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t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
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~ -L IJ all t ~~
~
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0 a a
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I
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(r-re)
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t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
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4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
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J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
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bull J~ ~~ ~~
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J J
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J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
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bull f-~lllrliila-_
I I
I I
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v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
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bullu~ E y =0 )114- P( Y Y)
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(~I ~L p)
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~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
o~tAI JJ ~~ ou 4gtl 6~ ~ ~ t) I ~JA 4S)b )~~J ~-J 1
~)1$ Js- ($4~ LajT )~ o~ 4gtl ~J-~ ~T JL=- J J~ IJT jl d I~
J ~ Cb J ~ c~lgt ~I ($Ib ($)~j )1 4S )1) ~~ u-l~1 ~~jb
($Jjl) jl 0) lA I ~ J ))1) IJj ~~ljT ~J- )) I J~ lt)I )lw Jsect bullu u~JI)t~J) ~ ($JJJ ($~ojlJJI~ ~~A ~ya~ ~
(~1~4) ~sectJgt- )4J) 0~ 4SJr ~~)) I t-J~t ~~~LjT
~b~)
L~ I) -J4A ~~ I~bv )~ J) s t) I r~~ ~~ 4 J) -
bullJ)U
bull1bull5 is J~J rJll~ I) ~Jl 4$ ~I JD- )1cI) )~Jl ~ ($Jd -y
~I ~t lAT ~ ~t ))~ ~ 4 4)) ~ ($Jd)IJA_
))b ~=-~ LA)~jl ~JbIlA ~ 4J) ~ ($Ji)IJA-f
I) ~T )) 4)~ 4S~~ ~fiJ I ~Igt~ J~)) ~ ($Jd )1lA- -~
)b ~ J)b
)r4$ ~Yr-J J) ~J4I~ ~ ~ ~
_ QQl Rl _ K Q Q Rl -R (Y-Y)FH -K -e- Rt -- l IRt-R I
~ I~ (~) )JjT($49)) J ~I ~tjT ~ t- li e
u ~fS
s l)b )~ ~)Ij o)Lil )y W ($l IJ ~-~ 4S~ K --l7 -J-rP bulll ~ u I f1T r 1~ mKS o~) ))
)Ilie~ ($)~~ ($Jr J~~Jyt l$))f J~Jt-1 ($If ~- Jlio
I I I
I
~---~------y
z
Ilgt I O(Y tOt) 4lU)) - c-be IU~ ~)~ ~ P( Y f) jUJ)
bullJi5 R =~+Ty+a6
It =Tac+a
X -v - + _ -Y bull 0 A 0 bullbull- middotmiddotya __ +F - _ (f) --abull ay - a tffJI X-X 0
1f ~~~-r
~ II JL- poundSu--~ lJulso tiaf)~PJ1 J~J)~ sJi Jfi ~jl~JJi 4tJ I 4lijl -ampiiJlJ) ~1)~ ~~u I) Q JtlAj~j
-gt)1)~~ Uaij) ~Jt ~~ Q )~ ~1 ~ U )JITQ J
rJP~~ 1J~JIJ JJ ~Jamp~ ~Jl ~lJ) rg J~ 4 ~I
IJj~ 4 ft ))lJ ISJJfl1JlJT ~t(i~tu-trtgtI~1) Q J
r--lr ~SJf ~Il-u J b Ui B ~~ t J ~ ~ Je~
(Y-f)
) ~ ~J~ 4j 0TJ) 45e-I ciJ JiJ-~ 1J-~ 44 ~)) ~~ I ampJ 1-- ~Jt ~
)Jrs- ))1-( C) lgt1- LSI~
ilzo jl RN J bullbullbullJ R s R lli)) --~J ~ qN J bullbullbull J qf J q lts1uuj)~ N 71 uLagt i~jl R ~l~ ~ -l)) lA)~ cr jl (IJSA x~~7 JI) ~~ LSJJd ~1_~J~i s ~If l~j Ev J bull J E - E ~pJl tSlA01~ -rJi~
0
~IT R uu ) LgtI~ 1iJ )1) u~JSJ1 J~ ~ ~ ~IJ ~ ~I cF- ~sect)
~~ EN sJ E s E ltslA~l~ ltSJb~ ~~
Q R-R Q( R-R E(R) =111pound R -Rd r + 111pound middotR-Rl+middotmiddotmiddotmiddot
= t ~ Qi(R-Ri) 1- f1180 fa -B I (Y -f)
------y
P( iT r O )1olAJ~~1( +r x 9_-)CJ (- TX ro-)C ($laJ~_Y Jl1e ~~w P( r r) 41aiiJ) I) E tJIl--~l -11 JI) JI P( fJ ) J
BJi~ flI1 J_ _ bullbull bullbullbull~----- t __ __ ~gt __q-_ __ -~_ ~__ pound bull [ - )( y - + bull] bull )(~ -+ J
bull d bull bull bull ~ bull bull -f shyyenftx-middot 11
middotmiddotAY-4 J) Jf~Fjll~ _14itdta1~ ~ J~Uraquo~~w
gt~ ~1J il t~c$ cSbull tItt J- jlJJlP~~Uat~
oul~ rAj I r)Jgt- jP1-~ ~JllSlA)~ ~ Lajl tSlgtLdeg1 1J~ o~J
d~ ~J~ 4$~~e-j-j I~ ) )~~I-~ ($4Aj)~ ul)~ Cj-jtS~~ u~
r~gt4)~ ~ W~ ~ b~ tSJ) Xly~ ~ ~~ ul~ o~J)~~ ~-~ o~b )~
t)IiS is ~U u~raquo~ J tSl~~)~ uhgt) 4 ~~j Ji- I) lA~T 6$l~ ou e- jy
) ~ J ~ J p ~b ) ~ ~H~- J PI ~ ) L~ uJ~ ~ ~Ii ~ I) lJb~ TLj J
)~ UJ~s~ u)~~ L~)~ c- jy ~~~ ~ L ~LiU ~~ Pfl U~gt
~~ ~~A IJ b- tSJ) ~ ~ 4 tSJJ ~ ~gt 4)~ ~-J- ) ~ J5~Iy u-
( -0)
~~ elm Pi ~T jl J elmO( P 1gtIJ J -)[m Ptl JgtIJ
jL ~~)~I)~ ~J ~J x~ ~-~ lAJ~ ~IJ)~ 4)~I~ ~j ~~
o~r)tgt~ t) ~- ~I9 ~I-j~ ~~~JI ~gt~ ~)y u~Jl)l 1~ jl t)
~~ P=(AI r)e- f Tsin () cosp uJl~~ U~lSj I)~ ~ljT lSLJ)~ Y-r Jlto
o~) ~l-gt I) J4 J5 ~I
f1J fTJoo Ae-fr Q= pdv= r sin(JcospmiddotrsinOmiddotdrdpd(0 Q 0J J
AlT ( ) - -fr 100
w _ Q=-lT-e =-middotA-1T r f A10
Jp 4 6$~ I ~9tgt 6 ~ J~ ~jl JPlgt u~J I ~I~ uu ~~I~ tSlr
-~y 4bAi ~J~ IJ ~Ifi~ ~Il UJ tl~ja)~ IJ ~~~ lJI ~IlA~
J rA~ k_~ ~)b ~y-J J4 4-5($I~l ~w tSlr IJ l~ UI J ~~Lgt- )zi
o~1 ($JI~I Js- ~w cJI
E(R) =--J p(R-R) dv ~)4 (~I Y-r) YlTE o IR -R IfI
E(R) ---I p(R -R) ds -YlTEo jR-Rj
E(R)=--I PI(R-~dl YlTE I JR-R 1o
---7
0)l4 ~T jl Utgt )JJ) j~ J ~ ~ Ji ($LA($$ JI~ ~b_ i-
1 J~ ~I~~~ ~)Jo)L IJ ~6 45 rJJ ($~) Ui-i J~)~ J ~y~u
~AJ Jk u1gt)~ ~~J jPlgt uS~ 1~I~ ~l iJ)I~($I~ i~ o~li
bull~laquos 0) lA 5j ugtJa J ~ ~
)JbJi 00 l -001 z~ 4SJJ P ~ t~ l~ ~IJ~ ~~IJ~~
~Il I ~0ljl jPlgt ~~ iJI~~u ~1gtU-shy
iLaiv ot~) 0~ ILl ))1) iJJLi z )Jgt- JJgt 4C- 4-)~I ~~
)~IIJdE iJI~A P 4A)pdz4 ~)~ 4 w~cs- ($141 O~)
4SJSIf
dE= -- p(R-R) dz fTfo IR -R l
R=ra+zaz R=zaz R-RI=(z-~)aZ+raT
IR-RI =Vrl+(z-zr
E(R) =~__J+oo(raT+(Z-Z)a)dZI flJEo [rl +(Z -ZfPl-co
+00E R P z -z
( )=f1Te [aT-r- rl Vrl+(z-zr -coo
(6-6)
cAJ1
E(R) P a ywe
o r
bull (OOz)
i(z -6) i (1- H
bullI
---------~_c-------------- _
(y-v)
aop Jaz 4AIJ J J4~ a ~~J~4AlJo (S1Jb Jw ~fiJl cJl~ ~if ~~
~A )IJ~ ~~ ~ J ~IJ ~~ r) ~ dl~1 ~lJI)jl sJL 0 ~JIL
u~ P bi u~1~ tSl~ rltA ~ 0 I~ pd ~~ J-I oJ Ju Il J j J Yo j1S1ilmiddotu~ u a ~~ bull~I -ultI lts (gtIl ~
~~l- J a
~J~ctgta 41 ~Igt ufw~ ~~ r-A J-ltolgt gtIlgt i- u~ Pbull ~ ltSI ~~ ltsIJI yz ~ )
e-I 4-ji J (j Q~ ~~L IJ ~ UI jl
~~ it I~ -i ~1~ ~IJI) ys ~j x doLi~ kli ~ _ 1
Y C( 0 l-e - UJ~~~ fSJ j lli ~~I ~~z ~J y ~j ~ liJ
u~ JJ~ z J) 4SlS~IA) Ja~ 4S ~ J~~ z s Y (SlAwJ ($IJI Ll~_f
lts (oJ s-- I 1- sIJ I bullji ~J bi-lt ~ gt Tgt ~ Ir bullrlt A )i
bullJ)tI ~ IJ J~tA la~ T z s yu-r ~ u~ ll~~x ~( b4 (Ez)x ~)J ~jJ ~b- ~I~ Ui
~l-gt 4-1U 0J~ IJ ~~ J ~r U )Z ) IJ p(x 0 0) ~ 44 ir-IJL T1
middot~U~ ~JuJ f~~~~i IJ ~)~ ~-dl ~lgt gt1J-~14 - ~ ~J -0 J rr~JtiJ dY ~JA~ l~ J j listAlJ -J ~ 1J ~ --~~ Jgt ~ jl U~I ~I~ ~u pdy Jf)f ulI ~Jfl ~Jl ~~
~ ~Jft ~ T x 4iJ 4SJ ~ tS pdY f1TE r lfi dY o
=f) -- (IT_+_17) Y7JE f Yo
--
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
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~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
I I I
I
~---~------y
z
Ilgt I O(Y tOt) 4lU)) - c-be IU~ ~)~ ~ P( Y f) jUJ)
bullJi5 R =~+Ty+a6
It =Tac+a
X -v - + _ -Y bull 0 A 0 bullbull- middotmiddotya __ +F - _ (f) --abull ay - a tffJI X-X 0
1f ~~~-r
~ II JL- poundSu--~ lJulso tiaf)~PJ1 J~J)~ sJi Jfi ~jl~JJi 4tJ I 4lijl -ampiiJlJ) ~1)~ ~~u I) Q JtlAj~j
-gt)1)~~ Uaij) ~Jt ~~ Q )~ ~1 ~ U )JITQ J
rJP~~ 1J~JIJ JJ ~Jamp~ ~Jl ~lJ) rg J~ 4 ~I
IJj~ 4 ft ))lJ ISJJfl1JlJT ~t(i~tu-trtgtI~1) Q J
r--lr ~SJf ~Il-u J b Ui B ~~ t J ~ ~ Je~
(Y-f)
) ~ ~J~ 4j 0TJ) 45e-I ciJ JiJ-~ 1J-~ 44 ~)) ~~ I ampJ 1-- ~Jt ~
)Jrs- ))1-( C) lgt1- LSI~
ilzo jl RN J bullbullbullJ R s R lli)) --~J ~ qN J bullbullbull J qf J q lts1uuj)~ N 71 uLagt i~jl R ~l~ ~ -l)) lA)~ cr jl (IJSA x~~7 JI) ~~ LSJJd ~1_~J~i s ~If l~j Ev J bull J E - E ~pJl tSlA01~ -rJi~
0
~IT R uu ) LgtI~ 1iJ )1) u~JSJ1 J~ ~ ~ ~IJ ~ ~I cF- ~sect)
~~ EN sJ E s E ltslA~l~ ltSJb~ ~~
Q R-R Q( R-R E(R) =111pound R -Rd r + 111pound middotR-Rl+middotmiddotmiddotmiddot
= t ~ Qi(R-Ri) 1- f1180 fa -B I (Y -f)
------y
P( iT r O )1olAJ~~1( +r x 9_-)CJ (- TX ro-)C ($laJ~_Y Jl1e ~~w P( r r) 41aiiJ) I) E tJIl--~l -11 JI) JI P( fJ ) J
BJi~ flI1 J_ _ bullbull bullbullbull~----- t __ __ ~gt __q-_ __ -~_ ~__ pound bull [ - )( y - + bull] bull )(~ -+ J
bull d bull bull bull ~ bull bull -f shyyenftx-middot 11
middotmiddotAY-4 J) Jf~Fjll~ _14itdta1~ ~ J~Uraquo~~w
gt~ ~1J il t~c$ cSbull tItt J- jlJJlP~~Uat~
oul~ rAj I r)Jgt- jP1-~ ~JllSlA)~ ~ Lajl tSlgtLdeg1 1J~ o~J
d~ ~J~ 4$~~e-j-j I~ ) )~~I-~ ($4Aj)~ ul)~ Cj-jtS~~ u~
r~gt4)~ ~ W~ ~ b~ tSJ) Xly~ ~ ~~ ul~ o~J)~~ ~-~ o~b )~
t)IiS is ~U u~raquo~ J tSl~~)~ uhgt) 4 ~~j Ji- I) lA~T 6$l~ ou e- jy
) ~ J ~ J p ~b ) ~ ~H~- J PI ~ ) L~ uJ~ ~ ~Ii ~ I) lJb~ TLj J
)~ UJ~s~ u)~~ L~)~ c- jy ~~~ ~ L ~LiU ~~ Pfl U~gt
~~ ~~A IJ b- tSJ) ~ ~ 4 tSJJ ~ ~gt 4)~ ~-J- ) ~ J5~Iy u-
( -0)
~~ elm Pi ~T jl J elmO( P 1gtIJ J -)[m Ptl JgtIJ
jL ~~)~I)~ ~J ~J x~ ~-~ lAJ~ ~IJ)~ 4)~I~ ~j ~~
o~r)tgt~ t) ~- ~I9 ~I-j~ ~~~JI ~gt~ ~)y u~Jl)l 1~ jl t)
~~ P=(AI r)e- f Tsin () cosp uJl~~ U~lSj I)~ ~ljT lSLJ)~ Y-r Jlto
o~) ~l-gt I) J4 J5 ~I
f1J fTJoo Ae-fr Q= pdv= r sin(JcospmiddotrsinOmiddotdrdpd(0 Q 0J J
