# Is $n=\sqrt{\mu_r\varepsilon_r}$ always true? even with complex value?

I have trouble deriving a supposedly "well-known" equation used in condensed matter physics: $$n^2=\mu_r\varepsilon_r+\frac{i\mu_r\sigma}{\varepsilon_r\omega}$$

I'm sure that $$n$$ and $$\sigma$$ are complex, and this equition is obtained by solving $$\mu_0\mu_r\sigma\frac{\partial}{\partial t}E+\mu_0\mu_r\varepsilon_0\varepsilon_r\frac{\partial^2}{\partial t^2}E-\nabla^2E =0$$

which is linked to my other question

However this seem to contradict with the well-known relationship:

$$n=\sqrt{\mu_r\varepsilon_r}$$

• You should provide references to the "well known" formula May 2, 2023 at 20:01
• In such cases you have to go back to Maxwell's equs and reason it through. Among other things you will need to state a definition of $n$. I am guessing it is $\omega/k = c/n$ where $k$ can be complex. May 12, 2023 at 10:32

## 1 Answer

I think you have to introduce a complex value for epsilon. You can understand that by using the Maxwell-Ampère equation. (I might be wrong, I am just a student)

• As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center.
– Community Bot
May 3, 2023 at 12:38