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Poynting's theorem

Poynting's theorem is a fundamental identity in classical electromagnetism that expresses the conservation of energy for electromagnetic fields, stating that the power delivered to charges within a volume by the fields equals the decrease in the stored electromagnetic energy minus the outward flux of energy through the bounding surface.^[1]^[2] Formulated by British physicist John Henry Poynting in his 1884 paper "On the Transfer of Energy in the Electromagnetic Field," the theorem derives directly from Maxwell's equations by manipulating Faraday's law and Ampère's law with Maxwell's correction.^[3]^[2] In its differential form, it is \nabla \cdot (\mathbf{E} \times \mathbf{H}) + \frac{\partial}{\partial t} \left( \frac{1}{2} \epsilon |\mathbf{E}|^2 + \frac{1}{2\mu} |\mathbf{H}|^2 \right) + \mathbf{E} \cdot \mathbf{J} = 0, where \mathbf{E} is the electric field, \mathbf{H} is the magnetic field strength, \mathbf{J} is the current density, \epsilon is the permittivity, and \mu is the permeability.^[1]^[4] The term \mathbf{S} = \mathbf{E} \times \mathbf{H}, known as the Poynting vector, quantifies the instantaneous power flux density, pointing in the direction of energy propagation perpendicular to both \mathbf{E} and \mathbf{H}.^[2]^[4] The integral form of the theorem, \oint_S (\mathbf{E} \times \mathbf{H}) \cdot d\mathbf{A} = -\frac{d}{dt} \int_V \left( \frac{1}{2} \epsilon E^2 + \frac{1}{2\mu} H^2 \right) dV - \int_V \mathbf{E} \cdot \mathbf{J}\, dV, highlights its role in balancing energy inflows, outflows, storage changes, and ohmic losses across any closed surface enclosing a volume V.^[1]^[4] This formulation applies universally to time-varying fields, encompassing both quasistatic approximations (electric or magnetic quasistatics) and full electrodynamic regimes.^[2] Poynting's theorem underpins key phenomena in electromagnetism, such as the energy transport in electromagnetic waves—where the time-averaged Poynting vector yields the intensity I = \frac{1}{2} c \epsilon_0 E_0^2 for plane waves in vacuum—and the analysis of power delivery in antennas, waveguides, and transmission lines.^[4] It also explains radiation pressure and momentum in fields, extending Poynting's original insights into mechanical effects of electromagnetic energy flow.^[3] In engineering contexts, it facilitates the design of efficient electromagnetic devices by quantifying losses and energy distribution.^[1]

Overview

Historical development

The development of Poynting's theorem traces its roots to James Clerk Maxwell's foundational work on electromagnetism in the mid-19th century. In his 1865 paper "A Dynamical Theory of the Electromagnetic Field," Maxwell introduced the concept of displacement current, which resolved inconsistencies in Ampère's law and enabled the propagation of electromagnetic waves through space, implying that energy could be carried by fields rather than solely through conductors. This laid the groundwork for understanding electromagnetic energy conservation, though Maxwell did not explicitly formulate a theorem for energy flux. The explicit statement of what became known as Poynting's theorem emerged in 1884 through the work of British physicist John Henry Poynting. In his paper "On the Transfer of Energy in the Electromagnetic Field," published in the Philosophical Transactions of the Royal Society, Poynting derived an expression for the flux of electromagnetic energy from Maxwell's equations, emphasizing that energy flows through the surrounding medium rather than along wires.^[3] He argued that "the surrounding medium contains at least a part of the energy, and it is capable of transferring it from point to point," highlighting the directional transfer of energy in the field.^[3] Poynting introduced the quantity now called the Poynting vector to represent this energy flow, marking a pivotal advancement in the theory of electromagnetic energy conservation. Independently, in 1885, Oliver Heaviside arrived at a similar formulation in his series of articles on electromagnetic theory published in The Electrician. Heaviside extended the concept to include more general forms, recognizing the role of the curl of an arbitrary vector field, and confirmed the energy conservation principle derived from Maxwell's equations.^[5] This parallel discovery underscored the theorem's significance, solidifying its place in electromagnetic theory shortly after Poynting's contribution.

