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Consider an electromagnetic wave in a lossy medium. This medium has a complex permittivity $\varepsilon_r=\varepsilon_r' +j \varepsilon_r''$ and a wave impedance given by $$Z=\frac{E}{H}=\sqrt{\frac{\mu_0}{\varepsilon_0 \varepsilon_r}}$$ where we assumed $\mu_r=1$. Hence, the complex permittivity results in a complex impedance and thus in a phase difference between $E$ and $H$.

The Poynting vector is given by $$\mathbf{S}=\frac{1}{2} \left( \mathbf{E}\times\mathbf{H}^* \right)$$ where the real part gives the net active power and the imaginary part gives the reactive power. So, when there is a phase difference between $E$ and $H$, there is also reactive power associated with the wave.

From this, we could conclude: every propagating electromagnetic wave in a lossy medium has also reactive power associated with it.

However, this conclusions sounds strange to me and I am not sure of it... Do I miss something or is there a mistake is the reasoning?

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