Using the complex susceptibility in E dot J

[MMS]

Homework Statement

Use the complex susceptibility

to prove that the dot product of E and J is related to the absorption term (the imaginary part - χ'') and independent of the real part (χ').
It is also stated that in order to do is, assume monochromatic field

and take the absorption time average of <E dot J>.
Remember that when using phasor notation temporal averaging is given by <AB>=0.5*Re{A⋅B*}. You may use the relation
Jpolarization=∂_t (P) and assume no external currents.

Homework Equations

(I)

(II)

The Attempt at a Solution

I tried a couple of things

1. I wrote E(ω)=E(t)*exp(-iωt), plugged it into the polarization equation (II) and then took the time derivative of it to calculate Jpolarization. I took the dot product of it with E(t), time averaged and still got dependence of
χ'.

2. I assumed (even though I'm pretty now it's wrong) P(t)=ε0*χ(ω)*E(t), plugged it the given electric field, took the time derivative and then time averaged it. I get an expression independent of χ' as required, however, I believe it's wrong as the expression P(ω)=ε0*χ(ω)*E(ω) is only correct in the frequency domain. In the time domain it isn't correct. Moreover, the expression P(ω)=ε0*χ(ω)*E(ω) is derived by taking the Fourier transform on both sides of (I) and what I did is like going backwards in an incorrect way.

3. I tried working out something with equation (I) and the given electric field but it just got more complicated.

Help would be much appreciated as this question is starting slowly kill me on the inside and already took me a lot of time.

Thanks in advance!

 

[Charles Link]

Your equation for $P(\omega)$ should prove useful. You also need to write the equation for $J(\omega)$ that you get from taking the F.T. of $J(t)=dP(t)/dt$.