AlT ( ) - -fr 100
w _ Q=-lT-e =-middotA-1T r f A10
Jp 4 6$~ I ~9tgt 6 ~ J~ ~jl JPlgt u~J I ~I~ uu ~~I~ tSlr
-~y 4bAi ~J~ IJ ~Ifi~ ~Il UJ tl~ja)~ IJ ~~~ lJI ~IlA~
J rA~ k_~ ~)b ~y-J J4 4-5($I~l ~w tSlr IJ l~ UI J ~~Lgt- )zi
o~1 ($JI~I Js- ~w cJI
E(R) =--J p(R-R) dv ~)4 (~I Y-r) YlTE o IR -R IfI
E(R) ---I p(R -R) ds -YlTEo jR-Rj
E(R)=--I PI(R-~dl YlTE I JR-R 1o
---7
0)l4 ~T jl Utgt )JJ) j~ J ~ ~ Ji ($LA($$ JI~ ~b_ i-
1 J~ ~I~~~ ~)Jo)L IJ ~6 45 rJJ ($~) Ui-i J~)~ J ~y~u
~AJ Jk u1gt)~ ~~J jPlgt uS~ 1~I~ ~l iJ)I~($I~ i~ o~li
bull~laquos 0) lA 5j ugtJa J ~ ~
)JbJi 00 l -001 z~ 4SJJ P ~ t~ l~ ~IJ~ ~~IJ~~
~Il I ~0ljl jPlgt ~~ iJI~~u ~1gtU-shy
iLaiv ot~) 0~ ILl ))1) iJJLi z )Jgt- JJgt 4C- 4-)~I ~~
)~IIJdE iJI~A P 4A)pdz4 ~)~ 4 w~cs- ($141 O~)
4SJSIf
dE= -- p(R-R) dz fTfo IR -R l
R=ra+zaz R=zaz R-RI=(z-~)aZ+raT
IR-RI =Vrl+(z-zr
E(R) =~__J+oo(raT+(Z-Z)a)dZI flJEo [rl +(Z -ZfPl-co
+00E R P z -z
( )=f1Te [aT-r- rl Vrl+(z-zr -coo
(6-6)
cAJ1
E(R) P a ywe
o r
bull (OOz)
i(z -6) i (1- H
bullI
---------~_c-------------- _
(y-v)
aop Jaz 4AIJ J J4~ a ~~J~4AlJo (S1Jb Jw ~fiJl cJl~ ~if ~~
~A )IJ~ ~~ ~ J ~IJ ~~ r) ~ dl~1 ~lJI)jl sJL 0 ~JIL
u~ P bi u~1~ tSl~ rltA ~ 0 I~ pd ~~ J-I oJ Ju Il J j J Yo j1S1ilmiddotu~ u a ~~ bull~I -ultI lts (gtIl ~
~~l- J a
~J~ctgta 41 ~Igt ufw~ ~~ r-A J-ltolgt gtIlgt i- u~ Pbull ~ ltSI ~~ ltsIJI yz ~ )
e-I 4-ji J (j Q~ ~~L IJ ~ UI jl
~~ it I~ -i ~1~ ~IJI) ys ~j x doLi~ kli ~ _ 1
Y C( 0 l-e - UJ~~~ fSJ j lli ~~I ~~z ~J y ~j ~ liJ
u~ JJ~ z J) 4SlS~IA) Ja~ 4S ~ J~~ z s Y (SlAwJ ($IJI Ll~_f
lts (oJ s-- I 1- sIJ I bullji ~J bi-lt ~ gt Tgt ~ Ir bullrlt A )i
bullJ)tI ~ IJ J~tA la~ T z s yu-r ~ u~ ll~~x ~( b4 (Ez)x ~)J ~jJ ~b- ~I~ Ui
~l-gt 4-1U 0J~ IJ ~~ J ~r U )Z ) IJ p(x 0 0) ~ 44 ir-IJL T1
middot~U~ ~JuJ f~~~~i IJ ~)~ ~-dl ~lgt gt1J-~14 - ~ ~J -0 J rr~JtiJ dY ~JA~ l~ J j listAlJ -J ~ 1J ~ --~~ Jgt ~ jl U~I ~I~ ~u pdy Jf)f ulI ~Jfl ~Jl ~~
~ ~Jft ~ T x 4iJ 4SJ ~ tS pdY f1TE r lfi dY o
=f) -- (IT_+_17) Y7JE f Yo
--
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
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(r-re)
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t fa
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
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4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
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J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
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bull J~ ~~ ~~
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bullu~ E y =0 )114- P( Y Y)
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csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
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rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
(Y-f)
) ~ ~J~ 4j 0TJ) 45e-I ciJ JiJ-~ 1J-~ 44 ~)) ~~ I ampJ 1-- ~Jt ~
)Jrs- ))1-( C) lgt1- LSI~
ilzo jl RN J bullbullbullJ R s R lli)) --~J ~ qN J bullbullbull J qf J q lts1uuj)~ N 71 uLagt i~jl R ~l~ ~ -l)) lA)~ cr jl (IJSA x~~7 JI) ~~ LSJJd ~1_~J~i s ~If l~j Ev J bull J E - E ~pJl tSlA01~ -rJi~
0
~IT R uu ) LgtI~ 1iJ )1) u~JSJ1 J~ ~ ~ ~IJ ~ ~I cF- ~sect)
~~ EN sJ E s E ltslA~l~ ltSJb~ ~~
Q R-R Q( R-R E(R) =111pound R -Rd r + 111pound middotR-Rl+middotmiddotmiddotmiddot
= t ~ Qi(R-Ri) 1- f1180 fa -B I (Y -f)
------y
P( iT r O )1olAJ~~1( +r x 9_-)CJ (- TX ro-)C ($laJ~_Y Jl1e ~~w P( r r) 41aiiJ) I) E tJIl--~l -11 JI) JI P( fJ ) J
BJi~ flI1 J_ _ bullbull bullbullbull~----- t __ __ ~gt __q-_ __ -~_ ~__ pound bull [ - )( y - + bull] bull )(~ -+ J
bull d bull bull bull ~ bull bull -f shyyenftx-middot 11
middotmiddotAY-4 J) Jf~Fjll~ _14itdta1~ ~ J~Uraquo~~w
gt~ ~1J il t~c$ cSbull tItt J- jlJJlP~~Uat~
oul~ rAj I r)Jgt- jP1-~ ~JllSlA)~ ~ Lajl tSlgtLdeg1 1J~ o~J
d~ ~J~ 4$~~e-j-j I~ ) )~~I-~ ($4Aj)~ ul)~ Cj-jtS~~ u~
r~gt4)~ ~ W~ ~ b~ tSJ) Xly~ ~ ~~ ul~ o~J)~~ ~-~ o~b )~
t)IiS is ~U u~raquo~ J tSl~~)~ uhgt) 4 ~~j Ji- I) lA~T 6$l~ ou e- jy
) ~ J ~ J p ~b ) ~ ~H~- J PI ~ ) L~ uJ~ ~ ~Ii ~ I) lJb~ TLj J
)~ UJ~s~ u)~~ L~)~ c- jy ~~~ ~ L ~LiU ~~ Pfl U~gt
~~ ~~A IJ b- tSJ) ~ ~ 4 tSJJ ~ ~gt 4)~ ~-J- ) ~ J5~Iy u-
( -0)
~~ elm Pi ~T jl J elmO( P 1gtIJ J -)[m Ptl JgtIJ
jL ~~)~I)~ ~J ~J x~ ~-~ lAJ~ ~IJ)~ 4)~I~ ~j ~~
o~r)tgt~ t) ~- ~I9 ~I-j~ ~~~JI ~gt~ ~)y u~Jl)l 1~ jl t)
~~ P=(AI r)e- f Tsin () cosp uJl~~ U~lSj I)~ ~ljT lSLJ)~ Y-r Jlto
o~) ~l-gt I) J4 J5 ~I
f1J fTJoo Ae-fr Q= pdv= r sin(JcospmiddotrsinOmiddotdrdpd(0 Q 0J J
AlT ( ) - -fr 100
w _ Q=-lT-e =-middotA-1T r f A10
Jp 4 6$~ I ~9tgt 6 ~ J~ ~jl JPlgt u~J I ~I~ uu ~~I~ tSlr
-~y 4bAi ~J~ IJ ~Ifi~ ~Il UJ tl~ja)~ IJ ~~~ lJI ~IlA~
J rA~ k_~ ~)b ~y-J J4 4-5($I~l ~w tSlr IJ l~ UI J ~~Lgt- )zi
o~1 ($JI~I Js- ~w cJI
E(R) =--J p(R-R) dv ~)4 (~I Y-r) YlTE o IR -R IfI
E(R) ---I p(R -R) ds -YlTEo jR-Rj
E(R)=--I PI(R-~dl YlTE I JR-R 1o
---7
0)l4 ~T jl Utgt )JJ) j~ J ~ ~ Ji ($LA($$ JI~ ~b_ i-
1 J~ ~I~~~ ~)Jo)L IJ ~6 45 rJJ ($~) Ui-i J~)~ J ~y~u
~AJ Jk u1gt)~ ~~J jPlgt uS~ 1~I~ ~l iJ)I~($I~ i~ o~li
bull~laquos 0) lA 5j ugtJa J ~ ~
)JbJi 00 l -001 z~ 4SJJ P ~ t~ l~ ~IJ~ ~~IJ~~
~Il I ~0ljl jPlgt ~~ iJI~~u ~1gtU-shy
iLaiv ot~) 0~ ILl ))1) iJJLi z )Jgt- JJgt 4C- 4-)~I ~~
)~IIJdE iJI~A P 4A)pdz4 ~)~ 4 w~cs- ($141 O~)
4SJSIf
dE= -- p(R-R) dz fTfo IR -R l
R=ra+zaz R=zaz R-RI=(z-~)aZ+raT
IR-RI =Vrl+(z-zr
E(R) =~__J+oo(raT+(Z-Z)a)dZI flJEo [rl +(Z -ZfPl-co
+00E R P z -z
( )=f1Te [aT-r- rl Vrl+(z-zr -coo
(6-6)
cAJ1
E(R) P a ywe
o r
bull (OOz)
i(z -6) i (1- H
bullI
---------~_c-------------- _
(y-v)
aop Jaz 4AIJ J J4~ a ~~J~4AlJo (S1Jb Jw ~fiJl cJl~ ~if ~~
~A )IJ~ ~~ ~ J ~IJ ~~ r) ~ dl~1 ~lJI)jl sJL 0 ~JIL
u~ P bi u~1~ tSl~ rltA ~ 0 I~ pd ~~ J-I oJ Ju Il J j J Yo j1S1ilmiddotu~ u a ~~ bull~I -ultI lts (gtIl ~
~~l- J a
~J~ctgta 41 ~Igt ufw~ ~~ r-A J-ltolgt gtIlgt i- u~ Pbull ~ ltSI ~~ ltsIJI yz ~ )
e-I 4-ji J (j Q~ ~~L IJ ~ UI jl
~~ it I~ -i ~1~ ~IJI) ys ~j x doLi~ kli ~ _ 1
Y C( 0 l-e - UJ~~~ fSJ j lli ~~I ~~z ~J y ~j ~ liJ
u~ JJ~ z J) 4SlS~IA) Ja~ 4S ~ J~~ z s Y (SlAwJ ($IJI Ll~_f
lts (oJ s-- I 1- sIJ I bullji ~J bi-lt ~ gt Tgt ~ Ir bullrlt A )i
bullJ)tI ~ IJ J~tA la~ T z s yu-r ~ u~ ll~~x ~( b4 (Ez)x ~)J ~jJ ~b- ~I~ Ui
~l-gt 4-1U 0J~ IJ ~~ J ~r U )Z ) IJ p(x 0 0) ~ 44 ir-IJL T1
middot~U~ ~JuJ f~~~~i IJ ~)~ ~-dl ~lgt gt1J-~14 - ~ ~J -0 J rr~JtiJ dY ~JA~ l~ J j listAlJ -J ~ 1J ~ --~~ Jgt ~ jl U~I ~I~ ~u pdy Jf)f ulI ~Jfl ~Jl ~~
~ ~Jft ~ T x 4iJ 4SJ ~ tS pdY f1TE r lfi dY o
=f) -- (IT_+_17) Y7JE f Yo
--
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
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ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
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D= PL YlTr
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middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
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L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
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o 0 0 a 0
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t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
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(Y-yo)
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~ -L IJ all t ~~
~
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bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
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bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
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(r-re)
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p(R)dVJ=f bull
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ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
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(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
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(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
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VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
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r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
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)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
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r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
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av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
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bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
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middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
