4. Solution in Quasi-static Approximation for the
Dipole and Slot Antennas
[18] For small electric dimension of the spheroid we take into account the
following representation of radial spheroidal functions
[Morse and Feshbah, 1953]
 | (11) |
Here the variables
h and
x take the values of the first and second
arguments of spheroidal functions in (6). As a result of solution of system
(6) we obtain for the transmission coefficients into the outer medium
expressions containing explicit dependency on the problem parameters.
[19] For the prolate plasma spheroid surrounding a source with fixed current
at the antenna clamps, we have the following expression for the
propagation coefficient
DI of the leading harmonic
 | (12) |
For the prolate spheroidal antenna covered by a plasma sheath with the
fixed voltage at the slot gap, the transmission coefficient
DV of the first
harmonics has the form
 | (13) |
where the fact is taken into account that the depleted ion layer of the
coating and the outer medium are vacuum:
e1 = e3 =1. The
geometrical parameters
a0 and
a characterize the relative thickness
of the depleted ion and plasma coatings, respectively.
[20] We will call the functions
KI (12) and
KV (13), what at the absence
of the plasma coating become equal to unity, functions of influence of the
two-layer plasma coating on current and voltage.
[21] In the case of a oblate plasma spheroidal coating for the same sources
considered above, the solution of the boundary problem constructed in
oblate spheroidal variables using oblate angular and radial spheroidal
functions leads for the 
and 
transmission coefficients to
expressions similar to (12) and (13) where
y(x),
G, and
D0V should
be replaced by
These
substitutions should be performed always transferring from the prolate
shape of the spheroid to the oblate one, so below we will present the
expression only for the prolate shape of the spheroid.
[22] The value of
D0V characterizing the radiation field of a slot antenna
without a coating located in the vacuum reaches the maximum value for
the spherical shape of the slot antenna with the radius
a0 (y=2/3),
because for strongly prolate spheroidal antenna this value decreases as
whereas for strongly oblate one (when 
it appears by a factor of three less than for the spherical one.
[23] It is worth noting that for the dipole antenna located within a plasma
spheroid, expression (12) does not allow for the limiting transition to the
case of a one-layer plasma coating ( a=0 )
[Bichoutskaia and Makarov, 2002, 2005].
At similar transition ( a0=1 ) expression (13)
becomes equal to the expression for the transmission coefficient for the
one-layer plasma coating of the slot antenna.
[24] We substitute transmission coefficients (12) and (13) into the
components of the electric field (1) in the outer medium. For the wave
zone of the source 
we use asymptotics of radial
spheroidal functions
[Morse and Feshbah, 1953]
and assuming
e3=1 we come to the electromagnetic field components
in the vacuum in a spherical coordinate system.
 | (14) |
where
D has the value of (12) or (13) for the problem with an electric
dipole or slot antenna.
[25] According to (14) the tangential component of the field
Er on the
Earth's surface ( cos q =0 )
becomes zero, as it should be for the infinite
conducting Earth's surface. On the finitely conducting underlying surface
at such distances the wave by its structure is close to a plain one, and
the tangent component of the field may be obtained from the vertical
component 
which is determined from formula (14)
multiplied by the attenuation function containing properties of the
propagation path.
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