Cuestiones y Ejercicios de Campos y Ondas Módulo 5
July 24, 2022 | Author: Anonymous | Category: N/A
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BWÆA MG GT]WMJH MG KA ATJBNA]WYA DAF_HT Z HNMAT
BYAMH
HYJGN]ADJHNGT _YÁD]JDAT _rh`. @êkjx Hrtjz Tándcgz
FHMWKH P (DWGT]JHNGT Z GEGYDJDJHT) HNMAT GKGD]YHFABNG]JDAT _KANAT ]gfa 11. Hnmas gkgdtrhfabnêtjdas pkanas sgfana : ]gfa 19. Jndjmgndja mg hnmas pkanas shirg pkanhs mg mjsdhntjnujmam mj sdhntjnujmam
sgfana 1? 1?
@ÊKJS HY]JU TÁNDCGU BYAMH GN JNBGNJGYÆA FGDÁNJDA BYAMH GN JNBGNJGYÆA GKÊD]YJDA BYAMH GN JNBGNJGYÆA GKGD]YÔNJDA JNMWT]YJAK Z AW]HFÁ]JDA AW]HFÁ]JDA
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
DWGT]JHNGT 1) _artjgnmh mg kas gduadjhngs mg Faxwgkk mgk rhtadjhnak y mg ka gduadjôn mg ka dhntjnujmam, hitgngr kas gduadjhngs mg Faxwgkk mg ka mjvgrbgndja. Thkudjôn Mg kas Gduadjhngs mg Faxwgkk sg mgmudg5 ?
G
?
I
I t
I
t M
E
I
P
t
]
t
?
M
M
?
9) Tg saig qug gn un fgmjh mjgkêdtrjdh ( 6 : ? , 6 ? , y 6 ? ): gk vakhr mg ka jntgnsjmam mg dafph fabnêtjdh gs5 C(z,t) 6 - ax ?'?4:3 dhs (1? t - z z ). ). Tg pjmg mgtgrfjnar gk vakhr mgk dafph gkêdtrjdh y mg ka dhnstantg mg hnma. Thkudjôn Khs vakhrgs mg khs dafphs shn5 C (z )
aˀx ? ' ?4:3 dhs(1?: t
C (z )
aˀx ? ' ?4:3g
G (z )
1
e z
C (z )
e
z) <
: ?,
aˀx Ag e z aˀy
A
?
,
?.
(1).
g e z . 9
C (z )
e
1
G (z ) ?
jbuakanmh (1) y (9)5 G(z )
aˀx
A
g
9
e z
(9).
?
?
4
4
? ?
aˀy 7g e 1?z .
9
d
1?. Dhn kh qug tgngfhs5
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
4) Mg un fatgrjak sg saig qug 6 01?2 T/f y r 6 1 ¸Duánth vakg ka khnbjtum mg hnma mg un un dafph dafph gkêdtrjdh gkêdtrjdh mg `rgdugndja 6 1?3 Cz qug sg transfjtg phr mjdch fatgrjak8 ¸Z ka prh`unmjmam mg pgngtradjôn8 Thkudjôn _ara un iugn dhnmudth dhnmudthr5 r5 1<
u p
9
9
`
1
? ' = 1? = f, y
`
0 ' 7 1? 0 f
`
.
9
=) Hitgnba ka gduadjôn mg hnmas para gk phtgndjak P. Thkudjôn Dhfh5 G
P
A
6 6> 6 6> G(z,t)6 a x 1?=dhs ((== 1? 1?3t-= (z-1/=)). L6
_ara mgtgrfjnar khs jnstantgs gn qug akdanza un fáxjfh gn ka phsjdjôn z 6 4/=, jbuakafhs a dgrh para gsg vakhr y mgspgeafhs t 5 t 6 6 19 t (n + + 1)1?-3 s para n 6 ?,1,9,...
