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In physics, the **Poynting vector** represents the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m^{2}). It is named after its discoverer John Henry Poynting who first derived it in 1884.^{ [1] }^{:132} 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.^{ [2] } 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 electric and magnetic fields.

In Poynting's original paper and in many textbooks, the Poynting vector is defined as^{ [3] }^{ [4] }^{ [5] }

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*.^{ [6] } 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** (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.^{ [7] } 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^{ [8] } discovered by Nikolay Umov in 1874 describes energy flux in liquid and elastic media in a completely generalized view.

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

where **J**_{f} is the current density of free charges and *u* is the electromagnetic energy density for linear, nondispersive materials, given by

where

**E**is the electric field;**D**is the electric displacement field;**B**is the magnetic flux density;**H**is the magnetic field.^{ [9] }^{:258–260}

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

where

*ε*is the permittivity of the material;*μ*is the permeability of the material.^{ [9] }^{:258–260}

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.^{ [9] }^{:262–264}

One consequence of the Poynting formula is that for electromagnetic field to do work, both magnetic and electric fields must be present. The magnetic field alone and the electric field alone can not do any work.^{ [10] }

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

where

*μ*_{0}is the vacuum permeability;**E**is the electric field vector;**B**is the magnetic field vector.

This is actually the general expression of the Poynting vector.^{ [11] } The corresponding form of Poynting's theorem is

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

where ε_{0} is the vacuum permittivity, and the notation **E**^{2} is understood to mean the dot product of the real vector **E**(t) with itself, thus the *square* of the vector norm ||**E**||. 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** = *μ*_{0}**H**. In all other cases, they differ in that **S** = (1/*μ*_{0}) **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.^{ [12] }

Since only the microscopic fields **E** and **B** occur in the derivation of **S** = (1/*μ*_{0}) **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.^{ [12] }

The above form for the Poynting vector represents the *instantaneous* power flow due to *instantaneous* electric and magnetic fields. More commonly, problems in electromagnetics are solved in terms of sinusoidally 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 instantaneous **E**(*t*) and **H**(*t*) used above, but rather a complex (vector) amplitude for each which describes a coherent wave's phase (as well as amplitude) using phasor notation. These complex amplitude vectors are *not* functions of time, as they are understood to refer to oscillations over all time. A phasor such as **E**_{m} is understood to signify a sinusoidally varying field whose instantaneous amplitude **E**(*t*) follows the real part of **E**_{m} *e ^{jωt}* where ω 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 2*ω*. But what is normally of interest is the *average* power flow in which those fluctuations are not considered. In the math below, this is accomplished by integrating over a full cycle *T* = 2*π* / *ω*. The following quantity, still referred to as a "Poynting vector", is expressed directly in terms of the phasors as:

where ^{∗} denotes the complex conjugate. The time-averaged power flow (according to the instantaneous Poynting vector averaged over a full cycle, for instance) is then given by the *real part* of **S**_{m}. The imaginary part is usually ignored, however it signifies "reactive power" such as the interference due to a standing wave or the near field of an antenna. In a single electromagnetic plane wave (rather than a standing wave which can be described as two such waves travelling in opposite directions), **E** and **H** are exactly in phase, so **S**_{m} is simply a real number according to the above definition.

The equivalence of Re(**S**_{m}) to the time-average of the *instantaneous* Poynting vector **S** can be shown as follows.

The average of the instantaneous Poynting vector **S** 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 of **E**_{m} and **H**_{m} refer to the *peak* fields of the oscillating quantities. If rather the fields are described in terms of their root mean square (rms) values (which are each smaller by the factor ), then the correct average power flow is obtained without multiplication by 1/2.

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.

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.^{ [13] }^{:61} 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.^{ [14] }^{:402} 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.^{ [15] }^{:454}

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 found as above to be:

where *E*_{m} is the complex amplitude of the electric field and η is the characteristic impedance of the transmission medium, or just η_{0} ≈ 377 Ω for a plane wave in free space. This directly follows from the above expression for the average Poynting vector using phasor quantities, and the fact that in a plane wave the magnetic field *H*_{m} is equal to the electric field *E*_{m} divided by η (and thus exactly in phase).

In optics, the time-averaged value of the radiated flux is technically known as the irradiance, more often simply referred to as the * intensity *.

The density of the linear momentum of the electromagnetic field is *S*/c^{2} 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

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 be counterintuitive, it is necessary to maintain conservation of angular momentum. Momentum density is proportional to energy flow density, so the circulating flow of energy contains an *angular* momentum.^{ [16] } 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.

The Poynting vector occurs in Poynting's theorem only through its divergence ∇ ⋅ **S**, 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 which 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.^{ [9] }^{:258–260}

However the theory of special relativity, in which energy and momentum are defined locally and invariantly via the stress–energy tensor, shows that the above-given expression for the Poynting vector is unique.^{ [9] }^{:258–260,605–612}

In physics, the **cross section** is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted *σ* (sigma) and is expressed in units of transverse area. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

In electromagnetism, there are two kinds of **dipoles**:

In physics the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity **v** in an electric field **E** and a magnetic field **B** experiences a force of

**Maxwell's equations** are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

**Flux** describes any effect that appears to pass or travel through a surface or substance. A flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

In electromagnetism and electronics, **inductance** is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the current, and follows any changes in current. From Faraday's law of induction, any change in magnetic field through a circuit induces an electromotive force (EMF) (voltage) in the conductors, a process known as electromagnetic induction. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated by Lenz's law, and the voltage is called *back EMF*.