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~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
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J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
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bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
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(y-ttv)
(Y-fA)
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~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
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l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
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p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
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~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
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~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
P( iT r O )1olAJ~~1( +r x 9_-)CJ (- TX ro-)C ($laJ~_Y Jl1e ~~w P( r r) 41aiiJ) I) E tJIl--~l -11 JI) JI P( fJ ) J
BJi~ flI1 J_ _ bullbull bullbullbull~----- t __ __ ~gt __q-_ __ -~_ ~__ pound bull [ - )( y - + bull] bull )(~ -+ J
bull d bull bull bull ~ bull bull -f shyyenftx-middot 11
middotmiddotAY-4 J) Jf~Fjll~ _14itdta1~ ~ J~Uraquo~~w
gt~ ~1J il t~c$ cSbull tItt J- jlJJlP~~Uat~
oul~ rAj I r)Jgt- jP1-~ ~JllSlA)~ ~ Lajl tSlgtLdeg1 1J~ o~J
d~ ~J~ 4$~~e-j-j I~ ) )~~I-~ ($4Aj)~ ul)~ Cj-jtS~~ u~
r~gt4)~ ~ W~ ~ b~ tSJ) Xly~ ~ ~~ ul~ o~J)~~ ~-~ o~b )~
t)IiS is ~U u~raquo~ J tSl~~)~ uhgt) 4 ~~j Ji- I) lA~T 6$l~ ou e- jy
) ~ J ~ J p ~b ) ~ ~H~- J PI ~ ) L~ uJ~ ~ ~Ii ~ I) lJb~ TLj J
)~ UJ~s~ u)~~ L~)~ c- jy ~~~ ~ L ~LiU ~~ Pfl U~gt
~~ ~~A IJ b- tSJ) ~ ~ 4 tSJJ ~ ~gt 4)~ ~-J- ) ~ J5~Iy u-
( -0)
~~ elm Pi ~T jl J elmO( P 1gtIJ J -)[m Ptl JgtIJ
jL ~~)~I)~ ~J ~J x~ ~-~ lAJ~ ~IJ)~ 4)~I~ ~j ~~
o~r)tgt~ t) ~- ~I9 ~I-j~ ~~~JI ~gt~ ~)y u~Jl)l 1~ jl t)
~~ P=(AI r)e- f Tsin () cosp uJl~~ U~lSj I)~ ~ljT lSLJ)~ Y-r Jlto
o~) ~l-gt I) J4 J5 ~I
f1J fTJoo Ae-fr Q= pdv= r sin(JcospmiddotrsinOmiddotdrdpd(0 Q 0J J
AlT ( ) - -fr 100
w _ Q=-lT-e =-middotA-1T r f A10
Jp 4 6$~ I ~9tgt 6 ~ J~ ~jl JPlgt u~J I ~I~ uu ~~I~ tSlr
-~y 4bAi ~J~ IJ ~Ifi~ ~Il UJ tl~ja)~ IJ ~~~ lJI ~IlA~
J rA~ k_~ ~)b ~y-J J4 4-5($I~l ~w tSlr IJ l~ UI J ~~Lgt- )zi
o~1 ($JI~I Js- ~w cJI
E(R) =--J p(R-R) dv ~)4 (~I Y-r) YlTE o IR -R IfI
E(R) ---I p(R -R) ds -YlTEo jR-Rj
E(R)=--I PI(R-~dl YlTE I JR-R 1o
---7
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~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
oul~ rAj I r)Jgt- jP1-~ ~JllSlA)~ ~ Lajl tSlgtLdeg1 1J~ o~J
d~ ~J~ 4$~~e-j-j I~ ) )~~I-~ ($4Aj)~ ul)~ Cj-jtS~~ u~
r~gt4)~ ~ W~ ~ b~ tSJ) Xly~ ~ ~~ ul~ o~J)~~ ~-~ o~b )~
t)IiS is ~U u~raquo~ J tSl~~)~ uhgt) 4 ~~j Ji- I) lA~T 6$l~ ou e- jy
) ~ J ~ J p ~b ) ~ ~H~- J PI ~ ) L~ uJ~ ~ ~Ii ~ I) lJb~ TLj J
)~ UJ~s~ u)~~ L~)~ c- jy ~~~ ~ L ~LiU ~~ Pfl U~gt
~~ ~~A IJ b- tSJ) ~ ~ 4 tSJJ ~ ~gt 4)~ ~-J- ) ~ J5~Iy u-
( -0)
~~ elm Pi ~T jl J elmO( P 1gtIJ J -)[m Ptl JgtIJ
jL ~~)~I)~ ~J ~J x~ ~-~ lAJ~ ~IJ)~ 4)~I~ ~j ~~
o~r)tgt~ t) ~- ~I9 ~I-j~ ~~~JI ~gt~ ~)y u~Jl)l 1~ jl t)
~~ P=(AI r)e- f Tsin () cosp uJl~~ U~lSj I)~ ~ljT lSLJ)~ Y-r Jlto
o~) ~l-gt I) J4 J5 ~I
f1J fTJoo Ae-fr Q= pdv= r sin(JcospmiddotrsinOmiddotdrdpd(0 Q 0J J
AlT ( ) - -fr 100
w _ Q=-lT-e =-middotA-1T r f A10
Jp 4 6$~ I ~9tgt 6 ~ J~ ~jl JPlgt u~J I ~I~ uu ~~I~ tSlr
-~y 4bAi ~J~ IJ ~Ifi~ ~Il UJ tl~ja)~ IJ ~~~ lJI ~IlA~
J rA~ k_~ ~)b ~y-J J4 4-5($I~l ~w tSlr IJ l~ UI J ~~Lgt- )zi
o~1 ($JI~I Js- ~w cJI
E(R) =--J p(R-R) dv ~)4 (~I Y-r) YlTE o IR -R IfI
E(R) ---I p(R -R) ds -YlTEo jR-Rj
E(R)=--I PI(R-~dl YlTE I JR-R 1o
---7
0)l4 ~T jl Utgt )JJ) j~ J ~ ~ Ji ($LA($$ JI~ ~b_ i-
1 J~ ~I~~~ ~)Jo)L IJ ~6 45 rJJ ($~) Ui-i J~)~ J ~y~u
~AJ Jk u1gt)~ ~~J jPlgt uS~ 1~I~ ~l iJ)I~($I~ i~ o~li
bull~laquos 0) lA 5j ugtJa J ~ ~
)JbJi 00 l -001 z~ 4SJJ P ~ t~ l~ ~IJ~ ~~IJ~~
~Il I ~0ljl jPlgt ~~ iJI~~u ~1gtU-shy
iLaiv ot~) 0~ ILl ))1) iJJLi z )Jgt- JJgt 4C- 4-)~I ~~
)~IIJdE iJI~A P 4A)pdz4 ~)~ 4 w~cs- ($141 O~)
4SJSIf
dE= -- p(R-R) dz fTfo IR -R l
R=ra+zaz R=zaz R-RI=(z-~)aZ+raT
IR-RI =Vrl+(z-zr
E(R) =~__J+oo(raT+(Z-Z)a)dZI flJEo [rl +(Z -ZfPl-co
+00E R P z -z
( )=f1Te [aT-r- rl Vrl+(z-zr -coo
(6-6)
cAJ1
E(R) P a ywe
o r
bull (OOz)
i(z -6) i (1- H
bullI
---------~_c-------------- _
(y-v)
aop Jaz 4AIJ J J4~ a ~~J~4AlJo (S1Jb Jw ~fiJl cJl~ ~if ~~
~A )IJ~ ~~ ~ J ~IJ ~~ r) ~ dl~1 ~lJI)jl sJL 0 ~JIL
u~ P bi u~1~ tSl~ rltA ~ 0 I~ pd ~~ J-I oJ Ju Il J j J Yo j1S1ilmiddotu~ u a ~~ bull~I -ultI lts (gtIl ~
~~l- J a
~J~ctgta 41 ~Igt ufw~ ~~ r-A J-ltolgt gtIlgt i- u~ Pbull ~ ltSI ~~ ltsIJI yz ~ )
e-I 4-ji J (j Q~ ~~L IJ ~ UI jl
~~ it I~ -i ~1~ ~IJI) ys ~j x doLi~ kli ~ _ 1
Y C( 0 l-e - UJ~~~ fSJ j lli ~~I ~~z ~J y ~j ~ liJ
u~ JJ~ z J) 4SlS~IA) Ja~ 4S ~ J~~ z s Y (SlAwJ ($IJI Ll~_f
lts (oJ s-- I 1- sIJ I bullji ~J bi-lt ~ gt Tgt ~ Ir bullrlt A )i
bullJ)tI ~ IJ J~tA la~ T z s yu-r ~ u~ ll~~x ~( b4 (Ez)x ~)J ~jJ ~b- ~I~ Ui
~l-gt 4-1U 0J~ IJ ~~ J ~r U )Z ) IJ p(x 0 0) ~ 44 ir-IJL T1
middot~U~ ~JuJ f~~~~i IJ ~)~ ~-dl ~lgt gt1J-~14 - ~ ~J -0 J rr~JtiJ dY ~JA~ l~ J j listAlJ -J ~ 1J ~ --~~ Jgt ~ jl U~I ~I~ ~u pdy Jf)f ulI ~Jfl ~Jl ~~
~ ~Jft ~ T x 4iJ 4SJ ~ tS pdY f1TE r lfi dY o
=f) -- (IT_+_17) Y7JE f Yo
--
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
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~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
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bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
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f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
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(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
E(R) ---I p(R -R) ds -YlTEo jR-Rj
E(R)=--I PI(R-~dl YlTE I JR-R 1o
---7
0)l4 ~T jl Utgt )JJ) j~ J ~ ~ Ji ($LA($$ JI~ ~b_ i-
1 J~ ~I~~~ ~)Jo)L IJ ~6 45 rJJ ($~) Ui-i J~)~ J ~y~u
~AJ Jk u1gt)~ ~~J jPlgt uS~ 1~I~ ~l iJ)I~($I~ i~ o~li
bull~laquos 0) lA 5j ugtJa J ~ ~
)JbJi 00 l -001 z~ 4SJJ P ~ t~ l~ ~IJ~ ~~IJ~~
~Il I ~0ljl jPlgt ~~ iJI~~u ~1gtU-shy
iLaiv ot~) 0~ ILl ))1) iJJLi z )Jgt- JJgt 4C- 4-)~I ~~
)~IIJdE iJI~A P 4A)pdz4 ~)~ 4 w~cs- ($141 O~)
4SJSIf
dE= -- p(R-R) dz fTfo IR -R l
R=ra+zaz R=zaz R-RI=(z-~)aZ+raT
IR-RI =Vrl+(z-zr
E(R) =~__J+oo(raT+(Z-Z)a)dZI flJEo [rl +(Z -ZfPl-co
+00E R P z -z
( )=f1Te [aT-r- rl Vrl+(z-zr -coo
(6-6)
cAJ1
E(R) P a ywe
o r
bull (OOz)
i(z -6) i (1- H
bullI
---------~_c-------------- _
(y-v)
aop Jaz 4AIJ J J4~ a ~~J~4AlJo (S1Jb Jw ~fiJl cJl~ ~if ~~
~A )IJ~ ~~ ~ J ~IJ ~~ r) ~ dl~1 ~lJI)jl sJL 0 ~JIL
u~ P bi u~1~ tSl~ rltA ~ 0 I~ pd ~~ J-I oJ Ju Il J j J Yo j1S1ilmiddotu~ u a ~~ bull~I -ultI lts (gtIl ~
~~l- J a
~J~ctgta 41 ~Igt ufw~ ~~ r-A J-ltolgt gtIlgt i- u~ Pbull ~ ltSI ~~ ltsIJI yz ~ )
e-I 4-ji J (j Q~ ~~L IJ ~ UI jl
~~ it I~ -i ~1~ ~IJI) ys ~j x doLi~ kli ~ _ 1
Y C( 0 l-e - UJ~~~ fSJ j lli ~~I ~~z ~J y ~j ~ liJ
u~ JJ~ z J) 4SlS~IA) Ja~ 4S ~ J~~ z s Y (SlAwJ ($IJI Ll~_f
lts (oJ s-- I 1- sIJ I bullji ~J bi-lt ~ gt Tgt ~ Ir bullrlt A )i
bullJ)tI ~ IJ J~tA la~ T z s yu-r ~ u~ ll~~x ~( b4 (Ez)x ~)J ~jJ ~b- ~I~ Ui
~l-gt 4-1U 0J~ IJ ~~ J ~r U )Z ) IJ p(x 0 0) ~ 44 ir-IJL T1
middot~U~ ~JuJ f~~~~i IJ ~)~ ~-dl ~lgt gt1J-~14 - ~ ~J -0 J rr~JtiJ dY ~JA~ l~ J j listAlJ -J ~ 1J ~ --~~ Jgt ~ jl U~I ~I~ ~u pdy Jf)f ulI ~Jfl ~Jl ~~
~ ~Jft ~ T x 4iJ 4SJ ~ tS pdY f1TE r lfi dY o
=f) -- (IT_+_17) Y7JE f Yo
--
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
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bull
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1 C C I
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AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
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L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
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~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
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r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
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i
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gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
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o 0 0 a 0
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r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
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pound=0
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u~u-~ jl ~ ~JJ~
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t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
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(Y-A)
