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Poynting vector

The Poynting vector \vec{S}, named after English physicist John Henry Poynting (1852–1914), represents the directional flux of electromagnetic energy, quantifying the power per unit area carried by an electromagnetic field.^[1]^[2] It is defined in SI units as \vec{S} = \vec{E} \times \vec{H}, where \vec{E} is the electric field and \vec{H} is the magnetic field strength, or equivalently \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} in free space, with \vec{B} denoting the magnetic flux density and \mu_0 the vacuum permeability.^[3]^[4] The units of \vec{S} are watts per square meter (W/m²), indicating instantaneous power density.^[1] Poynting derived this vector in 1884 from Maxwell's equations to describe energy propagation in electromagnetic fields, building on James Clerk Maxwell's earlier work on electromagnetism; a similar formulation was independently developed by Oliver Heaviside shortly thereafter.^[2]^[5] It forms the basis of Poynting's theorem, a local conservation law stating that the rate of change of electromagnetic energy density plus the work done on charges equals the negative divergence of \vec{S}, or in integral form, the power delivered to charges within a volume equals the decrease in stored field energy plus the outward flux of \vec{S} through the surface.^[6]^[7] This theorem underscores the Poynting vector's role in energy balance for both static and dynamic fields. In applications, the time-averaged Poynting vector \langle \vec{S} \rangle determines the intensity of electromagnetic waves, such as I = \langle S \rangle = \frac{1}{2} c \epsilon_0 E_0^2 for a plane wave in vacuum, where c is the speed of light and \epsilon_0 the vacuum permittivity; this is crucial for analyzing radiation from antennas and lasers.^[4]^[8] In circuit theory, \vec{S} reveals that energy flows through the space surrounding conductors rather than along the wires, as seen in the Poynting vector pointing radially inward toward a resistor's surface to account for Joule heating.^[1] Beyond classical electromagnetism, extensions like the complex Poynting vector apply to time-harmonic fields in optics and ionospheric physics, aiding in the study of energy dissipation in dispersive media.^[9]

Fundamentals

Definition

The Poynting vector, a fundamental quantity in classical electromagnetism, was introduced by British physicist John Henry Poynting in his 1884 paper addressing the transfer of energy within electromagnetic fields. In SI units, it is generally defined as

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

where \mathbf{E} is the electric field strength and \mathbf{H} is the magnetic field strength. In free space, this is equivalent to

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

where \mathbf{B} is the magnetic flux density and \mu_0 is the permeability of free space.^[10] This cross-product form arises naturally from the structure of Maxwell's equations and captures the instantaneous directional flux of electromagnetic energy.^[11] The Poynting vector emerges from the derivation of Poynting's theorem, which expresses the conservation of electromagnetic energy. In free space, starting from Maxwell's curl equations, \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} and \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}, one takes the appropriate dot products and applies vector identities to obtain the continuity equation

\nabla \cdot \mathbf{S} + \frac{\partial u}{\partial t} + \mathbf{J} \cdot \mathbf{E} = 0,

where u = \frac{1}{2} (\epsilon_0 E^2 + \frac{B^2}{\mu_0}) denotes the electromagnetic energy density in free space and \mathbf{J} \cdot \mathbf{E} represents the power density delivered to charges by the fields.^[10] This theorem identifies \mathbf{S} as the term ensuring local energy balance in the electromagnetic field. In media, the theorem is generalized using material properties \epsilon and \mu.^[10] Physically, the Poynting vector \mathbf{S} quantifies the energy flux density, representing the amount of electromagnetic power flowing through a unit area per unit time in the direction of propagation.^[11] Its magnitude has units of watts per square meter (W/m²), consistent with power per area.^[10] This formulation underscores the Poynting vector's role in linking field configurations to energy transport, as derived directly from the conservation principles embedded in Maxwell's equations.

