Excitation of the Cylindrical Plasma Columns with the Ring of Magnetic Currents

I n d ia n J. P h y a , 5lA , 343-349 (IDTl)

Excitation of the cylindrical plasma columns with the ring of magnetic currents

SuBHAS Chandra Sharmaand J . S. Vsr m a D e jia H m e n t o f P h y s io s ,

B i r l a hisUt'utv. o f T o c h n o lo y y a m i Scio.ncc, J ^ ila v i- X V M r M

[ R e c e iv e d 18 M a r c h 197(), revised 1) A u g u s t ll)7ti)

A cyliudi'ical geomolry haviiij^ a. c‘(mdiicl,or (radius ring sourco (radius ^{a ),} air coluuui (j'adius ^b), gas(‘(Uis f)lnsma (radius r). solid stiilo somiconductor plasma (radius d ) and gaH(M>ijs plasma eoluniu (radiirs e)

is studied. This mullilavorcMl syslcm gi\a\s riidiatioji peaks lor particular values o1’ b, r. d and e ^liilc^ llic source of (excitation Ixuug the ring of magnetic' (‘uriMuits. Tlu^ am]>litu(lc of tJics(e radiation peaks gets alTecdod with 1.h(e (Change in tliese ])arametcrs. Almost in aU th(i cases l adiation j>atteru lui})pons to ¹⁾⁰ of the same nature in the directions from 7 1 ‘" t(> 00'^’.

1. Introduction

Excitation of k^aky wavi-s on plasma columns was studied by Tamir ^{ei a l} (1902, 1963). They (considered the (^xcutation of kaiky waves on jdasma surfact^s.

Criipta ^{el a l} (1967) discussed tlu^ radiation jiatturn diu^ to ring soiii ee while plasma column was assumed to l)^^ (iylindrical shaped and extending upto infinity in axial direction. Thc^y also int-eiprettod this type (»f pattern diu^ to leaky Avaves.

Welonhancod peaks l>eJoro and ii(?ar critical angle wore obtained by DJiani Bam

ct a l (1972) when they included a central conductor along the i3-axis of tlu^ cylin­

drical coordinates. They assumed the source to be iniinorsed in plasma which unfortunately for exiierirnents posivs a serious problem. To Ix^ safe on tlu^

experimental side Sharma ^{et a l} (1975) discussed t-lui problem assuming the ring source situated outside the plasma column. In cuuitinuancc^ of the above cited work hero we have studied the plasma columns wit-h the holj; of ring of magnetic cut‘rents. Plasma columns are assumed to be homogeneous, losf^less, isotropic and in-compressiblc in nature while the semiconductor plasma column being lossy in nature. The columns are assumed lo be extending uplo infinity along ::-axis and the source is placed in — 0 plane.

2. ANALYSiS

Stating from the M axwelfs equations (Dhani Ram et a l 1972, and Sharma

et a l 1975) one can expand the inhomogeneous eqn. in cylindrical coordina^s

343

1144 Subhash Chandra Sharma, J . S. Venna

for tir* first asymmet rical mode and can come across witli the solution in the form o f bcssei's and hankoJ’s lunctions which can bo given as,

A, --- A i J t ( V a p ) \ -B iY ^ { v „ p ) < f) < a (la)

Aj Ji^ Y j^ V gp ) a < ^f) < ^b (lb)

A3 - A ^ J ^ { V fP )-\ B : i Y i ( V i p ) h < p < ^c (Ic) A4 - A i J i { v . j P -\~BnYy{>up) V < p < d (Id)

h,, -- A , J i ( r - i P ) \ B ; , Y i { v ^ i > ) d < p r (le)

r ^p (11)

whore ^k IS i‘ transform (jT ^{I I} given l)y

h{fj. kz) = J dkz

^ JO

and llu^ inverse* traiisioj’in is givmi by z) ■= i h{p,

—oo

J it . Y n , and aic rosptrctivoly bassel liinc.tjon of first, kind, 2nd kind and handers fnnetioji o f fir.si kind of' y/-th order

wliere^ n\ //„, Cj, and are*. rebspecti\xdy tlie se)urce frce|uency free spacei jununeability, jaainitt ivily, redalive peainit-t ivity of gaseous plasma, somiconduc-

|,e»r ])liisnia anel [Me>pagatie>n ve^cle>r.

