Page 226 - Electromagnetics

Now, by the Ampere–Maxwell law we can write for currents in free space

˜
˜ i
˜
˜ s
∇× H = J + J + jω 0 E. (4.82)
Considering the plasma to be a material implies that we can describe the gas in terms
c
of a complex permittivity dyadic ˜ ¯ such that the Ampere–Maxwell law is
˜
c
˜
˜ i
∇× H = J + jω ˜ ¯ · E.
˜
˜
Substituting (4.81) into (4.82), and defining the dyadic ¯ω c so that ¯ω c · E = ω c × E,we
identify the dyadic permittivity
2 2 2
ω 0 ω 0 ω
c p p p
˜ ¯ (ω) = 0 − 0 2 2 ¯ I + j 2 2 ¯ ω c + 2 2 2 ω c ω c . (4.83)
ω − ω ω(ω − ω ) ω (ω − ω )
c c c
Note that in rectangular coordinates
 
0 −ω cz ω cy
[ ¯ω c ] =  ω cz 0 −ω cx  . (4.84)
−ω cy ω cx 0
To examine the properties of the dyadic permittivity it is useful to write it in matrix
form. To do this we must choose a coordinate system. We shall assume that B 0 is aligned
along the z-axis such that B 0 = ˆ zB 0 and ω c = ˆ zω c . Then (4.84) becomes
 
0 −ω c 0
[ ¯ω c ] =  ω c 00  (4.85)
0 0 0
and we can write the permittivity dyadic (4.83) as

 
− jδ 0
 jδ
[ ˜ ¯ (ω)] = 0  (4.86)
0 0 z
where
2 2 2
ω ω 0 ω c ω
p p p
= 0 1 − 2 2 , z = 0 1 − 2 , δ = 2 2 .
ω − ω ω ω(ω − ω )
c c
Note that the form of the permittivity dyadic is that for a lossless gyrotropic material
(2.33).
Since the plasma is lossless, equation (4.49) shows that the dyadic permittivity must
be hermitian. Equation (4.86) confirms this. We also note that since the sign of ω c is
determined by the sign of B 0 , the dyadic permittivity obeys the symmetry relation

c
c
˜ (B 0 ) = ˜ (−B 0 ) (4.87)
ij ji
as does the permittivity matrix of any material that has anisotropic properties dependent
on an externally applied magnetic field [141]. We will find later in this section that the
permeability matrix of a magnetized ferrite also obeys such a symmetry condition.

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