Poynting vector: Definition from Answers.com

Dipole Radiation, Dipole parallel to the z-axis, electric field and poynting-vector in the x-z-plane.

In physics, the Poynting vector can be thought of as representing the energy flux (in W/m²) of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside independently co-discovered the Poynting vector. In Poynting's original paper and in many textbooks it is defined as

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

which is often called the Abraham form; here E is the electric field and H the auxiliary magnetic field.^[1]^[2] (All bold letters represent vectors.) Sometimes, an alternative definition in terms of electric field E and the magnetic field B is used, which is explained below. It is even possible to combine the displacement field D with the magnetic field B to get the Minkowski form of the Poynting vector, or use D and H to construct another.^[3] The choice has been controversial: Pfeifer et al^[4] admirably summarize the century-long dispute between proponents of the Abraham and Minkowski forms.

1 Interpretation
2 Formulation in terms of microscopic fields
3 Invariance to adding a curl of a field
4 Generalization
5 Examples and applications
6 Problems in certain cases
- 6.1 DC Power flow in a concentric cable
- 6.2 Independent E and B fields
7 Notes
8 Further reading

Interpretation

The Poynting vector appears in Poynting's theorem, an energy-conservation law^[2],

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

where J_f is the current density of free charges and u is the electromagnetic energy density,

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

where B is the magnetic field and D the electric displacement field.

The first term in the right-hand side represents the net electromagnetic energy flow into a small volume, while the second term represents the subtracted portion of the work done by free electrical currents that are not necessarily converted into electromagnetic energy (dissipation, heat). In this definition, bound electrical currents are not included in this term, and instead contribute to S and u.

Note that u can only be given if linear, nondispersive and uniform materials are involved, i.e., if the constitutive relations can be written as

$\mathbf{D} = \epsilon\mathbf{E}, \;\;\; \mathbf{H} = \mathbf{B}/\mu$

where ε and μ are constants (which depend on the material through which the energy flows), called the permittivity and permeability, respectively, of the material.^[2]

This practically limits Poynting's theorem in this form to fields in vacuum. A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms and the loss of their clear physical interpretation.^[2]

The Poynting vector is usually interpreted as an energy flux, but this is only strictly correct for electromagnetic radiation. The more general case is described by Poynting's theorem above, where it occurs as a divergence, which means that it can only describe the change of energy density in space, rather than the flow.

Formulation in terms of microscopic fields

In some cases, it may be more appropriate to define the Poynting vector $\mathbf{S}$ as

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

where $μ 0$ is the magnetic constant. It can be derived directly from Maxwell's equations in terms of total charge and current and the Lorentz force law only.

The corresponding form of Poynting's theorem is

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

where $\mathbf{J}$ is the total current density and the energy density $u$ is

$u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}^2 + \frac{\mathbf{B}^2}{\mu_0}\right)$

(with the electric constant $ε 0$ ).

The two alternative definitions of the Poynting vector are equivalent in vacuum or in non-magnetic materials, where $\mathbf{B}=\mu_0 \mathbf{H}$ . In all other cases, they differ in that $\mathbf{S}=1/\mu_0 \mathbf{E}\times\mathbf{B}$ and the corresponding $u$ are purely radiative, since the dissipation term, $-\mathbf{J}\cdot\mathbf{E}$ , covers the total current, while the definition in terms of $\mathbf{H}$ has contributions from bound currents which then lack in the dissipation term.^[5]

Since only the microscopic fields $\mathbf{E}$ and $\mathbf{B}$ are needed in the derivation of $\mathbf{S}=1/\mu_0 \mathbf{E}\times\mathbf{B}$ , assumptions about any material possibly present can be completely avoided, and Poynting's vector as well as the theorem in this definition are universally valid, in vacuum as in all kinds of material. This is especially true for the electromagnetic energy density, in contrast to the case above.^[5]

Invariance to adding a curl of a field

Since the Poynting vector only occurs in Poynting's theorem as a divergence $\nabla\cdot\mathbf{S}$ , the Poynting vector is arbitrary to the extent that the curl $\nabla\times\mathbf F$ of any field F can be added,^[2] because $\nabla\cdot\nabla\times\mathbf F=0$ for any field. Doing so is not common, though, and will lead to inconsistencies in a relativistic description of electromagnetic fields in terms of the stress-energy tensor.

Generalization

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^[6] discovered by Nikolay Umov in 1874 describes energy flux in liquid and elastic media in a completely generalized view.

Examples and applications

The Poynting vector in a coaxial cable

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) – so electric energy is flowing through the dielectric between the conductors. If the core conductor was replaced by a wire having significant resistance, then the Poynting vector would become tilted toward that wire, indicating that energy flows from the electromagnetic field into the wire, producing resistive Joule heating in the wire.

The Poynting vector in plane waves

In a propagating sinusoidal electromagnetic plane wave of a fixed frequency, the Poynting vector oscillates, always pointing in the direction of propagation. The time-averaged magnitude of the Poynting vector is

$\langle S \rangle = \frac{1}{2 \mu_0 c} E_0^2 = \frac{\epsilon_0 c}{2} E_0^2,$

where $\ E_0$ is the maximum amplitude of the electric field and $\ c$ is the speed of light in free space. This time-averaged value is also called the irradiance or intensity I.

