o-¥r'l(' l
TRANSCRIPT
~ tTic-J ~C51 q CC-Tlc I rs=t~n fl rIC~CrJ ~ Cit) 511 ~1C1c ~qf
4 r= ~~ IC5 i cit) 4 ~ Tctlr i~4 r= Sc 1 CrC~ctl tr S)~ CI ~rJ
bullbull ~I~ Ics~ ICwS tr rc$srmiddot lt1r=~q51 -rvct ~ ~~
It( r- ~r rtf r4 ~ rrlt lCIt)51(s=r~n C( lltt~ lt5 q C(
-J ~ ~~5 ~r ~~l~ l~rss~
middotr ~ bullbull r- Cfrl~ ~ lrr (I ~ ~$l~ sJ r ~f ~ ier1 CIC( 5 Vl1 ~ s
Ii-1(It) tr-f n i ~r ~ cr tt( ~ (I I~~C r sCcrr -n$V CI (15
r A-~ t ltjn q n f~ 5 CI C( 51 i C( IT ~n1c 1C 1 ~ [f1 er--
11= ~-n ci ~CSs1nJ~5 ~~ Vl11~ S1l 101 11 1C iq ~ i Ir C(~-J ltrgt=sr err ~ rr1--1 ~ C51 S ~- -iT (C
-It)C51 ltjwS C(s~tJ$V (lcl5 i(I~~lCIC ~r It( i1(~S
bullbull ~ errrl~ rr] ~r 1-
r=t ~ I~ r4cr ~~ -1 r C C-- ~ct ~ CS ~ t rJir S wS~rgt It( ~ -n~ IrJ -r I ts= i- lj~ q r151 Ie- j ~~ r($r -J ~O -
o-yenrl( lcrE~fl ~r lltwJ ~~S ~er
4ebl~r- ~Jgt~ tAJt~ ~T~))5r J Jj~~ JI ~I~ ~5j1 cSJ~
igt1 ~ l~ JgtI) )) )r-gt- ~ bullL I~ (5 t-J 4 x-H (S) tA cSjr da--~ li JA)ushy
5)~ cS)u- ~I~ ~I bullc(fSlJy-t Jt~1 ~ll- w~ ~))~J ~lgtOIJ) bull))J~ ~I J)u ampJlgt-~ ))1LAia cS)lAJ lAJ~ dr t)~u
~ cSlA Jgtb cSlA~ J-tN ~6-J) s1- lo~~ cS)lA Jgtb ) ~Ilshy
ltS1b (S)tA obtAi ~w J~) ~J~~ 4J ~I ~ l~iJTJ JcSJd u~ -lSu4l
~SJI iJ1-l-- bhgt~t rAC~ ~~~J~~ ~J Ji~fS ~~ 4 bullJiy4f ~jlgt 4 IJ cS)tA cb ~ )w ~)~~
fcjlAi ~lgt)) lA)uj1j ~jb ~r~J1 iJl~ 4 j) Ij (S)lA 4 ~J~
)4 cS)lA JgtI) J) J JJY~ ~ (s)u ck cSJ) ~ J ~ cSlA)~ _
~I~
c-I ~ cS)lA Jgtb )) ~JSj 101~ ~u_y
~I ~l ~~ cSl)b cS)lA ~ T
Jy~rgt ~J tJr- cS)tA da--~ )y ~)~4IS~J101~ 1 f
-t 44f 4l J j1~) lgt ~ 4 (S~)SJ 1 j6J j (S) tA Jgtb ~ -A
)
-1 ciI ~JI ~J~ 4)) 6 ~31 t$lu ~_ ~
u4gt-~ 4 ))yUI) ~I~ r J lgt-y (SlA(S)lA ))y)))1~ ~lh
~)lgt ~jl I) amp Jgtb ~ 01r ~ cs~lA ~4 4 Q)~~ oJ (shy~ lA 1--J lJb ~ ~~ )la)) I) cS) lA ~ da-- 4 J I ~--d ij~ bull) 4l Ji 1
bullliJJT~ )~J~ da--Jl~)) osL ~JI I~ ~~Jk-JL ~~ 1
tJI~ clJ u~~ (s)u da--~~ rlfl 4ScS)L Jgtb u~JI J-i~ ~I
middotJrAd Jgtb (S~JI J-i~ )~ J--J ~j~ (S)~ Cb11 JJ ~1cs~Jtgt
middotJY-if ~4j OJtAI1AplA uJlAgt[ ~lgt Lt11 bulliL U Co t
u )r 4~ hJ~ (f J ~~if I) ~~ ~) r I J-lS lSLAJ ~ ~~ )~
middotr~jbAU-- ~r~~ ~~~ Je~ ~tJl41~~leraquo
)~ I u l~ U-- 0 4 lSI I~) 4 0J~ lSJ ~lS~ 14l 4 (-y )J~ ~ b r- J uo ()lgt)~ cs~J5J I ~1~ v ~~ I q lS[sbAi cs~J~1 l~ 0J~ Jgtb ~4
Ji5~l-gt I) o dJ ~)b ~J~ J-J 0)gt J~~ -I-~ )ifJ~~ JJI olS~ ~
~4 q l~ 0~ gt~J~csA (qflTE~rf)aT rIr0~(bJ) iJ1-o ~ c-d)J~1
~)J~ ~ oJgt jl 0Jr )~ iJl~ 4~)~ )~J~ ~ oraquo~ lSJ)( -q It ld
JAoP 0J4gt jl (lgt)~ 0raquo ltk ($JJ (- q) 4 J q cgt14Ai~ 1 J~tgt ~I~ ~
~gtI~)k 01u~Jlgtek-cgtJJ (+q))~oil u~tf)~ 0J~ 0J)7J-l
~)Jis (q17TEor lt )ar r 1J t oS~)lgt)) ~1~ ~)~J )~J ($ ~j
p
-t 43Jf )I) J lgt~ cgt~~ 4)~ Q)~ 45JJ~)2)) t)(-l)~ Y-y JU4
)4 r q 4 4))Ji yen~Lgt ~ (SA 4-~ 0Ijl )J) ~ ~li )) q )~ J
~W u )IJ q r Q )~ 4)J~~ J J5~ ))IJ Q