Physical interpretation

Poynting's theorem expresses the conservation of energy in electromagnetic systems as a continuity equation for the energy density stored in the electric and magnetic fields. In this framework, the theorem describes how the rate of change of electromagnetic energy within a volume equals the negative of the energy flux through the surface enclosing that volume, plus the work performed by the fields on charges inside. The Poynting vector serves as the energy flux density, indicating the direction and magnitude of electromagnetic energy propagation perpendicular to both the electric and magnetic field lines. This interpretation underscores that electromagnetic energy is not confined to matter but flows dynamically through the fields themselves, analogous to the continuity equation for charge conservation.^[6]^[7] The physical significance of the theorem lies in its demonstration of energy balance: the power delivered by electromagnetic fields to charges (via the interaction term) precisely accounts for any decrease in the stored field energy, offset by the energy carried away through the Poynting flux across the boundaries. This reveals that in dynamic electromagnetic scenarios, such as wave propagation or circuit operation, energy is neither created nor destroyed but redistributed between field storage, mechanical work on particles, and radiative transport. For instance, in plane electromagnetic waves, the Poynting vector aligns with the direction of propagation, quantifying the intensity as the energy carried per unit area per unit time, which matches observed radiation pressure and power flow in antennas or lasers.^[7]^[8] A striking example of this energy flow occurs in a simple DC circuit, such as a battery connected to a resistor via conducting wires. Contrary to intuition, the Poynting vector shows that energy does not travel longitudinally through the wires alongside the current but instead flows transversely through the surrounding space, directed radially inward from the electric field between the wires and the azimuthal magnetic field encircling them. This results in energy streams that "circle" around the conductors, entering the resistor from all sides to dissipate as heat, while the wires themselves act primarily as guides for fields rather than conduits for energy. Such non-intuitive paths highlight how electromagnetic energy permeates the vacuum or medium outside material boundaries.^[9]^[10] Common misconceptions arise from assuming that electromagnetic energy follows the path of charge motion or current direction, as in classical circuit diagrams depicting power flowing "through" wires. In reality, the Poynting vector's direction, orthogonal to both \mathbf{E} and \mathbf{B}, often diverges sharply from particle trajectories, emphasizing that energy transport is a field-mediated process independent of matter's drift. This distinction resolves paradoxes in energy accounting, such as why ideal wires with no resistance appear lossless yet convey power via external fields.^[11]^[10]

Mathematical statement

Integral form

The integral form of Poynting's theorem states the conservation of electromagnetic energy over a fixed volume V enclosed by surface S:

\int_V \mathbf{E} \cdot \mathbf{J} \, dV + \frac{\partial}{\partial t} \int_V \left( \frac{1}{2} \mathbf{E} \cdot \mathbf{D} + \frac{1}{2} \mathbf{B} \cdot \mathbf{H} \right) dV = -\oint_S (\mathbf{E} \times \mathbf{H}) \cdot d\mathbf{A},

where \mathbf{E} is the electric field, \mathbf{H} is the magnetic field, \mathbf{J} is the electric current density, \mathbf{D} is the electric displacement field, and \mathbf{B} is the magnetic flux density.^[1] This formulation assumes linear, time-invariant, non-dispersive media with constitutive relations \mathbf{D} = \epsilon \mathbf{E} and \mathbf{B} = \mu \mathbf{H} (where \epsilon and \mu may be position-dependent), no magnetic current densities, and SI units.^[1] The first term on the left-hand side, \int_V \mathbf{E} \cdot \mathbf{J} \, dV, quantifies the rate of work done by the electromagnetic fields on free charges within V, corresponding to Joule heating or ohmic dissipation.^[1] The second term, \frac{\partial}{\partial t} \int_V \left( \frac{1}{2} \mathbf{E} \cdot \mathbf{D} + \frac{1}{2} \mathbf{B} \cdot \mathbf{H} \right) dV, represents the time rate of change of the total electromagnetic energy stored in the fields inside V, with \frac{1}{2} \mathbf{E} \cdot \mathbf{D} as the electric energy density and \frac{1}{2} \mathbf{B} \cdot \mathbf{H} as the magnetic energy density.^[1] The right-hand side, -\oint_S (\mathbf{E} \times \mathbf{H}) \cdot d\mathbf{A}, is the negative of the surface integral of the Poynting vector \mathbf{S} = \mathbf{E} \times \mathbf{H}, interpreted as the net electromagnetic power flowing out of V through S.^[1] Thus, the equation balances the energy supplied to charges and the increase in stored field energy against the net outward energy flux, ensuring overall conservation.^[3] An illustrative application is the charging of a parallel-plate capacitor with plate area A, separation d \ll \sqrt{A/\pi}, connected to a current source I. Inside the volume between the plates (idealized with uniform \mathbf{E} and negligible \mathbf{J}, assuming vacuum between plates), the left-hand side reduces to the rate of increase in electric field energy, \frac{\partial}{\partial t} \left( \frac{1}{2} \epsilon_0 E^2 \cdot A d \right) = I V, where V = E d is the voltage. The Poynting vector on the cylindrical side surface points radially inward (due to fringing fields from the charging wires), yielding a net inward flux through S that matches I V, demonstrating that energy flows into the capacitor via the surrounding electromagnetic fields rather than through the wires alone.^[10]