Derivation

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

$B_0 = \frac{E_0}{c},$

and their time and position dependences are

$E\left(t,{\mathbf r}\right) = E_0\,\cos\left(\omega\,t- {\mathbf k} \cdot {\mathbf r} \right),$

$B\left(t,{\mathbf r}\right) = B_0\,\cos\left(\omega\,t- {\mathbf k} \cdot {\mathbf r} \right),$

where $\ \omega$ is the frequency of the wave and $\mathbf{k}$ is wave vector. The time-dependent and position magnitude of the Poynting vector is then

$S(t) = \frac{1}{\mu_0} E_0\,B_0\,\cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r}\right) = \frac{1}{\mu_0 c} E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right) = \epsilon_0 c E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right).$

In the last step, we used the equality $\epsilon_0\,\mu_0 = {c}^{-2}$ . Since the time- or space-average of $\cos^2\left(\omega\,t-{\mathbf k} \cdot {\mathbf r}\right)$ is ½, it follows that

$\left\langle S \right\rangle = \frac{\epsilon_0 c}{2} E_0^2.$

Poynting vector and radiation pressure

S divided by the square of the speed of light in free space is the density of the linear momentum of the electromagnetic field. The time-averaged intensity $\langle\mathbf{S}\rangle$ divided by the speed of light in free space is the radiation pressure exerted by an electromagnetic wave on the surface of a target:

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

Problems in certain cases

The common use of the Poynting vector as an energy flux rather than in the context of Poynting's theorem gives rise to controversial interpretions in cases where it is not used to describe electromagnetic radiation. Two examples are given below.

DC Power flow in a concentric cable

Application of Poynting's Theorem to a concentric cable carrying DC current leads to the correct power transfer equation $P = V I$ , where $V$ is the potential difference between the cable and ground, $I$ is the current carried by the cable. This power flows through the surrounding dielectric, and not through the cable itself. ^[7]

However, it is also known that power cannot be radiated without accelerated charges, i.e. time varying currents. Since we are considering DC (time invariant) currents here, radiation is not possible. This has led to speculation that Poynting Vector may not represent the power flow in certain systems.^[8]^[9]

Independent E and B fields

Independent static $\mathbf{E}$ and $\mathbf{B}$ fields do not result in power flows along the direction of $\mathbf{E} \times \mathbf{B}$ .

For example, application of Poynting's Theorem to a bar magnet, on which an electric charge is present, leads to seemingly absurd conclusion that there is a continuous circulation of energy around the magnet.^[7] However, there is no divergence of energy flow, or in layman's terms, energy that enters given unit of space equals the energy that leaves that unit of space, so there is no net energy flow into the given unit of space.

Notes

^ Poynting, J. H. (1884). "On the Transfer of Energy in the Electromagnetic Field". Phil. Trans. 175: 277. doi:10.1098/rstl.1884.0016. http://www.archive.org/details/collectedscienti00poynuoft.
^ ^a ^b ^c ^d ^e John David Jackson (1998). Classical electrodynamics (Third ed.). New York: Wiley. ISBN 047130932X. http://worldcat.org/isbn/047130932X.
^ Kinsler, P.; Favaro, A.; McCall M.W. (2009). "Four Poynting theorems" (reprint). Eur. J. Phys. 30: 983. doi:10.1088/0143-0807/30/5/007. http://arxiv.org/abs/0908.1721.
^ Pfeifer, R.N.C.; Nieminen, T.A.; Heckenberg N. R.; Rubinsztein-Dunlop H. (2007). "Momentum of an electromagnetic wave in dielectric media". Rev. Mod. Phys. 79: 1197. doi:10.1103/RevModPhys.79.1197. http://link.aps.org/doi/10.1103/RevModPhys.79.1197.
^ ^a ^b Richter, F.; Florian, M.; Henneberger, K. (2008). "Poynting's theorem and energy conservation in the propagation of light in bounded media" (reprint). Europhys. Lett. 81: 67005. doi:10.1209/0295-5075/81/67005. http://arxiv.org/pdf/0710.0515v3.
^ Umov, N. A. (1874). "Ein Theorem über die Wechselwirkungen in Endlichen Entfernungen". Zeitschrift für Mathematik und Physik XIX: 97.
^ ^a ^b Jordan, Edward; Balmain, Keith (2003), Electromagnetic Waves and Radiating Systems (Second ed.), New Jersey: Prentice-Hall
^ Jeffries, Clark (Sep., 1992) (PDF). A New Conservation Law for Classical Electrodynamics. Society for Industrial and Applied Mathematics (SIAM Review). http://links.jstor.org/sici?sici=0036-1445%28199209%2934%3A3%3C386%3AANCLFC%3E2.0.CO%3B2-U. Retrieved 2008-03-04.
^ Robinson, F. N. H. (Dec., 1994). Poynting's Vector: Comments on a Recent Paper by Clark Jeffries. Society for Industrial and Applied Mathematics (SIAM Review). http://www.jstor.org/pss/2132722.

	electromagnetic momentum (electromagnetism)
	flux density
	John Henry Poynting (English physicist)

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