Aj)])Iying tliev propeu' boundary eemditions (Dluini Kain et <d 1972) and subst ituting ^{k^} - /r„ SiiW^ the^ magnitude o f the radiatiem field can be made kne)wn with the^ help e>r steepens! descent, method e>f saddle point integi*ation (Slianim cl at 1975). N n w | A o ( A ; z I inagnitude o f the radiat ion uie>eie^ of the structure^ analys(Ml here^ The expression for consists e»f bessefs fiinet iems, liaukors liinetions, r,, etc. and happens to be lengthy.

'JMie e'xpressieui l'e)r reads

I I I I ■^•^*711 l ' ‘ ‘ l " l A „ l

-^^74U‘ 7H •^‘*7g) /, _ -^^72^^33

It-jm — AN, p A³¹A⁷²

A .

Aai ri _ ^aaANa

^7U

-^74 —

•^72 ^

■^70 “ -^06 ==

■ ^aa ~

—

=

Ar.7 ^

^^:>5 - -

*^63-^60

■^70-^02 “F-^G4-^0B

*^20-^^22 28

'^25*^50

E x c i ta tio n o f p la s m a c o lu m n s

345

■^^{01-^22}

*^rj8-^V»i

■^^5-^51 ’^52-^57 y *^54/1rjr)

/*?4B

y^i 5" ^ j 1

y?j7/l^o(V

7^5] — y7^r,yi?2o -^^21-^11

/»?4 y^47 —

-^45 ~

^^^43 y^4i — y^Mi, -

/? I -**'4.'>-'*'4G

‘^‘*42 *¹“ ^{~ r»}

7^48

7^r7^30 -^40-^7

y^a-^aft

^*78 y»?7i y^fia i?rt4 y^Vi² y^«o y?r.B

y^riG y^^oi

yv.2 ^

y*^r.n - y*^

■^^71^^02 ~1“ -y^^GOy^OS yi?27y^22 ■yyasy^as y^23-y^28 y^2ftyy22

— y^59-y^24

;7yri0 y^gjjTy^i;

.^ 5 1 - y^osy^a?

l-'*^14

^40

J^w> yy.j^y?],

^y^oj 7i*n,7ii*j5 ]y*^2o ■““7^187^14

^48 4r»"'^47

yy.

y?.4fl

R ^ R \ \ y^ioy*^j, yy^yyii yy5 ²-yy⁸

^^44 7^42 7^40

yy««

- 7^41^*%

- 7^57^40

—^ 7?3g7?4 7ygfl72g7 - yy^y^g 7^47^7^^

y?37 — i t i ^ R R ^—R 9R 2

1

R . y R ^ y y ; "

7^36 — '^1 ^o('*'’l^)> 7^34 ~ ^"l*7o('*^1^'’)? 7^33 “ ^l^o7yo(^0®) /?32 — y7l('i^o^)i 7^31 “ Y ^(v ^c), 7*^30 7i(?^^e) 7^20 — <^1^2 • 7^28 t’x?^2«7l(^^2^y)- 7^27 ~ ^ 2 ^ 1 ^ o (^ l^ ) 7^20 ™ ^2"^1*7^o('^1^)? 7^25 - - 7^24 “- 7 i(v je Z )

7^23 ^ 7^22 *71(^2^ ) ’ 7?2j = 0(^1^^)

7^20 ^2^ⁱ'7^o(^^v 0 , ^R%1\ -— Cj ^2 ^ o ( *^2^) ’ 7^18 ” ^l'^2*7o(^^2^) y^i? = Y^iv^c)^ ^ 16 ” *71(^2^) ’ 7^16 “ ^ l( ^ l^ )

^ — v^Y^)(vjb), Ry2, ~

E Eq ^ J ii ih P ) ^ -"

— Y i('oJ}), /I'r, — J^(vyb),

340 Subhash Chandra Sharma, J . S. Verma

K ^ ^i(^o«) Ay — J x ( v J ) )

^ v ^ a Y ^ { v ^ ) - Y x ( v ^ a )

7^3 — Vf^aJ^y{o^^a) -'/j(vy;.). A^ -— i„(7;o«j), — */o(^o%)'

1 -<4(, I haw been eoiuimit^d wit-b tli('. help of IBM-llJiO computer for the different

\ralue.s of the various parameters e.g. a.y, a, b, r, d and ^f.

