# Highly conductive lossless medium

I've came across the term "highly conductive lossless medium" in the context of electromagnetic waves travelling in materials.

I'm wondering how to make sense of that statement? I thought "highly conductive" implies there's energy through ohmic heating. However how can that then be a "lossless medium"?

A good conductor is one for which $\frac{\sigma}{\omega\epsilon}\gg 1$ but a material is (nearly) lossless if the decay constant $$\alpha=\sqrt{\frac{\omega^2\mu\epsilon}{2}} \left[\sqrt{1+\frac{\sigma^2}{\omega^2\epsilon^2}}-1\right]^{1/2}$$ is small. Ultra-low frequency electromagnetic waves propagating in rock have $\sigma\sim 10^{-3}S/m$ and $\epsilon\sim 10\epsilon_0$, yielding $$\frac{\sigma}{\omega\epsilon}\sim 2\times 10^7$$ and thus a good conductor while $\alpha\sim 2\times 10^{-5}$, so rock is nearly lossless at that frequency.