Physical Interpretation

The Poynting vector \mathbf{S} quantifies the energy flux density in an electromagnetic field, pointing in the direction of energy propagation while its magnitude represents the power per unit area crossing a surface perpendicular to that direction.^[12]^[1] This vector nature arises from its definition as \mathbf{S} = \mathbf{E} \times \mathbf{H}, where the cross product ensures \mathbf{S} is orthogonal to both the electric field \mathbf{E} and magnetic field \mathbf{H}. In free space, this is \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}.^[10] In the context of plane waves, where \mathbf{E} and \mathbf{H} (or \mathbf{B}) are mutually perpendicular, \mathbf{S} aligns with the wave's propagation direction, perpendicular to the plane spanned by \mathbf{E} and \mathbf{H}.^[1] Poynting's theorem provides the foundational link between the Poynting vector and energy conservation, expressed differentially as -\nabla \cdot \mathbf{S} = \frac{\partial u}{\partial t} + \mathbf{J} \cdot \mathbf{E} in free space, where u = \frac{1}{2} (\epsilon_0 E^2 + \frac{1}{\mu_0} B^2) is the electromagnetic energy density and \mathbf{J} \cdot \mathbf{E} is the power density delivered to matter by the fields. In media, u = \frac{1}{2} (\mathbf{D} \cdot \mathbf{E} + \mathbf{B} \cdot \mathbf{H}).^[12]^[10] The term -\nabla \cdot \mathbf{S} thus represents the net power flowing into a volume from the surrounding fields, balancing the rate of change in stored field energy and the work done on charges or currents within that volume.^[11] In integral form, this becomes \oint \mathbf{S} \cdot d\mathbf{A} = -\frac{d}{dt} \int u \, dV - \int \mathbf{J} \cdot \mathbf{E} \, dV, confirming that outward flux of \mathbf{S} corresponds to energy leaving the fields or being transferred to matter.^[13] In free space, absent charges and currents (\mathbf{J} = 0), the theorem reduces to \nabla \cdot \mathbf{S} + \frac{\partial u}{\partial t} = 0, illustrating that the Poynting vector solely describes the transport of electromagnetic energy without dissipation or storage in matter.^[1] In materials or media, however, the \mathbf{J} \cdot \mathbf{E} term (conduction current) and generalized displacement current introduce energy exchange with matter, such that \mathbf{S} accounts for both propagating energy and the portions stored in fields or dissipated as heat through resistive currents; the full treatment involves material constitutive relations.^[14] This distinction highlights the Poynting vector's role in unifying field propagation and interaction with physical systems. Although powerful for far-field wave scenarios, the Poynting vector's physical interpretation can be counterintuitive in near-field regions, where field components may lead to energy flows that do not align with simple propagation pictures, or in static cases, where time-invariant fields yield \mathbf{S} = 0 despite significant stored energy.^[15]

Mathematical Formulations

Alternative Expressions

In linear media, the Poynting vector can be expressed as \mathbf{S} = \mathbf{E} \times \mathbf{H}, where \mathbf{H} = \mathbf{B}/\mu is the magnetic field strength and \mu is the permeability of the medium.^[15] This form is particularly useful in materials where the relationship between \mathbf{B} and \mathbf{H} is linear, allowing for straightforward computation of energy flux in non-vacuum environments.^[16] In Gaussian CGS units, the Poynting vector takes the form \mathbf{S} = \frac{c}{4\pi} \mathbf{E} \times \mathbf{H}, where c is the speed of light in vacuum.^[17] This expression accounts for the unit conventions in the CGS system, where the factor c/4\pi arises from the formulation of Maxwell's equations, ensuring consistency in energy flux calculations.^[18] An alternative representation in terms of the electric displacement \mathbf{D} and magnetic induction \mathbf{B} is \mathbf{S} = \frac{\mathbf{D} \times \mathbf{B}}{\epsilon_0 \mu_0}, which holds in vacuum where \mathbf{D} = \epsilon_0 \mathbf{E} and \mathbf{B} = \mu_0 \mathbf{H}.^[19] While equivalent to the standard form in free space, this variant is sometimes employed in contexts emphasizing symmetry between electric and magnetic contributions or in derivations of field momentum density.^[20] For time-harmonic fields represented by phasors, the time-averaged Poynting vector is given by \langle \mathbf{S} \rangle = \frac{1}{2} \operatorname{Re} (\mathbf{E} \times \mathbf{H}^*), where \mathbf{E} and \mathbf{H} are complex amplitudes and the asterisk denotes the complex conjugate. This complex formulation simplifies analysis of sinusoidal electromagnetic waves by directly yielding the average power flow without explicit time integration.^[21] The Poynting vector exhibits invariance under electromagnetic gauge transformations, remaining unchanged despite shifts in the scalar and vector potentials that define the fields.^[22] Specifically, since \mathbf{E} = -\nabla \phi - \partial \mathbf{A}/\partial t and \mathbf{B} = \nabla \times \mathbf{A}, the cross product \mathbf{E} \times \mathbf{H} (or equivalents) is unaffected by the gauge freedom \phi \to \phi + \Lambda and \mathbf{A} \to \mathbf{A} + \nabla \Lambda, preserving the physical energy flux.^[23]