3. Re s u l t s a n d Conoluston

Theoretical cojuputed values ni' 1^1,J h>i‘ the maj.^mtude of the radiation field hav(’ beiai ^iv(⁵ii fo?- numerical values of flu^ various parameters ^iven as

io - ²t7X¹⁰^‘^ rad s(H' -h A:j, — 27t/H c;tu“^. (jraseems plasma electron frequency -- 2x 10^“ rad see’ ’ , iVp,, — 1-57x10^^ rad s(^c~’ (for (j|a-As type semiconductoj) and ^Vy ((Jollision frequeiu*^' of tlu^ scuniconductor plasma column) =: iO X 10^'^

m>c A various features of tlu^ radiation pattern are following ; .*>, I o f r ( id a o f c o n d u c t o r a n d r i n g -source

The elianjj^e in and a does no¹ cause any importani- variation in fhe field pattern. Tlu^ pattern contains the {‘o,\v peaks and almost the same in te is ity of the radiation liekl beyond 72". One plot for l\ / i ^ 4-188 has boon given in^the figure with dot ted line. This plot, gives the gmieral bidiaviour of the field ampli- t.ude with k^/l i.e. a. The same kind of variation was computed in the radiation

field wlum th(^ change in occurs. Iliis is not reported herf^

11.2 R ff c c t o f the radiums o f the a i r c o l u m n

One plot is given for /cJj — (vl7 8 w ith dot. aaid cross in the figure. This shows the general nature o f the field pattern with change in ^b.

'fable 1. Kffect o f change in h on radiation pattern Values o f the various parameters

= 0-209.

3-874,

5-235

k „ f

= 7-330,

9-424 and V - 11-519

Dirocl ion of ra d ia iio n poak

(in dogrooH)

M agn itud e of tho ra d ia tio n fiold (in rel. un its)

/.-(,?»= 5.5r.

- 0.178

0.80

01 0.270743 0.310089 1.306500

02 0.39780f> 0.312701 1.100239

04 0.140932 1.637920 4.330998

71 2.406507 0.068649 0.003632

Excitation o f plasma colutTins

347

The peak occuih at 04 . Also in (1“ to 74" tlK're occui'k no rndiatiou.

Beyond 74 the maf^nitiulo of tlie field almost attains llio eonstant value upto {)0“.

Few more data are also givcui iji l.ahle I for change in h. These data show that peaks appear and diminish with change in h. The maximum intense peak appears in the direction ol 71" for ^{Ic^h —} fi'riO

^0R

‘i.3 E ffe c t o f chamje vn ra d iu s o f gaseous p la s m a co lu m n {u u ie r )

Tho change in k^f affects tlic ladiation pattern appreciably as <ixlhbitcd in tnblc 2. Ill diroclions 01", and 74" to 78' tlu^rc ocenrs the contimions

Table 2. Effect o f change in e on radiation pattern Diroctioii u(

radiation pcaK {in tlegroes)

Magniiiulo o n h o radiation fluid (in rol, uihIh)

V = 8.377 *(,c = !t.424 V lh47J

fil 0.947127 0. i62i:ir» 0.076136

62 5.27164ri 0.1398:i9 0.062010

71 0,187475 0.360986 0.198519

72 o..j:«76:; 1.000000 J.000000

74 to 78 l.OOUOOO Ko radiation No radiation

decrease in the amplitude of Ihe radiation field when increase iu c is caused.

In the region viewed through 74^' to 78^' the decrease in the amplitude occurs

348 Subhash Chandra Sharma, J . S. Verma

so drastically th at for k^c -- 9*424 and k^c — 10*471 the directions (74° to 78°) have no radiation field at all. The stroug(vst peak happens ^U be in the direction of 62" for V 8*377.