Differential form

The differential form of Poynting's theorem expresses the local conservation of electromagnetic energy in linear media as a continuity equation. It states that the rate of change of the electromagnetic energy density at a point, plus the divergence of the energy flux through that point, equals the negative of the power density delivered to charges by the fields. This is given by

-\mathbf{E} \cdot \mathbf{J} = \frac{\partial}{\partial t} \left( \frac{1}{2} \mathbf{E} \cdot \mathbf{D} + \frac{1}{2} \mathbf{B} \cdot \mathbf{H} \right) + \nabla \cdot (\mathbf{E} \times \mathbf{H}),

where \mathbf{E} is the electric field, \mathbf{H} is the magnetic field strength, \mathbf{J} is the current density, \mathbf{D} is the electric displacement, and \mathbf{B} is the magnetic flux density.^[1]^[3] Here, the term \frac{1}{2} \mathbf{E} \cdot \mathbf{D} + \frac{1}{2} \mathbf{B} \cdot \mathbf{H} represents the electromagnetic energy density u, comprising the electric energy density \frac{1}{2} \mathbf{E} \cdot \mathbf{D} and the magnetic energy density \frac{1}{2} \mathbf{B} \cdot \mathbf{H}. The vector \mathbf{S} = \mathbf{E} \times \mathbf{H} is the Poynting vector, denoting the instantaneous energy flux density (power per unit area). The term \mathbf{E} \cdot \mathbf{J} is the power density p at which the electromagnetic field performs work on the charges. Rewriting the equation as \frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = -p highlights its structure as a local continuity equation for energy, analogous to the charge continuity equation \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0, but governing energy balance rather than charge conservation.^[1]^[4] This form assumes linear, time-invariant, non-dispersive media and applies to instantaneous fields. Integrating the differential form over an arbitrary volume yields the corresponding integral form of the theorem. For time-harmonic fields in such media, the time average of \frac{\partial u}{\partial t} over one period vanishes, simplifying the theorem to \nabla \cdot \langle \mathbf{S} \rangle = -\langle \mathbf{E} \cdot \mathbf{J} \rangle, where \langle \cdot \rangle denotes the time average.^[4]

