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Consider a high frequency electromagnetic wave propagating, along the axis, through
a plasma with a longitudinal equilibrium magnetic field,
.
The equation of motion of an individual electron making up the plasma takes the
form

(1166) 
where the first term on the righthand side is due to the wave electric field,
and the second to the equilibrium magnetic field.
(As usual, we can neglect the wave magnetic field, provided that the
electron motion remains nonrelativistic.)
Of course,
, where is the electron
displacement from its equilibrium position. Suppose that all perturbed quantities vary with time
like
, where is the wave frequency.
It follows that



(1167) 



(1168) 
It is helpful to define
Using these new variables, Eqs. (1167) and (1168)
can be rewritten

(1171) 
which can be solved to give

(1172) 
where
is the socalled cyclotron frequency (i.e.,
the characteristic gyration frequency of free electrons in the
equilibrium magnetic fieldsee Sect. 3.7).
In terms of , the electron displacement can be written

(1173) 
where

(1174) 
Likewise, in terms of , the wave electric field takes the form

(1175) 
Obviously, the actual displacement and electric field are the real
parts of the above expressions. It follows from Eq. (1175)
that corresponds to a constant amplitude electric field which rotates anticlockwise
in the  plane (looking down the axis) as the wave propagates in the direction, whereas
corresponds to a constant amplitude electric field which rotates clockwise.
The former type of wave is termed righthand circularly polarized,
whereas the latter is termed lefthand circularly polarized.
Note also that and correspond to circular electron motion
in opposite senses.
With these insights, we conclude that Eq. (1172) indicates that
individual electrons in the plasma have a slightly different response to right and lefthand circularly polarized waves in the presence of a
longitudinal magnetic field.
Following the analysis of Sect. 9.7, we can deduce from Eq. (1172) that the dielectric constant of the plasma for right and
lefthand circularly
polarized waves is

(1176) 
respectively. Hence, according to Eq. (1154), the dispersion relation
for right and lefthand circularly polarized waves becomes

(1177) 
respectively.
In the limit
, we obtain

(1178) 
where
and
. In other words, in a magnetized plasma,
right and lefthand circularly polarized waves of the same frequency have
slightly different wavenumbers
Let us now consider the propagation of a linearly polarized electromagnetic
wave through the plasma. Such a wave can be constructed via a superposition
of right and lefthand circularly polarized waves of equal
amplitudes. So, the wave electric field can be written

(1179) 
It can easily be seen that at the wave electric field is aligned
along the axis. If right and lefthand circularly polarized waves
of the same frequency have the same wavenumber (i.e., if
) then the wave electric field will continue to be aligned
along the axis as the wave propagates in the direction:
i.e., we will have a standard linearly polarized wave. However, we have just
demonstrated
that, in the presence of a longitudinal magnetic field, the wavenumbers and
are slightly different. What effect does this have on the polarization of the
wave?
Taking the real part of Eq. (1179), and making use of Eq. (1178), and some standard trigonometrical identities,
we obtain

(1180) 
The polarization angle of the wave (which is a convenient measure of
its plane of polarization) is given by

(1181) 
Thus, we conclude that in the presence of a longitudinal magnetic field
the polarization angle rotates as as the wave propagates through the plasma. This effect is
known as Faraday rotation. It is clear, from the above expression,
that the rate of advance of the polarization angle with distance
travelled by the wave is given by

(1182) 
Hence, a linearly polarized electromagnetic wave which propagates through a
plasma with a (slowly) varying electron number density, ,
and longitudinal magnetic field, , has its plane of
polarization rotated through a total angle

(1183) 
Note the very strong inverse variation of
with .
Pulsars are rapidly rotating neutron stars which emit regular blips of
highly polarized radio waves. Hundreds of such objects have been found
in our galaxy since the first was discovered in 1967. By measuring
the variation of the angle of polarization, , of radio emission from a pulsar with frequency, , astronomers can effectively determine
the line integral of along the straightline joining the
pulsar to the Earth using formula (1183). Here, is the number
density of free electrons in the interstellar medium, whereas
is the parallel component of the galactic magnetic field. Obviously, in order
to achieve this, astronomers must make the reasonable assumption that the
radiation was emitted by the pulsar with a common angle of polarization, ,
over a wide range of different frequencies. By fitting Eq. (1183)
to the data, and then extrapolating to large , it is then possible to
determine , and, hence, the amount,
,
through which the polarization angle of the radiation has rotated, at a given frequency, during its passage to Earth.
Next: Propagation in a conductor
Up: Electromagnetic radiation
Previous: Dielectric constant of a
Richard Fitzpatrick
20060202