![]() | (7) |
where wr and wp are the resonant and plasma frequencies, respectively, and e0= 1+wp2/wr2 is the permittivity of the medium relative to the static field. It is worth noting that for such medium, a detailed study of the radiation even in the simplest case of a moving point charge was performed only in the recent years. In particular, Afanasiev and Kartavenko [1998] and Afanasiev et al. [1999] analyzed the radiation of a charge in an infinite resonantly dispersive dielectric, and Tyukhtin [2004b] considered the radiation in a waveguide filled in by a dielectric.
[12] Substituting (7) into (2), we obtain
![]() | (8) |
where
![]() |
![]() | (9) |
Condition (3) determining the range of the emitted frequencies is reduced to the following requirements:
![]() |
![]() | (10) |
(in the frequency region wr2 <w2 < wr2 + wp2, there can be no radiation at all, because condition (3) is not fulfilled due to negative value of e ).
[13] One can obtain relatively simple formulae for the boundary
frequencies and radiation energies at some limitations on the problem
parameters. We present below only the estimates for boundary frequencies.
We assume first that the resonant frequency of the medium is much less
than the oscillator frequency in the laboratory frame of reference:
wr w0 =w0'(1-b2)1/2.
Then it is easy to show that there exist two
frequency ranges in the radiation spectrum. The first one is determined by
the inequalities
w*<w<wr, where
w*
wr(1+ b2wp2w0-2)-1/2.
This
frequency range may be called a "resonant" one because it is located in the
vicinity of the resonance frequency
wr. The second emitted frequency
range is determined by the inequality
w1<w<w2, where
w1,2
(w0'
b(w0'2- wp2)1/2)/(1-b2)1/2.
This range may be called a
"proper"
one because at relatively small velocities it includes the oscillator
frequency
w0' (however, it should be borne in mind that at sufficiently high
values of
b, the lower boundary of this range becomes higher than
w0' ).
We emphasize that this frequency range in the radiation spectrum exists
only under condition
w0'>wp. It is worth also noting that the "resonant"
and "proper" frequency ranges at the condition
wr
w,
as a rule, are
located rather far from each other. One can show that the "resonant"
radiation is much weaker than the "proper" radiation (if the latter does
exist). We emphasize that in the case
w0 < wp,
the "proper" radiation
disappears, whereas the "resonant" radiation takes place at any relation
between the plasma frequency and proper frequency of the source.
[14] If
wr w0,
two principally different possibilities may be realized.
The first one takes place at
b(e01/2<1 when the dipole motion velocity
v is
less than the phase velocity of the low-frequency radiation
(c/(e0)1/2 ). In this
case there exist both the "proper" range of the emitted frequencies and the
"resonant" range adjacent to the frequency
wr.
For the "resonant"
radiation under the additional condition
w0(1-b2)1/2
wr(1-b2e0)1/2, we
obtain the frequency range
w*<w<wr,
where
w*
wr ((1-b2e0)/(1-b2))1/2.
For the "proper" radiation under the
additional condition
w0
wr(1-b(e0)1/2),
we obtain the frequency range
w1<w<w2,
where
w1,2
w0/(1
b (e0)1/2).
It is worth noting that in
these conditions both the "resonant" and "proper" radiations may prevail.
[15] In the case when
w0 wr but
b(e0)1/2>1,
the radiation spectrum
contains only one frequency range
w*<w<wr,
where
w*
w0/(1+b (e0)1/2).
One can show that in this situation, the total
radiation power depends very weakly on the oscillator frequency.
![]() |
Figure 3 |
[17] Figure 3c is typical for the case when
wr>w'0 and
wp <w'0.
In
this case three possibilities
may be realized. If the dipole motion velocity is
less than some value
b* ( b* 0.433 for the values of the parameters used
in Figure 3c), there are both the "resonant" and
"proper" (relatively low-frequency) ranges. If
b*< b< b** (in our case
b**
0.82
), there is only
one frequency range including the oscillator frequency and adjacent to the
resonant frequency. If the oscillator velocity is high enough
( b> b** ), then
(besides this range) there is one more range lying above the resonance
frequency. Figure 3d illustrates the case when
wr >w'0.
and
wp >w'0.
In
this case two possibilities can be realized: either there are two frequency
ranges (if the velocity is low enough), or there is only one range.
![]() |
Figure 4 |
[19] Concluding this section, we emphasize that the comparison of the obtained results with the results for the cases of nondispersive medium and cold plasma shows that the presence of the resonant dispersion leads to different, much more complicated, regularities characterizing the radiation of a moving oscillator.
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