jl J lgt ~~lJ iJ1y vu It cLl r tr xtis ~)rJ Q J~ q )~ ~~Jd ~
~)b)IJj tS~lJll jl ~)b q)~ 07 ~~ Q)~ )1~ )~J Q )l~ Jgt-- ))q)~
0~ bull5 (5 ))IJ Qr dJri ~~ q slaquo lil --4 w~ ~~I u il 1-- ($~l Jgt)))
ltS~~ ~)Lgt c-b ltSJ) (+Q)J cs6b ~ ltsJ (-Q))~ Wl ~L Q )~
R
r_y J~ r-r ~
q J~ j) ~ 11ze l )~ 1 ~~ loS) LA u~ j lgt amp lt$J j Q j ~ 1iJ )ss J Igt 3
~ I~ 4)u ~ )~) (51gt ~~1~ l I J J) (5 ) j IJ J ji q sJ) Q j l~ J ))Su
j~ Ir Jd 01 j Ju ~j~ ~ Ij) W)) s f~ j ~ 4 ~)JP~ Ij s) l-Al
bullt~~ j~J) jDIJ hgt )Jx1 sraquo ~I C4~ to)) is qQflTEoRY
bullJSss )) JJ Ji s)lA ~ J q j ~ 4~) f J-I -- li ~11
~L ~~f J~ r~f f-f
~I bullloli )~j) )Ijl ~JI)~ sl)b W~J ~ ~SJI ~~ 4 r-RS
~ j JP~ $ tS j t=i Jr )) 4- bullL j b J j~ ~~ j) s)~ j J t-I )j J L~ I
lA) y )~~ I j) ~~ ~-J ~) ~~ tJs j ~i J A I bb- j) Jr s LAo
~lamp- Cl64 J ~J sJj ~~J5JJ slAjJ I 45 ~~)I Jr j) 4~ J
ltS l J J-Ai lA~~ )j L ~ u~SJ 1 rP~ ~a L )) ~ J JJY u 4gt l ou J j
4i~ ~ bullul~ ~ ~T~Jl ~1JA~rP to~~ laquos ~~ ~S~~
bullu~ 4IJi )J~ J ~T j) s)ljT)l~
~jlgt u~rgtJ I~I~J) IS Ij(Jt~ ~) lAo ~ 4i -~ I ~ -0lj)
~ ~ ~ j~~ LA~J5JI 41 01 j) orlaquosr u )~ Ji~ ~jl
l~~T ~I Jl~ ~ 4i cJ1 slA~1 ~~l5 )ljJ) j) -=-~ )J)4 J jlo~
-Ie bulltS) ~ 1 cs~ j~ I ) I~- (f I j I ()Lgt) ~_A j J J) jP J )) 10~~
ZJ- ~Jj slr olmiddot~ l~1 orgt jf C)la j) ~j-ltJ I ~Il ~ J ooU uJ 41
~~I iJl~ ~~ d pJJ IJ iJTJ r-~ fuJJ IJ U 4j l OJL Jgt Jl ~~ 41_shy
~cr csMJr JJ~Y r tSJJ IJ j1~~1 ~Ih J ~J~ JI
~~L U~ ~ ~ U9~~J~f-f
tS~ ~SJI tSJfllG~J JJb JI ljitSla ~ ~JJ oJL rl4 r-d
~ JJ ~ J~ ~ ~ 4_su ~ ~J~~ ~I Jl_A J)Ili JJ-~J ~ JJ
~JJp~IJJ ~~OJf ~ IJ Rtk~ 01-q ~AJ1~ l~ J J-t)j)t4 ~ lAjJ5Jl
X)U4j~ j1-4- R t k jl ~Jtgt J~ ril ~f Jgt- J ~ 1 jt_~ L~J~ jAjSf
JJJT~ JYJ~ f ~Jtgt JJ ~~rSJ1 )1 ~ ~
ril ~ raquo ~JJI E ~IJS u5~rgtJ 1jl~ 4 JJ I ~t 0i 5 1Jl~
xJb ~J~ ~1~jJ~ Ix J E j1~A ~ ~~~ JJr-j ~~I ~~A Jt ltoSIJb 4
J~ (S J L~ 1E j IJ~ ir~ cj~J~ Jd
mQro
bull l c 0 z
- Electron Atom Electronic
polariution
eurot rooii -to 15 Q
+
Orientational H2O
polariution molecule
a pO -
-- + - --
- rmiddot- -Center of
electron cloud
ttJmiddot (a)
rmiddot (b)
UQOaJrmiddot - -Q t Representalion
P nQQd rmiddotd __Qt Representation
~ ~ ~~ lG ~llIgt ~~ W t-f J ~ J ~ ~ _~~)t tSJ~ 1 ss ~~ -uJ I
JylS I~ ~jl J Jjt~ j~ ~~ J- ltoStAJ~ ~ sf 1lJ
(- q) J (+q) J~JJ ~CoI 0-1 J~ ~JJ~ ~IJ) Lt d ~t ~I ~d
t~ jl ~ J tgt J) r= ~- 1 6 ~b~ ~~JJ~ X~ ci~ )I ~jl d Jpli~
bull
~J~ 1I ~ i -fl ~P- J~oIJ~ ~ JSu yY ~I~ (fiji ~Jb) R
~~s- E ~I~ ~i ~~ s~-JTu ~s~J~ ~~ J~
(u~) ~~IjJ~ 1gt~1 r-- j l IJ f l uA=J ~ lS~J~ ~u I~
-b jl ~ c1J~ ~f~ rj J~4 ~u-- d 4I~l 4-~ Ir ~~ sshy
r 1 w j1 su~ tlJfti J1t ~~ ~cs- )JlJ (+q) J~ ~ ~Jb 6J1~
~ ~~fis ~JIJ 6JTt uiJJ 1
qdE=-- (r- )
e fITpound aY o
(r-Y)
~11-- ~ ~t= r1 ~ tS1r- C XJ~ ~J~ ~SJI ~l- IJ qd -~J~
~~ a ~ J clis- ~ I ~J cski ~u I ~ l _~ J~ J ~ cs-- ~5J I
