Note on the positiveness of magnetic energy stored in the Wire-SRR metamaterial Pi-Gang Luan Wave Engineering Laboratory, Department of Optics and Photonics, National Central University, Jhungli 320, Taiwan Recently, a new energy density formula for electromagnetic waves in the metamaterial consisting of arrays of conducting wires and split-ring-resonators (SRR) has been derived [Phys. Rev. E 80, 046601 (2009)]. According to that formula, the positiveness is obvious for the electric part of the energy density but not clear for the magnetic part. In this paper, I show that the magnetic energy density is also positively definite. PACS numbers: 41.20.Jb, 03.50.De, 78.20.Ci In a recent paper, Luan derived a new expression for energy density of electromagnetic waves in the metamate- rial medium consisting of wires and split-ring-resonators (SRR) [1]. The motivation of that paper was to resolve the apparent contradictions between the results given by the equivalent circuit (EC) and electrodynamic (ED) ap- proaches [2, 3]. Luan pointed out a calculation error made in [2], which can be corrected so that EC approach can really give the correct formula. On the other hand, usually the ED approach [3] does not give unique result, and the analysis in [1] revealed that the expression of the energy density is in fact determined by the power loss we choose. For a lossy medium, Luan suggested to find the correct expression of power loss first, then the correct energy density formula will emerge automatically. The equivalence between EC and ED approaches was then confirmed by reproducing the EC energy formula after time averaging the ED formula. The positiveness of the electric energy density can be easily observed from the following expressions this property is not clear in the time averaged formula When ω > √$_{3}$$_{ω}$$_{0}$$_{, the above formula gives a}$ 〈 W$_{b}$ 〉 smaller than the vacuum value µ 0 | H | $^{2}$/ 4. Will this ex- pression always give a positive 〈 W$_{b}$ 〉 ? If we add the two terms in the square bracket directly, we get which is still hard to judge its sign. However, if we con- sider the identity the positiveness of the magnetic energy becomes very ob- vious (remember that both F and 1 − F are positive). According to this analysis, both the electric and mag- netic energy densities in the energy density formula of [1] are positively definite. The author gratefully acknowledge financial support from National Science Council (Grant No. NSC 98-2112- M-008-014-MY3) of Republic of China, Taiwan. W$_{e}$ = ϵ 0 E 2 2 + 1 2 ω 2 $_{p}$ϵ 0 ( ∂ P ∂t ) 2 , (1) 〈 W$_{e}$ 〉 = ϵ 0 | E | 2 4 ( 1 + ω 2 p ω 2 + ν 2 ) . (2) 〈 W$_{b}$ 〉 = µ 0 | H | 2 4 [ 1 + F ω 2 ( 3 ω 2 0 − ω 2 ) ( ω 2 0 − ω $^{2}$) 2 + ω $^{2}$γ2 ] . (5) 4 〈 W$_{b}$ 〉 µ 0 | H | 2 = (1 − F ) ω 4 + (3 F − 2) ω 2 $_{0}$ω 2 + ω 4 0 + ω $^{2}$γ2 ( ω 2 0 − ω $^{2}$)2 + ω $^{2}$γ2 , (6) ( ω 2 0 − ω $^{2}$)2 + ω $^{2}$γ2 + F ω $^{2}$(3 ω 2 0 − ω $^{2}$) = (1 F )( ω 2 − ω 2 $_{0}$) 2 + F ω 2 $_{0}$( ω 2 + ω 2 $_{0}$) + ω $^{2}$γ2 (7) W$_{b}$ = µ 0 (1 − F ) 2 H 2 + µ 0 2 F ( M + F H ) 2 (3) 2 + µ 0 2 ω 2 $_{0}$F ( ∂ M ∂t + F ∂ H ∂t + γ M ) , (4) ACKNOWLEDGEMENT Although the positiveness of magnetic energy density in the time-domain formula is obvious (note that 0 < F < 1 is assumed) [1] Pi-Gang Luan, Phys. Rev. E 80, 046601 (2009). [2] S.A. Tretyakov, Phys. Lett. A, 343, 231 (2005) [3] A. D. Boardman, K. Marinov, Phys. Rev. B 73, 165110 (2006) arXiv:1001.0350v2 [physics.optics] 5 Jan 2010