# electromagnetic energy density derivation

= ∂ The SI unit of the Poynting vector is the watt per square metre (W/m2). {\displaystyle \mathbf {S} \cdot d\mathbf {A} +\int _{V}\mathbf {J} \cdot \mathbf {E} dV}. 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. Taking the dot product of the Maxwell–Faraday equation with H: next taking the dot product of the Maxwell–Ampère equation with E: then, using the vector calculus identity: gives an expression for the Poynting vector: which physically means the energy transfer due to time-varying electric and magnetic fields is perpendicular to the fields, In a macroscopic medium, electromagnetic effects are described by spatially averaged (macroscopic) fields. Eric W. Weisstein "Poynting Theorem" From ScienceWorld – A Wolfram Web Resource. ∂ Using the divergence theorem, Poynting's theorem can be rewritten in integral form: − (Sanah Bhimani) Recall that for a parallel-plate capacitor, two plates close together create a constant electric field. The time derivative of the energy density (using the product rule for vector dot products) is, using the constitutive relations[clarification needed]. ∫ ∂ In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field, [clarification needed], in the form of a partial differential equation developed by British physicist John Henry Poynting. It can be described as the sum of kinetic energies of particles α (e.g., electrons in a wire), whose trajectory is given by rα(t): where Sm is the flux of their energies, or a "mechanical Poynting vector": Both can be combined via the Lorentz force, which the electromagnetic fields exert on the moving charged particles (see above), to the following energy continuity equation or energy conservation law:[7]. J Mathematically, this is summarised in differential form as: − V u v {\displaystyle \mathbf {J} =\rho \mathbf {v} } The electric field due to just one plate is where Q is the charge, A is the area of the plate, and ε0 is the permittivity of free space ( ). which is Poynting's theorem in differential form. The mechanical energy counterpart of the above theorem for the electromagnetic energy continuity equation is. J S in which B is the magnetic flux density. ∫ E [1] 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 (i.e. The partial time derivatives suggest using two of Maxwell's Equations. d Considering the statement in words above - there are three elements to the theorem, which involve writing energy transfer (per unit time) as volume integrals:[3], where ρ is the charge density of the distribution and v its velocity. Introduction to Electrodynamics (3rd Edition), D.J. Since For a parallel-plate capacitor, what is the force that one plate exert… an electrically charged object), through energy flux. It originates from the general derivation of the energy density contained in the electric and magnetic fields (starting from the Lorentz force and the definition of work). where um is the (mechanical) kinetic energy density in the system. In words, the theorem is an energy balance: A second statement can also explain the theorem - "The decrease in the electromagnetic energy per unit time in a certain volume is equal to the sum of work done by the field forces and the net outward flux per unit time". ⋅ u J Griffiths, Pearson Education, Dorling Kindersley, 2007, p.364, Learn how and when to remove these template messages, Talk:Poynting's_theorem § Inconsistencies_throughout_article, Learn how and when to remove this template message, "On the Transfer of Energy in the Electromagnetic Field". + + d From the theorem, the actual form of the Poynting vector S can be found. So by conservation of energy, the balance equation for the energy flow per unit time is the integral form of the theorem: and since the volume V is arbitrary, this is true for all volumes, implying. https://en.wikipedia.org/w/index.php?title=Poynting%27s_theorem&oldid=978507190, Wikipedia articles needing clarification from October 2017, All Wikipedia articles needing clarification, Articles needing expert attention with no reason or talk parameter, Articles needing expert attention from October 2017, Physics articles needing expert attention, Articles with multiple maintenance issues, Wikipedia articles needing clarification from May 2015, Creative Commons Attribution-ShareAlike License, The energy flux leaving the region is the, This page was last edited on 15 September 2020, at 09:40. = V ∇ ⋅ ∂ {\displaystyle -{\frac {\partial u}{\partial t}}=\nabla \cdot \mathbf {S} +\mathbf {J} \cdot \mathbf {E} }, where ∇•S is the divergence of the Poynting vector (energy flow) and J•E is the rate at which the fields do work on a charged object (J is the current density corresponding to the motion of charge, E is the electric field, and • is the dot product). ∂ In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field,[clarification needed], in the form of a partial differential equation developed by British physicist John Henry Poynting. ⋅ A {\displaystyle \scriptstyle \partial V} It is named after its discoverer John Henry Poynting who first derived it in 1884. Note how this equation is independent of the distance between the plates; as long as the charge and the area of a plate is constant, the electric field is the same no matter how much you pull the plates apart. {\displaystyle \partial V\!} t is the boundary of a volume V. The shape of the volume is arbitrary but fixed for the calculation. Oliver Heaviside also discovered it independently in the more general form that recognises the freedom of adding the … V V covering both types of energy and the conversion of one into the other. It is possible to derive alternative versions of Poynting's theorem. {\displaystyle -{\frac {\partial }{\partial t}}\int _{V}udV=} S 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.[4][5]. V In physics, the Poynting vector represents the directional energy flux (the energy transfer per unit area per unit time) of an electromagnetic field. t ⋅ where 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. [6] Instead of the flux vector E × B as above, it is possible to follow the same style of derivation, but instead choose the Abraham form E × H, 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. Planck’s Derivation of the Energy Density of Blackbody Radiation To calculate the number of modes of oscillation of electromagnetic radiation possible in a cavity, consider a one-dimensional box of side L. In equilibrium only standing waves are possible, and these will have nodes at the ends x … = ∂ In electrical engineering context the theorem is usually written with the energy density term u expanded in the following ways, which resembles the continuity equation: While conservation of energy and the Lorentz force law can give the general form of the theorem, Maxwell's equations are additionally required to derive the expression for the Poynting vector and hence complete the statement. The energy density u, assuming no electric or magnetic polarizability, is given by:[2]. E , the rate of work done by the force is. ρ d V