Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field
Electromagnetic wave transformation in plasma with a slowly changing external magnetic field has been considered in reference [1] and in suddenly formed plasma in the steady-state external magnetic f ield in [2,3]. In this work the electromagnetic transformation with changing in time both the extern...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2000 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/82379 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field / A.G. Nerukh, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2000. — № 3. — С. 105-108. — Бібліогр.: 7 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859665516237422592 |
|---|---|
| author | Nerukh, A.G. Khizhnyak, N.A. |
| author_facet | Nerukh, A.G. Khizhnyak, N.A. |
| citation_txt | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field / A.G. Nerukh, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2000. — № 3. — С. 105-108. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Electromagnetic wave transformation in plasma with a slowly changing external magnetic field has been considered in reference [1] and in suddenly formed plasma in the steady-state external magnetic f ield in [2,3]. In this work the electromagnetic transformation with changing in time both the external magnetic field and plasma density have been considered. Temporary changes of the magnetic field and plasma density are approximated by a succession of step functions. The field transformation for each temporal step is exactly defined by Volterra’s integral equation solution describing the magnetic field in magnetized plasma.
|
| first_indexed | 2025-11-30T10:58:33Z |
| format | Article |
| fulltext |
Problems of Atomic Science and Technology. 2000. N 3. Series: Plasma Physics (5). p. 105-108 105
UDC 538.566.2
ELECTROMAGNETIC SIGNAL TRANSFORMATION IN
NONSTATIONARY PLASMA AT TEMPORARY JUMP
OF EXTERNAL MAGNETIC FIELD
Nerukh A.G.
Kharkov State Technical University of Radio Electronics
Kharkov, 61059, Lenin Prospect, 14
Khizhnyak N.A.
National Scientific Center KPTI,
Kharkov 61108, Akademicheskaja, 1.
Electromagnetic wave transformation in plasma with a slowly changing external magnetic field has been
considered in reference [1] and in suddenly formed plasma in the steady-state external magnetic f ield in [2,3]. In this
work the electromagnetic transformation with changing in time both the external magnetic field and plasma density
have been considered. Temporary changes of the magnetic field and plasma density are approximated by a
succession of step functions. The field transformation for each temporal step is exactly defined by Volterra’s
integral equation solution describing the magnetic field in magnetized plasma.
INTRODUCTION
The electromagnetic field in uniform boundless
plasma being in the homogenous external unlimited
magnetic field satisfies the Volterra equation of the
second kind [4,5]:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫∫
′
∞
′′′′′′′−′′′′′+=
t
t
t
t
nnenn
n n
tdrtEttarrttKrdtdrtFE
rrrrrrrr
,,,,
4
,
2
π
ω
(1)
where ,
1
1
2
2
2 rr
rr
c
tt
tc
K rr
rr
′−
′−−′−
−∇∇=
δ
∂
∂
c is the light velocity, ( )δ t is the delta function, tn
is the moment of the n -th jump of the magnetic field,
ωe is the plasma frequency. The tensor of polarization
( )α n has components as follows [6]:
( ) ( ) ( ) ( ) ( ) ( )n
j
n
in
n
n
kikjn
n
ijn
n
n bbtbetttij
Ω
Ω
−+−Ω
Ω
+Ω
Ω
= sin
1
11cos
1
sin
1 δα (2)
where Ωn is the Larmor frequency,
( )b n is the unit
vector in the direction of the external magnetic field
after the n -th jump, δij is the Kronecher symbol eijk
is the antisymmetric tensor in the third’ dimention.
The free term of Eq. (1)
( )( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )∑ ∫ ∫ ∫ ∫
−
= ∞ ∞
+
′′′′′′+′′′′′′+
+=
1
1
0
1
,,,,,,,,
,,
n
k
t
t
t
t
k
r
k
n
k
k n
rtPrrttKrdtdrtPrrttKrdtd
rtFrtF
rrrrrrrrrr
rrrr
(3)
takes into account as the pre-history of interaction of the
electromagnetic field with plasma before changing of
the external magnetic field
( ) ( ) ( ) ( ) ( ),,
4
,,,,
0 2
0 rtEtttdrrttKrdtdrtF
t
e rrrrrr
′′′′′−′′′×′′′′= ∫ ∫ ∫
∞− ∞
′
∞−π
ω
(4)
as well the effect of a jump of the external magnetic
field on preliminary steps.
