## Transcribed Text

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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