Microscopic Formulation

The microscopic electric field \mathbf{E}_\text{mic} and magnetic field \mathbf{B}_\text{mic} incorporate all contributions from free charges, atomic bound charges, free currents, and atomic-scale currents, providing a complete description at scales comparable to atomic dimensions. The instantaneous microscopic Poynting vector, representing the local directional energy flux density, is given by

\mathbf{S}_\text{mic} = \frac{1}{\mu_0} \mathbf{E}_\text{mic} \times \mathbf{B}_\text{mic}.

This formulation arises directly from the microscopic Maxwell equations, analogous to the vacuum case but with all sources included explicitly. To obtain measurable quantities relevant to continuum descriptions, \mathbf{S}_\text{mic} must be spatially averaged over volumes that encompass many atoms yet remain small relative to the wavelength of the fields, yielding the macroscopic Poynting vector \mathbf{S} = \langle \mathbf{S}_\text{mic} \rangle. This averaging eliminates rapid spatial oscillations induced by the discrete nature of bound charges and atomic currents, effectively smoothing the field variations into a coarse-grained energy flow. In dielectric and magnetic materials, the polarization \mathbf{P} and magnetization \mathbf{M} significantly influence this averaged flux by accounting for the fields generated by bound charges and currents. The averaging procedure demonstrates that the macroscopic energy flux incorporates cross terms involving \mathbf{P} and \mathbf{M}, resulting in an effective form \mathbf{S} = \mathbf{E} \times \mathbf{H} (with \mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}) that reflects the additional energy transport mediated by the material's response to the applied fields. A notable challenge in interpreting the microscopic formulation arises near sources, where near-field effects cause \mathbf{S}_\text{mic} to exhibit counterintuitive directions, such as local flows opposing the overall propagation due to dominant reactive (non-radiating) field components and interference from atomic-scale contributions. These local anomalies average out in the macroscopic limit but underscore the limitations of intuitive energy flow pictures at subwavelength scales.

Applications in Wave Propagation

Plane Waves

In uniform plane electromagnetic waves propagating in vacuum, the electric and magnetic fields can be expressed for a monochromatic wave as \mathbf{E} = \mathbf{E}_0 \cos(\mathbf{k} \cdot \mathbf{r} - \omega t) and \mathbf{B} = \frac{1}{c} \hat{k} \times \mathbf{E}, where \mathbf{E}_0 is the amplitude vector perpendicular to the wave vector \mathbf{k}, c is the speed of light, and \hat{k} is the unit vector in the direction of propagation.^[24] These fields satisfy Maxwell's equations in free space, ensuring transverse propagation with |\mathbf{E}| = c |\mathbf{B}|.^[25] The instantaneous Poynting vector for such a plane wave is given by \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} = \frac{E_0^2}{\mu_0 c} \cos^2(\mathbf{k} \cdot \mathbf{r} - \omega t) \hat{k}, revealing its oscillatory nature with twice the frequency of the fields due to the squared cosine term.^[26] This vector points in the direction of wave propagation, parallel to \hat{k}, indicating the local flow of electromagnetic energy along the propagation axis.^[24] The magnitude of the instantaneous Poynting vector relates directly to the energy density u of the wave, satisfying |\mathbf{S}| = c u, where the instantaneous energy density is u = \frac{1}{2} \left( \epsilon_0 E^2 + \frac{B^2}{\mu_0} \right).^[26] For plane waves in vacuum, the electric and magnetic contributions to u are equal, so u = \epsilon_0 E^2.^[27] This relation underscores that the energy flux equals the product of the energy density and the wave speed. Polarization affects the magnitude of the Poynting vector in plane waves: for linear polarization, \mathbf{E}_0 oscillates in a fixed plane, yielding a peak |\mathbf{S}| proportional to E_0^2; in contrast, circular polarization decomposes into two orthogonal linear components each with amplitude E_0 / \sqrt{2} to yield the same time-averaged intensity, resulting in a constant |\mathbf{S}| equal to half the peak value of the linearly polarized case (where the peak field for circular is E_0 / \sqrt{2}).^[28] This difference arises because the time-averaged intensity stems from the orthogonal components' contributions in the cross product.^[29]

