## Abstract

It is shown that, when all macroscopic currents associated with the electric and magnetic polarizability are properly accounted for, the standard expression for the Poynting vector and the average work exerted by the electric field on the electric charges provide exactly the same value for the heating rate. Therefore, there is no contradiction between negative refraction and thermodynamics.

©2009 Optical Society of America

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### Equations (7)

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(1)
$$q=\frac{1}{4\pi}\u3008\mathbf{E\xb7}\frac{\partial \mathit{D}}{\partial t}+\mathbf{H}\xb7\frac{\partial \mathit{B}}{\partial t}\u3009$$
(2)
$$q=\u3008\mathbf{J}\mathbf{\xb7}\mathbf{E}\u3009.$$
(3)
$$q=\u3008\mathbf{E}\xb7\frac{\partial \mathbf{P}}{\partial t}+c\mathbf{M}\xb7\nabla \times \mathit{E}\u3009.$$
(4)
$$\int \int \int \mathit{qdv}=\int \int \int \mathbf{E\xb7}\left(\frac{\partial \mathbf{P}}{\partial t}+c\nabla \times \mathbf{M}\right)\mathrm{dv}+c\int \int \mathbf{E\xb7}\left(\mathbf{M}\times \mathbf{n}\right)\mathit{ds}$$
(5)
$${\mathbf{J}}_{v}=\frac{\partial \mathbf{P}}{\partial t}+c\nabla \times \mathbf{M}$$
(6)
$${\mathbf{J}}_{s}=c\mathbf{M}\times \mathbf{n}.$$
(7)
$$\delta \mathbf{m}\equiv \int \int {\int}_{\mathit{\delta V}}\mathbf{M}\mathit{dv}=\frac{1}{2}\int \int {\int}_{\mathit{\delta V}}\mathbf{r}\times \left(\nabla \times \mathbf{M}\right)\mathit{dv}+\frac{1}{2}\int {\int}_{\mathit{\delta \sum}}\mathbf{r}\times \left(\mathbf{M}\times \mathbf{n}\right)\mathit{ds}$$