The Ionosphere and Atmosphere - Ultra and Extremely Low Frequency Electromagnetic Fields - page 53

0 1

" ? 00
0" ? 0
00" k

@ A ;

" D (2.18)

with the components

! pe
! 2 ;

c 2
V A

" ? D and " k D (2.19)

where c is the speed of light in vacuum and ! pe is the squared plasma frequency

ne 2
m e " 0

1=2

! p e D : (2.20)

Now we study the plasma wave properties which come from the anisotropical
character of the plasma conductivity. Consider a homogeneous plasma immersed
in a uniform magnetic field B 0 .Letı B be the small perturbation of the mag-
netic field B 0 , so that ıB B 0 . If all perturbed quantities are considered
to vary as exp .i k r i!t/, where k denotes the wave vector, then Maxwell's
equations ( 2.17 ) and ( 1.2 ) are reduced to the following equations

!
c 2 "

k ı B D E ; (2.21)

k E D !ı B ; (2.22)

whence it follows that

! 2
c 2 "

k . k E / D E : (2.23)

As long as ! ! pe the absolute value of the parallel plasma dielectric
permittivity in Eq. ( 2.23 ) is much greater than unity and thus can be assumed to
be infinite in this frequency range. The field-aligned component " k E k in Eq. ( 2.23 )
is finite, however, that means that the field-aligned electric field E k must be zero.
Applying Eq. ( 1.55 ) for the triple cross product to Eq. ( 2.23 ) and rearranging
yields

k 2

! 2
V A

k ? . k ? E ? / E ? D 0: (2.24)

where k ? and E ? are the components perpendicular to the unperturbed magnetic

field B 0 .