**Problem 1** [30 points]
Consider as a simplified model for the frequency dependent permittivity function

ε(ω)/ε_{0} = 1 + | ω_{p}^{2} ω_{0}^{2} − ω^{2} |

(This is the model discussed in lecture in the limit that the damping force vanishes.) Assume ω_{0} < ω_{p}.
a) Using the dispersion relation k^{2} = (ω^{2}/c^{2})(ε(ω)/ε_{0}), make a sketch of k vs ω for electromagnetic wave propagation.

b) Re-draw this sketch as ω vs k, and show that for each value of k > 0 there are two allowed values (modes) of ω for the electromagnetic waves. Show that at both small k and at large k one of these two modes has a dispersion relation characteristic of an electromagnetic wave in the vacuum, i.e. ω = v_{p}k where v_{p} is only weakly dependent on k. Show that the other mode has a frequency ω that is, to lowest order, independent of k and so like some internal atomic mode of vibration. Show that these two modes exchange their characteristic behavior as one crosses from small k to large k. In this intermediate region the modes have a mixed character and are referred to as "polaritons".

c) Show that, for a given value of k, the higher frequency mode has a phase velocity that satisfies v_{p} > c, while the lower frequency mode satisfies v_{p} < c. Show that for both modes the group velocity always satisfies v_{g} < c. You may demonstrate this either algebraically or graphically.

**Problem 2** [20 points]
The longitudinal oscillations of the electric field in a conductor at frequency ω = ω_{p} can be viewed physically in terms of the following simple model:

Consider a slab of conductor in which the charges consist of a fixed uniform background density of positive ionic charge, +eN, and an equal, but moveable, uniform density of negative electonic charge, −eN. When the negative charge density sits exactly on top of the positive charge density the system is neutral everywhere and the electric field inside the conductor is **E** = 0. Now displace the center of mass of the negative charge density by an amount δx, as shown in the sketch below.

a) What surface charge density σ builds up on the ends of the conductor?

b) What is the **E** field created inside the conductor by this σ? (hint: it is just like a parallel plate capacitor) Express **E** as a function of δx and the density of electrons N.

c) Assuming that the electrons move together as a rigid body, what is the total force on the electrons due to the electric field **E** of part (b)? Write Newton's equation of motion for the center of mass of the electrons and show that the solution is an oscillation of δx with frequency ω_{p}, where ω_{p} is the plasma frequency discussed in lecture.

This oscillation in δx results in an oscillation in the uniform **E** of part (b), as well an an oscillation in the surface charge σ. This is called the plasma oscillation.