Time-Averaged Poynting Vector

The time-averaged Poynting vector, denoted as \langle \mathbf{S} \rangle, quantifies the average rate of energy flow per unit area carried by an electromagnetic field over one period of oscillation, which is particularly relevant for time-harmonic or periodic fields in alternating current (AC) systems.^[30] For a general periodic field with period T, it is defined as \langle \mathbf{S} \rangle = \frac{1}{T} \int_0^T \mathbf{S}(t) \, dt, where \mathbf{S}(t) = \mathbf{E}(t) \times \mathbf{H}(t) is the instantaneous Poynting vector. In the phasor domain for monochromatic fields, where \mathbf{E}(\mathbf{r}, t) = \Re \{ \tilde{\mathbf{E}}(\mathbf{r}) e^{-i\omega t} \} and similarly for \mathbf{H}, the time-averaged form simplifies to \langle \mathbf{S} \rangle = \frac{1}{2} \Re \{ \tilde{\mathbf{E}} \times \tilde{\mathbf{H}}^* \}, with the asterisk denoting the complex conjugate; this expression arises from the orthogonality of the exponential time factors over the period.^[21] For a plane electromagnetic wave in free space propagating in the \hat{\mathbf{k}} direction with electric field amplitude E_0, the time-averaged Poynting vector is \langle \mathbf{S} \rangle = \frac{E_0^2}{2 \mu_0 c} \hat{\mathbf{k}}, where \mu_0 is the permeability of free space and c is the speed of light; its magnitude equals the constant intensity I of the wave, representing the average power flux through a unit area perpendicular to the propagation direction.^[30] This intensity scales quadratically with the field amplitude and linearly with frequency for fixed energy density, underscoring the wave's steady-state energy transport in vacuum. In lossy media, the phasor-based \langle \mathbf{S} \rangle becomes complex, with the real part indicating the time-averaged active power flow (dissipative energy transport) and the imaginary part corresponding to reactive power associated with energy storage and oscillation without net dissipation, such as in near-field regions or standing waves.^[31] The magnitude and direction of this vector depend on the complex wave impedance Z = \sqrt{\mu / \epsilon} of the medium, where for a plane wave, \langle \mathbf{S} \rangle = \frac{1}{2} \frac{|\tilde{E}|^2}{|Z|} \hat{\mathbf{k}} when Z is real (non-dispersive, low-loss case), linking the average power density directly to the field strength and material properties.^[32] In practical measurements, such as in antennas or waveguides, the time-averaged Poynting vector enables the calculation of detectable average power by integrating its normal component over a closed surface enclosing the device; for instance, the total radiated power from an antenna equals the surface integral of \langle \mathbf{S} \rangle, providing a basis for efficiency assessments and far-field power reception in communication systems.