bullbl~cs--
(t-t)
bull1Sif J li~ ~ IA ($1 J - I~ OI 41 I eC ~ cs-- f I r~ V AS
J qd r1~ ~T Ii 4-)~f o~b ~t~i P )br ~ u~J~ ~~I ~t
~11A4)~ u~~1 ubiJ~ 4ol1 bull1~(S ~ J~ -)=~ uA )~ jl ~T~~
~)~ ~~~f IS ~JIJ T JJlt4 ~Tr ~r JI cs~~~JI
T=PxE (r-Y)
~ jl ~lgt- u~rSJ1 ~I~ dAcJTJ1 ~ J ~~ U ~JJJ~ ~ 4ebl 1 Ji ~~u-- JIJi ~Jt ~JA ~Tjl JJ~ ~I J~ IJ uJ=iJ)
~~f~~~r-tJ
~~J csA J0 ltljl d ~l- v~ 1uJJ lSJL J~J~ jl~SJ ~J) It
JJ) ~ ~I J) I) ulU J) 4ol ~~5J I ~~ J~r~J1 ~I~ ~Igtu-
(-~)J~ (r~d) ~ ~~
EP (field point) (a) P r-(z) ~E
~ - -- bull ~--- E ~r r I 1 -_~~ dlq
(~o)
~ - ----- --- - I-q~ I r) (y)
(a) (II)
d R~r--eos8 r
q ( ) V(r)= -- shyfl1fo R s
deos8 d Y
rlt -eosf8 f
~4J ~~ -L df
rlt ~ - COS 8 f
(r--o)
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
4ebl~r- ~Jgt~ tAJt~ ~T~))5r J Jj~~ JI ~I~ ~5j1 cSJ~
igt1 ~ l~ JgtI) )) )r-gt- ~ bullL I~ (5 t-J 4 x-H (S) tA cSjr da--~ li JA)ushy
5)~ cS)u- ~I~ ~I bullc(fSlJy-t Jt~1 ~ll- w~ ~))~J ~lgtOIJ) bull))J~ ~I J)u ampJlgt-~ ))1LAia cS)lAJ lAJ~ dr t)~u
~ cSlA Jgtb cSlA~ J-tN ~6-J) s1- lo~~ cS)lA Jgtb ) ~Ilshy
ltS1b (S)tA obtAi ~w J~) ~J~~ 4J ~I ~ l~iJTJ JcSJd u~ -lSu4l
~SJI iJ1-l-- bhgt~t rAC~ ~~~J~~ ~J Ji~fS ~~ 4 bullJiy4f ~jlgt 4 IJ cS)tA cb ~ )w ~)~~
fcjlAi ~lgt)) lA)uj1j ~jb ~r~J1 iJl~ 4 j) Ij (S)lA 4 ~J~
)4 cS)lA JgtI) J) J JJY~ ~ (s)u ck cSJ) ~ J ~ cSlA)~ _
~I~
c-I ~ cS)lA Jgtb )) ~JSj 101~ ~u_y
~I ~l ~~ cSl)b cS)lA ~ T
Jy~rgt ~J tJr- cS)tA da--~ )y ~)~4IS~J101~ 1 f
-t 44f 4l J j1~) lgt ~ 4 (S~)SJ 1 j6J j (S) tA Jgtb ~ -A
)
-1 ciI ~JI ~J~ 4)) 6 ~31 t$lu ~_ ~
u4gt-~ 4 ))yUI) ~I~ r J lgt-y (SlA(S)lA ))y)))1~ ~lh
~)lgt ~jl I) amp Jgtb ~ 01r ~ cs~lA ~4 4 Q)~~ oJ (shy~ lA 1--J lJb ~ ~~ )la)) I) cS) lA ~ da-- 4 J I ~--d ij~ bull) 4l Ji 1
bullliJJT~ )~J~ da--Jl~)) osL ~JI I~ ~~Jk-JL ~~ 1
tJI~ clJ u~~ (s)u da--~~ rlfl 4ScS)L Jgtb u~JI J-i~ ~I
middotJrAd Jgtb (S~JI J-i~ )~ J--J ~j~ (S)~ Cb11 JJ ~1cs~Jtgt
middotJY-if ~4j OJtAI1AplA uJlAgt[ ~lgt Lt11 bulliL U Co t
u )r 4~ hJ~ (f J ~~if I) ~~ ~) r I J-lS lSLAJ ~ ~~ )~
middotr~jbAU-- ~r~~ ~~~ Je~ ~tJl41~~leraquo
)~ I u l~ U-- 0 4 lSI I~) 4 0J~ lSJ ~lS~ 14l 4 (-y )J~ ~ b r- J uo ()lgt)~ cs~J5J I ~1~ v ~~ I q lS[sbAi cs~J~1 l~ 0J~ Jgtb ~4
Ji5~l-gt I) o dJ ~)b ~J~ J-J 0)gt J~~ -I-~ )ifJ~~ JJI olS~ ~
~4 q l~ 0~ gt~J~csA (qflTE~rf)aT rIr0~(bJ) iJ1-o ~ c-d)J~1
~)J~ ~ oJgt jl 0Jr )~ iJl~ 4~)~ )~J~ ~ oraquo~ lSJ)( -q It ld
JAoP 0J4gt jl (lgt)~ 0raquo ltk ($JJ (- q) 4 J q cgt14Ai~ 1 J~tgt ~I~ ~
~gtI~)k 01u~Jlgtek-cgtJJ (+q))~oil u~tf)~ 0J~ 0J)7J-l
~)Jis (q17TEor lt )ar r 1J t oS~)lgt)) ~1~ ~)~J )~J ($ ~j
p
-t 43Jf )I) J lgt~ cgt~~ 4)~ Q)~ 45JJ~)2)) t)(-l)~ Y-y JU4
)4 r q 4 4))Ji yen~Lgt ~ (SA 4-~ 0Ijl )J) ~ ~li )) q )~ J
~W u )IJ q r Q )~ 4)J~~ J J5~ ))IJ Q
jl J lgt ~~lJ iJ1y vu It cLl r tr xtis ~)rJ Q J~ q )~ ~~Jd ~
