Calculate wave propagation properties of the medium in the previous question but for a
10GHz wave. i.e. the wave frequency is equal to the resonant frequency of the molecules.
Show that the medium is very strongly absorbing. Can you understand this result physically?
SOLUTION:
In this case:
since 
.
From previous working we have:
The wave propagation factor is:
The phase velocity is then:
The wave field varies like:
The absorption coefficient 
.
So the attenuation length (the distance over which the amplitude falls by a factor 
) is:
and:
not very big!!
Here we are not really describing a propagating wave. It is so massively attenuated.
So if energy is lost from the incident wave where does it go? There is no mechanism here for
absorption in to internal degrees of freedom of the molecules. There must be an equilibrium
set up between the rate of absorption of radiation by the electrons and re-radiation.
The predominant importance of the 
factor in the expression for 
shows the
radiation reaction force plays a dominant role in the motion of the electrons.
The radiation reaction force depends on 
whereas normally, radiation depends
on the acceleration 
. So the re-radiated wave must cancel the incident wave
in the direction of its propagation. The re-radiated energy must go off in other directions.