Poynting's theorem

Poynting's theorem In electrodynamics, Poynting's theorem is a statement of conservation of energy for electromagnetic fields developed by British physicist John Henry Poynting.[1] It states that in a given volume, the stored energy changes at a rate given by the work done on the charges within the volume, minus the rate at which energy leaves the volume. It is only strictly true in media which is not dispersive, but can be extended for the dispersive case.[2] The theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation.

Contents 1 Definition 1.1 Integral Form 1.2 Continuity Equation Analog 2 Derivation 3 Poynting vector in macroscopic media 4 Alternative forms 5 Modification 6 Complex Poynting vector theorem 7 References 8 External links Definition Poynting's theorem states that the rate of energy transfer per unit volume from a region of space equals the rate of work done on the charge distribution in the region, plus the energy flux leaving that region.

Mathematically: {displaystyle -{frac {partial u}{partial t}}=nabla cdot mathbf {S} +mathbf {J} cdot mathbf {E} } where: {displaystyle -{frac {partial u}{partial t}}} is the rate of change of the energy density in the volume. ∇•S is the energy flow out of the volume, given by the divergence of the Poynting vector S. J•E is the rate at which the fields do work on charges in the volume (J is the current density corresponding to the motion of charge, E is the electric field, and • is the dot product). Integral Form Using the divergence theorem, Poynting's theorem can also be written in integral form: {displaystyle -{frac {partial }{partial t}}int _{V}u~mathrm {d} V=} {displaystyle scriptstyle partial V} {displaystyle mathbf {S} cdot mathrm {d} mathbf {A} +int _{V}mathbf {J} cdot mathbf {E} ~mathrm {d} V} where S is the energy flow, given by the Poynting Vector. {displaystyle u} is the energy density in the volume. {displaystyle partial V!} is the boundary of the volume. The shape of the volume is arbitrary but fixed for the calculation. Continuity Equation Analog In an electrical engineering context the theorem is sometimes written with the energy density term u expanded as shown.[citation needed] This form resembles the continuity equation: {displaystyle nabla cdot mathbf {S} +epsilon _{0}mathbf {E} cdot {frac {partial mathbf {E} }{partial t}}+{frac {mathbf {B} }{mu _{0}}}cdot {frac {partial mathbf {B} }{partial t}}+mathbf {J} cdot mathbf {E} =0,} where ε0 is the vacuum permittivity and μ0 is the vacuum permeability. {displaystyle epsilon _{0}mathbf {E} cdot {frac {partial mathbf {E} }{partial t}}} is the density of reactive power driving the build-up of electric field, {displaystyle {frac {mathbf {B} }{mu _{0}}}cdot {frac {partial mathbf {B} }{partial t}}} is the density of reactive power driving the build-up of magnetic field, and {displaystyle mathbf {J} cdot mathbf {E} } is the density of electric power dissipated by the Lorentz force acting on charge carriers. Derivation For an individual charge in an electromagnetic field, the rate of work done by the field on the charge is given by the Lorentz Force Law as: {displaystyle {frac {dW}{dt}}=qmathbf {v} cdot mathbf {E} } Extending this to a continuous distribution of charges, moving with current density J, gives: {displaystyle {frac {dW}{dt}}=int _{V}mathbf {J} cdot mathbf {E} ~mathrm {d} ^{3}x} By Ampère's circuital law: {displaystyle mathbf {J} =nabla times mathbf {H} -{frac {partial mathbf {D} }{partial t}}} (Note that the H and D forms of the magnetic and electric fields are used here. The B and E forms could also be used in an equivalent derivation.)[3] Substituting this into the expression for rate of work gives: {displaystyle int _{V}mathbf {J} cdot mathbf {E} ~mathrm {d} ^{3}x=int _{V}left[mathbf {E} cdot (nabla times mathbf {H} )-mathbf {E} cdot {frac {partial mathbf {D} }{partial t}}right]~mathrm {d} ^{3}x} Using the vector identity {displaystyle nabla cdot (mathbf {E} times mathbf {H} )= (nabla {times }mathbf {E} )cdot mathbf {H} ,-,mathbf {E} cdot (nabla {times }mathbf {H} )} : {displaystyle int _{V}mathbf {J} cdot mathbf {E} ~mathrm {d} ^{3}x=-int _{V}left[nabla cdot (mathbf {E} times mathbf {H} )-mathbf {H} cdot (nabla times mathbf {E} )+mathbf {E} cdot {frac {partial mathbf {D} }{partial t}}right]~mathrm {d} ^{3}x} By Faraday's Law: {displaystyle nabla times mathbf {E} =-{frac {partial mathbf {B} }{partial t}}} giving: {displaystyle int _{V}mathbf {J} cdot mathbf {E} ~mathrm {d} ^{3}x=-int _{V}left[nabla cdot (mathbf {E} times mathbf {H} )+mathbf {E} cdot {frac {partial mathbf {D} }{partial t}}+mathbf {H} cdot {frac {partial mathbf {B} }{partial t}}right]~mathrm {d} ^{3}x} Continuing the derivation requires the following assumptions:[2] the charges are moving in a medium which is not dispersive. the total electromagnetic energy density, even for time-varying fields, is given by {displaystyle u={frac {1}{2}}(mathbf {E} cdot mathbf {D} +mathbf {B} cdot mathbf {H} )} It can be shown[4] that: {displaystyle {frac {partial }{partial t}}(mathbf {E} cdot mathbf {D} )=2mathbf {E} cdot {frac {partial }{partial t}}mathbf {D} } and {displaystyle {frac {partial }{partial t}}(mathbf {H} cdot mathbf {B} )=2mathbf {H} cdot {frac {partial }{partial t}}mathbf {B} } and so: {displaystyle {frac {partial u}{partial t}}=mathbf {E} cdot {frac {partial mathbf {D} }{partial t}}+mathbf {H} cdot {frac {partial mathbf {B} }{partial t}}} Returning to the equation for rate of work, {displaystyle int _{V}mathbf {J} cdot mathbf {E} ~mathrm {d} ^{3}x=-int _{V}left[{frac {partial u}{partial t}}+nabla cdot (mathbf {E} times mathbf {H} )right]~mathrm {d} ^{3}x} Since the volume is arbitrary, this can be cast in differential form as: {displaystyle -{frac {partial u}{partial t}}=nabla cdot mathbf {S} +mathbf {J} cdot mathbf {E} } where {displaystyle mathbf {S} =mathbf {E} times mathbf {H} } is the Poynting vector.