Derivation

From Maxwell's equations in vacuum

Poynting's theorem in its differential form can be derived algebraically from Maxwell's equations in vacuum, where there are no free charges or magnetization effects beyond the vacuum permittivity \epsilon_0 and permeability \mu_0. The relevant starting equations are the curl forms: Faraday's law \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} and Ampère's law with Maxwell's correction \nabla \times \mathbf{H} = \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}, where \mathbf{B} = \mu_0 \mathbf{H} and the current density \mathbf{J} represents any sources. These equations describe the time-varying electromagnetic fields in free space without material media. To begin the derivation, take the dot product of \mathbf{E} with Ampère's law: \mathbf{E} \cdot (\nabla \times \mathbf{H}) = \mathbf{E} \cdot \mathbf{J} + \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t}. Next, take the dot product of \mathbf{H} with Faraday's law: \mathbf{H} \cdot (\nabla \times \mathbf{E}) = -\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}. Subtract the second equation from the first to eliminate the curl terms on the left: \mathbf{E} \cdot (\nabla \times \mathbf{H}) - \mathbf{H} \cdot (\nabla \times \mathbf{E}) = \mathbf{E} \cdot \mathbf{J} + \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} + \mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}. This step isolates the interaction terms involving the fields and sources. Now apply the vector identity \nabla \cdot (\mathbf{E} \times \mathbf{H}) = \mathbf{H} \cdot (\nabla \times \mathbf{E}) - \mathbf{E} \cdot (\nabla \times \mathbf{H}), which rearranges the left side to -\nabla \cdot (\mathbf{E} \times \mathbf{H}). Substituting this into the previous equation yields -\nabla \cdot (\mathbf{E} \times \mathbf{H}) = \mathbf{E} \cdot \mathbf{J} + \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} + \mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}. The term \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} recognizes the time derivative of the electric energy density, since \frac{\partial}{\partial t} \left( \frac{1}{2} \epsilon_0 E^2 \right) = \epsilon_0 \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t}. Similarly, for the magnetic part, \mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t} = \mathbf{H} \cdot \frac{\partial (\mu_0 \mathbf{H})}{\partial t} = \mu_0 \mathbf{H} \cdot \frac{\partial \mathbf{H}}{\partial t} = \frac{\partial}{\partial t} \left( \frac{1}{2} \mu_0 H^2 \right); this form arises because the magnetic energy density in vacuum is \frac{1}{2} \mathbf{B} \cdot \mathbf{H}, and substituting \mathbf{B} = \mu_0 \mathbf{H} equivalently gives \frac{1}{2\mu_0} B^2 or \frac{1}{2} \mu_0 H^2, reflecting the linear relation in free space. Thus, the equation simplifies to the differential form of Poynting's theorem:

-\nabla \cdot (\mathbf{E} \times \mathbf{H}) = \mathbf{E} \cdot \mathbf{J} + \frac{\partial}{\partial t} \left( \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2} \mu_0 H^2 \right),

where \mathbf{E} \times \mathbf{H} is the Poynting vector representing energy flux, \mathbf{E} \cdot \mathbf{J} is the rate of work done by the fields on charges, and the time derivative term is the rate of change of electromagnetic energy density. This local form expresses energy conservation at every point in vacuum.

Local conservation law

The differential form of Poynting's theorem provides a pointwise statement of energy conservation in electromagnetic fields, asserting that the local rate of change of electromagnetic energy density plus the divergence of the energy flux density equals the negative of the power density delivered to charges by the electric field.^[2] This form, \nabla \cdot \mathbf{S} + \mathbf{E} \cdot \mathbf{J} + \frac{\partial u}{\partial t} = 0, where \mathbf{S} is the Poynting vector representing energy flux, u is the electromagnetic energy density, \mathbf{E} is the electric field, and \mathbf{J} is the current density, holds at every point in space and time, enabling the analysis of energy balance on arbitrarily small scales.^[12] The "local" character distinguishes it from global statements, as it captures instantaneous, position-dependent interactions without averaging over volumes or times.^[13] To derive the integral form and reveal global conservation, integrate the differential equation over an arbitrary volume V bounded by surface S:

\int_V \left( -\mathbf{E} \cdot \mathbf{J} - \frac{\partial u}{\partial t} \right) dV = \int_V \nabla \cdot \mathbf{S} \, dV.

Applying the divergence theorem to the right-hand side yields the surface integral of the outward energy flux:

\int_V \left( -\mathbf{E} \cdot \mathbf{J} - \frac{\partial u}{\partial t} \right) dV = \oint_S \mathbf{S} \cdot d\mathbf{A}.

Rearranging terms gives

\int_V \mathbf{E} \cdot \mathbf{J} \, dV + \frac{\partial}{\partial t} \int_V u \, dV + \oint_S \mathbf{S} \cdot d\mathbf{A} = 0,

where the first term represents the total power delivered to charges within V, the second is the rate of change of stored electromagnetic energy in V, and the third is the net power flowing out through S.^[2] For an isolated system with no external currents (\mathbf{J} = 0) and no energy flux across a closed surface enclosing all fields, the equation implies zero net change in total energy, confirming conservation.^[13] This structure mirrors the continuity equation in fluid dynamics, \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = s, where \rho corresponds to the energy density u, \mathbf{j} to the flux \mathbf{S}, and the source term s = -\mathbf{E} \cdot \mathbf{J} acts as a sink representing energy transfer to matter (e.g., ohmic heating).^[12] Just as the continuity equation enforces local mass conservation in fluids, Poynting's theorem enforces local energy conservation in electromagnetism, with the integral form emerging naturally from volume averaging to describe macroscopic balances.^[2]