Energy Flow in Materials and Devices

Coaxial Cable Example

A coaxial cable provides a practical example of how the Poynting vector describes electromagnetic energy flow in a guided structure, specifically in the transverse electromagnetic (TEM) mode where fields are confined between the conductors. The geometry consists of a central inner conductor of radius a and a surrounding outer cylindrical sheath with inner radius b > a, assuming vacuum or a dielectric filler between them for simplicity. In this setup, the electric field points radially from the inner to the outer conductor, while the magnetic field circulates azimuthally around the inner conductor. The electric field \mathbf{E} in the region a < r < b is radial and given by

\mathbf{E} = \frac{V}{r \ln(b/a)} \hat{r},

where V is the voltage difference between the conductors and r is the radial distance from the axis. The magnetic field \mathbf{H} is azimuthal and expressed as

\mathbf{H} = \frac{I}{2\pi r} \hat{\phi},

with I denoting the current flowing along the inner conductor (and returning via the outer sheath). These field expressions arise from Gauss's law for the electric field and Ampère's law for the magnetic field in the TEM approximation, valid for low frequencies or steady-state conditions.^[33] The Poynting vector \mathbf{S} = \mathbf{E} \times \mathbf{H} then evaluates to

\mathbf{S} = \frac{V I}{2\pi r^2 \ln(b/a)} \hat{z},

assuming a right-handed coordinate system where the cable axis is along \hat{z} and power flows in the +\hat{z} direction toward a load. This vector points along the cable axis and describes energy propagation through the space between the conductors. To verify the total power flow, integrate the z-component of \mathbf{S} over the annular cross-sectional area between a \leq r \leq b:

P = \int_a^b \int_0^{2\pi} S_z \, r \, d\phi \, dr = V I.

This result matches the electrical power V I, confirming the Poynting theorem's accounting of energy transport.^[33] The key insight from this example is that electromagnetic energy propagates through the region between the conductors via the Poynting vector, rather than along the conducting wires themselves, countering the common intuition that power flows through the current-carrying paths. This spatial separation highlights the Poynting vector's role in revealing non-local energy flux in electromagnetic devices.

Resistive Dissipation

The Poynting theorem, derived from Maxwell's equations, expresses the conservation of electromagnetic energy in a volume V as -\nabla \cdot \mathbf{S} = \mathbf{J} \cdot \mathbf{E} + \frac{\partial u}{\partial t}, where \mathbf{S} is the Poynting vector, \mathbf{J} \cdot \mathbf{E} represents the rate of energy dissipation per unit volume due to Joule heating in resistive materials, and \frac{\partial u}{\partial t} is the time rate of change of the electromagnetic energy density u = \frac{1}{2} (\epsilon |\mathbf{E}|^2 + \frac{1}{\mu} |\mathbf{B}|^2).^[34] This form indicates that a negative divergence of \mathbf{S} (inward flux) supplies the energy for both dissipation and storage within the volume.^[35] In steady-state conditions, where fields and currents are time-independent, the storage term vanishes (\frac{\partial u}{\partial t} = 0), simplifying the theorem to -\nabla \cdot \mathbf{S} = \mathbf{J} \cdot \mathbf{E}. Integrating over a volume V enclosing a conductor and applying the divergence theorem yields \oint \mathbf{S} \cdot d\mathbf{A} = -\int_V \mathbf{J} \cdot \mathbf{E} \, dV, demonstrating that the net influx of Poynting vector through the surface provides the total power dissipated as heat within the volume.^[34] This influx arises from the electromagnetic fields surrounding the conductor, rather than directly from the current flow inside.^[35] Consider a cylindrical resistor of length L and cross-sectional radius a, carrying a uniform steady current I driven by an axial electric field \mathbf{E} = \frac{\mathbf{J}}{\sigma} = \frac{I}{\pi a^2 \sigma} \hat{z}, where \sigma is the conductivity and \mathbf{J} is the current density. The associated azimuthal magnetic field outside the resistor is \mathbf{B} = \frac{\mu_0 I}{2\pi r} \hat{\phi} for r > a, leading to a Poynting vector \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} that points radially inward toward the resistor surface. Integrating \mathbf{S} over the cylindrical surface gives a total inward power flow of I^2 R, where R = \frac{L}{\sigma \pi a^2} is the resistance, matching the Joule heating P = I^2 R = \int_V \mathbf{J} \cdot \mathbf{E} \, dV.^[36] This illustrates how energy from external fields permeates the resistor to account for the dissipation.^[37] For alternating currents in conductors, the skin effect confines the current to a thin layer near the surface, with depth \delta = \sqrt{\frac{2}{\omega \mu \sigma}}, where \omega is the angular frequency. In this case, the Poynting vector penetrates only to approximately the skin depth \delta, delivering energy primarily to the outer region where dissipation occurs, while the interior remains largely field-free.^[38] The time-averaged Poynting vector \langle \mathbf{S} \rangle may be referenced for such AC scenarios to quantify net power flow.^[39] In general conducting media, the divergence of the time-averaged Poynting vector relates to the real part of the complex conductivity \sigma_r, with -\nabla \cdot \langle \mathbf{S} \rangle = \frac{1}{2} \sigma_r |\mathbf{E}|^2, representing the average Joule dissipation per unit volume.^[34] This connection highlights how the Poynting vector framework unifies energy transport and loss in lossy materials across frequency regimes.