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(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
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bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
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u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
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~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
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rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
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ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
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lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
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d gt ~ rj~ ($-1
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lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
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bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
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qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
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p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
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~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
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av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
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middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
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~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
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~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
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q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
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JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
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~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
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WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
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(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
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~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
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~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
R=ra+zaz R=zaz R-RI=(z-~)aZ+raT
IR-RI =Vrl+(z-zr
E(R) =~__J+oo(raT+(Z-Z)a)dZI flJEo [rl +(Z -ZfPl-co
+00E R P z -z
( )=f1Te [aT-r- rl Vrl+(z-zr -coo
(6-6)
cAJ1
E(R) P a ywe
o r
bull (OOz)
i(z -6) i (1- H
bullI
---------~_c-------------- _
(y-v)
aop Jaz 4AIJ J J4~ a ~~J~4AlJo (S1Jb Jw ~fiJl cJl~ ~if ~~
~A )IJ~ ~~ ~ J ~IJ ~~ r) ~ dl~1 ~lJI)jl sJL 0 ~JIL
u~ P bi u~1~ tSl~ rltA ~ 0 I~ pd ~~ J-I oJ Ju Il J j J Yo j1S1ilmiddotu~ u a ~~ bull~I -ultI lts (gtIl ~
~~l- J a
~J~ctgta 41 ~Igt ufw~ ~~ r-A J-ltolgt gtIlgt i- u~ Pbull ~ ltSI ~~ ltsIJI yz ~ )
e-I 4-ji J (j Q~ ~~L IJ ~ UI jl
~~ it I~ -i ~1~ ~IJI) ys ~j x doLi~ kli ~ _ 1
Y C( 0 l-e - UJ~~~ fSJ j lli ~~I ~~z ~J y ~j ~ liJ
u~ JJ~ z J) 4SlS~IA) Ja~ 4S ~ J~~ z s Y (SlAwJ ($IJI Ll~_f
lts (oJ s-- I 1- sIJ I bullji ~J bi-lt ~ gt Tgt ~ Ir bullrlt A )i
bullJ)tI ~ IJ J~tA la~ T z s yu-r ~ u~ ll~~x ~( b4 (Ez)x ~)J ~jJ ~b- ~I~ Ui
~l-gt 4-1U 0J~ IJ ~~ J ~r U )Z ) IJ p(x 0 0) ~ 44 ir-IJL T1
middot~U~ ~JuJ f~~~~i IJ ~)~ ~-dl ~lgt gt1J-~14 - ~ ~J -0 J rr~JtiJ dY ~JA~ l~ J j listAlJ -J ~ 1J ~ --~~ Jgt ~ jl U~I ~I~ ~u pdy Jf)f ulI ~Jfl ~Jl ~~
~ ~Jft ~ T x 4iJ 4SJ ~ tS pdY f1TE r lfi dY o
=f) -- (IT_+_17) Y7JE f Yo
--
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
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I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
u~ P bi u~1~ tSl~ rltA ~ 0 I~ pd ~~ J-I oJ Ju Il J j J Yo j1S1ilmiddotu~ u a ~~ bull~I -ultI lts (gtIl ~
~~l- J a
~J~ctgta 41 ~Igt ufw~ ~~ r-A J-ltolgt gtIlgt i- u~ Pbull ~ ltSI ~~ ltsIJI yz ~ )
e-I 4-ji J (j Q~ ~~L IJ ~ UI jl
~~ it I~ -i ~1~ ~IJI) ys ~j x doLi~ kli ~ _ 1
Y C( 0 l-e - UJ~~~ fSJ j lli ~~I ~~z ~J y ~j ~ liJ
u~ JJ~ z J) 4SlS~IA) Ja~ 4S ~ J~~ z s Y (SlAwJ ($IJI Ll~_f
lts (oJ s-- I 1- sIJ I bullji ~J bi-lt ~ gt Tgt ~ Ir bullrlt A )i
bullJ)tI ~ IJ J~tA la~ T z s yu-r ~ u~ ll~~x ~( b4 (Ez)x ~)J ~jJ ~b- ~I~ Ui
~l-gt 4-1U 0J~ IJ ~~ J ~r U )Z ) IJ p(x 0 0) ~ 44 ir-IJL T1
middot~U~ ~JuJ f~~~~i IJ ~)~ ~-dl ~lgt gt1J-~14 - ~ ~J -0 J rr~JtiJ dY ~JA~ l~ J j listAlJ -J ~ 1J ~ --~~ Jgt ~ jl U~I ~I~ ~u pdy Jf)f ulI ~Jfl ~Jl ~~
~ ~Jft ~ T x 4iJ 4SJ ~ tS pdY f1TE r lfi dY o
=f) -- (IT_+_17) Y7JE f Yo
--
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
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I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
y
---_
E P L E P x as=- =- -J (1-A)yeo 1e x 1amp
o
~ ~ li Ju ~)~ ~ Ifltcent )~ 1~ )~ ~ 4j1 ~l ~I~ olxl ~ J-L~pJ~ ~~ ~1l=1 )~ cJT~1Ae1 J ~I cJl~1 JA-- 6 p (Eo ~ ~I I
middot~4~)~
J ~I~ r ~ ~SJ I gtLt r ~ r 0 ~()~ cSJ o~~ )~ Y-f J(o ~ ftr jl rgtraquo0 41p~ 4bA 4)) ~SJIcJl-yen ooU ~I JJJ P0 uJi~
bullJ$- ~ lJ-
)J pdv )~ 4 ~ ~I~JJ~~ jl JPlgt ~J~~ J116 -J~
4 ~IJ I~ )jJgtJ~ 4kA
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
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p _ deg +_O-X-rl
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
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1
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p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
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csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
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(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
bull bull
p0 (r T sin Dd(Jd(jJtir)dE =dEcosa--= 6 -- -_~_ Cosa
bull f~toRY
41) U-Irl ~yoi ltsr-f JI~I y~ ~I)]I u-~ Jgt Eb ltsIJI
~ Iics 10_5 jiU J f= r 1 R rA-1 --JI I)
RY =zY+r Y - rzr COS 8
zY+r Y-RY (Y- 0)cos IJ = 6--__
fzrbull
bulllA er 1 Z J Iamp 8 ltsr-fJ~I )l~u ~ cl u-S ~ ~ cJl~ y~ ~) il ($~ ~ fi1ft ~
bull (J sill RilRSID -fI- =_-shyzr6 (r- )
~~u -~ 1raquo R 1=1J- bull c 8 0 7T IJiI 1-0- ltS 1fl 6
z-ltRltz+r r-- tSJJlSJI~Il (1-) ltl-l (1-11)1 (T_le) Ygt lSJl~ J ~
f r ]E ==_[middotEo Po
bull WEmiddot zto
- cJ I4S -f
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
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(r-re)
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
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4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
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J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
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bull J~ ~~ ~~
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bullu~ E y =0 )114- P( Y Y)
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p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
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csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
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y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
E=ro
Po a re r
-
z
P
-L-l__~_~-+ y
-
i
dEz Q ---- (0 O z)
CI
R
r -1 Jh f-V JI
~T tw-JJ(~)~~J=w- r- k bullu t1L1 (Faraday V -Af ) 0)1)1 ~ Jar ~I~ JJ- r~
d-01r ~pJ101~ taij($I~~) ~~_Ju-P (~)Jagt ~ 010shy
us~~ 101~Jb middotw4 u--~~ cJl~)IlIcJ T)JoI A) ~ )~~ ~ ~I
lilo))JI )~J~ 1) IAcJT6 ~~I ~lA)4 ~ ~1~1 cJJ~ tSLA~r~
-U~ is cJ I~ ~ J ~u p ~ lA~T) 1JAi J j ~~ J~ bullJiu--rgt )) laquos ~ ~)~ ~ J ~)lgt ~)~jJ ~JI cJl~ Jp
cJl~ uu jb)) cJ~ rsis r)r I) ~~J) ($lA~l~ JaQ ~cj1))
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
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Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
)~ ~t- 1P- ~j j ~~ ~lr ~ ~ 4cf ~d ~~J ~L E ~SJI
bullu~ u-- cJT x 4AJ~ cgt1-- Y W ~J IJ ~
dyEy dx =Ex
y
I
~ --------~-s
))~T-)~ lJ y=(x) ~ ~J iJl1-~J~ ~J
)~d ~ u~L ~I dJ)- E=fXSz--tyay u~Jt )11-- JL
ltJ~T~)~
dy Ey -fy Y-=-- =-shydx Ex Ix x
ay -ax - c -= ~ loy= -Inx+lnc ~ y==- xy =c y xmiddot middotx
aJT~~ iJl~ JtJ ~Ifi) ~~)~ J-jll) ~t~s-~J~
ax dy dz _ _-shyE - E y - Emiddot
dr rdf-dz E= E =E
dr rtl r sinN--------=shyE - E - E
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
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~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
o~fJtS ~lojT r_ 0
oJJ 45~ gt~I ~b ~~jl uSL lS~gtJ1 ($lA~l~ JyJ~ oIJl 45u~LjT
