Consider a bound atomic electron of charge q in a dielectric medium...

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Consider a bound atomic electron of charge q in a dielectric medium with a complex, fre- quency dependent, atomic polarizability a(w). We can write O = a1 + where a1 and a are the real and imaginary parts of O respectively. The dielectric function may be taken to be €(w) = 1 + where N is the density of polarizable atoms. We take = 1 for this medium. Note: for this problem you should not assume any particular functional form for a(w). An oscillating electric field E(t) = Re ] exerts a force on the bound electron cloud. Assume Ew is a real valued vector and s is an arbitrary phase factor. a) [5 pts] The resulting velocity of the electron cloud center of mass can be written as = . Find Vw in terms of Ew and a(w). b) [10 pts] If W is the work done on the electron by the electric field in one period of oscillation T, then find the average work done over one period, i.e. W/T, where T = 2TT/W. This is the energy absorbed by the electron. Express your answer in terms of Ew and a(w). c) [10 pts] Consider a linearly polarized electromagnetic plane wave traveling in the +2 direction. The electric field of this wave can be written as, = = where the complex wavenumber k = K1 + ik2 is related to the complex dielectric function E = E1 + iE2 by k2 = (w2/e2)E(w). Assume the material has a cross-sectional area A in the xy plane. Using your results from (b), find the total energy per unit cross-sectional area, per period of oscillation, that is absorbed by the medium in the half space Z > 0. Express your answer in terms of Ew, E, w and k. d) [10 pts] The magnetic field of this electromagnetic wave can be written as - wt + where k = Ki} + K2 and the phase shift P = tan (k2/k1). Taking the Poynting vector for this wave as S = (c/4m)E X B, show that the total energy per unit cross-sectional area, per period of oscillation, which flows through the plane at Z = 0 is just equal to the energy absorbed by the half-space of the material at Z > 0, as computed in part (c). This is as it must be, since the energy dissipated in the half-space Z > 0 must equal the energy flowing into the half-space through the plane at Z = 0.

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