Poynting vector

Definition and properties

The Poynting vector \mathbf{S}, introduced by John Henry Poynting in his seminal 1884 paper, quantifies the directional energy flux density in an electromagnetic field. In vacuum, it is defined as the cross product of the electric field \mathbf{E} and the magnetic field strength \mathbf{H}, expressed as \mathbf{S} = \mathbf{E} \times \mathbf{H}, or equivalently \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} where \mathbf{B} = \mu_0 \mathbf{H} and \mu_0 is the permeability of free space.^[3]^[14] The units of \mathbf{S} are watts per square meter (W/m²), corresponding to power per unit area.^[14] The direction of \mathbf{S} is always perpendicular to both \mathbf{E} and \mathbf{H}, following the right-hand rule for the cross product. In the context of plane electromagnetic waves propagating in vacuum, \mathbf{S} aligns with the direction of the wave vector \mathbf{k}, making it orthogonal to the plane containing \mathbf{E} and \mathbf{H}. For such waves, the magnitude relates to the fields via the impedance of free space \eta = \sqrt{\mu_0 / \epsilon_0} \approx 377 \, \Omega, where |\mathbf{H}| = |\mathbf{E}| / \eta. The instantaneous magnitude is |\mathbf{S}| = E H \sin\theta, with \theta = 90^\circ yielding the maximum value, but for time-varying fields, the physical intensity I is given by the time average \langle |\mathbf{S}| \rangle.^[14]^[15] In vacuum, \mathbf{S} represents the instantaneous Poynting flux, derived as the term describing energy transport in Poynting's theorem. For sinusoidal fields, the time-averaged Poynting vector is \langle \mathbf{S} \rangle = \frac{E_0 H_0}{2} \hat{\mathbf{k}} = \frac{E_0^2}{2 \eta} \hat{\mathbf{k}}, providing the average power flow per unit area along the propagation direction. This average is crucial for quantifying energy transport in applications such as waveguides and antennas, where \mathbf{S} qualitatively indicates the flow of electromagnetic energy from source to load through the surrounding vacuum.^[14]^[15]

Behavior in macroscopic media

In macroscopic media, the Poynting vector retains its form as \mathbf{S} = \mathbf{E} \times \mathbf{H}, representing the electromagnetic energy flux density, while the electromagnetic energy density is adjusted to account for material responses: u = \frac{1}{2} (\mathbf{D} \cdot \mathbf{E} + \mathbf{B} \cdot \mathbf{H}).^[16]^[12] For linear media, the constitutive relations simplify this to \mathbf{D} = \epsilon \mathbf{E} and \mathbf{B} = \mu \mathbf{H}, where \epsilon is the permittivity and \mu is the permeability, both exceeding their vacuum values \epsilon_0 and \mu_0 due to polarization and magnetization effects.^[16] Poynting's theorem in this context takes the differential form

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

where \mathbf{J}_\mathrm{free} denotes the free current density, excluding bound currents from material polarization or magnetization.^[12] Rearranged, it expresses energy conservation as the negative of the power delivered to free charges, -\mathbf{E} \cdot \mathbf{J}_\mathrm{free}, equaling the rate of change of stored energy plus the divergence of the energy flux: -\mathbf{E} \cdot \mathbf{J}_\mathrm{free} = \frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S}.^[16] This formulation incorporates material contributions, with the electric term \frac{1}{2} \mathbf{D} \cdot \mathbf{E} capturing energy stored in electric polarization and the magnetic term \frac{1}{2} \mathbf{B} \cdot \mathbf{H} accounting for magnetization energy.^[12] Compared to the vacuum case, where u = \frac{1}{2} (\epsilon_0 E^2 + \frac{B^2}{\mu_0}) and no material storage occurs, macroscopic media introduce effective \epsilon > \epsilon_0 and \mu > \mu_0, enhancing stored energy.^[16] In dielectrics, for instance, the additional energy \frac{1}{2} \mathbf{P} \cdot \mathbf{E} (where \mathbf{P} = (\epsilon - \epsilon_0) \mathbf{E}) represents work done to align molecular dipoles against thermal disorder.^[12] In conductors, the term \mathbf{E} \cdot \mathbf{J}_\mathrm{free} = \sigma E^2 (via Ohm's law, \mathbf{J}_\mathrm{free} = \sigma \mathbf{E}) quantifies ohmic losses as heat dissipation in the material lattice.^[16] This adaptation assumes linear, non-dispersive media where \epsilon and \mu are frequency-independent scalars, and \mathbf{J}_\mathrm{free} includes only conduction or external currents, omitting bound currents that are already embedded in \mathbf{D} and \mathbf{H}.^[12]