Let the external magnetic field after inclusion in
zero moment abruptly changes in time, but remaining
constant at each step. Then the expression of a
polarization vector
( )P n after the n -th jump of the
external magnetic field obtained from equations of
motion for plasma particles is expressed by the ratio
106
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
dt
tPd
tttPtP
tdtEtttPtP
n
n
n
n
n
nn
r
t
t
nnen
r
n
n
1
1
2
,
4
−
− −+=
′′′−+= ∫
r
rr
rrr
α
α
π
ω
(5)
where
( )Pr
n in the remainder of the plasma polarization.
Polarization to the zero time momentum ( )t1 0= is
defined by the expression
( )( ) ( ) ( ) ( ) 0,
4
2
0 =′′′−= ∫
∞−
n
r
t
e PtdtEtttP
rr
π
ω
(6)
The equation (1) solution is realized by the
resolventa method. For its consideration let us studied
one step of the external magnetic field. In this case
( )n = 1 of the field equations (1) resolventa is external
magnetic field direction is along
( )( )
( )
( ) ( ) ( ) ( )rrkittp
i
i
ekpTkddp
i
rrttR ′−+′−
∞
∞
∞−
∫∫=′′
rrrrr
,
2
1
,,, 1
4
!
π
(7)
Then it may be supposed, that
( ) { }z b, , ,1 0 0 1= .
Then in (7)
( ) ( ) ( )
( ) ( ) ( ) ( )T p k
c
G p k
T p T
p
c
T
p
c
p Ie
e e
1
2 2
1
1
1 2
1
2
2 3
1
2
2
2 2,
,
r
r=
−
+ + + +
ω
ϕ ωΩ (8)
where ϕ ωe ep c k I= + +2 2 2 2 , - unit matrix, and
( )Tj
1 matrices are:
( )T p
k k k
p
k k
k k k
p
k k
k k k k
p
k
Á
e
e
e
1
1 2
1
2
1 2
1
2
2 1 3
2 1 2
2 1
2
2 2 3
3 1 3 2
1
2
2 3
2
1
1
1
=
+
+
+
Ω
Ω
Ω
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
(9)
( ) ( )T
k k k k k k
k k k k k
k k k k
T
p
p
e
e
e
e
e
e
e
e
e e
e
2
1
1 2 2
2
2
3
2
2
2 3
2
2
2
3
2
2
2
2
3
2
3
3 2 3 1
3
1
1
1
1
2
0
0 0
0 0
0 0
=
− + −
− −
−
= −
ω
ϕ
ω
ϕ
ω
ϕ
ω
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
,
Ω
Ω
Ω
ϕ = +p c k2 2 2 . The hyromagnetic polynomial in (8)
has the following form
( )( ) ( )( ) ( ) ( )[ ]2
3
22222222222
1
222222221 , kckcppkcpkcpppkpG eeee ωωωω ++++Ω++++=
In case of stepwise changing of the external
magnetic field it is necessary to transfer gradually the
initial temporary moment from step to step. The field
for each step is defined by the same resolventa (7) for
which it is necessary only to substitute the value of the
Larmor frequency Ωn . It is necessary to modify the
free term of (3) in the equation (1) adding to it after
each step the integral for the previous time integral.
Thus, after the n -th jump of the external magnetic
field the formula for the electrical field in
magnetoactive plasma will be of the form
107
( )( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∫
∞
′′′′′′+=
t
nnnn rtFrrttRrdtdrtFrtE
0
,,,,,,
rrrrrrrrr
(10)
TRANSFORMATION OF
ELECTROMAGNETIC OSCILLATIONS
1) Transformation of a plane wave.
Let before initial changing of the external magnetic
field in plasma there is only one of proper waves e.g.
the plane wave is ( ) ( )[ ]r r r r r
E t r E i t s r
0 0
, exp= −ω ,
where ( )s c e= −−1 2 2
1 2
ω ω . From (4), (2) and (10)
we find the expression of the transformed electrical field
for the external magnetic field arbitrarily oriented:
( ) ( ) ( ) ( )( ) ( )( )∫
∞
∞−
−−+=
i
i
rsipt dpepÔspTI
i
rtE
rrrrrr
111 ,
2
1
,
π
(11)
where ( ) ( ) ( ) ( ) ( ) ( )( )[ ]r r r r r r
Ô p
i
p c s
c s i p E p A E c s s A E1
2 2 2
2 2
0
2 1
0
2 1
0
=
−
+
− + +
ω
ω ,
( )
( )A
p p
p p
p p
p
1
2 2
1
2
2
1
1
2
2
1
2
1
0
0
0 0
=
+
−
+
Ω
Ω
Ω
Ω
.