Relativistic and Advanced Considerations

Radiation Pressure

The electromagnetic field possesses a linear momentum density given by \mathbf{g} = \frac{\mathbf{S}}{c^2}, where \mathbf{S} is the Poynting vector representing the energy flux and c is the speed of light in vacuum. This relation stems from special relativity, as the energy density u of the field corresponds to an equivalent mass density u/c^2, and the momentum density follows from the field's energy flow at speed c. Radiation pressure arises from the transfer of this field momentum to a material surface upon interaction with an electromagnetic wave. For normal incidence on a perfect absorber, where the wave's momentum is fully deposited, the time-averaged pressure is P = \frac{\langle S \rangle}{c}, with \langle S \rangle denoting the magnitude of the time-averaged Poynting vector. On a perfect reflector, the wave's momentum reverses direction, doubling the transfer and yielding P = \frac{2 \langle S \rangle}{c}. These expressions can be derived using the Maxwell stress tensor \mathbf{T}, which quantifies the momentum flux due to electromagnetic fields:

T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right),

where \epsilon_0 and \mu_0 are the vacuum permittivity and permeability, respectively.^[40] The force \mathbf{F} on a volume V is \mathbf{F} = \int_V (\nabla \cdot \mathbf{T}) \, dV, which, by the divergence theorem, becomes a surface integral \oint \mathbf{T} \cdot d\mathbf{A}.^[40] This force balances the rate of change of total momentum, linking to the field momentum density via the conservation law

\frac{\partial}{\partial t} (\mathbf{g}_\text{mech} + \mathbf{g}_\text{field}) + \nabla \cdot \mathbf{T} = 0,

with \mathbf{g}_\text{field} = \frac{\mathbf{S}}{c^2}.^[40] For steady-state radiation pressure, the time derivative vanishes, and the surface integral of \mathbf{T} directly gives the momentum transfer rate. In the case of a plane wave normally incident on a surface, the radiation pressure equals the incident momentum flux \langle S \rangle / c for absorption, as the wave delivers momentum at this rate per unit area; reflection doubles it by reversing the outgoing flux.^[40] Practical applications harness this momentum transfer: solar sails employ large reflective membranes to generate thrust from solar radiation pressure, enabling propellantless propulsion for deep-space missions.^[41] Optical tweezers utilize focused laser beams to exert radiation pressure forces on microscopic particles, enabling their precise trapping and manipulation in biological and colloidal studies.^[42]