4
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
0) Qug hnma sg atgnõa fás rápjmh gn un fatgrjak dhn /( ) >>> 1, aqugkka dhn khnbjtum mg hnma dhrta h aqugkka dhn khnbjtum mg hnma karba8 k arba8 Eustj`jdar ka rgspugsta. Thkudjôn
Gn un iugn dhnmudthr,
>>> 1 , y ka prh`unmjmam mg pgngtradjôn gs5 6
9
._hr
tanth sg atgnõan fás kas hnmas mg fgnhr khnbjtum mg hnma (fayhr `rgdugndja).
2) Mgtgrfjnar ka vgkhdjmam mg `asg mg una hnma pkana mg afpkjtum 1? -7fP/f y `rgdugndja 1 BCz saijgnmh qug sg fugvg gn un fgmjh dhn r 61, 6 1?- e e ?'??1 ?'??1 @/f y 6 ?.?1 T/f. Thkudjôn 1< gxprgsjôn gxprgsjôn gn ka qug ka pgrfjtjvjmam gs ddhfpkgea hfpkgea 6 ‚ + e ‚‚, ‚ ‚, gsth nhs
jnmjda qug gstafhs gn prgsgndja mg un ajskantg dhn pgqugñas pêrmjmas, gn gstg dash ka dhnstantg mg `asg vakg5 vgkhdjmam mg `asg u ` sgrá5 u `
Jf( ) w
1 1
1
jbuak a ?‚49 f/s.
=
9
1 3 1 3
ram/f, y phr kh tanth ka 9
f/s qug gn nugstrh dash gs
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
3) Wna hnma pkana jndjmg dhn un ánbukh mg 4?µ shirg una supgr`jdjg mg sgparadjôn mg mhs fgmjhs 1 y 9 dhn paráfgtrhs dhnstjtutjvhs ( r16 =, r16 ? < r9 6 1, r9 6 ?). ¸Quê sg pugmg a`jrfar mgk ánbukh mg jndjmgndja8 ¸ Z sj gk ánbukh `ugsg mg =7µ, dhfh sgræa ka hnma rg`radtama8 Thkudjôn
_ugsth qug ak apkjdar ka Kgy mg Tngkk5 hitjgng qug
t
9
sgn
t
r 1
sgn
j
r 9
, dhn dhn khs maths sufjnjstramhs, sg
, y gstg rgsuktamh jfpkjda qug nugstrh áánbukh nbukh mg jndjmgndja mg
4?µ, gs gk ánbukh drætjdh, ka hnma rg`radtama rgsiaka phr ka supgr`jdjg. Tj gk ánbukh mg jndjmgndja `ugsg mg =7µ ak sgr fayhr qug gk ánb ánbukh ukh drætjdh nh gxjstjræa rg`raddjôn, rg`raddjôn, thma ka hnma jndjmgntg sg rg`kgearæa.
:) Gk pkanh y6 ?, sgpara un iugn dhnmudthr (y ; ?) ?) mgk ajrg ( y > ?) . Tj gn gk pkanh gxjstg una mgnsjmam mg darba 6 L 1g sgnwx , y una mgnsjmam supgr`jdjak mg darba Es 6 az L 9 g dhs x . Mgtgrfjng G y C gn khs mhs fgmjhs. Thkudjôn Tg apkjdan dhnmjdjhngs gn ka `rhntgra gn khs fgmjhs 1 (ajrg) y 9 (dhnmudthr) y shkh qugman mjstjntas mg dgrh kas dhfphngntgs nhrfak mgk Dafph Gkêdtrjdh gn gk fgmjh 1, G 1n y ka dhfphngntg tanbgndjak mgk Dafph Fabnêtjdh C t . Tj gkjeh ka nhrfak gn ka `rhntgra gn ka mjrgddjôn S , qugma5 G1n dhfphngntgs mgk dafph vakgn dgrh.
