Phase difference between current and voltage related to the frequency of the applied field I know that in a plasma the complex electrical conductivity is:
$$ \sigma _n = \frac{Nq^2}{m(\tau^{-1} - i\omega)}$$
My book suggests that voltage and the current in the material have a phase difference dependent on the frequency of the oscillating electric field. However, I do not understand what is the relationship between the frequency and this phase difference. 
My question is:
How can I derive the formula for the phase difference angle between the current and the voltage in the plasma?
 A: The phase difference (or phase angle) in a 'load' is a function of the load's complex impedance and also the frequency of the applied voltage.  So in your case plug in the numbers and write conductivity
$$\sigma _n = \frac{Nq^2}{m(\tau^{-1} - i\omega)}$$
as a complex number and then convert it to an impedance $R +iZ$. (There is a reciprocal relationship between the two.) Then the phasor's angle will be the phase shift between the voltage and the current at frequency $\omega$ .
The imaginary component of conductance arises from either capacitance or inductance in the impedance (or conductance), and the impedance of either of these is a function of frequency. Since v=iZ (re. v=iR), a phase angle in Z causes a phase shift between i and v. 
Google impedance, and see impedance of inductors and capacitors.  For example:
https://www.allaboutcircuits.com/textbook/alternating-current/chpt-5/series-r-l-and-c/
The impedance of an inductor is
$$Z_L = 2\pi fL$$
The impedance of a capacitor is
$$Z_C = 1/(2\pi fC)$$.
These are both imaginary components of $R+iZ$.
To see why there is a phase shift between voltage, $V$, and current, $I$, use Ohm's Law:
$$V=I(R+iZ)$$ .
The angle of the phasor $R+IZ$ is the phase difference between $I$ and $V$, and Z is a function of frequency.