Uniqueness and Static Fields

In source-free regions of space, the electromagnetic fields \mathbf{E} and \mathbf{B} can be uniquely decomposed using the Helmholtz theorem into irrotational and solenoidal components, with the decomposition determined solely by the boundary conditions on the fields. This uniqueness extends to the Poynting vector \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}, which represents the energy flux density and is thus fully specified by the same boundary conditions without ambiguity in vacuum or source-free media.^[43] In purely electrostatic fields, where \mathbf{B} = 0, the Poynting vector vanishes everywhere (\mathbf{S} = 0) due to the absence of a crossed \mathbf{E} \times \mathbf{B} term. Similarly, in ideal magnetostatic configurations without associated electric fields, \mathbf{E} = 0 outside current sources, again yielding \mathbf{S} = 0. However, in static configurations with both static \mathbf{E} and \mathbf{B} fields present (e.g., a static charge near a permanent magnet), \mathbf{S} may be non-zero but circulates in closed loops with no net energy transport.^[1] In steady-state configurations with ongoing currents and dissipation, such as resistive DC circuits, the Poynting vector is non-zero and steady, directing energy flow from the surrounding fields into the resistor to account for Joule heating; this is discussed further in the section on energy flow in materials and devices.^[1] Relativistically, the Poynting vector \mathbf{S} forms the spatial components of the energy flux in the electromagnetic energy-momentum tensor T^{\mu\nu}, alongside the energy density T^{00}, ensuring the overall structure is a rank-2 tensor that transforms invariantly under Lorentz transformations. This tensorial form guarantees the conservation laws for energy and momentum hold in all inertial frames, with \mathbf{S}/c appearing as the momentum density, linking field energy flow to relativistic mechanics.^[44]^[45] Alternative expressions for energy flux density, such as those involving the scalar and vector potentials (e.g., \mathbf{S}' = -\frac{\partial}{\partial t} (\phi \mathbf{A}) or gauge-dependent variants), fail to uniquely represent physical energy flow because they depend on the choice of gauge and lack locality or gauge invariance. Unlike the gauge-invariant \mathbf{E} \times \mathbf{B} form, these alternatives do not consistently satisfy Poynting's theorem or electromagnetic momentum conservation, rendering them non-physical for describing observable energy transport. Additionally, they violate Lorentz invariance in their transformation properties, further underscoring the uniqueness of the standard Poynting vector.^[15]^[46]