In classical electromagnetism, **Ampère's circuital law** relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it using hydrodynamics in his 1861 published paper "On Physical Lines of Force" In 1865 he generalized the equation to apply to time-varying currents by adding the displacement current term, resulting in the modern form of the law, sometimes called the **Ampère–Maxwell law**, which is one of Maxwell's equations which form the basis of classical electromagnetism.

The **magnetic moment** is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.

In electrodynamics, **Poynting's theorem** is a statement of conservation of energy for the electromagnetic field,, in the form of a partial differential equation developed by British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution, through energy flux.

**Magnetic vector potential**, **A**, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field: . Together with the electric potential *φ*, the magnetic vector potential can be used to specify the electric field **E** as well. Therefore, many equations of electromagnetism can be written either in terms of the fields **E** and **B**, or equivalently in terms of the potentials *φ* and **A**. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

In physics, the **gyromagnetic ratio** of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol γ, gamma. Its SI unit is the radian per second per tesla (rad⋅s^{−1}⋅T^{−1}) or, equivalently, the coulomb per kilogram (C⋅kg^{−1}).

In classical electromagnetism, **magnetization** or **magnetic polarization** is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Diametric. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field. Paramagnetic materials have a weak induced magnetization in a magnetic field, which disappears when the magnetic field is removed. Ferromagnetic and ferrimagnetic materials have strong magnetization in a magnetic field, and can be *magnetized* to have magnetization in the absence of an external field, becoming a permanent magnet. Magnetization is not necessarily uniform within a material, but may vary between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector **M**.

In classical electromagnetism, **reciprocity** refers to a variety of related theorems involving the interchange of time-harmonic electric current densities (sources) and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints. Reciprocity is closely related to the concept of Hermitian operators from linear algebra, applied to electromagnetism.

The **Maxwell stress tensor** is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impossibly difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

In relativistic physics, the **electromagnetic stress–energy tensor** is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions.

The **covariant formulation of classical electromagnetism** refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

The **gradient theorem**, also known as the **fundamental theorem of calculus for line integrals**, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the fundamental theorem of calculus to any curve in a plane or space rather than just the real line.

**Dielectric loss** quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the **loss angle***δ* or the corresponding **loss tangent** tan *δ*. Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

**Magnetic energy** and electrostatic potential energy are related by Maxwell's equations. The potential energy of a magnet or magnetic moment in a magnetic field is defined as the mechanical work of the magnetic force on the re-alignment of the vector of the magnetic dipole moment and is equal to:

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*European Journal of Physics*.**30**(5): 983. arXiv: 0908.1721 . Bibcode:2009EJPh...30..983K. doi:10.1088/0143-0807/30/5/007. S2CID 118508886. - ↑ Pfeifer, Robert N. C.; Nieminen, Timo A.; Heckenberg, Norman R.; Rubinsztein-Dunlop, Halina (2007). "Momentum of an Electromagnetic Wave in Dielectric Media".
*Reviews of Modern Physics*.**79**(4): 1197. arXiv: 0710.0461 . Bibcode:2007RvMP...79.1197P. doi:10.1103/RevModPhys.79.1197. - ↑ Umov, Nikolay Alekseevich (1874). "Ein Theorem über die Wechselwirkungen in Endlichen Entfernungen".
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*Modern Electrodynamics*. Cambridge University Press. p. 508. ISBN 9780521896979. - 1 2 Richter, Felix; Florian, Matthias; Henneberger, Klaus (2008). "Poynting's Theorem and Energy Conservation in the Propagation of Light in Bounded Media".
*EPL*.**81**(6): 67005. arXiv: 0710.0515 . Bibcode:2008EL.....8167005R. doi:10.1209/0295-5075/81/67005. S2CID 119243693. - ↑ Harrington, Roger F. (2001).
*Time-Harmonic Electromagnetic Fields*(2nd ed.). McGraw-Hill. ISBN 978-0-471-20806-8. - ↑ Hayt, William (2011).
*Engineering Electromagnetics*(4th ed.). New York: McGraw-Hill. ISBN 978-0-07-338066-7. - ↑ Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (2008).
*Foundations of Electromagnetic Theory*(4th ed.). Boston: Addison-Wesley. ISBN 978-0-321-58174-7. - ↑ Feynman, Richard Phillips (2011).
*The Feynman Lectures on Physics*. Vol. II: Mainly Electromagnetism and Matter (The New Millennium ed.). New York: Basic Books. ISBN 978-0-465-02494-0.`|volume=`

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- Becker, Richard (1982).
*Electromagnetic Fields and Interactions*(1st ed.). Mineola, New York: Dover Publications. ISBN 978-0-486-64290-1. - Edminister, Joseph; Nahvi, Mahmood (2013).
*Electromagnetics*(4th ed.). New York: McGraw-Hill. ISBN 978-0-07-183149-9.

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