Phase difference between polarization and field?

[Mixer]

Homework Statement

Atomic losses can be described in the Lorentz model by adding into the equation of motion a damping term proportional to velocity. The equation of motion is then$\partial ^2x(t) / \partial t^2 + \gamma \partial x(t) / \partial t + \omega_o^2 x(t) = qE/m$

Consider the optical field $E(t) = E_o e^{-i\omega t}$ Solve the equation of motion by using a trial and calculate the polarization of the material. What is the phase difference between the polarization and the field for very large and very small frequencies and on resonance? Calculate also the real and imaginary parts of the index of refraction by assuming that the material is rare (low density). Show that the imaginary part leads to the attenuation of the field as a function of distance (absorption).

Homework Equations

The Attempt at a Solution

My trial was $x(t) = x_0 e^{-i\omega t}$

I was able to solve everything but calculating phase difference is the part I have not been able to do. How is it done? Expression for polarization I got is:

$p(t) = {E(t)(q^2 / m) N} /( \omega _0 ^2 - \omega ^2 - i \omega \gamma )$

 

[Mixer]

I was able to find out (it was written somewhere) that on resonance the phase difference is 90° But the material did not explain why it is so. What on Earth should I do?

Ok, I know that when $\omega ≈ \omega_0$ the polarization is imaginary.But how does that help me (how do I regognize that phase difference is 90°?) ?