a) If the including magnetic field is oriented along
the wave vector of the initial wave
( ) { } { }r r r
b s s E E1
0 0
0 0 0 0= =, , , , , , then the
transformed field contains three couples of waves with
frequencies p l those are the roots of the polynomial
( ) ( ) ( ) ( )H p p p p c s1 2 2 2
2
1
2 2 2 2
2
= + + +ω Ω , (12)
and vector amplitudes
( ) ( )
( ) ( ) ( ) ( ) ( ){ }1
0
1
1 2
1
2 2 2 2 2
1
2
ω
ω ω ω ω ω
dH p
dp
p i Q p i p c s i p p i
l
l l e l e l l+ − + − +Ω Ω, , (13)
where ( ) ( ) ( )Q p p p c s1 4 2
1
2 2
1
2 2 2= + + +ω Ω Ω .
In each couple of waves transverse waves have the same
wave vectors as the initial ones and propagate in
opposite directions.
â) If the external magnetic field is perpendienlar to
the wave vector i.e., { }r
s s= , ,0 0 ,
{ }r
E E E
0 2 3
0= , , , then the transformed electrical
field is determined by
( ) ( ) ( ) ( )[ ] ( ) ( )[ ]r r r r r rr
E t r E i t sr E ip t is r
m l
m
l
lm
1 1 1
1
2
0
1
1, exp exp= − + − −
⊥
==
∑∑ω (14)
where ( ) { }r
E E1
3
0 0= , , ,
( ) ( )[ ] ( )[ ]( ){ }r
E p i p p i E
mn e
m
l
m
l l e e⊥
= − − + − + − + + −1
1
2 2
1
2
1
2 2
2
1 1 0Ω Ω Ωω ω ω ω ω ω, ,
and new frequencies equal to
( ) ( )p l e
l
t e= + + + − + + +
−
2 1 4
1
2 2 2
1
2 2 2
1
2
2
2
1
2
1
2
ω ω ω ω ωΩ Ω Ω .
In this case the component of the initial wave parallel to
the vector
( )r
b 1 does not change. And the component
perpendicular to
( )r
b 1 ,forms two couples of waves.
These waves have transverse and longitudinal
components and correspond to rapid and slow unusual
waves [5].
It should be noted that in all of the cases the sharp
jump of the external magnetic field transforms a linear
spectrum of the initial electromagnetic field into linear
that.
2) Transformation of plasma oscillations.
Having been substituted ω ω= e in formulas
obtained above we find the expression for
transformation of plasma oscillations for including of
the magnetic field. If
( )r
b 1 is perpendicular to the
electrical field of oscillations the latter ones are
transformed into two elliptically polarized oscillations:
108
( ) ( ) ( ) ( )r
E t
E
t t
e l
e l l
e
l
l
l
1 0
2 2
1
2
2 2
1
2
1
2
1
2
2
0=
− +
− +
=
∑
ω ω
ω ω ω
ω
ω
ω
Ω
Ω Ωcos , sin , (15)
where ( )( )ω ω ωl e
l
e
2 2
1
2 1
1
2
1
2
1
2
2 1 4= + + − +−Ω Ω Ω .
In case of the weak magnetic field Ω
1
<< ωe both
oscillations have small different frequencies,
ω ω ωl e e
2 2
1
≈ ± Ω and nearly circular polarization:
( ) ( )
( )
( ) ( )
r
E t
E
t t
l
e
l
l
l
e
l
l
1 0
1
1
1 1
1
1
2
1 2
1 1 1 0=
− +
× − + −
−
− −
=
∑
ω
ω
ω
ω
Ω
Ω
cos , sin ,
In case of a strong magnetic field Ω
1
>> ωe
oscillation swith the frequency ω
1 1
≈ Ω will be
cyclotron ones with circular polarization and a small
amplitude equal ( )≈ ωe Ω
1
2
. The second oscillation
has the plasma frequency of ω ω
2
≈ e and linear
polarization. When the external magnetic field is
parallel to the electrical one of oscillations those do not
change.
