}\mathbf{B},$

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.

Adding the curl of a vector field

Since the Poynting vector occurs in Poynting's theorem only through its divergence ∇ ⋅ S, and 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. The theory of special relativity, however, in which energy and momentum are defined locally and invariantly via the stress–energy tensor, shows that the given expression for the Poynting vector is unique.

It is generally argued that Maxwell equations are manifestly Lorentz covariant while the electromagnetic (EM) stress-energy tensor follows from the Maxwell equations; thus the EM momentum defined from the EM tensor certainly respects the principle of relativity. However a recent article indicates that “such an argument is based on an incomplete understanding of the relativity principle”, and states that the EM stress-energy tensor is not sufficient to define EM momentum correctly. The study also claims that Poynting vector does not represent the EM power flow (energy flux density vector) for a plane wave in an anisotropic medium if the Poynting vector is not parallel to the wave vector, and states “this conclusion is clearly supported by Fermat’s principle and special theory of relativity”.

Formulation in terms of microscopic fields

The "microscopic" 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

$\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B},$

where

• μ₀ is the vacuum permeability;
• E is the electric field;
• B is the magnetic flux density.

The corresponding form of Poynting's theorem is

$\frac{\partial u}{\partial t} = - \nabla \cdot \mathbf{S} -\mathbf{J} \cdot \mathbf{E},$

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

$u = \frac{1}{2}\! \left(\varepsilon_0 \mathbf{E}^2 + \frac{1}{\mu_0} \mathbf{B}^2\right)\! ,$

where ε₀ 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 B = μ₀H. In all other cases, they differ in that S = 1/μ₀ E × B and the corresponding u are purely radiative, since the dissipation term −J ⋅ E 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 S = 1/μ₀ E × B, assumptions about any material present are completely avoided, and Poynting vector and theorem are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the "macroscopic" form E x H.

Time-averaged Poynting vector

For time-periodic sinusoidal electromagnetic fields, the average power flow per unit time is often more useful, and can be found by using the analytic representation of the electric and magnetic fields as follows (the subscript "a" denotes an analytic signal, the underbar with the subscript "m" a complex amplitude, and the superscript " * " a complex conjugate):

$\begin{align}\mathbf{S} &= \mathbf{E} \times \mathbf{H}\\ &= \operatorname{Re}\! \left(\mathbf{E_\mathrm{a}}\right) \times \operatorname{Re}\!\left(\mathbf{H_\mathrm{a}} \right)\\ &= \operatorname{Re}\! \left(\underline{\mathbf{E_m}} e^{j\omega t}\right) \times \operatorname{Re}\!\left(\underline{\mathbf{H_m}} e^{j\omega t}\right)\\ &= \frac{1}{2}\! \left(\underline{\mathbf{E_m}} e^{j\omega t} + \underline{\mathbf{E_m^*}} e^{-j\omega t}\right) \times \frac{1}{2}\! \left(\underline{\mathbf{H_m}} e^{j\omega t} + \underline{\mathbf{H_m^*}} e^{-j\omega t}\right)\\ &= \frac{1}{4}\! \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m^*}} + \underline{\mathbf{E_m^*}} \times \underline{\mathbf{H_m}} + \underline{\mathbf{E_m}} \times \underline{\mathbf{H_m}} e^{2j\omega t} + \underline{\mathbf{E_m^*}} \times \underline{\mathbf{H_m^*}} e^{-2j\omega t}\right)\\ &= \frac{1}{4}\! \left[\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m^*}} + \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m^*}}\right)^* + \underline{\mathbf{E_m}} \times \underline{\mathbf{H_m}} e^{2j\omega t} + \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m}} e^{2j\omega t}\right)^*\right]\\ &= \frac{1}{2} \operatorname{Re}\! \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m^*}}\right) + \frac{1}{2}\operatorname{Re}\! \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m}} e^{2j\omega t}\right)\! . \end{align}$

The average over time is given by

$\langle\mathbf{S}\rangle = \frac{1}{T} \int_0^T \mathbf{S}(t)\, dt = \frac{1}{T} \int_0^T\! \left[\frac{1}{2} \operatorname{Re}\! \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m^*}}\right) + \frac{1}{2} \operatorname{Re}\! \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m}} e^{2j\omega t}\right)\right]dt.$

The second term is a sinusoidal curve

$\operatorname{Re}\! \left(e^{2j\omega t}\right) = \cos(2\omega t)$

and its average is zero (for large T), giving

$\langle \mathbf{S}\rangle = \frac{1}{2} \operatorname{Re}\! \left(\underline{\mathbf{E_m}} \times \underline{\mathbf{H_m^*}}\right) = \frac{1}{2} \operatorname{Re}\! \left(\underline{\mathbf{E_m}} e^{j\omega t} \times \underline{\mathbf{H_m^*}} e^{-j\omega t}\right) = \frac{1}{2} \operatorname{Re}\! \left(\mathbf{E_\mathrm{a}} \times \mathbf{H_\mathrm{a}^*}\right)\! .$

Examples and applications

Coaxial cable

File:Poynting vector coaxial cable.svg

Poynting vector in a coaxial cable, shown in red.

