Poynting vector

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 can be thought of as representing the directional energy flux density (the rate of energy transfer per unit area, in W/m²) of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented 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 magnetic field. (All bold letters represent vectors.)

Occasionally an alternative definition in terms of electric field E and the magnetic flux density B is used. It is even possible to combine the 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 another. The choice has been controversial: Pfeifer et al. summarize the century-long dispute between proponents of the Abraham and Minkowski forms.

Interpretation

The Poynting vector appears in Poynting's theorem, an energy-conservation law,

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

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 flux density 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

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

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.

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.

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 S as

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

where μ₀ 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

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

(with the electric constant ε₀).

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.

Since only the microscopic fields E and 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.

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 one can add a field curl $\nabla\times\mathbf F$, since $\nabla\cdot(\nabla\times\mathbf F)=0$ for an arbitrary field F. 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.^{[citation needed]}

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

Time-averaged Poynting vector

For time-harmonic (sinusoidal) electromagnetic fields, the average power flow over time is often more useful, and can be found as follows,

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

$= \operatorname{Re}\left(\mathbf{E} e^{j\omega t}\right) \times \operatorname{Re}\left(\mathbf{H} e^{j\omega t}\right)$

$= \frac{1}{2}\left(\mathbf{E} e^{j\omega t} + \mathbf{E}^* e^{-j\omega t}\right) \times \frac{1}{2}\left(\mathbf{H} e^{j\omega t} + \mathbf{H}^* e^{-j\omega t}\right)$

$= \frac{1}{4}\left(\mathbf{E} \times \mathbf{H}^* + \mathbf{E}^* \times \mathbf{H} + \mathbf{E} \times \mathbf{H} e^{2j\omega t} + \mathbf{E}^* \times \mathbf{H}^* e^{-2j\omega t}\right)$

$= \frac{1}{4}\left(\mathbf{E} \times \mathbf{H}^* + \left(\mathbf{E} \times \mathbf{H}^*\right)^* + \mathbf{E} \times \mathbf{H} e^{2j\omega t} + \left(\mathbf{E} \times \mathbf{H} e^{2j\omega t}\right)^*\right)$

$= \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H}^*\right) + \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H} e^{2j\omega t}\right)$

The average over time is given as

$\bar{\mathbf{S}} = \frac{1}{T}\int_0^T \mathbf{S}(t)dt = \frac{1}{T}\int_0^T \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H}^*\right) + \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H} e^{2j\omega t}\right) dt$

The second term is a sinusoidal curve ($\operatorname{Re}\left(e^{2j\omega t}\right) = \operatorname{cos}2\omega t$) whose average will be zero, which gives

$\bar{\mathbf{S}} = \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H}^*\right)$.

Examples and applications

In a coaxial cable

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 is flowing entirely through the dielectric between the conductors. No energy flows in the conductors themselves, since the electric field strength is zero. 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. See Hayt page 402 for the definition and computation of the speed of light in a conductor. 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.

In 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

$\langle S \rangle = \frac{1}{2 \mu_0 c} E_0^2 = \frac{\varepsilon_0 c}{2} E_0^2,$

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

$B_0 = \frac{1}{c}E_0,$

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

$S(t) = \frac{1}{\mu_0} E_0\,B_0\,\cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r}\right) = \frac{1}{\mu_0 c} E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right) = \varepsilon_0 c E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right).$

In the last step, we used the equality $\varepsilon_0\,\mu_0 = {c}^{-2}$ . Since the time- or space-average of $\cos^2\left(\omega\,t-{\mathbf k} \cdot {\mathbf r}\right)$ is $\textstyle\frac{1}{2}$, it follows that

$\left\langle S \right\rangle = \frac{\varepsilon_0 c}{2} E_0^2.$

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:

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

In static fields

Poynting vector in a static field, where E is electric field, H is magnetic field, and S is 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, $q \cdot (\vec v \times \vec B)$. 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 proves to be absolutely 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.

References

• Harrington, Roger F. (1961). Time-Harmonic Electromagnetic Fields. McGraw-Hill 
• Hayt, William (1981). Engineering Electromagnetics (4th ed.). McGraw-Hill. ISBN 0070273952 
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993). Foundations of Electromagnetic Theory (4th ed.). Addison-Wesley. ISBN 0201526247 

Further reading

• "Poynting Vector" from ScienceWorld (A Wolfram Web Resource) by Eric W. Weisstein
• Richard Becker & Sauter, F (1964). Electromagnetic fields and interactions. New York: Dover. ISBN 0486642909. http://worldcat.org/isbn/0486642909. 
• Joseph Edminister (1995). Schaum's outline of theory and problems of electromagnetics. New York: McGraw-Hill Professional. p. 225. ISBN 0070212341. http://books.google.com/?id=xV97IDOqBZIC&pg=PA225&dq=%22Poynting+vector%22.