Poynting vector in macroscopic media In a macroscopic medium, electromagnetic effects are described by spatially averaged (macroscopic) fields. The Poynting vector in a macroscopic medium can be defined self-consistently with microscopic theory, in such a way that the spatially averaged microscopic Poynting vector is exactly predicted by a macroscopic formalism. This result is strictly valid in the limit of low-loss and allows for the unambiguous identification of the Poynting vector form in macroscopic electrodynamics.[5][6] Alternative forms It is possible to derive alternative versions of Poynting's theorem.[7] Instead of the flux vector E × H as above, it is possible to follow the same style of derivation, but instead choose E × B, the Minkowski form D × B, or perhaps D × H. Each choice represents the response of the propagation medium in its own way: the E × B form above has the property that the response happens only due to electric currents, while the D × H form uses only (fictitious) magnetic monopole currents. The other two forms (Abraham and Minkowski) use complementary combinations of electric and magnetic currents to represent the polarization and magnetization responses of the medium.[7] Modification The derivation of the statement is dependent on the assumption that the materials the equation models can be described by a set of susceptibility properties that are linear, isotropic, homogenous and independent of frequency.[8] The assumption that the materials have no absorption must also be made. A modification to Poynting's theorem to account for variations includes a term for the rate of non-Ohmic absorption in a material, which can be calculated by a simplified approximation based on the Drude model.[8] {displaystyle {frac {partial }{partial t}}{mathcal {U}}+nabla cdot mathbf {S} +mathbf {E} cdot mathbf {J} _{text{free}}+{mathcal {R}}_{dashv int }=0} Complex Poynting vector theorem This form of the theorem is useful in Antenna theory, where one has often to consider harmonic fields propagating in the space. In this case, using phasor notation, {displaystyle E(t)=Ee^{jomega t}} and {displaystyle H(t)=He^{jomega t}} . Then the following mathematical identity holds: {displaystyle {1 over 2}int _{partial Omega }Etimes H^{*}cdot d{mathbf {a} }={jomega over 2}int _{Omega }(varepsilon EE^{*}-mu HH^{*})dv-{1 over 2}int _{Omega }EJ^{*}dv,} where {displaystyle J} is the current density.

Note that in free space, {displaystyle varepsilon } and {displaystyle mu } are real, thus, taking the real part of the above formula, it expresses the fact that the averaged radiated power flowing through {displaystyle partial Omega } is equal to the work on the charges.

References ^ Poynting, J. H. (December 1884). "On the Transfer of Energy in the Electromagnetic Field" . Philosophical Transactions of the Royal Society of London. 175: 343–361. doi:10.1098/rstl.1884.0016. ^ Jump up to: a b Jackson, John David (1999). Classical Electrodynamics (3rd ed.). John WIley & Sons. pp. 258–267. ISBN 978-0-471-30932-1. ^ Griffiths, David J. (1989). Introduction to electrodynamics (2nd ed.). Englewood Cliffs, N.J.: Prentice Hall. pp. 322–324. ISBN 0-13-481367-7. ^ Ellingson, Steven. "Poynting's Theorem". LibreTexts. Retrieved 3 December 2021. ^ Silveirinha, M. G. (2010). "Poynting vector, heating rate, and stored energy in structured materials: a first principles derivation". Phys. Rev. B. 82: 037104. doi:10.1103/physrevb.82.037104. ^ Costa, J. T. , M. G. Silveirinha, A. Alù (2011). "Poynting Vector in Negative-Index Metamaterials". Phys. Rev. B. 83: 165120. doi:10.1103/physrevb.83.165120. ^ Jump up to: a b Kinsler, P.; Favaro, A.; McCall M.W. (2009). "Four Poynting theorems" (PDF). European Journal of Physics. 30 (5): 983. arXiv:0908.1721. Bibcode:2009EJPh...30..983K. doi:10.1088/0143-0807/30/5/007. ^ Jump up to: a b Freeman, Richard; King, James; Lafyatis, Gregory (2019), "Essentials of Electricity and Magnetism", Electromagnetic Radiation, Oxford: Oxford University Press, doi:10.1093/oso/9780198726500.001.0001/oso-9780198726500-chapter-1#oso-9780198726500-chapter-1-displaymaths-20, ISBN 978-0-19-872650-0, retrieved 2022-02-18 External links Wikiversity has a lesson on Poynting's theorem Eric W. Weisstein "Poynting Theorem" From ScienceWorld – A Wolfram Web Resource. hide Authority control National libraries IsraelUnited States Other Faceted Application of Subject Terminology Categories: ElectrodynamicsPhysics theoremsCircuit theorems