7
l sgn x g < C
ˀy lg dhs x , a
kas mgfás
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
1?) Mgmuzda kas gduadjhngs nh chfhbêngas mg Cgkfchktz para una sjtuadjôn gkgdtrhfabnêtjda dhn mgpgnmgndja arfônjda gn gk tjgfph. ¸Quê sudgmg sj ka khnbjtum mg hnma gs gs fuy branmg `rgntg a ka mjstandja Y mhnmg sg mgtgrfjna gk dafph8 Thkudjôn _ábjnas 90:-901 mgk tgxth mg M. L. Dcgnb, gspgdjakfgntg gk `jnak mg ka pábjna 901.
11) Wn dafph G xˀ7?g z P/f vjaea phr djgrth fatgrjak mg dhnstantgs5 r 6 :, r 6 1 y 6 ?‚= T/f. Tj ka `rgdugndja gs 9‚7= Bcz mgtgrfjng ka atgnuadjôn gn mgdjigkgs phr fgtrh. Thkudjôn
Tg trata mg un mjgkêdtrjdh mjsjpatjvh, y gn gkkhs
e
1
.
e
Hpgranmh dhn khs maths qugma 6 6 19=‚71 19=‚71 + e + e 9?9‚=9. 9?9‚=9. Ka atgnuadjôn gn mgdjigkjhs/f sg hitjgng mgk dhdjgntg gntrg kas jntgnsjmamgs5 9
1? khb
x z ?
9? khb g 19='71 .
9 x
z 1
19) Mgfugstrg qug una hnma pkana unj`hrfg nh tjgng dhfphngntgs mgk dafph gn su mjrgddjôn mg prhpabadjôn. Thkudjôn Gn `hrfa `ashrjak ka gduadjôn mg hnmas chfhbênga, 9G¨ l 9G ? , sg dhnvjgrtg gn trgs gduadjhngs, una para dama dhfphngntg mg G , phr gegfpkh, para ka dhfphngntg 9
x 5
x
9
9
y
9
9 9
z
l 9 G x
?.
Wna hnma pkana dhnsta mg un dafph gkgdtrhfabnêtjdh dhn `áshrgs G y C unj`hrfgs a kh karbh mg khs pkanhs pgrpgnmjdukargs a ka mjrgddjôn mg prhpabadjôn (supugsta z ) G y C nh
varæan dhn x g y . Gsth sjbnj`jda qug kas mgrjvamas
0
Gx
G x
x
y
? ,y ka
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas 9
gduadjôn mg hnmas sg rgmudg a5 Gy , C x y C y ).
? (gduadjhngs sjfjkargs para
l 9 G x
z 9
_ara vgr qug kas dhfphngntgs rgstantgs ( Gz y C z ) shn nukas usafhs
kas gduadjhngs mg Faxwgkk G
e
Cy
e
C
G,
mg ka prjfgra tgngfhs, phr gegfpkh, qug (para ka
d
dhfphngntg z ))55 ˀz
G y G x x
gduadjôn mg Faxwgkk, para
C
C z kh
ˀz e
y
qug mgfugstra qug C z
, hitgnmrgfhs qug G z
? . Wsanmh ka
? , dqm.
14) Mgfugstrg qug gn ka dhnmjdjôn mg nh rg`kgxjôn gn una supgr`jdjg supgr`jdjg mg sgpar sgparadjôn, adjôn, ka sufa mgk ánbukh mg Irgwstgr y gk ánbukh mg rg`raddjôn vakg :?µ, para phkarjzadjôn parakgka ( 1 9 , 1 6 9 ). Thkudjôn Tg saig qug5 dhs
1 t
dhs
i
dhn
, y qug
1
9
9
dhs t 9 dhs i , kkafanmh a 1 dhs 9 t 1 dhs 9 i 1
9
amgfás sg saig qug5 sgn
1
9
vgr qug
t
9
i
1
1
9
i
1
9
t
1
1
phr kh tanth
a
mgtgrfjnh gk vakhr mg dhs(
i
)
a
dhs9
i
1
a
. _ara
9
Wtjkjzanmh gk vakhr mg dhs i sg tjgng5 dhs dufpkg qug dhs(
a
1 t
?, dqm.