~)b)IJj tS~lJll jl ~)b q)~ 07 ~~ Q)~ )1~ )~J Q )l~ Jgt-- ))q)~
0~ bull5 (5 ))IJ Qr dJri ~~ q slaquo lil --4 w~ ~~I u il 1-- ($~l Jgt)))
ltS~~ ~)Lgt c-b ltSJ) (+Q)J cs6b ~ ltsJ (-Q))~ Wl ~L Q )~
R
r_y J~ r-r ~
q J~ j) ~ 11ze l )~ 1 ~~ loS) LA u~ j lgt amp lt$J j Q j ~ 1iJ )ss J Igt 3
~ I~ 4)u ~ )~) (51gt ~~1~ l I J J) (5 ) j IJ J ji q sJ) Q j l~ J ))Su
j~ Ir Jd 01 j Ju ~j~ ~ Ij) W)) s f~ j ~ 4 ~)JP~ Ij s) l-Al
bullt~~ j~J) jDIJ hgt )Jx1 sraquo ~I C4~ to)) is qQflTEoRY
bullJSss )) JJ Ji s)lA ~ J q j ~ 4~) f J-I -- li ~11
~L ~~f J~ r~f f-f
~I bullloli )~j) )Ijl ~JI)~ sl)b W~J ~ ~SJI ~~ 4 r-RS
~ j JP~ $ tS j t=i Jr )) 4- bullL j b J j~ ~~ j) s)~ j J t-I )j J L~ I
lA) y )~~ I j) ~~ ~-J ~) ~~ tJs j ~i J A I bb- j) Jr s LAo
~lamp- Cl64 J ~J sJj ~~J5JJ slAjJ I 45 ~~)I Jr j) 4~ J
ltS l J J-Ai lA~~ )j L ~ u~SJ 1 rP~ ~a L )) ~ J JJY u 4gt l ou J j
4i~ ~ bullul~ ~ ~T~Jl ~1JA~rP to~~ laquos ~~ ~S~~
bullu~ 4IJi )J~ J ~T j) s)ljT)l~
~jlgt u~rgtJ I~I~J) IS Ij(Jt~ ~) lAo ~ 4i -~ I ~ -0lj)
~ ~ ~ j~~ LA~J5JI 41 01 j) orlaquosr u )~ Ji~ ~jl
l~~T ~I Jl~ ~ 4i cJ1 slA~1 ~~l5 )ljJ) j) -=-~ )J)4 J jlo~
-Ie bulltS) ~ 1 cs~ j~ I ) I~- (f I j I ()Lgt) ~_A j J J) jP J )) 10~~
ZJ- ~Jj slr olmiddot~ l~1 orgt jf C)la j) ~j-ltJ I ~Il ~ J ooU uJ 41
~~I iJl~ ~~ d pJJ IJ iJTJ r-~ fuJJ IJ U 4j l OJL Jgt Jl ~~ 41_shy
~cr csMJr JJ~Y r tSJJ IJ j1~~1 ~Ih J ~J~ JI
~~L U~ ~ ~ U9~~J~f-f
tS~ ~SJI tSJfllG~J JJb JI ljitSla ~ ~JJ oJL rl4 r-d
~ JJ ~ J~ ~ ~ 4_su ~ ~J~~ ~I Jl_A J)Ili JJ-~J ~ JJ
~JJp~IJJ ~~OJf ~ IJ Rtk~ 01-q ~AJ1~ l~ J J-t)j)t4 ~ lAjJ5Jl
X)U4j~ j1-4- R t k jl ~Jtgt J~ ril ~f Jgt- J ~ 1 jt_~ L~J~ jAjSf
JJJT~ JYJ~ f ~Jtgt JJ ~~rSJ1 )1 ~ ~
ril ~ raquo ~JJI E ~IJS u5~rgtJ 1jl~ 4 JJ I ~t 0i 5 1Jl~
xJb ~J~ ~1~jJ~ Ix J E j1~A ~ ~~~ JJr-j ~~I ~~A Jt ltoSIJb 4
J~ (S J L~ 1E j IJ~ ir~ cj~J~ Jd
mQro
bull l c 0 z
- Electron Atom Electronic
polariution
eurot rooii -to 15 Q
+
Orientational H2O
polariution molecule
a pO -
-- + - --
- rmiddot- -Center of
electron cloud
ttJmiddot (a)
rmiddot (b)
UQOaJrmiddot - -Q t Representalion
P nQQd rmiddotd __Qt Representation
~ ~ ~~ lG ~llIgt ~~ W t-f J ~ J ~ ~ _~~)t tSJ~ 1 ss ~~ -uJ I
JylS I~ ~jl J Jjt~ j~ ~~ J- ltoStAJ~ ~ sf 1lJ
(- q) J (+q) J~JJ ~CoI 0-1 J~ ~JJ~ ~IJ) Lt d ~t ~I ~d
t~ jl ~ J tgt J) r= ~- 1 6 ~b~ ~~JJ~ X~ ci~ )I ~jl d Jpli~
bull
~J~ 1I ~ i -fl ~P- J~oIJ~ ~ JSu yY ~I~ (fiji ~Jb) R
~~s- E ~I~ ~i ~~ s~-JTu ~s~J~ ~~ J~
(u~) ~~IjJ~ 1gt~1 r-- j l IJ f l uA=J ~ lS~J~ ~u I~
-b jl ~ c1J~ ~f~ rj J~4 ~u-- d 4I~l 4-~ Ir ~~ sshy
r 1 w j1 su~ tlJfti J1t ~~ ~cs- )JlJ (+q) J~ ~ ~Jb 6J1~
~ ~~fis ~JIJ 6JTt uiJJ 1
qdE=-- (r- )
e fITpound aY o
(r-Y)
~11-- ~ ~t= r1 ~ tS1r- C XJ~ ~J~ ~SJI ~l- IJ qd -~J~
~~ a ~ J clis- ~ I ~J cski ~u I ~ l _~ J~ J ~ cs-- ~5J I
bullbl~cs--
(t-t)
