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PHY 415: Electromagnetic Theory I
Prof. S. Teitel stte@pas.rochester.edu  Fall 2010
Problem Set 6
Due Wednesday, November 24, in my mailbox
 Problem 1 [10 points]
Two infinite parallel wires carrying currents I_{1} and I_{2} are separated by a distance d. Compute the flux of electromagnetic momentum ∫daT⋅n passing through an infinite plane half way between the wires; the normal n to the plane is in the direction d. Consider both the cases where the currents are parallel and antiparallel. Interpret your answer.
 Problem 2 [10 points]
Consider a spherical conducting shell of radius R that has a total charge Q. Compute the total force on the northern hemisphere of the shell.
 Problem 3 [15 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=mc^{2}. 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 4 [15 points]
Consider as a very simplified model for the dielectric function
ε(ω) = 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}ε(ω)/c^{2}, make a sketch of k vs ω for electromagnetic wave propagation.
b) Redraw 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, at 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.