For example, the Poynting vector within the dielectric insulator of a coaxial cable is nearly parallel to the wire axis (assuming no fields outside the cable and a wavelength longer than the diameter of the cable, including DC). Electrical energy delivered to the load is flowing entirely through the dielectric between the conductors. Very little energy flows in the conductors themselves, since the electric field strength is nearly zero. The energy flowing in the conductors flows radially into the conductors and accounts for energy lost to resistive heating of the conductor. No energy flows outside the cable, either, since there the magnetic fields of inner and outer conductors cancel to zero.

Resistive dissipation

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. Once the Poynting vector enters the conductor, it is bent to a direction that is almost perpendicular to the surface. 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.

Plane waves

In a propagating sinusoidal linearly polarized electromagnetic plane wave of a fixed frequency, the Poynting vector always points in the direction of propagation while oscillating in magnitude. The time-averaged magnitude of the Poynting vector is

$\langle S\rangle = \frac{1}{2 \mu_0 \mathrm{c}}E_\mathrm{m}^2 = \frac{\varepsilon_0 \mathrm{c}}{2} E_\mathrm{m}^2$

where E_m is the amplitude of the electric field and c is the speed of light in free space. This time-averaged value is called irradiance and denoted E_e in radiometry, or is called intensity and denoted I in other fields.

Derivation

In an electromagnetic plane wave, E and B are always perpendicular to each other and the direction of propagation. Moreover, their amplitudes are related according to

$B_\mathrm{m} = \frac{1}{\mathrm{c}}E_\mathrm{m}$

and their time and position dependences are

$E(\mathbf{r}, t) = E_\mathrm{m} \cos(\omega t - \mathbf{k} \cdot \mathbf{r})$

$B(\mathbf{r}, t) = B_\mathrm{m} \cos(\omega t - \mathbf{k} \cdot \mathbf{r})$

where ω is the angular frequency of the wave and k is wave vector.

The time-dependent and position magnitude of the Poynting vector is then

$S(\mathbf{r}, t) = \frac{1}{\mu_0}E_\mathrm{m}B_\mathrm{m} \cos^2(\omega t - \mathbf{k} \cdot \mathbf{r}) = \frac{1}{\mu_0 c}E_\mathrm{m}^2 \cos^2(\omega t - \mathbf{k} \cdot \mathbf{r}) = \varepsilon_0 \mathrm{c}E_\mathrm{m}^2 \cos^2(\omega t - \mathbf{k} \cdot \mathbf{r}).$

In the last step, we used the equality ε₀μ₀ = 1/c². Since the time- or space-average of cos²(ωt − k ⋅ r) is 1/2, it follows that

$\langle S\rangle = \frac{1}{2\mu_0 \mathrm{c}}E_\mathrm{m}^2 = \frac{\varepsilon_0 \mathrm{c}}{2}E_\mathrm{m}^2.$

File:Cosine squared graph, or half of one plus the cosine of twice x.svg

average value of (cos(x))^2 is a half, unlike the average of cos(x) which is zero

It will be appreciated that quantitatively the Poynting vector is evaluated only from a prior knowledge of the distribution of electric and magnetic fields, which are calculated by applying boundary conditions to a particular set of physical circumstances, for example a dipole antenna. Therefore the E and H field distributions form the primary object of any analysis, while the Poynting vector remains an interesting by-product.

Radiation pressure

The density of the linear momentum of the electromagnetic field is S/c² 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

$P_\mathrm{rad} = \frac{\langle S\rangle}{\mathrm{c}}.$

Static fields

Poynting vector in a static field, where E is the electric field, H the magnetic field, and S the Poynting vector.

The consideration of the Poynting vector in static fields shows the relativistic nature of the Maxwell equations and allows a better understanding of the magnetic component of the Lorentz force, q(v × B). To illustrate, the accompanying picture is considered, which describes the Poynting vector in a cylindrical capacitor, which is located in an H 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 nonsensical or paradoxical, it is necessary to maintain conservation of momentum. Momentum density is proportional to energy flow density, so the circulating flow of energy contains an angular momentum. This is the cause of the magnetic component of the Lorentz force which occurs when the capacitor is discharged. During discharge, the angular momentum contained in the energy flow is depleted as it is transferred to the charges of the discharge current crossing the magnetic field.