bull1Sif J li~ ~ IA ($1 J - I~ OI 41 I eC ~ cs-- f I r~ V AS
J qd r1~ ~T Ii 4-)~f o~b ~t~i P )br ~ u~J~ ~~I ~t
~11A4)~ u~~1 ubiJ~ 4ol1 bull1~(S ~ J~ -)=~ uA )~ jl ~T~~
~)~ ~~~f IS ~JIJ T JJlt4 ~Tr ~r JI cs~~~JI
T=PxE (r-Y)
~ jl ~lgt- u~rSJ1 ~I~ dAcJTJ1 ~ J ~~ U ~JJJ~ ~ 4ebl 1 Ji ~~u-- JIJi ~Jt ~JA ~Tjl JJ~ ~I J~ IJ uJ=iJ)
~~f~~~r-tJ
~~J csA J0 ltljl d ~l- v~ 1uJJ lSJL J~J~ jl~SJ ~J) It
JJ) ~ ~I J) I) ulU J) 4ol ~~5J I ~~ J~r~J1 ~I~ ~Igtu-
(-~)J~ (r~d) ~ ~~
EP (field point) (a) P r-(z) ~E
~ - -- bull ~--- E ~r r I 1 -_~~ dlq
(~o)
~ - ----- --- - I-q~ I r) (y)
(a) (II)
d R~r--eos8 r
q ( ) V(r)= -- shyfl1fo R s
deos8 d Y
rlt -eosf8 f
~4J ~~ -L df
rlt ~ - COS 8 f
(r--o)
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
middotJY-if ~4j OJtAI1AplA uJlAgt[ ~lgt Lt11 bulliL U Co t
u )r 4~ hJ~ (f J ~~if I) ~~ ~) r I J-lS lSLAJ ~ ~~ )~
middotr~jbAU-- ~r~~ ~~~ Je~ ~tJl41~~leraquo
)~ I u l~ U-- 0 4 lSI I~) 4 0J~ lSJ ~lS~ 14l 4 (-y )J~ ~ b r- J uo ()lgt)~ cs~J5J I ~1~ v ~~ I q lS[sbAi cs~J~1 l~ 0J~ Jgtb ~4
Ji5~l-gt I) o dJ ~)b ~J~ J-J 0)gt J~~ -I-~ )ifJ~~ JJI olS~ ~
~4 q l~ 0~ gt~J~csA (qflTE~rf)aT rIr0~(bJ) iJ1-o ~ c-d)J~1
~)J~ ~ oJgt jl 0Jr )~ iJl~ 4~)~ )~J~ ~ oraquo~ lSJ)( -q It ld
JAoP 0J4gt jl (lgt)~ 0raquo ltk ($JJ (- q) 4 J q cgt14Ai~ 1 J~tgt ~I~ ~
~gtI~)k 01u~Jlgtek-cgtJJ (+q))~oil u~tf)~ 0J~ 0J)7J-l
~)Jis (q17TEor lt )ar r 1J t oS~)lgt)) ~1~ ~)~J )~J ($ ~j
p
-t 43Jf )I) J lgt~ cgt~~ 4)~ Q)~ 45JJ~)2)) t)(-l)~ Y-y JU4
)4 r q 4 4))Ji yen~Lgt ~ (SA 4-~ 0Ijl )J) ~ ~li )) q )~ J
~W u )IJ q r Q )~ 4)J~~ J J5~ ))IJ Q
jl J lgt ~~lJ iJ1y vu It cLl r tr xtis ~)rJ Q J~ q )~ ~~Jd ~
~)b)IJj tS~lJll jl ~)b q)~ 07 ~~ Q)~ )1~ )~J Q )l~ Jgt-- ))q)~
0~ bull5 (5 ))IJ Qr dJri ~~ q slaquo lil --4 w~ ~~I u il 1-- ($~l Jgt)))
ltS~~ ~)Lgt c-b ltSJ) (+Q)J cs6b ~ ltsJ (-Q))~ Wl ~L Q )~
R
r_y J~ r-r ~
q J~ j) ~ 11ze l )~ 1 ~~ loS) LA u~ j lgt amp lt$J j Q j ~ 1iJ )ss J Igt 3
~ I~ 4)u ~ )~) (51gt ~~1~ l I J J) (5 ) j IJ J ji q sJ) Q j l~ J ))Su
j~ Ir Jd 01 j Ju ~j~ ~ Ij) W)) s f~ j ~ 4 ~)JP~ Ij s) l-Al
bullt~~ j~J) jDIJ hgt )Jx1 sraquo ~I C4~ to)) is qQflTEoRY
bullJSss )) JJ Ji s)lA ~ J q j ~ 4~) f J-I -- li ~11
~L ~~f J~ r~f f-f
~I bullloli )~j) )Ijl ~JI)~ sl)b W~J ~ ~SJI ~~ 4 r-RS
~ j JP~ $ tS j t=i Jr )) 4- bullL j b J j~ ~~ j) s)~ j J t-I )j J L~ I
lA) y )~~ I j) ~~ ~-J ~) ~~ tJs j ~i J A I bb- j) Jr s LAo
~lamp- Cl64 J ~J sJj ~~J5JJ slAjJ I 45 ~~)I Jr j) 4~ J
ltS l J J-Ai lA~~ )j L ~ u~SJ 1 rP~ ~a L )) ~ J JJY u 4gt l ou J j
4i~ ~ bullul~ ~ ~T~Jl ~1JA~rP to~~ laquos ~~ ~S~~
bullu~ 4IJi )J~ J ~T j) s)ljT)l~
~jlgt u~rgtJ I~I~J) IS Ij(Jt~ ~) lAo ~ 4i -~ I ~ -0lj)
~ ~ ~ j~~ LA~J5JI 41 01 j) orlaquosr u )~ Ji~ ~jl
l~~T ~I Jl~ ~ 4i cJ1 slA~1 ~~l5 )ljJ) j) -=-~ )J)4 J jlo~
-Ie bulltS) ~ 1 cs~ j~ I ) I~- (f I j I ()Lgt) ~_A j J J) jP J )) 10~~
