Frequency dependent dielectric constant

The polarization of a dielectricum P(r,t) is given by calculating the mean of the induced dipole moment of the bound charge in a small volume V(r) in the neighbourhood of r, i.e P(r,t) = (1/V(r))∑j pj(t)

In a simple model for a linear dielectric we make the assumption that there are N charged particles (with mass m and charge q) per unit volume, which are bound to their equilibrium positions rj´ through a harmonic potential φ(rj) = a² (rj - rj´)². An electric field E(r,t) acts such that it tends to push the particles out of their equilibrium position

In addition there is there is a damping force acting, Fj = -b²drj/dt
which is proportional to velocity.

a) Give the equation of motion for rj(t).

b) Calculate the polarisation P(r,t) for an electric field with time dependence exp(-iωt). What is the complex dielectric constant ε(ω) in this case

c) Sketch the real and imaginary parts of ε as a function of frequency