(SJJ J ~L ~s- A e1-j~ ~J~ ~ (StAJ u- l~ IJ lAeT~ ($la 4$($b jSJ rA
O=-j~ tJ ~Jlgt o~ ($J) ~~I)~ J Jbl ~ ~rSJ1 J~ ~gtb o~
~JJT~~~ IJ Jj ~li J o~bJI ~J ~J eT e~f
4S ~I tSl~JI J~ I~ ojlxl jbJjl ~J~ ofri tSJJ JI t~-
e-I ob JI uLl~ bulls- ($J)
~~ ~~ eT ~N J ~J~ ~ dJ 0-~b ~J~gt~ ~l_y
J ~ is JgtI~ of J ~
JSlts ($~ uJmiddotJ~ o~ J~ ofJ ~ ~~ fti ~-l
JSltS ltgtJJei ltSiJ~ of($JJ J~-J~ t~~j ~-y
~d)~~u ~1 o-J~ ~ d~~ ~ ll 4$bullbullbullbullJ 4~~ e~l)li ~I jl
~Lmiddot~~PIJWl)J~~LJ1JIli lAT t~JoJ~0~ O~~
--rP 4$ Q==KlJ ~Yf~ ltSLJ ~)~- elyL-S tJ ~lJl ~Lo~T
J~ ~ gt-IJ ~ I mks o1S~ J~ k ~L
Jw~ b cs~)lgt ot~ J a craquo-b o t~ r-d Q =tJ ~
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
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I
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~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
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(r-re)
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t fa
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p _ deg +_O-X-rl
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
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J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
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bull J~ ~~ ~~
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VXEmiddotAs=o
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dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
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bull f-~lllrliila-_
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I I
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s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
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(~t PL Ps) ~JT~~~
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bullu~ E y =0 )114- P( Y Y)
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(~I ~L p)
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p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
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csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
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yr ~jT~b ~)))~ )~ )~)1)
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b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
bull oJ)~b ek tSJ) (-Q))~ )I)li ~tjT ~ ~~ (+Q))~ ~Ib ~I~
eh ~ J c)6=- u-lgtb bull oJ)~ jl s~sJI~Ii ~ ~ oJ~4f~ uiJ~
ol ~ U Q ft lft )J~ ))IJ ~ bullf
~~sJli~~ t- )J l~ti-S u~J uJ1i ~~ oJ ~~J)) uSr~ltSt (J~) ~I)
)) J J4 ~ QI f1la~ u-0J)J bullJ ~l ~gtJ 1 tSJIi ull~ )I) ~tjT
oJA ull JI)ti ~ LejT)) uJ~ uJS) ~ ou~lS Q f1lbl ~)l-f~
~)uJ~ D ~I) ~sJI tSt J~ ou4U Qf1lr t r1J r t~~ u=-Jt
D-- Q af ifr
d-- o)r)tgt~ ~ tSl~)4 ~r- ~ ~~ a )b ~ ~ tJlJiU-- ) Y4 ~
~ ~I J1r tSlJu)4~ ~Ir ~~I ~ti J ~
(y- r)
D=eEo
gt1 ~J) ltftl )ljTlaquogtla)) ~oJf I J ~)f~~jU E jl JA ) Do~J ~~
5 ~I Igt~) tJ)J rJl- ~~ ~ - b J J) JJ (A ~ lSi~~ I )ii
oJ)ss j~J) lItT ~Jlpoundj +A))II ~ ))~ IJ )JI iJ
-r Uli r- f 1 Q ~J 11gtli 45 ~I ~ I )JuA J ~ jl) ~ LajT jll$Ut-I ~
UI J ~tJT Jgtb ~~l )~ JS l ~I J IJ u~ clgt- C- ~~
~Y fl j -gt)J4 I) JT tJ Iu-- J ~l -~ ~yli ~
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
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p _ deg +_O-X-rl
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
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1
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J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
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bull J~ ~~ ~~
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uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
(T- A)
D u~_i1A J eh ~ ~ W ~ lgt u~ u~ cs D ds DII bull ds ()
ds eb ojlt1
l~ J ~ JI~I jl ~Iy~ I Y J~ ut s-~ J~ J~4 Q ~) 1
~ o-T~J~ ~ I oJ~ ~jy~ 4S($I~l tS-J
I~lt-middot~ Jb~ PL cgt1~ ~~jl ~6- u~31 ~I~ ~J Y-9 Jto
JS~Lgt
r- As ~~ - ~l Jb~ -til) ~ e~ lk s cs-fl~ 11 l ~ ~~ ~I~ (S1raquoJ~ ~ J-Yshy
~Dds=fDmiddotds+fDmiddotds==D Y1m1f=PLllf ~~ ~ o~liJ~~
D= PL YlTr
~r~ u ~~f t-1 f ~ymiddotlJS=f ~I ~~s ~yl ~ J~J ~S3J ~I~ ~IJ ~Icl ~J ~ laquo)T ~I titgt ~~ ~I ou ~ t c$~ cslft t~~t lts141ii J~ 4 ltsI I
J 1
tJJtgt~ LAI ~ Ja1J ~ ~~ ampSJ~ ($lA~~ I J~ Q ltgt1sbAi ~ 4f~~)3) ( s oJgtJ) ~ ~4 (Y_ ) ~~ lb
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
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dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
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(~
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w1=S
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I l4i ~ ~Y Y u~SjI~ ) ~ (~
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)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
z ~ ~IG ~~
~t J~
bull
bull I
-shy - reIshy
~- -- 1 D D
1 C C I
~ I Jbullbullbull l~L bull ~ I bull ~~S bullIi ~i bull II16bull-~~ middot~middotr _I 1 middotmiddot _~ middotmiddotmiddotmiddot~~ bull -rwr- ~
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
AS ~ ds JJ I) JT d Q uZii~ufj 4rt~J )~ f) r t l~ ~If JS1 4f)b cJ ~ cJliu-- uf ) L~
ds ds -( a bullB =--y a bullB =d Qr
Of L J~ deg
middot middotmiddotmiddotmiddotmiddotmiddottmiddotmiddotmiddotQt= fshy ~ tJ=-Qefgt fitbull bulle
u -1 I gtIi 1iI ~0 = hi 4S j
J~ ~ I)~ uc~ lJTjf cJ- Q 4f~~~I 4JjJ-- ekr I
1iJ
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
ISI ~ J D s_~ IISII ~JS I bulluI )1~11 Ibullbull 4 ltJsect liJ
-~ 1J 1 L ~ d I csltltJ 1 J ~ J I-LA I u ~~ -~ ~J SSJ 1A c-raquoify JgtJ u t - Dsl ) I L S
L JoU IJ) u ~ J J~ J j J JP- Sr-d ltS 1
1
e1 L J1 u P- ISI rYi J bullu ~ - 1gtraquo Llt ISIJ I u Jgt
~I Jjd---T I (~p do-)lt5 1gt1IS1- ek- IS)J J D -lt ~ _ 1 IJ J 1Js tr ~ lti I ~ Ij T l 1 ltJ l T_~ l 1 ~
bull~ J-J J J~ ) Is J I)~il - is j I)r cJ 6shyu=--i~ 4S~ ~ l 1($ J1 I U ~d--- 4J IJ-~ ~ ~ ~I J ~~ D _
JIJ~I 1- j I IJ oJi 1 (u l (J I D J I -i D lis T 1
~ CJ 6shyI J 41 b4 JP J Llt ISIJ I u sr-Slt5 rL1J l ~ br
J 6J e-I 4 ltssect d-~ 1S14ii J Llt J 6J l uL 4
Jbs ~ d--- 1IS I JiJ 14 ~P do- ltSJ~ ilJ fA
middotu~u )~ I IJS =-gtJ ~ I~ 4S~ tl)L I ~ I-S 4l- 4J t4i ~ ~
-J~ ~ IJSJ ~~gt- iJ I~ ~ ~~ j-j I 4--1A) J JS($rN (lA~)
~ ~~ ~~~ ~fJI) r~a 4gt( cgtJ~~~ o~)~ Y-V J~ rf
P=POT (cm()a
yen~lgt lQ lt4 ~J) I) ~JI ~It-- ~~ ~~($
~J 4- ($J~~fbblA ~ ~I ~If ~I ~ ~5i4-~ ~~J) JIr ek ~ loSJ) 4J1~ ~5co1~~ 1iJ JiJb J )~~ )~- ~
ISJ b s T a 1 0 -1 ~ t Jo l J h tl t I
~JIli )~J craquo~ 4JTJ) 4S rgtlaquo 4gt1~) ~I~ J~
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
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t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
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~ -L IJ all t ~~
~
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0 a a
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I
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(r-re)
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t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
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tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
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4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
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J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
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bull J~ ~~ ~~
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J J
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J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
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bull f-~lllrliila-_
I I
I I
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v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
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bullu~ E y =0 )114- P( Y Y)
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(~I ~L p)
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~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~ - ~
r~ tJ~J) r tJA~tSJJU5cle~ VliVtIJ) rlta ~l (~I
~ U )100amp d)ll4 J daJ ~ 01 D ~amp ~ J ($J) ) (~dW -)
- - ~fiJiti~ ~ D~ D(r)a ~ ~-J
pDiJs=PD~d~=~dS~D(fltr)
~ Jb)~s-~ ~Irf~J)~ u~s)~ fJP cJylirJ) w 1lJ ~I ~)~ cpoundj~ iJ~J u~~ rlta t L-~ ($1bull1
i
~)) J
gt6 r rlta D(VwrC)=P---Y(V) ~ D=p-
bQ ba
p)) r t~4f $Jf~r~ ~ ji ~t cnJ~ gta ~1i (
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
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- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~ ~ ampJ ~~ J ~t tS)lJA- DIif~I tSJ) ~~~
ggtn ds =9iP -D fdS =D(wr~)
)~ 1aAi rlta ~li)~ ~~~I ~U)IArgta~)~ ~~ ~Jl )~
-~ U~amp Jgt-b)~)~ ~ ~(rgta) ~li ~I)~ r J~ ~J ~J ~J1