3.4 I^ffa ci o f c h a n g e i n r a d i u s o f the s e m ic o n d u c t o r p l a s m a (d )

This is shown ir table 3. No rtigulai* foaturt! can be pointed out by viewing this tables. For some directions, flie inei‘eas(^ in Vr causes the decrease in ampli-

Tal)le 3. Effect o f cliaiigc in ^d on j adiation patterji Dinxition of

radiatioTi pjxik (ia dogrocH)

Maguitudo of tho radiation (in rol, unitH)

hold Jc^d = 0.7:i8 ^{^} 10.053 Jc^,d - 10.995

(il 1 . 20o;n7 0.0952770 0.288057

02 :i.058.S28 0.0179770 0.042601

57 0.107812 0.0530900 18.441030

f)8 0.159423 10.8036100 0.218740

70 0.482713 2.3968930 0.520023

71 1.751459 0.0939700 0.02799.3

tude o f th(^ hold whik^ for some otluu' Olios, increase in that.. Th(? maximum intense radiation pc^ak occurs at 57° for■ V ^ 10*905.

3.5 E f f e c t o f c h a n g e i n r a d i u s o f the o u te r g a s e o u s p la s m a , c o l u m n C

For ^{k^^c}— 11*833 tlu^ plot is given with solid line I h r peak o('cnr at 62°

for this value o f In 61°, 62°. 71° and 72° direcLions the increase in ^e gives tlio (continuous increase in tlu^ magnitude o f the ladiaticui fi(^ld while no change^

ill magnitude is found a i 72°. Another im portant feature o f the pattern is the oteurreuce o f vanishing amplitude of the field in the region viewed through 74°

to 79° directions.

Table 4. Effect of change in ^e on radiation pattern

‘Direction of meUation poak

(in degroeH)

Magnitude o f tho ra,diaiion field (in rol. units)

K e = 11.833 hoe = 12.147 V - 12.461

61 0.562196 0.679040 0.775576

62 1.653001 2.231304 2.887645

71 0.172674 0.175839 0.180503

72 1.000000 1.000000 l.OOOOOO

74 to 79 radiation No radiation No radiation

Excitation of plama columns

349

Having a glance at the abov^e results one (‘an eoiiclude th at tlio theoretically calculated radiation pattern is appreciably ailbclcd A\ ith thc^ change ixi fe, o, d

and c. For particular values of these paraineiiu's ther(' ot*cin‘s certain peaks of radiation which m ay happen to be useful while seiuliug the signal in some jm - ferred directions. The geometry analysed heri^ comes into cxrsteiuu^ AAhile a radiating antenna having a central conductor, scinix‘ouductt>i cohiinn ajul ring source (jxlaced in between them) encounters lh(' ionospheric ]dasnia Mhi( h consti­

tutes the inner and out^r layer of the gas(H)us plasma l(\aving thi* ring in the air The physical explanation can be borrowed Iroxn Tamil rt a l (11Hi2 I9();i) and Dhani Ham et a l (1972) to excite such giuiinctries Aviih

sources

A^okmow^^ledcjmen 'r

The

author (SA^.S.)

sim‘cr«^-

(K

lor aAvarding

oJ'

referet^

tor ])ointing out few

errors in

original

R^efeuisnces

D hani R am & V orm a J . S. 1972 fn.dia.7i J . J^7jre d' AppL Pliya.

10.

G upta K . C. &. iSingh A. 1907 I n t . J . TJlccfrov. 23,

S h arm a Subhash C handra. R am Ohuiirlra k A^onna 9. S. l ‘)7r» h u h a n -f P v r e tl‘ Appl. Phija.

13, 820.

S h arm a S ub hash C handra & A^^rma .1. S. 1975 fnditni J . Had, d' Ppttcr /Vo/.v. 4 282.

T am ir T. & Olinor A. A. 1902 IHE Trails. Antenna Pro'pag.

AP-10,

T. &

A. A.

llO,