2
1
t
i
)
y sgn a
dhs
t
dhs
i
a t
1
a
sgn
t
sgn i .
dhn kh duak sg
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
1=) Mgfugstrg qug mama una hnma mgk tjph Gx cakkar
G x z
az
Gx g ?
z ) ,
dhs(wt
gs kh fjsfh
qug fuktjpkjdar phr ka dhnstantg mg prhpabadjôn dafijama dafijama mg sjbnh.
Thkudjôn
G x z Gx g
Gx g ?
z
z
dhs( t
dhs( t
?
z) z)
z
Gx g ?
dhs( t
sgn( t
z)
9 )
z
_asanmh a nhtadjôn @ashrjak5 ˀ G x
Gˀx g
z
z
g
e z
g
e z e
?
g
9
Gˀx g
(
e )z
?
(
e )
Gˀx g
z
?
(
17) Gsdrjia ka gxprgsjôn jnstantánga P (t ) para gk `ashr Pˀ = rg`grgndjas sgnh y dhsgnh. ¸Duák mg khs mhs vhktaegs gstá amgkantamh8 Thkudjôn
P
=
e4
10
e tan
:g
14 =
a) Yg`grgndja dhsgnh5 P (t ) i) Yg`grgndja sgnh5 P (t )
7g e 40,:µ =
1
)
Gˀx .
e 4 ,
gn kas mhs
?
Yg 7g e 40,:?g e
t
e
Yg 7g e 40,:?g e t g
3
7 dhs 9
7 dhs
40, :µ
t t
<
190, :µ
, amgkantamh /9 /9 .
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
10) Mgtgrfjng ka `rgdugndja a ka duak ka prh`unmjmam mg pgngtradjôn mg una hnma gn gk far gs mg 17 f. ¸Duák gs ka dhrrgsphnmjgntg khnbjtum mg hnma8 ¸Duánth vakg ka khnbjtum mg hnma gn gk ajrg8 6 6 = T/f. Thkudjôn
Gk abua mg far gs iugn dhnmudthr. 1
1
`
931,, ==< 931
` far
9 9
ajrg
l
9
`
9
d ? ?
:=, 9= f.<
`
1?07:=2f.
12) Hitgnba gk vakhr mg ka mgnsjmam mg phtgndja fgmja ramjama _ fgm6 19 Yg(G C*), para una hnma pkana unj`hrfg, mg `rgdugndja qug sg prhpaba gn ka mjrgddjôn z gn un fgmjh dhn pêrmjmas. Thkudjôn
_ prhf
19 Yg G C* , dhfh sg tjgng qug khs dafphs shn (`ashrjakgs)5
G( z )
a x G? g ( e ) z y
_ prhf
G?9 z e g g a y Yg 9 d
C( z)
d
ax
G ?
d
g
( e e d ) z
G ?9 z g dhs a y 9 d
d
:
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
13) ¸Iaeh quê dhnmjdjôn khs dhg`jdjgntgs mg transfjsjôn y rg`kgxjôn shn jbuakgs para mhs hnmas, una phkarjzama parakgkafgntg y ka htra phkarjzama pgrpgnmjdukarfgntg8 Thkudjôn
Gn mhs dashs5 a) Tj ka jndjmgndja gs nhrfak 9
1
<
, 1
ύ
9 ,
9
1
9
i) Tj gk sgbunmh fgmjh gs un dhnmudthr ,
1<
ύ
,
?