~T~ ~~J
rC af Q= f pdv = ftr f middotf tl Po ~r( sin Bdrd (Jdp= Po -(V1T)
o 0 0 a 0
~
)~~JlAi ~ bull~1 ~~ D ~1~ r =a ~J~ jf)~ ~- ~I)~ I$~JJ~~~
JlUi ~~~ u-SJ ~I bullu ~)~)~ Il ~I) rdJ ~~ dll~J))) ~~
~I bullu JI~)J
-~ ~J)~ ~I)I~ rlta ~L ~ljT~lAt)~ ~141-1~~ )~ y-A JIio P=-po(afyb) u1~~ ~)~ tSl)l~ r=bampJ Po ~I~ ~~
~L I) l4 J lAi 4JS)~ ~5J J 1~ ~l bull1 l~ ~
~
-4i1~1 ~ ($J) ~I~ ~I ($)~ ~)Ui ~I)I)~ rlta ~l (~I
~ t 4i1~1 01 cs-I~ amp ~J) ~)IlA J ~w -i1~1 amp fl bull)1 cri
~ ~I
ggtraquodS= fDds+fDmiddotds= fDr ds sx Draquo YwrL ~~ fta- 1AooAi u ~~ eJa-
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
Jnmiddotels =Dmiddot YlTTL
r-J~ cs-)ii)~ I) lts1~11~~ 4ji ~i ~I)~ rgtb ~l (c s~3 J oltrlta Jli)) ~)~ ~ ))) 3~J )~J) ~I~I Lf-I ~I) sraquo
J1-a- ~~ I) 1-1 J 1) UiA1~ t~ JjJ ~IJ) r =b 1) ~J~
r
Imiddot ~
b
r-A Jt - Js
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
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~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
D(Y1TrL) = 0
D= 0 rgtb
pound=0
J lU ltgt1gtI 1-- r = b gth -=-1(-l) J D 1-- r = a gt ~bullbullJ- -poaltTb ~ ~~I foSJ) ~)~ ul~ ~I ~)J JtuJ )IJAJ~
JYJ u~ )~ (f 45~J ~4Jy1 4 ~1 ~J r- r~ ~~ c~csshy~ uS~ rIJ~ J~l JliJ ~~f Jl4amp1 foSlJJ D t$Js- 4J1e u~ ~I)
u~u-~ jl ~ ~JJ~
~tili (4SfdUi) ~1~_ifA~ JC r_ ~
~A I ltgt14i ~5J IgtIltgt-gt ~5J 11-10 rltlJMult ltgt lAJ 10gtbull x~ bull1TJM gt f ~l I u jU J gt L ltgt1gt ji I uI u Ifu
~ ~j IfJ IfP iJli ~JAJf ~I)($J if i~ LtJ )J ~ )JA
igtIY 4f ~s ~ ~ ~amp ~ ~gt)~ ~ 4 ($J rt~csA ~bull)) ~
~D-pv shy(Y- v)
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
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E=-vv
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s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
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p=VmiddotD
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(~I ~L p)
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p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
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csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
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JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
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a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
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P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
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t II
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z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
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E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
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yr ~jT~b ~)))~ )~ )~)1)
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Po (e ) - fIT mC
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p( 00) pCo 0 )
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V= a(cosp+~ r( r
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( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
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(d 0 d) J (0 d d) (~
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q C B-------q
1
1
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J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
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dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
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y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
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l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
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~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
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E=E (~_a)a o r Y r
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bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~~~ ~~ IJ dW ~gt~ ~J~r gtL J 6t~ ~IJ 4lJL ~I ~J~~wJ ~ltsfJM ltsrJ~ ($Ift (1JJ ~IJ ~r ~I ~gt~ tJ~ p ~~ 4b J)
t JJ~~ ~~ ~I~u- IJlAJ~ t~middot c)-J ci~ ~~ ~ ~ ~1
1MJ Q J J ~ )u )C ~ lts~I1 lts~ lts~ ( r) lIp C ~
~ ~ ~j ~J~~ laquoJji~-IJ (Xo Yo Zo)t
(Y-A)
p = PL 3x)8(y)
(Y-yo)
f +OO _8(xdx=
- x = x 8x-xJ= -Ishy 0 bull X -r- X (Y-l )
~fiJIJ~ JI~ ~JJJ~ftJjcr-~ ~~I~f~)ljT($UiJ) _ J~ lts1middotfJJ )~r~SJfJ~~J~~t ~JI~IJ tJl~ 0 Ji1U4
~ -L IJ all t ~~
~
Of tlta =+- P=VmiddotD=e - -a (y p _ r ) = Vrp0
bull r T (Jr eoa a0 0
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
Eo rgta p=- -a ( r C p a) -0
rC ar 0 E rlf( shyo
olr Yrp Y1T p (a)Q= f pdv f (f1Tr()dr= 0_0
0 a a
~ ~)Ij ~JJ~ )J I) r =a e1~ ~~ )~ cJ)JSI~ lSl~ s))~~
))Ix )~ ~($)~ )41iJ ~I ~~~ UI)) ~I~ 4~Lr-IJ) P
I) u=ili l~Ii J (~lJ UT 1 ($)b~~ ~s ~JSJI cJl~ ~ cJsli
u~)~ ~)t~ ul~ ~~I cJl~ cent ~IJ) J~JSJ1 ~Il-- ~bo($l~
41- J~ 1 1_s j)1) )tgtJr ($)I)r cSlA~~ OJIrJII 4f~~) ~)c)lA J
~~)) JI ~ Igt is S~ I )Ycs- Js u J- 4 ~1JJ ~ b cJJ tA
~T iSJ~ jlJJ1 J ~~ ~gt4 jl ti J~~ )tc1 ~)4 I$ri5~~ I) ($~~
~)~ I ~l~~u 01~~ )~I ~)~I)I 0)lQ1 ~A) ~~jl J J~ o)
rsu ~~ ~SJI ($jl tr~ ~ U ~~I ~)JT
~J5J 1 ~Ila ~)) ($Iuui )~4 cJ~b d ~ ~~ joy ($iJ1
ltS~JI cJl~)) q ~LjT l- rS1 ~5JJ cJl~~u ~jU J~
~-) ~T ~Ir- ) cs- ))IJcJT r- F=qmiddotE r-l~ ~JA )$Jl~ E jl ~tgt lS~ ~Jb- J r1~ ufJ~ ~~~)r-gt o~ cJJ~ U q 4 riJy~
lS ~dk_~ ou r~1 )tgt )tgt ~N ~ J F =-qE u=~ ~I (i cJl~
4 e-Ifllfi dIJ)~ l---~J))~ cJ)l) dr- lts1~ ~)b
I~ C -A 4 ~J E ~Il~)) q ~tjT J~ d J-~ j~ tgtulS)~ J
~ 1
w=- fqEodl (r-)
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
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( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
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dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
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(~
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I l4i ~ ~Y Y u~SjI~ ) ~ (~
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p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
4sJf ~~ J~ r- v q == ~ ~ 1 bullu~~Ii q ~LjTJ~~~ 4$~~~~ IJ~ c)~1 IJ -tW 4~ i-lA ~t jl ~IJ)~ ~I JI dJ ($IJ rj~ ($-1 1 )Iu w~
~u ~lJ V ~ iscs- ~~ wzA j~ ~ u~31 J-il w~l
VIB==V-VB== -JEdl (Y-Yr)
d gt ~ rj~ ($-1
)Ii jU) ~~ 1~~ ~~( $r-- ~~ T)~ ~~ 1J-~ lgtI ~ bullu~csA J~) 44j~ 4imiddot~dl 1gt1 tS4LAi ~4 ~)b
~I (JJjeYfi) ~~I ~l ~ f I) tS 41U 4 f 1 jJ4AA ~~4 j~r Jtl- )
lgtmiddot1 tS41aimiddot 4) ~~J~I-t ~~uwmiddot~T~~ IJjli~) ~~
(~ y L)IJ I JtJr J~IaA ~ S-uILJ u ~ TJo-II s-rshy~Lwl ~J~ ~ )15 ~lgtwmiddot1 (J~ ~tilts) )J) JI~ ~(j J~ 44 ~ J-u-~-oJ I l1A Jo-I1 J r1J ~ I lyJJ 1-1 t J
bull a _C middotIL I ~ l I f Iii LlJ wL t l lt=(( bull Jil)rY~cr- bull v- ~~rv-y ~ _ bullJ1t
tSlth4i~ 4 jl Jb ~~ ~11- Y_o JIia ~-IJIJ (ljTtS l4i J) lts1ltlbA J~ 4jl Jo6 s~1 lo 11
bull4J
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
dR=tJlcos
bullbullmiddot=4 cosl
Jrr - _ _ f 4 ~R -flre- II
1(middotmiddotmiddotmiddot )y- - - - _~ It
f4A ~y lt4 ~~~ JJIWgtiWJllMll_bull rt_
u-VJM~middotUIbull 4it
I
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~~J J Jy J qJ RB J R ~ ~ sbAiJJ ~ J-il w~1 _y
~IJJmiddot U~ ~~~ u~1 ~~ I Ir JTj I r d IiJ A UaAJ q ltS 14U ~~ clb ~~ LrIr~
qV c =----==----shy
Y11E rlt IR - RII (t-T~)
~ 4 bu) ~i~ qN J bullbullbull q J q ltslA~ ~ 4bA4 N ~ljitSLa) I jl R ~J ~ tS1bAi J ~~ c4 ~b IJi R J bullbullbull J R J R J RB r t
~ ~I 1 ~~ il-e
(r-re)
u~~ c~ ~~ ~~jl ~I ~J~~ Jly~ lA~ jl4rl Jl
~Tu-- ~J~ flj u J1s~4Imiddot ~ I ~~ )a) P
p(R)dVJ=f bull
fWE fR -Rfv p=gt c
PsR)dsv=fs (r-rv)
V=f PL(R)dL Ll Y11E~ IIi- R I (T-TA)
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
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I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
ott ~ JtJI Jr-ilc ampttl)-t1 it tA1)1~ ~ A~)LI 1tIJJ ~I ~ ~ -
J4t ~ lJJ~~lU
J~~ tSl~ ~rJ1J~ u1jT(j LJJ) (IJf ~lf1JJ y Jt J~ ~~ I~J) J-i~ Jf) )Iri Q tt l - ~tJ- jl P= P(ra)
)Ushy
Y=jmiddotW1 Jj1~lli (Ull _ Jr-i1~ ~Ijl~-l ~ (~
~_~-
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
rgta fnmiddotds=f pdv
(~
D =por l
t fa
v -fEdl= -fmiddot Edr-fO Edr =poalx~ a
00 bull fE r ccgt o
p _ deg +_O-X-rl
fa Eo r bull
~~~jiSJfuf~u~~l~f ~lgt f-A jl ~~~5JI ~1~ JJ 4bA sraquo ~~~ ~~I sfb ~Li J~ ~JJ
~ s~J ~bi~ tft fE middotdl JI~1 ~ lJIIr--~sIuA J~ 4
(l o)L1 ~ W JJzilA) IJ tS I~ d- SlA)~ ~-~ J~-s- ampi u J1u ~
J-i~ ~gtI 4$~~~ ~~ 1iJ ciJ JiziJ~ tSl~tS lAJ~jl ($I~ ~~Iu
tSlJ ~~~ ~I jl JIX ~~ ~~ ~J~ tlU~ ~ ~SJl
r--r1 s~J ~ltEd1 JI~I lt5u ci bullu~ s1gt ~l
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
