Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 11

become

∂ B
∂t ,

1
c

∇ × E = − ∇ · B =0 , (1.48)

1
c

∂ D
∂t ,

∇ × B = ∇ · D =0 . (1.49)

In order to use these equations, the way in which D and j depend on E and
B must be given.
Theoretical considerations in this topic are carried out mainly within the
framework of the linear electrodynamics which is valid for the small wave
fields, when the conductivity and dielectric permeability are independent on
the wave E and B . The non-linear phenomena appear only in suciently
strong fields, and will be discussed in Chapter 13 on man-made generation of
the ULF-waves.
In the general case, the state of the system depends not only on the field
in a given time instant t and at a given point r , but also on their values at
previous moments and in remote points. Therefore, generally, it is necessary
to use in the material relations which connect D and J with E and B taking
into account the time and spatial dispersion of the medium. Therefore, the
material relations may be written as

dt d r ε ij ( r , r ,t

t

t ) E j ( r ,t ) , D i ( r ,t )= − (1.50)

−∞

t

dt d r σ ij ( r , r ,t t ) E j ( r ,t ) , j i ( r ,t )= − (1.51)

−∞

where i, j are the tensor indices 1 , 2 , 3 corresponding to the axes x = x 1 ,
y = x 2 , z = x 3 .
These are the most general forms of the linear relations between the electric
induction D and the electric current j with the electric field E taking into
account the causality principle (the electric induction at a given time point
t is determined by the electric field at t ≤ t ) and the time homogeneity of

plasma properties.
In numerous important cases the kernels ε ij and σ ij of the integral op-
erators in (1.50), (1.51) vanish fast with distance d in the neighborhood of
the point r . Then integrating over d r can be done only at r | d .Ifthe

electric field E varies slowly at the scale d , it can be factored out from the
integral over d r . Then we obtain from (1.50)-(1.51)

| r −

t

dt ε ij ( r ,t t ) E j ( r ,t ) ,

D i ( r ,t )= − (1.52)

−∞

t

dt σ ij ( r ,t t ) E j ( r ,t ) .

j i ( r ,t )= − (1.53)

−∞

The effects of spatial dispersion are neglected here. This approximation
is applicable for plasmas with rather low temperatures (cold plasma