1:) Tj ka prh`unmjmam mg pgngtradjôn mgk bra`jth a 1?? FCz gs mg ?½10 ff, Mgtgrfjng5 a) Ka dhnmudtjvjmam mgk bra`jth y, i) Mjstandja mg prhpabadjôn gn gk bra`jth mg una hnma mg ?½7 BCz, BCz, antgs mg qug ka jntgnsjmam mgk da dafph fph sg rgmuzd rgmuzdaa 4?mI (gk bra`jth gs fuy iugn dhnmudthr) Thkudjôn 1
1
a) i) m
`
?, 10 1?
14:27 Np/f <
`
4, =7
4
m
:3:= T/f.
f
dhfh
4?mi6
?, 97 ff.
1?
4,=7
Np
m
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
9?) Mg una dafph gkgdtrhfabnêtjdh qug sg prhpaba phr una buæa mg hnmas rgdtanbukar mg mjfgnsjhngs a 6 6 9.4 df y i 6 1 df, sg saig qug `hrfa un paqugtg dgntramh gn ka `rgdugndja phrtamhra mg 2 BCz. Ka dhnstantg mg `asg mg mjdca hnma gs 9 9 ? ?
a
. Mgtgrfjng kas vgkhdjmamgs mg `asg y bruph mg gsta hnma y
dhfpárgkas dhn ka vgkhdjmam mg `asg mg una hnma pkana unj`hrfg mg mjdca `rgdugndja gn gk vadæh.
Thkudjôn
Maths5 1= 1?: ram s-1, ?, ?.
Dhfh ka dhnstantg mg hnma fg ka man gn gk gnundjamh, sustjtuygnmh khs vakhrgs gn gkka hitgnbh5 9 ? ?
u `
a
:3 ' :4 ram f -1 . Dhn
kh
qug
ka
vgkhdjmam
mg
`asg
gs
= ' =7 1? 3 fs -1 , qug dhfh sg pugmg vgr gs supgrjhr a ka vgkhdjmam mg ka kuz
gn gk vadæh d . Ka vgkhdjmam mg bruph gs u b
1
m
, y sg pugmg dhfprhiar mgrjvanmh qug
m
d 9
dhn kh qug sg hitjgng `jnakfgntg qug u b
u `
gs fgnhr qug d .
11
m
u `
m
d 9
,
9 ' ?9 ?9 1?3 fs -1 , qug dhfh mgig sgr
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
91) Wna hnma pkana unj`hrfg mg afpkjtum G ? ? 6 1 P/f y `rgdugndja =?? FCz, Jndjmg mgsmg gk ajrg shirg una supgr`jdjg mg drjstak ( ?, r 6 =, r 6 1) gn jndjmgndja nhrfak. Mgtgrfjng khs dhg`jdjgntgs mg rg`kgxjôn y transfjsjôn, asæ dhfh ka razôn mg hnma gstadjhnarja. Tj jndjmjgra dhn una jndkjnadjôn mg 4?µ ¸Duák sgrá gk ánbukh mg rg`raddjôn8
Thkudjôn
3 1?3 Cz, ?, ?, r9, r9.
Maths5
hitgngfhs
gn
afihs
fgmjhs
khs
Dhfh gn fgmjhs sjn pêrmjmas sjbujgntgs
vakhrgs5
1
3 ram f -1 y 4
10 -1 4 ram f . Mg ka fjsfa fjsfa `hrfa sg saig qug ka jfpgmandja jfpgmandja jntrænsgd jntrænsgdaa mg un
1
fgmjh sjn pgrmjmas gs5
, dhn kh duak gn nugstrhs mhs fgmjhs tgngfhs kas
sjbujgntgs jfpgmandjas,
1
y
19?
. Dhn gsths maths sg pugmgn
0?
9
mgtgrfjnar khs dhg`jdjgntgs mg rg`kgxjôn y transfjsjôn 5 9
1
9
1
1 < 4
ύ
9 4
1
.
Ka razôn mg hnma gstadjhnaræa T gs gs gnthndgs5 1
T
9.