ZJ- ~Jj slr olmiddot~ l~1 orgt jf C)la j) ~j-ltJ I ~Il ~ J ooU uJ 41
~~I iJl~ ~~ d pJJ IJ iJTJ r-~ fuJJ IJ U 4j l OJL Jgt Jl ~~ 41_shy
~cr csMJr JJ~Y r tSJJ IJ j1~~1 ~Ih J ~J~ JI
~~L U~ ~ ~ U9~~J~f-f
tS~ ~SJI tSJfllG~J JJb JI ljitSla ~ ~JJ oJL rl4 r-d
~ JJ ~ J~ ~ ~ 4_su ~ ~J~~ ~I Jl_A J)Ili JJ-~J ~ JJ
~JJp~IJJ ~~OJf ~ IJ Rtk~ 01-q ~AJ1~ l~ J J-t)j)t4 ~ lAjJ5Jl
X)U4j~ j1-4- R t k jl ~Jtgt J~ ril ~f Jgt- J ~ 1 jt_~ L~J~ jAjSf
JJJT~ JYJ~ f ~Jtgt JJ ~~rSJ1 )1 ~ ~
ril ~ raquo ~JJI E ~IJS u5~rgtJ 1jl~ 4 JJ I ~t 0i 5 1Jl~
xJb ~J~ ~1~jJ~ Ix J E j1~A ~ ~~~ JJr-j ~~I ~~A Jt ltoSIJb 4
J~ (S J L~ 1E j IJ~ ir~ cj~J~ Jd
mQro
bull l c 0 z
- Electron Atom Electronic
polariution
eurot rooii -to 15 Q
+
Orientational H2O
polariution molecule
a pO -
-- + - --
- rmiddot- -Center of
electron cloud
ttJmiddot (a)
rmiddot (b)
UQOaJrmiddot - -Q t Representalion
P nQQd rmiddotd __Qt Representation
~ ~ ~~ lG ~llIgt ~~ W t-f J ~ J ~ ~ _~~)t tSJ~ 1 ss ~~ -uJ I
JylS I~ ~jl J Jjt~ j~ ~~ J- ltoStAJ~ ~ sf 1lJ
(- q) J (+q) J~JJ ~CoI 0-1 J~ ~JJ~ ~IJ) Lt d ~t ~I ~d
t~ jl ~ J tgt J) r= ~- 1 6 ~b~ ~~JJ~ X~ ci~ )I ~jl d Jpli~
bull
~J~ 1I ~ i -fl ~P- J~oIJ~ ~ JSu yY ~I~ (fiji ~Jb) R
~~s- E ~I~ ~i ~~ s~-JTu ~s~J~ ~~ J~
(u~) ~~IjJ~ 1gt~1 r-- j l IJ f l uA=J ~ lS~J~ ~u I~
-b jl ~ c1J~ ~f~ rj J~4 ~u-- d 4I~l 4-~ Ir ~~ sshy
r 1 w j1 su~ tlJfti J1t ~~ ~cs- )JlJ (+q) J~ ~ ~Jb 6J1~
~ ~~fis ~JIJ 6JTt uiJJ 1
qdE=-- (r- )
e fITpound aY o
(r-Y)
~11-- ~ ~t= r1 ~ tS1r- C XJ~ ~J~ ~SJI ~l- IJ qd -~J~
~~ a ~ J clis- ~ I ~J cski ~u I ~ l _~ J~ J ~ cs-- ~5J I
bullbl~cs--
(t-t)
bull1Sif J li~ ~ IA ($1 J - I~ OI 41 I eC ~ cs-- f I r~ V AS
J qd r1~ ~T Ii 4-)~f o~b ~t~i P )br ~ u~J~ ~~I ~t
~11A4)~ u~~1 ubiJ~ 4ol1 bull1~(S ~ J~ -)=~ uA )~ jl ~T~~
~)~ ~~~f IS ~JIJ T JJlt4 ~Tr ~r JI cs~~~JI
T=PxE (r-Y)
~ jl ~lgt- u~rSJ1 ~I~ dAcJTJ1 ~ J ~~ U ~JJJ~ ~ 4ebl 1 Ji ~~u-- JIJi ~Jt ~JA ~Tjl JJ~ ~I J~ IJ uJ=iJ)
~~f~~~r-tJ
~~J csA J0 ltljl d ~l- v~ 1uJJ lSJL J~J~ jl~SJ ~J) It
JJ) ~ ~I J) I) ulU J) 4ol ~~5J I ~~ J~r~J1 ~I~ ~Igtu-
(-~)J~ (r~d) ~ ~~
EP (field point) (a) P r-(z) ~E
~ - -- bull ~--- E ~r r I 1 -_~~ dlq
(~o)
~ - ----- --- - I-q~ I r) (y)
(a) (II)
d R~r--eos8 r
q ( ) V(r)= -- shyfl1fo R s
deos8 d Y
rlt -eosf8 f
~4J ~~ -L df
rlt ~ - COS 8 f
(r--o)
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
R
r_y J~ r-r ~
q J~ j) ~ 11ze l )~ 1 ~~ loS) LA u~ j lgt amp lt$J j Q j ~ 1iJ )ss J Igt 3
~ I~ 4)u ~ )~) (51gt ~~1~ l I J J) (5 ) j IJ J ji q sJ) Q j l~ J ))Su
j~ Ir Jd 01 j Ju ~j~ ~ Ij) W)) s f~ j ~ 4 ~)JP~ Ij s) l-Al
bullt~~ j~J) jDIJ hgt )Jx1 sraquo ~I C4~ to)) is qQflTEoRY
bullJSss )) JJ Ji s)lA ~ J q j ~ 4~) f J-I -- li ~11
~L ~~f J~ r~f f-f
~I bullloli )~j) )Ijl ~JI)~ sl)b W~J ~ ~SJI ~~ 4 r-RS
~ j JP~ $ tS j t=i Jr )) 4- bullL j b J j~ ~~ j) s)~ j J t-I )j J L~ I
