Electric Current Homework: Solving Equations of Motion & Complex Conductivity

[johnaphun]

Homework Statement

The model for the resistivity of metals can be described differently by adding the scattering term into the electron equation of motion.

mx'' + (2m/t)x' = qE

Where x is a mean quantity.

DC conductivity is found by considering the steady state when E is constant. Show that this results in the above equation leading to an equation for mean velocity of qEt/2m and that this equation in turn results in a more familiar equation of conductivity.
Now consider the case where an oscillating electric field is applied, by writing E as E^eiωt . Solve the equation of motion in this case and show that this leads to a complex AC conductivity varying with frequency as

s(w) = s(0)/1+iωt/ 2

The Attempt at a Solution

I'm ok with the first two parts of the question but I'm really stuck on the oscillating field bit, any help would be much appreciated!

 

[ideasrule]

This is a well-known differential equation. The solution is x=a*e^iwt, where a is a complex number (so phase is included). Plug it into the differential equation and you'll see why it's a good ansatz.