Problem 1 [20 points]
Consider, as a classical model of an electron, a uniformly charged spherical shell
with charge e and radius R, spinning with angular velocity ω.
a) Compute the total energy contained in the electromagnetic fields.
b) Compute the total angular momentum contained in the electromagnetic fields.
If Π is the electromagnetic momentum density, then r×Π is the angular momentum density.
c) According to Einstein, the rest energy of a particle is related to its rest mass by E=mc2. If one assumes that all the rest mass m is due to the energy
of the electron's electromagnetic field computed in (a), compute the radius R of the electron.
d) Assuming that the total angular momentum computed in (b) is equal to the intrinsic angular momentum of the electron, /2, compute the angular velocity
ω of the electron.
e) Are your results in (c) and (d) physically reasonable for the electron?
Problem 2 [40 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 is a constant.
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) = Eωe-iωt,
where Eω is in the xy plane.
Assume that there is a restoring force on the electron, Frest = -mω o2r, but assume the damping force is negligible. The solution for the electron's position will then have the form,
r(t) = rωe-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 rω in terms of Eω. Show that rω is not in general parallel to Eω. Show that rω will be parallel to Eω when Eω is circularly polarized, i.e. when Eω = Eω(ex±iey), where (+) and (-) refer to right and left handed polarizations respectively, and ex,y are unit vectors in the x and y directions. For these two circular polarizations, show that the atomic polarizations can be written as,
pω = -e rω = α±(ω)Eω, where α+ ≠ α-.
b) [5 points] Using the above result, show that the two different (±) circularly polarized waves travel through the medium according to different dispersion relations,
c2k±2 = ω2µε±(ω),
where the dielectric functions, ε±(ω) = 1+4πNα±(ω), are related to the atomic polarizabilities in the usual way. N is the density of polarizable atoms.
c) [10 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, 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.
Hint: To do this part, recall that any linearly polarized wave can always be written as a superposition of counter rotating circularly polarized waves.