**Problem 1** [50 points]
In 1845, Faraday made the experimental discovery that a linearly polarized light wave (i.e. EM wave) could have its direction of polarization rotated by passing it through a static magnetic field in a dielectric. This was an important step in demonstrating that light was in some way an electromagnetic phenomenon. The purpose of this problem is to derive this result theoretically. For this problem, assume that µ = μ_{0}.

a) [20 points] Start by considering how the polarization of atoms in the dielectric is influenced by the presence of a uniform static magnetic field. Consider an electron in a uniform magnetic field
**B** oriented in the z-direction, and an oscillating electric field due to an EM wave,

**E**(t) = **E**_{ω}e^{-iωt},

where **E**_{ω} is in the xy plane.
Assume that there is a restoring force on the electron, **F**_{rest} = -mω _{o}^{2}**r**, but assume the damping force is negligible. The solution for the electron's position will then have the form,

**r**(t) = **r**_{ω}e^{-iωt}.

Write down the equations of motion for the x and y components of the electrons position, substitute in the above form of the solution, and solve for **r**_{ω} in terms of **E**_{ω}. Show that **r**_{ω} is not in general parallel to **E**_{ω} (i.e. **r**_{ω} does not point in the same direction as **E**_{ω}). Show, however, that **r**_{ω} will be parallel to **E**_{ω} when **E**_{ω} is circularly polarized, i.e. when **E**_{ω} = E_{ω}(**e**_{x}±i**e**_{y}), where (+) and (-) refer to right and left handed polarizations respectively, and **e**_{x,y} are unit vectors in the x and y directions. For these two circular polarizations, show that the atomic polarizations can be written as,

**p**_{ω} = -e **r**_{ω} = α_{±}(ω)**E**_{ω}, where α_{+} ≠ α_{-}.

b) [10 points] Using the above result, show that the two different (±) circularly polarized waves travel through the medium according to *different* dispersion relations,

k_{±}^{2} = ω^{2}µ_{0}ε_{±}(ω),

where the permittivities, ε_{±}(ω) = ε_{0}[1+Nα_{±}(ω)/ε_{0}], are related to the atomic polarizabilities in the usual way. N is the density of polarizable atoms.

c) [20 points] Consider now a slab of the dielectric of thickness L (the surfaces of the slab are perpendicular to the z axis). Suppose a plane wave, linearly polarized in the x direction, **E**_{ω}=E**e**_{x}, enters the slab at z=0. Show that when the wave exits the slab at z=L, the direction of polarization has been rotated. What is the angle of rotation? Has it rotated clockwise or counterclockwise? You may ignore reflections at the interfaces. You may assume that the magnitude of **B** is small.

To do this part, recall that any linearly polarized wave **E**_{ω}=E_{x}**e**_{x}+E_{y}**e**_{y} can always be written as a superposition of counter rotating circularly polarized waves, **E**_{ω}=E_{+}(**e**_{x}+i**e**_{y})+E_{-}(**e**_{x}-i**e**_{y}), where E_{+}=(1/2)(E_{x}-iE_{y}] and E_{-}=(1/2)(E_{x}+iE_{y}). Decompose the entering wave into a superposition of the two circular polarizations, let these circularly polarized waves travel throught the slab each according to their own phase velocity, and then recombine them when they exit to get the polarization of the wave when it exits the slab.