Extensions

Alternative formulations

One alternative formulation of Poynting's theorem incorporates hypothetical magnetic sources, such as magnetic charge density \rho_m and magnetic current density \mathbf{J}_m, to emphasize the duality between electric and magnetic phenomena. In this dual form, derived from the symmetrized Maxwell's equations, the local energy conservation law becomes

-\nabla \cdot (\mathbf{E} \times \mathbf{H}) = \mathbf{E} \cdot \mathbf{J} + \mathbf{H} \cdot \mathbf{J}_m + \frac{\partial}{\partial t} \left( \frac{\epsilon_0 E^2}{2} + \frac{B^2}{2\mu_0} \right),

where the additional term \mathbf{H} \cdot \mathbf{J}_m represents the power delivered by the magnetic field to magnetic currents, mirroring the \mathbf{E} \cdot \mathbf{J} term for electric currents. This extension maintains the structure of energy balance but highlights the symmetry in source terms, applicable in theoretical contexts exploring magnetic monopoles or dual electrodynamics.^[17] In Lagrangian and Hamiltonian formulations of classical field theory, Poynting's theorem emerges as a consequence of time-translation invariance via Noether's theorem. The electromagnetic Lagrangian density \mathcal{L} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} - A_\mu J^\mu (in relativistic notation, with F_{\mu\nu} the field strength tensor and A_\mu the 4-potential) yields the conserved energy-momentum tensor T^{\mu\nu}, whose \nu=0 components encode the total field energy \int \left( \frac{\epsilon_0 E^2}{2} + \frac{B^2}{2\mu_0} \right) dV. The Hamiltonian formulation, treating electric and magnetic fields as conjugate variables, further confirms this energy expression and links the Poynting vector to the energy flux in the conservation law.^[18] A relativistic invariant formulation expresses Poynting's theorem through the divergence of the electromagnetic stress-energy tensor T^{\mu\nu} = \frac{1}{\mu_0} \left( F^{\mu\lambda} F^\nu{}_\lambda - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right), satisfying \partial_\mu T^{\mu\nu} = -f^\nu, where f^\nu = F^{\nu\lambda} J_\lambda / c is the 4-force density on sources. The energy flux 4-vector, with components S^\mu = (c u, \mathbf{S}) (where u is the energy density and \mathbf{S} the Poynting vector), transforms covariantly under Lorentz transformations, ensuring conservation in all inertial frames within classical electrodynamics. This form unifies energy and momentum conservation without deriving full quantum or general relativistic extensions.^[18] Other variants address formulations in dispersive media, where energy conservation via Poynting's theorem remains robust, though momentum density expressions differ in the Abraham-Minkowski controversy—pitting \mathbf{g}_A = \mathbf{S}/c^2 (Abraham) against \mathbf{g}_M = \mathbf{D} \times \mathbf{B} (Minkowski). Focus on energy flux avoids this debate, as the Poynting vector consistently describes power flow.^[19]

Complex form for time-harmonic fields

For time-harmonic electromagnetic fields, which vary sinusoidally in time at a single angular frequency \omega, the analysis simplifies using phasor representation, where the real fields are the real parts of complex phasors multiplied by e^{j\omega t}.^[20] This convention assumes linear media and steady-state conditions, eliminating transient terms.^[21] The complex form of Poynting's theorem emerges by substituting phasors into Maxwell's equations and taking appropriate dot products, yielding a frequency-domain conservation law for complex power.^[22] The complex Poynting vector is defined as \mathbf{S}_c = \frac{1}{2} \mathbf{E} \times \mathbf{H}^*, where \mathbf{E} and \mathbf{H} are the complex phasor amplitudes of the electric and magnetic fields, and \mathbf{H}^* denotes its complex conjugate.^[20] The real part \operatorname{Re}(\mathbf{S}_c) represents the time-averaged power flux density, while the imaginary part \operatorname{Im}(\mathbf{S}_c) relates to reactive power associated with energy storage in the fields.^[21] In integral form, for a volume V bounded by surface S, the theorem states:

\frac{1}{2} \int_V \operatorname{Re}(\mathbf{E} \cdot \mathbf{J}^*) \, dV = -\frac{1}{2} \oint_S \operatorname{Re}(\mathbf{E} \times \mathbf{H}^*) \cdot d\mathbf{A},

where \mathbf{J} is the current density phasor.^[22] This follows from applying the divergence theorem to \nabla \cdot (\mathbf{E} \times \mathbf{H}^*) and using the phasor forms of Ampère's and Faraday's laws; the time-derivative term vanishes in the steady-state average, leaving a balance between ohmic dissipation (left side) and net power outflow (right side).^[22] The full complex version includes a reactive term j\omega \int_V (\mathbf{E} \cdot \mathbf{D}^* - \mathbf{H}^* \cdot \mathbf{B}) \, dV / 2, where \mathbf{D} and \mathbf{B} are displacement and induction phasors, quantifying stored energy differences.^[21] This formulation finds applications in analyzing average power delivery and reactive energy in AC systems, such as transmission lines or resonant structures in circuit design, where \operatorname{Im}(\mathbf{S}_c) indicates energy oscillation between sources and loads.^[20] In antenna design, it quantifies radiation efficiency by separating radiated real power from stored reactive power near the structure.^[22] For example, consider a plane wave normally incident from medium 1 (intrinsic impedance \eta_1) onto medium 2 (\eta_2); impedance mismatch yields a reflection coefficient \rho = (\eta_2 - \eta_1)/(\eta_2 + \eta_1), reducing the transmitted time-averaged Poynting flux to (1 - |\rho|^2) times the incident value, illustrating power conservation across the interface.^[23]

References

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The conservation theorem includes the effects of both displacement current and of magnetic induction. The EQS and MQS limits, respectively, can be taken by ...
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A space containing electric currents may be regarded as a field where energy is transformed at certain points into the electric and magnetic kinds.
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POYNTING, and independently discovered and interpreted a little later by myself in an extended form. It will be observed that in my mode of proof above ...
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[PDF] 20 Poynting theorem and monochromatic waves
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(a) The time-averaged Poynting vector is related to the amplitude of the electric field by ... fields make up an electromagnetic wave in vacuum? 7. What ...Missing: theorem | Show results with:theorem
[16]
[PDF] Energy in the Fields Linear Momentum - Rutgers Physics
∇ ·. (. E × ~H. ) + E ·. ∂ D. ∂t. + ~H ·. ∂ B. ∂t. + J · E = 0. Let us assume the medium is linear without dispersion in electric and magnetic properties, that ...
[17]
[PDF] Poynting's Theorem with Magnetic Monopoles 1 Problem 2 Solution
Heaviside (1885) used the term duplex to describe his version of Maxwell's equations that included magnetic charges and currents as well as electric charges and ...Missing: Koenigsberger | Show results with:Koenigsberger
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[PDF] Noether's theorem applied to classical electrodynamics
Nov 8, 2017 · Noether's theorem states that every differentiable symmetry of the action corresponds to a conserved quantity. Considering classical ...
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[PDF] Revisiting the Abraham-Minkowski Dilemma - arXiv
Abraham-Minkowski controversy has ... electromagnetic instabilities that lead to the spontaneous conversion of kinetic energy into electromagnetic energy.
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### Summary of Complex Form of Poynting’s Theorem for Time-Harmonic Fields
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https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_(Staelin)/02%3A_Introduction_to_Electrodynamics/2.07%3A_Power_and_energy_in_the_time_and_frequency_domains%2C_Poynting_theorem
[22]
[PDF] EECS 217
Complex Poynting Theorem. We derived the Poynting Theorem for general electric/magnetic fields. We'd like to derive the Poynting. Theorem for time-harmonic ...
[23]
[PDF] EECS 117 - University of California, Berkeley
Complex Poynting Theorem (II). Using our knowledge of circuit theory, P ... Where φη is the angle of the complex wave impedance,. 45◦ in this particular ...