References

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Jun 5, 2014 · In terms of the full stress-tensor tensor, D×B is part of the field momentum density, while E×H is part of energy flux density. The latter is ...Poynting vector direction - Physics Stack ExchangeWhat does the magnitude of the Poynting vector mean in practice?More results from physics.stackexchange.com
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[PDF] 5.6 Electromagnetic Power Density - 5.6.1 Poynting Vector - BYU
Complex Poynting vector (phasor domain): S = E × H. ∗. The time-average power can be obtained from the complex Poynting vector using Sav = 1. 2. Re{E × H.
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[PDF] Lecture Notes on Electromagnetism and Gauge Invariance
Sep 3, 2020 · 2.2 Invariance of electromagnetic fields under a gauge transformation ... where the Poynting vector (or Poynting flux) is defined as. S ≡ E ...
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Comparison of Poynting's vector and the power flow used in ...
Aug 25, 2022 · Poynting's theorem describes that the total energy flux out of a volume is equal to the change in stored energy subtracted by the energy losses.III. POYNTING'S VECTOR... · IV. POWER FLOW... · VI. REWRITING POYNTING'S...
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[PDF] Chapter 13 Maxwell's Equations and Electromagnetic Waves - MIT
Figure 13.6.2 Poynting vector for a plane wave. As expected, points in the direction of wave propagation (see Figure 13.6.2). S. The intensity of the wave, I ...
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[PDF] physics 232 – chapter 32: electromagnetic waves
For a plane wave in vacuum,. E = cB , c = 1. √. ²oµo. ~E ⊥ ~B ⊥ (direction of propagation). Sinusoidal plane wave traveling in the +x direction,. E = Emax sin ...
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[PDF] LECTURE NOTES 5 - High Energy Physics
Monochromatic EM plane waves propagating in free space/the vacuum are sinusoidal EM ... Instantaneous Poynting's Vector Associated with an EM Plane Wave (Free ...
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[PDF] 1 Unit 4-4: Energy and Momentum of Electromagnetic Waves in a ...
Apr 4, 2021 · The Poynting vector of the wave is,. S(r,t) = c. 4π. E(r,t) × B(r,t) = c. 4π h. Ek × (k × Ek) i cos2(k · r − ωt). (4.4.38). = c. 4π k |Ek|2 cos2 ...
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[PDF] Plane Waves, Polarization and the Poynting Vector - Sandiego
As the wave propagates along z, the ellipse spreads out to an elliptical helix. This is called elliptical polarization. For the special case of. , we have ...
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[PDF] PLANE ELECTROMAGNETIC WAVES - UT Physics
the unit wave vector k points into your eye. • But in the Particle Physics convention, we look at the ...
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16.3 Energy Carried by Electromagnetic Waves - OpenStax
Oct 6, 2016 · The time average of the energy flux is the intensity I of the electromagnetic wave and is the power per unit area.
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Details for IEV number 705-02-10: "complex Poynting vector"
Note 1 – The real part of the complex Poynting vector is the average of the Poynting vector during one cycle. Note 2 – The imaginary part of the complex ...<|control11|><|separator|>
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Poynting Vector - an overview | ScienceDirect Topics
The Poynting vector refers to the energy flux density of an electromagnetic wave, representing the rate of energy transmission per unit area. AI generated ...Missing: CGS | Show results with:CGS
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[PDF] Flow of Energy and Momentum in a Coaxial Cable - Kirk T. McDonald
Mar 31, 2007 · The Poynting vector equals the electromagnetic energy density times the (vector) wave velocity, which confirms the interpretation of S as ...Missing: mode | Show results with:mode
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[PDF] Continuity Equation (Charge Conservation), Poynting's Theorem ...
is the Dirac delta function. Now, in the process of deriving Poynting's theorem (above), we used Griffith's Product Rule # 6 to obtain ( ) ( ) (. ) E. B. B. E.
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[PDF] Where Does the Power Become AC in an AC Power Source?
Aug 28, 2020 · The Poynting vector at the cylindrical surface of the resistor points inward, Sr = −EzHφ = −ρI2/2π2a3. Since the. Poynting vector vanishes ...
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Energy flow from a battery to other circuit elements: Role of surface ...
Nov 1, 2010 · A qualitative description of energy transfer from a battery to a resistor using the Poynting vector was recently published.
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[PDF] 7 Lecture: pp:308-314 7-5:6
EE335 - Skin Effect & Power Flow. 7 Lecture: pp:308-314 7-5:6 ... Direction of the Poynting vector is the direction of the power flow, the magnitude is the.
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[PDF] LECTURE NOTES 7 EM WAVE PROPAGATION IN CONDUCTORS
At = 60 : 5 10 !! 5 10. 3.87 10 !! 0.129. 60 AC skin depth: 2.05 10 2.05 ... Instantaneous Poynting's Vector for EM waves propagating in a poor conductor:.
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[PDF] Radiation Pressure of a Monochromatic Plane Wave on a Flat Mirror
Oct 10, 2009 · Consider the various perspectives of the momentum density in the wave, the Maxwell stress tensor, the Lorentz force, and the radiation reaction ...
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Theory of radiation pressure on a diffractive solar sail
Get Citation. Copy Citation Text. Grover A. Swartzlander, "Theory of radiation pressure on a diffractive solar sail," J. Opt. Soc. Am. B 39, 2556-2563 (2022).
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Optical trapping and manipulation of neutral particles using lasers
The technique of optical trapping and manipulation of small neutral particles by lasers is based on the forces of radiation pressure. These are forces arising ...
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[PDF] The Helmholtz Decomposition and the Coulomb Gauge
Apr 20, 2023 · The flow of energy in the electromagnetic field is described by the Poynting vector S = (c/4π)E × B, which can be given a Helmholtz ...
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What is the significance of the Poynting vector in a static ...
Jun 25, 2014 · The significance of the Pointing vector in static electromagnetic fields is thus that it helps to preserve sacred conservation laws, and thereby ...
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The electromagnetic energy tensor - Richard Fitzpatrick
The right-hand side of Eq. (1588) represents the rate per unit volume at which energy is transferred from the electromagnetic field to charged particles.Missing: relativistic | Show results with:relativistic
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[PDF] The Electromagnetic Energy-Momentum Tensor
May 7, 2020 · So we've got the Poynting vector and the off-diagonal components of the Maxwell stress tensor, but we don't quite have u, Txx, Tyy and Tzz yet.
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(PDF) On the uniqueness of the vector for energy flow density in ...
Aug 6, 2025 · Second, Poynting's vector is used to formulate the electromagnetic momentum, and any alternative energy flow vectors would not. These two ...