Problem 1 [15 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.
Problem 2 [15 points]
Using our results from lecture for the reflection coefficients:
R_{⊥} =  
μ_{b}k_{Iz}−μ_{a}k_{Tz} μ_{b}k_{Iz}+μ_{a}k_{Tz}  ^{2}, 

R_{} =  
ε_{b}k_{Iz}−ε_{a}k_{Tz} ε_{b}k_{Iz}+ε_{a}k_{Tz} 
^{2} 
and the transmitted amplitudes
E_{T⊥} = 
2μ_{b}k_{Iz} μ_{b}k_{Iz}+μ_{a}k_{Tz} 
E_{I⊥}, 

H_{T} = 
2ε_{b}k_{Iz} ε_{b}k_{Iz}+ε_{a}k_{Tz} 
H_{I} 
and defining the transmission coefficients by
T = 
<S_{T}>⋅n <S_{I}>⋅n 
where <S> = (1/μ)<E×B> = <E×H> is the time averaged Poynting vector in the corresponding material, and n is the unit normal to the interface, show that
T_{⊥} + R_{⊥} = 1 and T_{} + R_{} = 1.