EM: relative permittivity After having defined the relative permittivity in lectures, I have problem understanding why it is a complex quantity.
Thank you for your help,
 A: The imaginary part of the relative permittivity accounts for absorption in materials. The refractive index can be defined as
\begin{equation}
n = \sqrt{\epsilon / \epsilon_0}.
\end{equation}
Once you have this quantity, you're in a better position to start thinking about what happens when EM waves pass through a material. When loss is neglected $\epsilon / \epsilon_0$ and $n$ are real, and the EM wave gains a phase shift with propagation distance $z$ from the refractive index according to:
\begin{equation}
\exp \left\{ i2\pi z n/\lambda\right\}
\end{equation}
However, if you now include the imaginary part then you find the following:
\begin{equation}
\exp \left\{ i2\pi z (n_{real} + in_{imag})/\lambda\right\} = \exp \left\{ i2\pi z n_{real}/\lambda\right\}\exp\left\{ -2\pi n_{imag}z/\lambda\right\}
\end{equation}
and so the EM wave is attenuated exponentially with propagation distance. The complex permittivity therefore includes information regarding both the phase and amplitude modulation on the EM wave from the material.
