Conductivity from dielectric permittivity If I have a typical Drude dielectric permittivity:
$$\epsilon(\omega) = \epsilon_{\infty} - \frac{\omega_p^2 \tau}{\omega^2\tau + i\omega}$$
Now, decomposing $\Re{(\epsilon(\omega))}$ and $\Im{(\epsilon(\omega))}$:
$$\Re{(\epsilon(\omega))}=-\frac{\omega_p^2}{\omega^2+1/\tau^2} + \epsilon_\infty$$
$$\Im{(\epsilon(\omega))} = \frac{\omega^2_p}{\omega^3 \tau+1/\tau}$$
Now, the imaginary part is the important one, because it relates the permittivity to conductivity (which is what I want).
$$\epsilon_{\text{im}}(\omega)= \frac{\sigma(\omega)}{\omega \epsilon_0}$$
So, why does equating the imaginary part and the above equation not yield the correct answer?
$$\sigma(\omega)=\frac{\omega_p^2\epsilon_0}{\omega^2\tau+1/\tau}$$
Should be:
$$\sigma_D(\omega) = \frac{\epsilon_0\omega_p^2 \tau}{1-i\omega\tau}$$
Where am I going wrong?
 A: You’re going wrong by considering the optical conductivity to be real only. Conductivity is complex just as permittivity is! It can be related to epsilon through Ampere’s Law:
$$\nabla\times H = J + \frac{dD}{dt}$$
For time-harmonic fields, we get
$$\nabla\times H = (\sigma - i \omega \epsilon)E.$$
So the complex relative permittivity is related  to the complex conductivity as
$$\epsilon_r=\frac{\sigma}{i \omega \epsilon_0} - 1.$$
This is general (although there might be a sign discrepancy depending on your phasor convention) and, depending on your mood, can be taken as a matter of definition. If you leave out the imaginary part of $\sigma$, then you lose half of the information contained in the Drude model!
A: Not really answering your question. However, if the only thing you are really wanting, is a derivation of the complex conductivity. You should start off with the drude model of conductivity. Namely, the differential equation
$m\frac{d^2\vec{x}}{dt^2} = q\vec{E_{0}}e^{-i\omega t} - \frac{m}{T} \frac{d\vec{x}}{dt}$
This models a single electron, moving at an instantaneous velocity $ \frac{d\vec{x}}{dt}$
Substituting this definition of instantaneous velocity of a single electron into the definition of current density. Acts as though this velocity is a VELOCITY FIELD, aka , at a SINGLE point in space, the velocity of an electron follows the instantaneous velocity of an electron moving under the differential equation above
The complex conductivity is a "Steady state solution" to this differential equation, so technically isn't accurate for low t values.
The way you solve this is via substitution in the form of a complex exponential. To find the transient solution, use the same substitution on the homogenous equation, and add it with the steady state solution
Drude models the E field as a complex number to make solving via substitution easier. THE ACTUAL CONDUCTIVITY is the REAL part