lA) y )~~ I j) ~~ ~-J ~) ~~ tJs j ~i J A I bb- j) Jr s LAo
~lamp- Cl64 J ~J sJj ~~J5JJ slAjJ I 45 ~~)I Jr j) 4~ J
ltS l J J-Ai lA~~ )j L ~ u~SJ 1 rP~ ~a L )) ~ J JJY u 4gt l ou J j
4i~ ~ bullul~ ~ ~T~Jl ~1JA~rP to~~ laquos ~~ ~S~~
bullu~ 4IJi )J~ J ~T j) s)ljT)l~
~jlgt u~rgtJ I~I~J) IS Ij(Jt~ ~) lAo ~ 4i -~ I ~ -0lj)
~ ~ ~ j~~ LA~J5JI 41 01 j) orlaquosr u )~ Ji~ ~jl
l~~T ~I Jl~ ~ 4i cJ1 slA~1 ~~l5 )ljJ) j) -=-~ )J)4 J jlo~
-Ie bulltS) ~ 1 cs~ j~ I ) I~- (f I j I ()Lgt) ~_A j J J) jP J )) 10~~
ZJ- ~Jj slr olmiddot~ l~1 orgt jf C)la j) ~j-ltJ I ~Il ~ J ooU uJ 41
~~I iJl~ ~~ d pJJ IJ iJTJ r-~ fuJJ IJ U 4j l OJL Jgt Jl ~~ 41_shy
~cr csMJr JJ~Y r tSJJ IJ j1~~1 ~Ih J ~J~ JI
~~L U~ ~ ~ U9~~J~f-f
tS~ ~SJI tSJfllG~J JJb JI ljitSla ~ ~JJ oJL rl4 r-d
~ JJ ~ J~ ~ ~ 4_su ~ ~J~~ ~I Jl_A J)Ili JJ-~J ~ JJ
~JJp~IJJ ~~OJf ~ IJ Rtk~ 01-q ~AJ1~ l~ J J-t)j)t4 ~ lAjJ5Jl
X)U4j~ j1-4- R t k jl ~Jtgt J~ ril ~f Jgt- J ~ 1 jt_~ L~J~ jAjSf
JJJT~ JYJ~ f ~Jtgt JJ ~~rSJ1 )1 ~ ~
ril ~ raquo ~JJI E ~IJS u5~rgtJ 1jl~ 4 JJ I ~t 0i 5 1Jl~
xJb ~J~ ~1~jJ~ Ix J E j1~A ~ ~~~ JJr-j ~~I ~~A Jt ltoSIJb 4
J~ (S J L~ 1E j IJ~ ir~ cj~J~ Jd
mQro
bull l c 0 z
- Electron Atom Electronic
polariution
eurot rooii -to 15 Q
+
Orientational H2O
polariution molecule
a pO -
-- + - --
- rmiddot- -Center of
electron cloud
ttJmiddot (a)
rmiddot (b)
UQOaJrmiddot - -Q t Representalion
P nQQd rmiddotd __Qt Representation
~ ~ ~~ lG ~llIgt ~~ W t-f J ~ J ~ ~ _~~)t tSJ~ 1 ss ~~ -uJ I
JylS I~ ~jl J Jjt~ j~ ~~ J- ltoStAJ~ ~ sf 1lJ
(- q) J (+q) J~JJ ~CoI 0-1 J~ ~JJ~ ~IJ) Lt d ~t ~I ~d
t~ jl ~ J tgt J) r= ~- 1 6 ~b~ ~~JJ~ X~ ci~ )I ~jl d Jpli~
bull
~J~ 1I ~ i -fl ~P- J~oIJ~ ~ JSu yY ~I~ (fiji ~Jb) R
~~s- E ~I~ ~i ~~ s~-JTu ~s~J~ ~~ J~
(u~) ~~IjJ~ 1gt~1 r-- j l IJ f l uA=J ~ lS~J~ ~u I~
-b jl ~ c1J~ ~f~ rj J~4 ~u-- d 4I~l 4-~ Ir ~~ sshy
r 1 w j1 su~ tlJfti J1t ~~ ~cs- )JlJ (+q) J~ ~ ~Jb 6J1~
~ ~~fis ~JIJ 6JTt uiJJ 1
qdE=-- (r- )
e fITpound aY o
(r-Y)
~11-- ~ ~t= r1 ~ tS1r- C XJ~ ~J~ ~SJI ~l- IJ qd -~J~
~~ a ~ J clis- ~ I ~J cski ~u I ~ l _~ J~ J ~ cs-- ~5J I
bullbl~cs--
(t-t)
bull1Sif J li~ ~ IA ($1 J - I~ OI 41 I eC ~ cs-- f I r~ V AS
J qd r1~ ~T Ii 4-)~f o~b ~t~i P )br ~ u~J~ ~~I ~t
~11A4)~ u~~1 ubiJ~ 4ol1 bull1~(S ~ J~ -)=~ uA )~ jl ~T~~
~)~ ~~~f IS ~JIJ T JJlt4 ~Tr ~r JI cs~~~JI
T=PxE (r-Y)
~ jl ~lgt- u~rSJ1 ~I~ dAcJTJ1 ~ J ~~ U ~JJJ~ ~ 4ebl 1 Ji ~~u-- JIJi ~Jt ~JA ~Tjl JJ~ ~I J~ IJ uJ=iJ)
~~f~~~r-tJ
~~J csA J0 ltljl d ~l- v~ 1uJJ lSJL J~J~ jl~SJ ~J) It
JJ) ~ ~I J) I) ulU J) 4ol ~~5J I ~~ J~r~J1 ~I~ ~Igtu-
(-~)J~ (r~d) ~ ~~
EP (field point) (a) P r-(z) ~E
~ - -- bull ~--- E ~r r I 1 -_~~ dlq
(~o)
~ - ----- --- - I-q~ I r) (y)
(a) (II)
d R~r--eos8 r
q ( ) V(r)= -- shyfl1fo R s
deos8 d Y
rlt -eosf8 f
~4J ~~ -L df
rlt ~ - COS 8 f
(r--o)