4 lt5 j Sl ~~ I 1Jj lj ~rSlJA 4)I s-e-I W I j~ uI lti ~~ ~middotbJ ~ t-r- 4bAmiddot~ Jels J- ~ l ~- jl V s ~~ cSr-~ r-shy1 1Jr- ~ w- JCo J r lt$31 1$ ci ~ J lJj I oS31
1
bullJ~f (~I-rS) LSi~f tgtlA~IA IJ
JJlsectl- ~rgtJf ~I~ JJYJ)bAt ~pb U 41$~b ~Y ~~ ~~ ) C~I l~ ~ J ~~ ~ fJ U1u~ ~ tj ~~ ~I~JS I - ~ I
J ~J I il bullu~ w TJ li rL J- I s-e-I J JJl ~ ~ )~ u--i ~II--o jl ~jr-il~ rA) d ~ eh ampHI fS-J JJ ~J I
-r-- ehr ~fiJ ~I~ ~JJ- EJdlsS~I VJIgtU 4-jY J Edl= 0
bull J~ ~~ ~~
~Ji iJJj~ cr)~J~ ~
JEmiddotdl =1 (VXmiddotE)ds= 0
VXEmiddotAs=o
1iJJ VXE=o
-ul~~ u)L ~J 1ltslA~l~ ltsJ~ J-)L etY)l-jl ~) ~ 4bl- laquojl
~JbJ JJ-Ji ~ tSlb sectL ~~I iJ1~ et-u 4-5c)~ ~(j u)~ u
iJ1~ y~ amp 4$))pu ~~ T)l~a v D =p Aj)1-p7 )iJ) ~ ~ ~
u~u-- ul- ~~IJ~ ~ - ~ JL u~pJ1 AJ) tSJbfi Jf71 ~t~) JJI Jai ltSJbfi~lT ~J) f) ~jl ~A)ua ~~ ~ ~J) -l~(S) 1(1 ~ iJ~)fJ)gt )I T )) jP JlAi
J J
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
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ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~fOf J-i~ ~fO ~f~ ~~ ~fJ y_ bullCo ~ ) t1 ~4 J- Ip tS)lJ ~4 srSJ I 11 sr-C~ ii
r-~~ -S)r s ck ~~ ~~I-1 Lgtlfl eIJJ~ ~Sli )bull srSJ I I - ~ s bulls11 110- ~ J- l do I) JI
J I~il -l~ ~ 1Juzs~ ~ J-i t ~~ I ~ Iju- cI ~~ ~ -JEodl
~rSJ IA ( j ybullJ ~s srsJ I) ~ (jy ~ J l 4 I) (
)~~~ I~ ~J ~l ~I~v (0Tlti~)
V-J p(R)dv - YweolR-RI
)bull ) srSJ I) ~ (jyO -1 ~rSJ 10-0 Y ~ oF I-o bull J
~ 4)fJ~T~~ LgtI~ IJ ~1gtU-- 1iJ J ur4u-- ~ ~J~ I ~t ~Je )J~T-~ 1)~5j ~I~ J-i~ ~I~ U=-I) ~ tJl~~ ~Tjl e)(i1
r1J1E ~SJ ~I~ J) tlJ ~s- )~ J1 4 r J) J-~ u~
~ ~ I
av= -Edl= -EA cosS
i1v -=-Ecos91
dv Ii dv 8-d= ImA=-Ecos AI - 0 -11
WUJJ ~Jb ~J tJlJ- LJ~ ~Jlj~ ~ ~~ ~ J-i~ ~IJ~ ~ bullCo ~ ltbiTJ ~J 110- J r-ltflo rH Lr-t bulle~ 8 = u
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
E=dYjtil lUX
~ ~+AV I I
bull f-~lllrliila-_
I I
I I
I I
=~I==~~r _A)_~~bullbullt_~~
~r~~~
~~Sshy
middotmiddotmiddotmiddotmiddotmiddotmiddot~~~~~t
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
E=-vv (ry-y)
~
alt y=v -cosO o rlt
av dV dVE= -vY= --8r - - -8- -aaT r a8 rsinO 3p
f
E=V r (ycos8a+sinlJa)
y - (J pR)dv) (y-rf)V -V f1tE IR- R1 o
~J~ L71 ~~ J u~u-- R ~ ~ JIPIJ RI ~~ ~~ 4laIJ J~
~ J ~J ~r IJ ~~T cS~ ~1iU-- ~ -OJbullbull ~jl Ji-
vv=J v( PeR) )dv fmolR-R 1
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
p(R) ) _ p(R) ( )(v YTEoIR-R1 - YTE V IR-RI o
R-R VIR-Ri == -IR-il
v == -f peR) (R-R) dv (Y-fY)V Y1TEo IR-R ITti
E=-vv
ufL~~IIw~J~ o~ 0JP~ ~jjf r-ro 6 qN J bullbullbullbull J q J q ltS14bA ~~ ~ jl ~ ~ 4)~ oe ogt~ ltgtJI I~ I ltSj iJo 45~I ~ ~S~ ~~r- ~ooVJI) ~IJJ lai jl ~~JlAi~)
WltS~lS ~IJA ~ ~I r1r ~~I~) ou~~ ltSJ-il ~IJA)-~~ 1ft jl
~ ~jJ JIgt~6- ltSIM bullJJ~ ~IJi iJ~~r lA~~ ~I I)~ ~~i1 ~I ~jJ
J )~I~ )JJJ La J) ltS~~ ~ I~ I ~) AS~~~ iJ~ ~r ~~ ~ iJ)) ~IJ
~)~~I~ ~~J J )~JT(~J) ~lamp-lts) ~J) ~ sJpL jl ~~~ I~ 1AJ~
~A (~JT~ I~ q ~~ tJ ~JJ~ qN J bullbullbull J qT J q ~~ l)lA~~ ~d)
bull)~~ ~~i1 ~ ltS~lSJil bullcs)~IJ Jd ~~ ~Ir 4$ ~l4~) (S~~I iJ1l-shy
~jJ JlS qTVT ojllil~ ILJ ))1) )JJ-J l3~) q ~~ ~~JT~ I~ qTJ
~~ ~ V n) )~ O)JJT~ ~ r~ JJ) ~ 4J~ jl q ~~ lj )y ~l=il ~I
bullbullmiddotiJb~JalO- q J q r~Jr IJ qrJ~ uJJ ( bull)~b ~yiJgt q ~~ ~J (y) 4Ai
)~JIJ (r) 4kAi~) qr J~ t )J ~~I ~I ~jJ ~lS qrV+qrVn jil~ lil
q J~ ~J (r) 64 J-I~ V )~b ~~ q )~ ~J (t) 4hA J-i~ Vr)
()~b JJagt
JIgt ~JJ) J ~))Ij iJ~~r I lA)~ ~W Ii r-~ )1 ~J ~ 1
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~ltsIJ jSlts jl ~J JJ~4t It bullurl=U1 ltslAJtgtt~ rfIJ bullu rbu1
~ T~JJ )j JJ~4t lAJ~middot ~~
(y-ttv)
(Y-fA)
~Jb (Y-ttA) (y-tty) ~IJ~ cJ~f~ ~
WE = (Vf+V+V+ )q+ (Y-tt~ )
~ ~ ltslA1~ rW jl ~b-~~ )11 FI~ A JgtI ltsLA~~ 4P~ 4)
bullu~U--1lA jl ~~ JJ 4A cJTJ) J~
~Tjl
(Y-t 0 )
lts~~ J JT~J~ ~~ Y~ Iics- IJ c~ J~ ~~ bullu ~~ lSjll
~~~Jlt lSl~ ~J~ IJ PLdl ~ Psds ~ pdV4S~lJAJ) )l qi
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
l TIgt JI)-I J_~ ~ J~ lt$Ir (Y-f o ) ~ ltLIJ 11
WE == f p V bulldv ~
s ~~ ~ V ~-fr ~ ~J -r I y~ ltLIJ
~~ u ~ p ul ~~~ ~~ t$1IJ
p=VmiddotD
w == ~JV(VD) dv- ~ jneVVdvcrmiddot r
IJ JI JI)(I JJJ I=- L 0-1 E= -VV ~J oJo ~JI~It Jt-
w== 95s(VD)ds+ fDEmiddotdv ~
I 1iJ 111 II T orb I 10 ~ (W 0 -1 0$1 ~ )1 JII
lA I-U u II-U J- J 1)-I I l--I ~ Ifu-f 11I ~ I J~ lt$1 11gt ltLW bull-1 oA S c do-- lt$1 --s- -=-1 lt$11
oA ~ 11B ~ _lIoTu b-1iJloA ~ U SocBY I Doc IIBYoc I R
o)(S ~ JI~I )P-~ J~ - JI~middotJJj JI~ JJ) ~1fJ~ ~)J
ct trl ~
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~ a c5lAtW~ r- c514i11JJ lts~A ~~- oil cJj~ ~ - Jc J~ Vo ~ ~~ cJj~ v~1 ~I bullu J~ ~ljT~lai~ (Qltb)L JJ s b
~~~ ~ l~cf~ sraquo jl I cJjb- bullu r-~ fSjJ1 ~I ~J
J Q b J 8 0Yo =-poundmiddot=Ps- In- Ps= b E = b
81 a aln- r Inshya tl
Jl1 JJ
WIlt= ~f pYd= ~f pYdV= ~ Yo -r YffaL Qln-
Il
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
rJ~ J-JJ
WE V L WE =-fDmiddotEdv= -E f V 0 0 0ywrLdr=
(b) bY ~-
Y In~ Ina
~J~ ~~~ J-JJJ~ jO jlJiS-s- ~~ ~~ bullu ~ ~j~1 4-0~ ~ hJli4e J-JJ J~ 451A~ ~L ~ ~ ~-
~I J~ (jj~ tSJJ JI~I J-JJ uJJt Ju-- ~ ~ ~ J~A 4S~Jb ~~J ~ ~liJ ~I J JI p-J ~Il-e 4 ~I ~ tSJJ JI~I J-JJ ceraquo JJ
~~~Ift ja 4S~I u Uti f~1 j r-=-~ ~ltsjV~ x-~JtA
~ I bullsss IJ
~I~~ 4 ~ ~y ~ ~ ~5J1 ~1~J~u JtI~ j 4-6J -
u~ 4Jr1 ~J ET J E ~I~ ~J 1 iJ- ~Iuli1 ~J ~lijl Jj~ ~~~$a ltsjj t~flIJ J4Jtl-J~ bullu o~~ LSjt
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
- shy
)k (cm)PL ukgt ~JI)~ R t~~~ tS1r b jl 9 4Jlj~ U tSJ) f ampoJ 1 j I JoP ~ E u~JSJ I ~ I1A ~J ~~J )114- ~I 01 tj ~1~
~J~) )Y- lSJ) lS1AbA)~ 11 E ~ JS- Lgt IJ b j) JA ~ IJ ~) ~
~~ ~~ J-t- J jl d dij~ J (oJb r jl OJ I ~~
~~6- 1lA ~I1~ I) V
z= -a z=a z= 0
LJI~ ~J ~~ 4u - - P -P YP ~~ ~ Lk )~ sI1b ~J~
P ~-L LQj klAi ~5 1i IJ ~b lSl11~ Lr-~Ijl JP~ E uSSJ1 ~I ~t
~)~ ~ lgt x= 0 ~)~ z= ydb P ~)~ JA lgt z= 0 4gt~ 5 Jj 1tAi)~ I) E ~11A J~~ (0 0 d) sldA )~ (+q) lSl41Ai J~ J PL
(~t PL Ps) ~JT~~~
(ood)4A(~1
(0 d d) (~
(d 0 d) (~
( - q f) J ( +q) lSlAl)~ lSl)b ~~~ 4-S1J rA~ t s~lA 11)ofJ~ V uJJ~ ri5~ 1J~ d dPlj ~ ~jl J o~b cJW ~~ ~ xt~ d dPlj ~
1)U-- J Irc-~~ ~) lJ ~t 4
~Ai)~ Qf 14 J P(Y 00) 4-laA)J Q ~JSJ1 )~ JljT lSLai)J A qw)J L w~ ~ ~ ~ Qf J Q 1ft 4l= 11 Ji1I) )11- Pf( - T 0 0)
bullu~ E y =0 )114- P( Y Y)
P= P cos 8
bull-=5-l_ 11 01) j) r- )~ J ~ cJ 1j I J1_gt u~~ I ~ 1 ~u -U 4is
(~I ~L p)
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
tSfJr sI)I bullc o~b ~~)~ )t~)t~ Xy 4gtu)~ ~ljT($La)~ 0
J4Lgto I) ~~tf ~)IJ ~~ 1 (1-lSJtA I ~
P ( 0) 0) IlkAi )) Q == 0) y pC
P(o) Y 0) 44)~ Q== o)f pC
p r( 0) r o)V) 4Ai )~ Qr = ) ( pC
Pf( ol - 0 r) 4hti ~ Qf= - pC
)~ x )~~ ($J) Q =- l pC )~ J 11 )~ Q == ~ pC U~J~lI )L~
csJ Ii riamps-~~I )~ ~jljU 4~)~ I) QC )t~ Ji bl x = 0 m )~ ))IJ ~T ~
J~~ Po cs~4gtI~ r ~($l)b atW~~ cSIfl b ~J ~ middotr j~ ~ jl vP J ~yW )gt=V jl ill ~ iSJ ) )~Lrljl Jpb- ~JI ~Ilz-
JIr iJ1l ~I JS a ~ 00 t~ J~I ~ ~~ ~~ J ~~ -t- I ~T
~gtcs Pol TEo
a~~ t~YllSJ- ~~ ~ i4)~ Q J -Q J Q ~SJI ~~ ~ 4 Lgt- I) ~l r )~ ~SJ I SIJ~ ~~ bulll Iti~bull)I
c-j~ l tb ~~I )~ ($I)b L J1J a t~s~ lSl4iI-1 ~~ ~
I iilyl ojamp-ti j~ rJ~ u~SJI ~I~ ~~ J~cs Po J~s ~I~
JlJl-gt
~ bl) P ( - 0) 0 l 0) 0 0) If 0 0 ) ltlkA)~ Q =00 DC iSl~b~ 4 0 JS~l-gt I ~j ($LA)~ )I)~)~ it~ ~)IJ ($Jr
P(o)ol o)0l -ooY) JQ== yoILC (~I
P C( - 0) 0 - 0 J 0 r 0 J ) ) Q =yo pC (~
e Q )~J) (~
~JltJI~ ~ljT($La)~ a tlA- J~ 4 oampt ~J ($JJ~~ ($J ~
~)~ t)o~ o~t j~J~ ~r~H J-1~ ~I)I p J ~~ u=h
t II
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
J~~ ~JIJI~ 1lyJ=d Jb I$J) x=o u~J~JagtJ--J~ v u~ ~ )~ u~J1 ~~ d x J~ ~ ~ J~ liJIJJI~ PL =+X
(~I ~ u X) ~J1J IJ ~I)N r4t
E=xa+yaT+zabull
bull~JJT ~~~ IJ J-i~ rA cb- -1 JI)r
~~~ ~ rJ ~r~JLA ~JJ ~ Js ~~ q= Ync ~SJIJ~
~~Jc= CJ BJ A jij)~ ~PSI I~~u jllil ~I 4JJJI
f~l
J~ ~ ~J~)J u~ 4lJlI J~I cJl-- ~ lIU-- rt) I~ ~ T fo bullIt )T~~ I)TJJ ~ t ~
ul~ ~)~($IJb L J)4t r=D cSl4iIYICz ltsI4iI-I~J)J
bullJ)l IJ 4l1~1 fir J) ~5J1 I1--~u u~U-- P cosgJ
raquo _
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~ J~ ~SJI cJIL--~cJiS Lgt ~I OJ)I~ z)~ ~J
bullmiddot1 P( 0 t bto)
E=raz+fa
J R t~4~ ~I-Iri (S14i11 ~jl 15~SJ I (S-J JI1it )Jb JYJJ
oltcplt1f --LltzltL
rlta ~I p=pora ~Iot o~b Jj ~Jyt~ ~gt~ J1jT~LaJ~
~ t- rgtYa ~ Ir P=0 J a ltr ltsa ~Ir P =PoP J
I~~I ~~ ~J I la JlA0 4SJ~ ~gtJI cJI~~c (AJI
ftc
1 sectr~ J ra]raquo tLt~ ~Ibull)~ tSJ)5~jl 6 ~~I~)i (
( -C-A ~ u J) lie fJ J PJ csshy ~J lgt ~~
~c ~ ~5J1 cJl~ )IjT~l4i J) f~
E= xaz +y8)+za
Jt J R t ~ J Z )~~ ~14i1~1 J~ U craJ~ ~~I )~ JS (~1
bull)b )J J L
~Jgt ~~ 1l-- f r~ J R t lA~ ~I bull s-~I~ IJ (~I)--i (
z J-~ J R t 1~ J L J~~ ~l4jl1 ~L~ ~ jl 6 cf5JI~t (~
bull)JffS ~ 11-- j~~
bull
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
fT o~8lt-R
y
~)jfIS
~)Jt ~ xy ~)~ R t1aJ cSlbullJI~ ~ )1 C ~~J cS)i (
(~ ~)~J v cSl~ ~ J ~I cSlaI$)I (~