1
_hr htrh kamh mg ka Kgy mg Tngkk sg saig qug5 sgn
t
sgn
1 j
9
sgn
?
sgn
1 =
j
j
? r 9
r 9
19
t
1=µ 93½ 4: ' ?=½½ .
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
99) ¸Mg quê dkasg gs ka jfpgmandja mg gntrama mg una kænga mg transfjsjôn mg //33 mg khnbjtum sj gstá gstá tgrfjnama 1µ) gn ddhrthdjrdujth hrthdjrdujth y 9µ) gn djrdujth aijgrth8 Thkudjôn
9
Ka dhnstantg mg `asg gs
ram f-1 , y ka khnbjtum gs k
3
. Dhn gsths maths sg
pugmg dakdukar ka jfpgmandja mg gntrama para una kænga mg transfjsjôn gn dhrthdjrdujth z d h gn djrdujth aijgrth z a , pugs sg saig qug5 9 3 9 eY? tan 3
eY? dht k
zd za
eY? dht
eeY Y? tan k
?
jfpgmandja dapadjtatjva.
?
jfpgmandja jnmudtjva.
94) _hkarjzadjôn mg hnmas pkanas Thkudjôn
_ábjnas 934-937 mgk tgxth mg Mavjm L. Dcgnb.
9=) Mgtgrfjng ka gxprgsjôn tgfphrak tgfphrak mgk dafph dafph gkêdtrjdh
ea y g elz ,
G ax
kas gxprgsjhngs `ashrjakgs y tgfphrakgs mgk dafph fabnêtjdh ashdjamh. Thkudjôn a ) Gˀ(z )
(a x e
Yg a xg G (z, t )
eay )g elz t lz
a x dhs
e
Yg a y g t
Y Ygg Gˀ(z )g e
G (z, t )
lz
t lz
t
9
a y sgn
t
lz
14
e
Yg a xg
t lz
e
t lz
ea yg
asæ dhfh
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
i) Dhfh ka mjrgddjôn mg G
Cˀ (z )
1
az
1
Gˀ(z )
?
C (z, t )
1
gs z, sg tjgng qug5
ay
ea x g elz
C (z, t )
Yg Cˀ(z )g e t
?
a y dhs
t
a x sgn
lz
t
lz
?
97) Tg qujgrg ajskar una caijtadjôn mg jntgr`grgndjas ramjhgkêdtrjdas dhn `rgdugndjas gntrg 1?? lCz y 1BCz dhn djgrth fatgrjak mg dhnstantgs dhnstantgs ( ?, ?, 7,3 1? 2 T). _ara fayhr ajskafjgnth sg qujgrg qug gk brhshr mg kas káfjnas mgk fatgrjak sga mg 1? vgdgs ka prh`unmjmam mg pjgk mgk fatgrjak ajskantg. ¸Quê brhshr mgig tgngr ka káfjna8 (dhfprugig prgvjafgntg qug gk fatgrjak utjkjzamh dhfh ajskantg gs iugn
dhnmudthr) Thkudjôn _ara ka `rgdugndja fás iaea ` 1?7 lCz
1?14
(gs dhirg). Ka prh`unmjmam mg pjgk ,, gs gnthndgs5
1 , gs fuy iugn dhnmudthr 1
?, 91 ff , kugbh gk
`
gspgshr qug sg hs pjmg gs jbuak a 9,1 ff, qug phr supugsth vakg tafijên para ka `rgdugndja fás akta.
1=
WNJPGYTJMAM NADJHNAK MG GMWDADJHN A AMJT]ANDJA Gsdugka ]êdnjda Tupgrjhr mg Jnbgnjgrhs Jnmustrjakgs Gqujph Mhdgntg mg Dafphs y Hnmas
90) Jnmjqug ka phkarjzadjôn y mjrgddjôn mg prhpabadjôn para kas hnmas pkanas unj`hrfgs sjbujgntgs qug sg prhpaban phr gk vadæh5 4
G ax
G
tan
a x g
e
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