CONCLUSION
The plane electromagnetic wave and also plasma
oscillation transformation the temporary jump of the
arbitrary oriented external magnetic field were
considered. It is shown that if the turning magnetic field
is directed to the primary wave propagation then latter is
transformed into three couples of waves with different
frequencies, with that in each couple waves remain
transverse ones conserve the wave vector and propagate
in opposite directions. In case of the perpendicular
orientation of the magnetic field waves have both
transverse and longitudinal components and
corresponds to rapid and slow unusual waves [7].*
Plasma oscillations in case of inclusion of
magnetized plasma perpendicular to the electric field
are transformed in two elliptically polarized oscillations
with different frequencies. In case of the strong
magnetic field one of three oscillation has almost
circular polarization and frequency near cyclotron one,
the second oscillations has almost linear polarization
and frequency near plasma that. Inclusion of the
magnetic field parallel to the electrical one of
oscillations does not effect on the latter. In all of these
cases transformed field in the result of inclusion of
magnetization has the discrete spectrum.
REFERENCES
1. Í.Ñ. Ñòåïàíîâ. // ÆÝÒÔ, 1967, (53), 6(1), ññ.2186-
2193.
2. J.H. Lee, D.K. Kalluri. // IEEE Trans. on Plasma
Science, 1998, (26), No 1, pp. 1-6.
3. B.V. Stanic, I.N. Draganic. // IEEE Trans. on
Antennas and Propag., 1996, 44, No 10, pp. 1394-
1398.
4. Í.À. Õèæíÿê. Èíòåãðàëüíûå óðàâíåíèÿ
ìàêðîñêîïè÷åñêîé ýëåêòðîäèíàìèêè. Êèåâ:
«Íàóêîâà äóìêà», 1986.
5. À.Ã. Íåðóõ, Í.À. Õèæíÿê. Ñîâðåìåííûå
ïðîáëåìû íåñòàöèîíàðíîé ìàêðîñêîïè÷åñêîé
ýëåêòðîäèíàìèêè, Õàðüêîâ, ÍÏÎ «Òåñò-ðàäèî»,
1991.
6. Â.Ë. Ãèíçáóðã, À.À. Ðóõàäçå. Âîëíû â ìàãíèòî-
àêòèâíîé ïëàçìå. Ìîñêâà: «Íàóêà», 1975.
7. À.È. Àõèåçåð è äð. Ýëåêòðîäèíàìèêà ïëàçìû.
Ìîñêâà: «Íàóêà», 1974.
|
| id | nasplib_isofts_kiev_ua-123456789-82379 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T10:58:33Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Nerukh, A.G. Khizhnyak, N.A. 2015-05-29T07:39:22Z 2015-05-29T07:39:22Z 2000 Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field / A.G. Nerukh, N.A. Khizhnyak // Вопросы атомной науки и техники. — 2000. — № 3. — С. 105-108. — Бібліогр.: 7 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/82379 538.566.2 Electromagnetic wave transformation in plasma with a slowly changing external magnetic field has been considered in reference [1] and in suddenly formed plasma in the steady-state external magnetic f ield in [2,3]. In this work the electromagnetic transformation with changing in time both the external magnetic field and plasma density have been considered. Temporary changes of the magnetic field and plasma density are approximated by a succession of step functions. The field transformation for each temporal step is exactly defined by Volterra’s integral equation solution describing the magnetic field in magnetized plasma. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Рlasma Dynamics and Plasma-Wall Interaction Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field Article published earlier |
| spellingShingle | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field Nerukh, A.G. Khizhnyak, N.A. Рlasma Dynamics and Plasma-Wall Interaction |
| title | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field |
| title_full | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field |
| title_fullStr | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field |
| title_full_unstemmed | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field |
| title_short | Electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field |
| title_sort | electromagnetic signal transformation in nonstationary plasma at temporary jump of external magnetic field |
| topic | Рlasma Dynamics and Plasma-Wall Interaction |
| topic_facet | Рlasma Dynamics and Plasma-Wall Interaction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82379 |
| work_keys_str_mv | AT nerukhag electromagneticsignaltransformationinnonstationaryplasmaattemporaryjumpofexternalmagneticfield AT khizhnyakna electromagneticsignaltransformationinnonstationaryplasmaattemporaryjumpofexternalmagneticfield |