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 and in the more general form that recognises the freedom of adding the curl of an arbitrary vector field to the definition.^{ [2] }

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's auxiliary 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 field **B** (described later in the article).

It is also possible to combine the electric displacement field **D** with the magnetic field **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 field;**H**is the magnetic auxiliary 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}

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;**B**is the magnetic field.

This is actually the general expression of the Poynting vector.^{ [10] } 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. 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.^{ [11] }

Since only the microscopic fields **E** and **B** occur in the derivation of **S** = (1/*μ*_{0}) **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^{[ clarification needed ]} for the electromagnetic energy density, in contrast to the "macroscopic" form **E** × **H**.^{ [11] }

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 is understood to signify a sinusoidally varying field whose instantaneous amplitude **E**(*t*) follows the real part of 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 . 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 . 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 is simply a real number according to the above definition.

The equivalence of 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 and 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.^{ [12] }^{: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.^{ [13] }^{: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.^{ [14] }^{: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} 377Ω 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 is equal to the electric field 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 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.^{ [15] } 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 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. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (*c*) in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who published an early form of the equations that included the Lorentz force law between 1861 and 1862. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

In radio-frequency engineering, a **transmission line** is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

**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 through a conductor creates a magnetic field around the conductor, whose strength depends on the magnitude of the current. A change in current causes a change in the magnetic field. From Faraday's law of induction, any change in magnetic field through a circuit induces an electromotive force (EMF) (voltage) in the conductors; this is known as electromagnetic induction. So the changing current induces a voltage in the conductor. This induced voltage is in a direction which tends to oppose the change in current, so it is called a *back EMF*. Due to this back EMF, a conductor's inductance opposes any increase or decrease in electric current through it.

In mathematics and classical mechanics, the **Poisson bracket** is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called *canonical transformations*, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.

In electromagnetism, **permeability** is the measure of the resistance of a material against the formation of a magnetic field, otherwise known as distributed inductance in transmission line theory. Hence, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically represented by the (italicized) Greek letter *μ*. The term was coined in September 1885 by Oliver Heaviside. The reciprocal of magnetic permeability is magnetic reluctivity.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

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.

The term **magnetic potential** can be used for either of two quantities in classical electromagnetism: the *magnetic vector potential*, or simply *vector potential*, **A**; and the *magnetic scalar potential**ψ*. Both quantities can be used in certain circumstances to calculate the magnetic field **B**.

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

In quantum mechanics, the **probability current** is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics.

In electromagnetism, **current density** is the amount of charge per unit time that flows through a unit area of a chosen cross section. The **current density vector** is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI base units, the electric current density is measured in amperes per square metre.

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

**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.

When an electromagnetic wave travels through a medium in which it gets attenuated, it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

**Multipole radiation** is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.

**Magnetic resonance** is a phenomenon in quantum mechanics that affects a magnetic dipole when placed in a uniform static magnetic field. Its energy is split into a finite number of energy levels, depending on the value of quantum number of angular momentum. This is similar to energy quantization for atoms, say ^{}e^{−}_{} in H atom; in this case the atom, in interaction to an external electric field, transitions between different energy levels by absorbing or emitting photons. Similarly if a magnetic dipole is perturbed with electromagnetic field of proper frequency(), it can transit between its energy eigenstates, but as the separation between energy eigenvalues is small, the frequency of the photon will be the microwave or radio frequency range. If the dipole is tickled with a field of another frequency, it is unlikely to transition. This phenomenon is similar to what occurs when a system is acted on by a periodic force of frequency equal to its natural frequency.

**Magnetic current** is, nominally, a current composed of fictitious moving magnetic monopoles. It has the dimensions of volts. The usual symbol for magnetic current is which is analogous to for electric current. Magnetic currents produce an electric field analogously to the production of a magnetic field by electric currents. **Magnetic current density**, which has the units of V/m², is usually represented by the symbols and . The superscripts indicate total and impressed magnetic current density. The impressed currents are the energy sources. In many useful cases, a distribution of electric charge can be mathematically replaced by an equivalent distribution of magnetic current. This artifice can be used to simplify some electromagnetic field problems. It is possible to use both electric current densities and magnetic current densities in the same analysis.

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Wikiquote has quotations related to: Poynting vector |

- 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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