Here {\psi }_{\alpha }=[\varphi ,{\bf{A}},{\bf{P}},{\bf{Q}}] and {\pi }_{\alpha }=\left[0,-{\bf{D}},{\partial }_{t}{\bf{P}}/{\omega }_{p}^{2}{\varepsilon }_{0},{\bf{H}}+\tfrac{{\bf{M}}}{F}\right] are the dynamical variables and their corresponding canonical momenta, and \tfrac{\delta H}{\delta {\pi }_{\alpha }} and \tfrac{\delta H}{\delta {\psi }_{\alpha }} denote the functional derivatives. The electrons in Drude model are assumed to be non-interacting among themselves, accelerated by the applied electric field, and dissipate their momenta and energies through the collisions with phonons, dislocations, and defects in the ions background. For either system, application of Legendre transformation leads to a, The power loss and electromagnetic energy density of a metamaterial consisting of arrays of wires and split-ring resonators are investigated. 2 075016. He described the relationship between mass and energy accurately using his theory of relativity. The defining relations {\bf{E}}=-{\rm{\nabla }}\varphi -{\partial }_{t}{\bf{A}} and {\bf{B}}={\rm{\nabla }}\times {\bf{A}} still hold, but the magnetic induction B now contains the magnetization M (i.e.,{\bf{B}}={\mu }_{0}({\bf{H}}+{\bf{M}})), which is lacking in the Drude model. Commun. Now we want to address two problems we left before. Whenever there is a chemical reaction, breakage and formation of new bonds take place. The same time-domain formulation of energy density also applies to the bianisotropic medium proposed by Zhang et al. This fact leads to an equality {({\partial }_{t}{\bf{P}})}^{2}/{\omega }_{p}^{2}{\varepsilon }_{0}={\mu }_{0}{{\bf{M}}}^{2}/F, which implies that any term proportional to {({\partial }_{t}{\bf{P}})}^{2} in the Lagrangian or the dissipation function is not different from a term proportional to M2. Leave a Reply Cancel reply Derivation. The radioactivity of various elements is based on the theory of mass-energy equivalence. This feature will also hold in the following examples of dispersive-absorptive metamaterials as can be checked later. simulations an d experiments,” J. O pt. Number 7, 1 Department of Optics and Photonics, National Central University, Jhongli District 32051, Taoyuan City, Taiwan, Republic of China, Pi-Gang Luan https://orcid.org/0000-0003-1691-1952, Received 25 May 2018 In addition, since the Gauss' law {\rm{\nabla }}\cdot {\bf{B}}=0 for the B field and the Faraday's induction law {\rm{\nabla }}\times {\bf{E}}+{\partial }_{t}{\bf{B}}=0 hold automatically by assuming the defining relations {\bf{B}}={\rm{\nabla }}\times {\bf{A}} and {\bf{E}}=-{\rm{\nabla }}\varphi -{\partial }_{t}{\bf{A}}, where \varphi and A are the scalar and vector potential respectively, here we need only to derive the Gauss' law and the Ampere's law. It works on Einstein’s equation. requiring that all the dynamic equations for the fields (Eq. It is obvious that the SRR structure contributes a partial Lagrangian of the form {{ {\mathcal L} }}_{{\rm{SRR}}}=\alpha {{\bf{M}}}^{2}-\beta {{\bf{Q}}}^{2}. Index Terms – energy density, quasimonochromatic electromagnetic wave, energy velocity, group velocity. Over the past two decades, artificial structures named metamaterials composed of 'artificial atoms' made of split rings or helical resonators have aroused great interest due to their potential utilities in the microwaves and photonics technologies. In addition, equation (2) is a Drude-model type dispersion, while equation (17) corresponds to a Lorentz-model dispersion of resonance frequency {\omega }_{0}, therefore a 'potential energy' term proportional to P2 or Q2 must be included in the Lagrangian density. equations,” Int. Here we consider the electromagnetic energy density propagated on and dissipated at real metal–dielectric surfaces, including the important surface plasmon polariton, the wave guided by the interface. Therefore, only one of these two terms will show up in the Lagrangian and the dissipation function. According to the field of applied mechanics, the sum of all these energies is smaller than the product of the mass of the object and square of the speed of light. Note that the magnetization M is proportional to the charge current and the cross section area of each SRR, therefore the first order differential equation (10) also represents the second order differential equation for the charge stored in the capacitance of the SRR. Revisions: 1 In section 4, we derive the LHDs for the more subtle system of single-resonance chiral metamaterial. In section 3, we derive the LHDs for the wire-SRR metamaterial step by step. The double-negative metamaterial considered here is the 'wires and split-rings' periodic structure, while the chiral metamaterial is the 'single-resonance helical resonators' array. In addition, the dissipation function density is still one half of the power loss of the system, just like we have already learned in the Drude model problem. Using the Poynting theorem for a quasimonochromatic electromagnetic wave concurrently interacting with electric and magnetic dipoles, exact formulas for the average total energy density and energy velocity v_s (ω) of the wave in a dispersive medium with losses are derived as functions of the refractive index n(ω). (9) for the harmonic fields. A stationary body does not have kinetic energy. (3)) for the displacement field leads to, is proporti onal to the c harge current and the, due to the work done by the electric field to the moving, must be included in the Lagrangian density. All third party content is fully copyright protected and is not published on a gold open access basis under a CC BY licence, unless that is. Applying Legendre transformation to these, cases, the resultant Hamiltonians are iden, we obtained previously based on Poynting t, systems of metamaterials for the further research of the quasiparticles, I would like to thank my colleague C. S. Tang for some useful discussions. The organization of this paper is as follows. It is obvious that the Lagrangian density should, themselves, together with the interaction energy term, similar to the power of Joule heat. It means the magnitude is squared. February 27, 2013 Share this: Twitter; Facebook; Like this: Related. . An important problem about dispersive media is how to identify the electromagnetic energy density in them [1, 4, 5]. (2) And, the dimensional formula of volume = [M 0 L 3 T 0] . . For the second problem, note that {\rm{\nabla }}\varphi \cdot {\bf{D}}={\rm{\nabla }}\cdot (\varphi {\bf{D}})-\varphi {\rm{\nabla }}\cdot {\bf{D}}, where the divergence term {\rm{\nabla }}\cdot (\varphi {\bf{D}}) contributes nothing to the bulk Hamiltonian H of equation (27), and thus it has no effect on the equations of motion in equation (28). Published by IOP Publishing Ltd, blem has practical significance because it may provide relevant, Though we have fix ed these ambigui ties in [14,15], Drude model can be derived from Eq. It is obvious that the Lagrangian density should contain the term {{ {\mathcal L} }}_{0}=\tfrac{{\varepsilon }_{0}}{2}{E}^{2}-\tfrac{{\mu }_{0}}{2}{H}^{2} for the electromagnetic fields themselves, together with the interaction energy term {{ {\mathcal L} }}_{\mathrm{int}}={\bf{P}}\cdot {\bf{E}} for the dipole-field interaction, and the kinetic energy term of the form {{ {\mathcal L} }}_{k}=\eta {({\partial }_{t}{\bf{P}})}^{2} for the moving electrons, here \eta is a constant to be determined. To find out more, see our, Browse more than 70 science journal titles, Read the very best research published in IOP journals, Read open access proceedings from science conferences worldwide, © 2018 The Author(s).

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