Effective Conductivity of a Cloudy Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 305

an isolated circle inhomogeneity placed into crossed electric E and magnetic
B 0 fields. The total dipole moment of the ensemble of such inhomogeneities
is easily derived. Let us suppose that there are N 1 inhomogeneities of radius
a and conductivity σ 1 , and N 2 inhomogeneities of radius b and conductivity
σ 2 . Assuming that the whole region is occupied by non-intersecting inhomo-
geneities and denoting x 1 = N 1 a 2 , x 2 = N 2 b 2 , we see that x 1 and x 2 must
satisfy

x 1 + x 2 =1 .

The total polarization caused by the inhomogeneities should vanish. After
some algebraic manipulations with expressions in brackets of (10.2), we can
write a system of equations to find σ e P and σ e H :

σ e P 2

+ σ e H − σ Hi 2 σ Pi −

x j =0 , (10.35)

σ e P

+ σ Pi 2 + σ e H −

σ Hi 2 i =1 , 2

σ e H

σ Hi −

x j =0 . (10.36)

σ e P + σ Pi 2

+ σ e H − σ Hi 2

i =1 , 2

Here j =2 , 1 respectively for i =1 , 2.
The 3D case of isotropic media ( σ H 1 = σ H 2 = 0) was studied by Landauer
[20] for spherical inhomogeneities.
Figure 10.3 shows the dependency of the ratio of effective Pedersen con-
ductivity σ e P on the magnetization

parameter β e . The conductivities σ P 1 ( σ H 1 )and σ P 2 ( σ H 2 ) refer to the local
Pedersen (Hall) conductivities of the inhomogeneities of the 1-st- and 2-nd
kind, respectively. One of the curves with x 1 =0 . 5 relates to the case when
the areas of the two components are equal. The curve with x 1 =0 . 1shows
σ e P

to average Pedersen conductivity σ P

in which 10% of the whole area is occupied by a highly
conductive component with σ 1 , whereas the rest of the mixture is σ 2 =0 . 9 σ 1 .
The effective conductivity is 5 times larger than the average conductivity, even
when the medium is weakly perturbed. The ratio σ e P

( β e ) / σ P ( β e )

( β e ) /<σ P ( β e ) > can

be broadly estimated from

σ e P ( β e )

∼

β e ε, (10.37)

σ P ( β e )

where ε is defined in (10.1).
Using the same approach used to derive (10.3) for an isolated inhomogene-
ity, we can construct 'an effective bounded medium' in a strong magnetic field.
By virtue of the fact that in this case the dependency on magnetic field drops
out, for we can use (10.35) putting σ Hi = σ e H σ eff =0 . Equation (10.35)