JljJ ~I)T Lii)~J)cS~J1--lutt~R~ )~ J~ t-)Ji fV
lt)_y costl + sin (($ a- a+ a a (~I r liD
(cS l4if-) E= sinfll+ Sbl+Vd (
Jf1Jf Z J ~ cS J)~)lt 4It I )) P( f ) da J) cJ5J t ~J~ bullrA ~ ~I
E=a+a bullJ4tJ t1J N e )WiJt wl~ ~u-- M-- shy
10) Iri Llt1J))~JtQt Jlt middottCt(~middot1
lat I ~ ita edt )II ~ J) II (~
o ltzltoo deglt1ltfllfJ ~r r-- cS~ Lf-I e~ dt cSJ) ~j$Jl J~ )~ ~))1
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
Jl~~ 0ltLu~JIJ~~ YO a($ltL~ fJ~lt5)lAfJJ~ ~i middotfo t-q vaJlgt ~gtJIJ~ lt51)b Ao1 ~~~Jfl ~Iow J~ P= -polr
(~L ~I lt5JIJ~ pJ ~~Lgt c~
IJ ofJ u~ E (~I
J) ~ ofJJ ~ J~~ ~~I liu~ JJ~ q JIJA- (-
J r-S~J ~~ J jti ~ ~J~ IJ oJ$J) J J~ q= YP o 1Ta s1 (~
1S-Lgt 1- o~J) ~ ~~ LJI~ ~ J) of~ ~ JLv J~
I z =+ 0- ~6aaJJ JIItSliUJJ I~J~~ J-tgt~JLA f~ J~ ~J~ f vJ~ ~ ~JsJl ~Il-- )~I ~J ~ ~ J~ 4 JiJI~
E =( oozY+zI 0 0 )a
J$~ b J5csshy
IJ ~J~~~ J~ ul~ (~t
I ~~ ~ J~~ ~~I (~
~ ~~ J ~~J rA~ Jjl h ~ 41--J~ IJ ~JJJ I (~
~- I ~JJ ~ ~~ uSSJ1 ~Il-- JS rN ~JJ
~-5 l-IJLgtJLAJ~ lt5Ju~J I~l~ ~ S lt5)lA vbiP ~) 1 (J
j)JJ~ xb ~J~J ~11lt5La J~ (altb) b a ~lAtl- Ffr middotfJJ fr JbJ4 - q jllil~ I LAo_) 1 ~~rA J o~I~IJi +q ~SIIJ~ LAof
~Lgt~cs-
I La raquo lA 4JS~ E (~I
IJ Lai j tAj ~JJ ~5J 1J-i~ (
IJ lAofjl ~J cb tSJ) cs~~IJ~ (~
4SiJ~ 0J~ 1 ~JSJ 1 J~ JI~ ~ riSJ ~~ IJ ~ of~ 1 (~ ~~ef ~IJ gtT~ It ~6-
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
Ji5- ~ L Ai ~lSlr I) ~SJ I ~ t J
V(Z=plusmnOO)= 0)
Jzlgtd
JI ra sa a lSlAt LJ~ )IjTl5U i J~ fr rA JAlSlS)lA lSJf-~ ~ ff lS1J 1) Yaltrltra ~li p=poTa ~ J ~ lSI)) rlta lSliJ lJ)1)
~I p= 0 btAi ~ JJ - u~~ p= -(YPoVf-) ~ u~rSJ1)~
~-L IJ j Ai ~)J D (aJ I
IJ lSJf 4-oX A C)ar lSJJ )~ ) 1lA- (
r = Yo r =Yo ~b -J laquoJ ~~ w~1 (~
tdegf fir ~JI ~~ (J
J ~ ~ ~rsJ I )~ JAu rlta ~lJ l5Jr- ~J )J JljT lSL4i J~ r~
JpoundS-L u~U-- J~ iJJ~ rgta ~li - p= por
rgta ra ~li JJ ~JI ~1Je~u (~I
v(oo) = 0) rlta ~l JJ ~rsJ1 J-i~ (
~b~JJJ~~ rA)IlSJlAcS-f~n~r= Ya ~JJI (~
JfS ~ iJT
uI~~ ~I~ ~~5J I J~ ~ -aya ~1 JljTcSlM)J f~
bullJi5~ Lgt ~ I bullu t Po
IJ ~tAi 11S)J ~rSJl iJ~~u (aJi
J~J O91T at~~ ~I-Irf umiddot~Cb-jl4S~JI~JJ (
1J J)jgs- z J~ s - LltzltL
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
J-~ y= -a ~ J poa ~~ ~ ~)~ Jo~ y=a 4~J1 (~
blAi ~)) ~~ ~1~ ~J ~~ y~~ )~fl oJ) -poa ~)~
bull1)JT~)4J I) ai-
E=~a (~) r T m
~I)Ir )ljTlts~)) ~Jf~~ L~) ))
t$J) ) ~ J~ ~laquoa~)Ii r = em ~J LS LA ltSJJs~~ 4 r~ I (~I
Yr-u-- cr sr~b d=shy~ )) ($) lA ($14i11 uk 4 J1 (~
-Y emltzltY em deg ltrpltYft r zx rem
yr ~jT~b ~)))~ )~ )~)1)
J~- ~)~ ($)1) rltm 4gt1 (1
(~ ) = fPo in P IT
Po (e ) - fIT mC
bullJS~Lgt I))J ~4i~~ ~--i r x tJ m d=-~~ ~~gtI bullu~~
b tSJJ PL = f nero ubgt )~J ~La il--)) q= Y DC SI~)~ n ~J ~~ ~~ 1iJblj Z uyen)) y= m
p( 00) pCo 0 )
J~L ~~I)~ lts1)b J )Ijl ltsLz rlta ~t tSJ-5 ~))) 0 Ji5~l-gtu~u-~~ gt)~ ~~SJIlts)lAaltrltYa otgtjJpQra
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
)J~ ~4i~ ~ ~ (r= o)middotf~- ~JI J-ili (~I
lS~lA ~ ~ ISJ- Y~~JI-~ (
~aJTJgtb ~ sJj j~ -~ ufJILS~1Af-4 r=a j ~I (~
y~
flildt-I xy ~-~ ~ ~Uu t$l~ -~-~ f bull ~ J~ xy ~-~ chJ 1)1l-tJ ~jSJl ~Ic (~I
~JJT)4f I)~d J~
~JJT~~4 1- Traquod Jr ~ P(r I )diiJ ~~ (
yL- J- gtJu-- cJt r =a t~ If~ jJ 45~SJI Ji (~
~
t-------
d
~-------4
------ p bull r bullI I I I I
I I I I I
------~
d
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~caJ j1sJ f r A
E= - Yxy a - (+xC) (+xf) aS
yen l-gt 7 lS 1A r-It lS r 1J
z= 0 y=xl (~I
z= o y=x (~
(~~) f~~Sis lS1~ ~ -y~ utA~J~1 (~
J~ ~ Jl- ~~ ~JI JI- altrltb ~u J~ ~IT ($LQ J~ ff ~JI J-ili ul~-Sit p= 0 jfA Jl- J~ J ~Jl~ ~~J p=(poaCrC)
~ -l-gt IJ j fA 4J)J~- shyJJP~ 4J1r1 4l~~ J~ ~JI J-il ~~ ~IT($WU J~ f
V= a(cosp+~ r( r
IJ E (~I
b ~Jlgt t ~ J a Jgtb tL ~ (SJ~gt Y 1-1 J~ ~ Lgt J~ JIlAe (
8~ J J L Jk~ Z Jgtrolt JJgt
~Jb IJ e1di ~sect t ~ ~J~ V (~
~I ~ ( ~ 1-1 J~ 010 ~) c~JJ ~j1 (~
uJ~~~lS~J-1gt 0 lt9~ 11 z= 0 J ~rlta Jt~~ u-P) middotf~0
~JJT~~- ~ljT lS~ J~liV( 0 0 z) ~ P - pJrfa()
)agt J~ Jet-gt x= 0 ~ J~ z= yd Jagt P ~ J~ J-b- z= 0 ~
I~J1 ~~ ~gtI u~u (0 0 d) ~J~ +q lS14bAi J~ J PL
( bulll-A ~ l ft~ u PL Ps) ~JJT~~~ 7 j lA lS1)
( 0 d d) J (0 0 r d) (~ I
(d 0 d) J (0 0 rd) (
(d 0 d) J (0 d d) (~
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
)1 (S~ I ~~I rj) )l$)~ j ~)~~ ~J~(j I )~ fJ )))I ($IJ fV l)JJ )J ~t-Smiddot~ tA)4 J) wI ($J-Jd I) J)~ )IT -q lS~)~
~J-5J I lSJJIJL J~ l-gt~ ~1 )I ~ I) o-L 0 Jgt~ lSjj1 ~
1)(S ~lgt- lA)~ -)~ t ~r j~ J J YO tl~ lSlo~ jl ~
Y~I )Jbbull
q C B-------q
1
1
A gt~lt -q
o q A x
J-a-l
~ ~ lSJ) -vYq~ J X= UW)) X ~ ($J) q ~~I)~ fA bulllib) Ij X = - 4Ai )
~ J t) )T41) tM J o)f~ I fSJ ~ lS La )) v= 0 ~ (~I
(S~SJ I )~ )~ ~) Iri v= 0 amp~ ~b ~ lgti fS)tA~l (~
~)Yff ~)TCb JA fSJ)
lSLAi)) ~jl m J- 4 ~ti ~ fJ = fILe J q =YILe ~SJI LJ) f~
4i~ ~ltiu rIJ q(=YILe~ JtAi1 fSlJ rjtsiloflj )Ijl
J$~lgt I q J q ) ~J) ~ bJ
P=PoYaf ul ~~ ~ (SSt~ I)~ ltyltZ4J-Pti) y)~ fSJ) ~o0
I) tA x )JA fSJ) ~~ ~ ) ~ 0-1 jl JPLgt ~SJI ~~ )I))lj
~JT~)~
x= 0 ) J Z )~ ~ljl~ P J~~ ~pI~~ (j-j ~ ~ )~J) 0 1) x= +0 raquogt o))-Yp uI~~A) ~L~~1) J y= +a1
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
dgt ~ loS Ir -- I )ij -- ~ lta t t- ~s Jsl I J I~ ~I T -gt lai )gt bull~r LL a~ I) ~J gtJIL tVj
- -II ~J bull shy
(~
lL at r=a~ lt 5~1 ~Ila ~I ~IJ 4SlgtJ~ oJl ~ lgt _ ~ ~-
~ J ~=-Ja ~ ll l l ~b lf~ =-ltJ I )~ J-lgt eb ~ ~ ~ )L-li ~ ~
~5 ~t- ~tl I r = a
vbI)~ljTloSLa )-bJaSLAt~~ ~-k ~14i1J=-1 tS~LA4~ J~ -or ~IJ 1 li1 ~ ul~ ~ ebJ ~ J-i~ wgtI 1
y f ~ q l J ~4U p= Pc r] af ~( u~~ i4 J u r lta 4gtl -Of ~ rgtmiddotQ ~t ~ ~jSJl ~l-t ~u l~ l~ q I~ i) f ~ j~ I bullluJT~~ I rlta ~l)gt J1~~l -Jl-- ir-IJ~ ~~J
~ --gt 1 L-b _J~~ )~ JL J1D- AJI j d)lJb 0 ltL r = a -gt-_ c--- - T~ rmiddot
Lgt u~UV= x+v~+z JJ p~ ~J~ ~~ ~ljT ~Lai J~ -~
w1=S
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
I ~(JI ~I~ ~u (~I
I l4i ~ ~Y Y u~SjI~ ) ~ (~
)IJr ~~ )~ ~~I ~~ ~Ijl ~I~ (c
)raz+Ya+ra
)I) ~ sraquo Jlt~ t~ ~ J r = r t ~~ J J~j ptA bull I (~
()~~) f~yu ~lgt ~I~)~ cS-~~ ~ rfb~
~)jgcSZ)~J r=Y tl~ ($1II~middot~ebj1 45~J~I tSI (~
)l~ ~ (bgta) bs a ($ Lat k ~ ~ iJJ~ j)f r- cS)1A J~ ~ ~w ~~
Ji5Lgt ~I bulllo J~ p=poa t1 r lt u~~ ~L ~ ~rJI
l b J a iS J~J) cJ~ ~JI ~~ gt-I (LAJI
eb fJ lSr) I) J) t~ tSJJ ~ WI ~ iSt-Jlt J ~ (- (J~ )2iJ~ ~Jlgt - ~b
u~ ~ )l~ J r~J l~ u)jL Jl- ~ ~J~ 1)~J~j1 (c oJi)Lgt I) ~J~ ~ ~~ ~I~ 6 Jdai ~J)
LgtI~)~ p(r Y Y) 4ii- p(Y r )4kCi jl q=Ync~rSJl4 ~V
~l1 JfiJi
lSlAt~~ )~~rA cSJlA)Iraquo~~ rf )r~cgtI4iII~J~ ~lji($~J~ ~
J4 I~ ($I~middotII~J)~I~ u~raquo~4 1iJblj (agtb) b J a
E=E (~_a)a o r Y r
~~bA ~ I ~f)~II)
I) 4i 11 JJ u~ ~~J~ J ~ (~I
I 4i1~1 JJ ~ ~~ j~1 (
1) jlAi J L )) ~J 1JI~ (~
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
p J~ u 4JJj1 ~~I)~ iJJ~ R t t~ ~~lA bulls~ )~ J5 ~tb ~
ut)~ Q )~ J I 0J )1) qf J q J~J) lAgt laquoi 1)f)~ J J )~ 1
bullJi5~ u~ ~I) )I ~~LA Jr-f J jl r ~ R
~~ ))1 Q q q ~LA)~ J s-~J~ (~I
lAbulligt cb- (5J) s lts~lA r-~~)lgt cb- ltgt) u~SJ I)~ JIu (
~r~~
r-ffTH--+----~Q
r
ub)~ Jo~ lA))~J PI=Yncm J~~ Jagt)~ Jetgt lA x)~ 90 ~~w u~~ PI= -y ncm cJ~~
I) xy ~)~ ~ J~l ~ lob- (JlI
I) xy ~J~ y= xb ltS) ~j5J ~I~ ~ (
~ y= +a j)zgt rSJJ x = 0 ~~)~ PI= - Y ncm ~~~ ~)~J~ ~
($J) W 4J~)~ PI= -Yncm J~~ ~~ ~ )t~J~J y= -bJ
bullJiI4jJJI) y= +b y= -a j~
bullJ)~~ I) P(x 0 z) 4laA)) ~SJI eJl~~w (~I
~W ~ I) cJTJ~ J o~)JT~~~ I) ~ ~~ ck IblA (
)JA- rA~)I) ~ J-i~ cb- ~)~ Ijt )~ lts~1A cn ~JI (~
f~~J ue bull tAl cJT~ jA lSJ) Jk lgt )) J~
~I )It ltSJ-5 O~))~ rj ~u~ ~JI eJ1l- ~ljT ltS~)) ~r
E=Aor(rf+af) art rlta ~li J)
E= A o afr(a
T rgta ~ li ))
~~I d~1 cJl~ ~ cJl~ cr 4SJ5~t (~I
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
~~ JiI)JS )~I I u~~ ~~gt ~ u~ tStA~ u J ~ (
~
bulll=Sl~ I )~) ~~4i~ ~ ~ Sr ~JI ~l (~
J)~ 4SJ Q)~ 1 ~~)ai)) J~ Ulk a t ~- tS)tA ~H ~ ~
4kA ) ~~I J-i~ l~Lgt ~L bullt c-jj-i ~I~J~ ~T
1 A( 0 0 k) I
A
x ~ )~ I s a tS ut -~ tSJJk~) R t ~~ J ~ ~JJ tS)tA bull~)) ~
=f J) q J q uSpOr tSlA)~ LA~ r) (l- j ~ ~UaA) ~l
bull1S~ l-gt -1 I LA~ Cb tSJ J tS)u f~ tSJ ~ J~ ~~ (~I
I) tS)A bullc)lgt )) ~~I ~~~w (~
I ~ jt Jgtb ~J I ~Il--~c (~
I q J q tS LA~ fl ))IJ 4SJ~ ()
~ ~~ r-)lA 1 ~S f)J~ tS)LA bullf41 ~)i qe ~~JI (A
fosshy
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used
1 Hayt W H Jr laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981shy
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing
Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticss McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley amp Sons Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo 3d ed bull John Wiley
amp Sons Inc NewYork 1949 There are a large number of well shydrawn field maps of various charge distributions jncludtng the
dipole field Vector analysis is not used