Page 227 - Electromagnetics

We can let ω → ω − jν in (4.81) to obtain the secondary current in a plasma with
collisions:
2
0 ω (ω − jν)
˜ s p ˜
J (r,ω) = jω − E(r,ω)+
2
2
ω[(ω − jν) − ω ]
c
2
0 ω (ω − jν)
p
˜
+ j ω c × E(r,ω) +
2
2
ω(ω − jν)[(ω − jν) − ω )]
c
2
0 ω (ω − jν)
ω c p
˜
+ ω c · E(r,ω) .
2
2
2
(ω − jν) ω[(ω − jν) − ω ]
c
From this we find the dyadic permittivity
2 2
0 ω (ω − jν) 0 ω
c p ¯ p
˜ ¯ (ω) = 0 − I + j ¯ ω c +
2
2
2
2
ω[(ω − jν) − ω ] ω[(ω − jν) − ω )]
c
c
1 0 ω 2 p
+ ω c ω c .
2
2
(ω − jν) ω[(ω − jν) − ω ]
c
Assuming that B 0 is aligned with the z-axis we can use (4.85) to find the components of
the dyadic permittivity matrix:
2
ω (ω − jν)
c c p
˜ (ω) = ˜ (ω) = 0 1 − , (4.88)
xx yy 2 2
ω[(ω − jν) − ω ]
c
2
ω ω c
c c p
˜ (ω) =−˜ (ω) =− j 0 , (4.89)
xy yx 2 2
ω[(ω − jν) − ω )]
c
2
ω
c p
˜ (ω) = 0 1 − , (4.90)
zz
ω(ω − jν)
and
c
c
˜ c zx = ˜ c xz = ˜ zy = ˜ yz = 0. (4.91)
c
We see that [˜ ] is not hermitian when ν = 0. We expect this since the plasma is lossy
c
when collisions occur. However, we can decompose [ ˜ ¯ ] as a sum of two matrices:
[ ˜ ¯σ]
c
[ ˜ ¯ ] = [ ˜ ¯ ] + ,
jω
where [ ˜ ¯ ] and [ ˜ ¯σ] are hermitian [141]. The details are left as an exercise. We also note
that, as in the case of the lossless plasma, the permittivity dyadic obeys the symmetry
c
c
condition ˜ (B 0 ) = ˜ (−B 0 ).
ij ji
4.6.3 Simple models of dielectrics
We define an isotropic dielectric material (also called an insulator) as one that obeys
the macroscopic frequency-domain constitutive relationship
˜
˜
D(r,ω) = ˜ (r,ω)E(r,ω).
Since the polarization vector P was defined in Chapter 2 as P(r, t) = D(r, t) − 0 E(r, t),
an isotropic dielectric can also be described through
˜
˜
˜
P(r,ω) = (˜ (r,ω) − 0 )E(r,ω) = ˜χ e (r,ω) 0 E(r,ω)
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