Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 110

this current corresponds to negative current moment. In our study the CG current
moment is positive since we use the upward directed vertical axis z ,asshown
in Fig. 4.1 .
Let R e and R i D R e C d be the radii of the Earth and the ionosphere, where d is
the height of the non-conductive atmosphere. The upward-directed lumped element
m situated in the atmosphere has the coordinates r D R and D 0 as shown in
Fig. 4.1 . In such a case the Fourier transform of the current density j s produced by
the lumped source can be written as follows:

m.!/
2r 2 sin ı.r R/ı./;

j s .r;;!/ D (4.9)

where ı.x/ is the so-called Dirac's function/delta-function, which equals zero for
all the x, except for x D 0, and by definition

Z

ı.x/dx D 1: (4.10)

1

The arrangement of the formula ( 4.9 )forj s is in accord with the requirement that
the integral of j s over the space gives the total current moment m.!/. Indeed, the
elementary current moment is j s dV, where dV D r 2 sin dd'dr is the volume
element/differential in the spherical coordinates. Multiplying Eq. ( 4.9 )bydV and
integrating the elementary current moment over the whole space yields the value
m.!/, which is required to be proved.
Substituting Eq. ( 4.9 )forj s into Eq. ( 4.8 ) gives a differential equation which
contains the delta-function. In order to derive a boundary condition at point r D R
one should integrate Eq. ( 4.8 ) over r between R Ǜ and R C Ǜ, and then formally
take Ǜ ! 0. Taking into account the continuity of U at r D R , we thus obtain the
following condition at r D R

m.!/ı./
2i" 0 !R 2 sin ;

Œ@ r .rU/ D (4.11)

where the square bracket denotes the jump of the function, that is, Œf .x/ D
f .x C 0/ f .x 0/.InEq.( 4.11 ) the square bracket stands for the jump of the
function @ r .rU/ at r D R.
Boundary conditions ( 4.11 ) may be simplified because the point current element
occurs at the small altitude above the ground, that is, in the case R R e d.
If the Earth is supposed to be a perfect conductor, the tangential component of the
electric field E becomes zero at the terrestrial surface. Whence it follows from
Eq. ( 4.7 ) that @ r .rU/ D 0 at r D R e . Combining this condition with Eq. ( 4.11 )
in the extreme limit case R ! R e we come to the following boundary condition
at r D R e