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
~~I iJl~ ~~ d pJJ IJ iJTJ r-~ fuJJ IJ U 4j l OJL Jgt Jl ~~ 41_shy
~cr csMJr JJ~Y r tSJJ IJ j1~~1 ~Ih J ~J~ JI
~~L U~ ~ ~ U9~~J~f-f
tS~ ~SJI tSJfllG~J JJb JI ljitSla ~ ~JJ oJL rl4 r-d
~ JJ ~ J~ ~ ~ 4_su ~ ~J~~ ~I Jl_A J)Ili JJ-~J ~ JJ
~JJp~IJJ ~~OJf ~ IJ Rtk~ 01-q ~AJ1~ l~ J J-t)j)t4 ~ lAjJ5Jl
X)U4j~ j1-4- R t k jl ~Jtgt J~ ril ~f Jgt- J ~ 1 jt_~ L~J~ jAjSf
JJJT~ JYJ~ f ~Jtgt JJ ~~rSJ1 )1 ~ ~
ril ~ raquo ~JJI E ~IJS u5~rgtJ 1jl~ 4 JJ I ~t 0i 5 1Jl~
xJb ~J~ ~1~jJ~ Ix J E j1~A ~ ~~~ JJr-j ~~I ~~A Jt ltoSIJb 4
J~ (S J L~ 1E j IJ~ ir~ cj~J~ Jd
mQro
bull l c 0 z
- Electron Atom Electronic
polariution
eurot rooii -to 15 Q
+
Orientational H2O
polariution molecule
a pO -
-- + - --
- rmiddot- -Center of
electron cloud
ttJmiddot (a)
rmiddot (b)
UQOaJrmiddot - -Q t Representalion
P nQQd rmiddotd __Qt Representation
~ ~ ~~ lG ~llIgt ~~ W t-f J ~ J ~ ~ _~~)t tSJ~ 1 ss ~~ -uJ I
JylS I~ ~jl J Jjt~ j~ ~~ J- ltoStAJ~ ~ sf 1lJ
(- q) J (+q) J~JJ ~CoI 0-1 J~ ~JJ~ ~IJ) Lt d ~t ~I ~d
t~ jl ~ J tgt J) r= ~- 1 6 ~b~ ~~JJ~ X~ ci~ )I ~jl d Jpli~
bull
~J~ 1I ~ i -fl ~P- J~oIJ~ ~ JSu yY ~I~ (fiji ~Jb) R
~~s- E ~I~ ~i ~~ s~-JTu ~s~J~ ~~ J~
(u~) ~~IjJ~ 1gt~1 r-- j l IJ f l uA=J ~ lS~J~ ~u I~
-b jl ~ c1J~ ~f~ rj J~4 ~u-- d 4I~l 4-~ Ir ~~ sshy
r 1 w j1 su~ tlJfti J1t ~~ ~cs- )JlJ (+q) J~ ~ ~Jb 6J1~
~ ~~fis ~JIJ 6JTt uiJJ 1
qdE=-- (r- )
e fITpound aY o
(r-Y)
~11-- ~ ~t= r1 ~ tS1r- C XJ~ ~J~ ~SJI ~l- IJ qd -~J~
~~ a ~ J clis- ~ I ~J cski ~u I ~ l _~ J~ J ~ cs-- ~5J I
bullbl~cs--
(t-t)
bull1Sif J li~ ~ IA ($1 J - I~ OI 41 I eC ~ cs-- f I r~ V AS
J qd r1~ ~T Ii 4-)~f o~b ~t~i P )br ~ u~J~ ~~I ~t
~11A4)~ u~~1 ubiJ~ 4ol1 bull1~(S ~ J~ -)=~ uA )~ jl ~T~~
~)~ ~~~f IS ~JIJ T JJlt4 ~Tr ~r JI cs~~~JI
T=PxE (r-Y)
~ jl ~lgt- u~rSJ1 ~I~ dAcJTJ1 ~ J ~~ U ~JJJ~ ~ 4ebl 1 Ji ~~u-- JIJi ~Jt ~JA ~Tjl JJ~ ~I J~ IJ uJ=iJ)
~~f~~~r-tJ
~~J csA J0 ltljl d ~l- v~ 1uJJ lSJL J~J~ jl~SJ ~J) It
JJ) ~ ~I J) I) ulU J) 4ol ~~5J I ~~ J~r~J1 ~I~ ~Igtu-
(-~)J~ (r~d) ~ ~~
EP (field point) (a) P r-(z) ~E
~ - -- bull ~--- E ~r r I 1 -_~~ dlq
(~o)
~ - ----- --- - I-q~ I r) (y)
(a) (II)
d R~r--eos8 r
q ( ) V(r)= -- shyfl1fo R s
deos8 d Y
rlt -eosf8 f
~4J ~~ -L df
rlt ~ - COS 8 f
(r--o)
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
bull
~J~ 1I ~ i -fl ~P- J~oIJ~ ~ JSu yY ~I~ (fiji ~Jb) R
~~s- E ~I~ ~i ~~ s~-JTu ~s~J~ ~~ J~
(u~) ~~IjJ~ 1gt~1 r-- j l IJ f l uA=J ~ lS~J~ ~u I~
-b jl ~ c1J~ ~f~ rj J~4 ~u-- d 4I~l 4-~ Ir ~~ sshy
r 1 w j1 su~ tlJfti J1t ~~ ~cs- )JlJ (+q) J~ ~ ~Jb 6J1~
~ ~~fis ~JIJ 6JTt uiJJ 1
qdE=-- (r- )
e fITpound aY o
(r-Y)
~11-- ~ ~t= r1 ~ tS1r- C XJ~ ~J~ ~SJI ~l- IJ qd -~J~
~~ a ~ J clis- ~ I ~J cski ~u I ~ l _~ J~ J ~ cs-- ~5J I
bullbl~cs--
(t-t)
bull1Sif J li~ ~ IA ($1 J - I~ OI 41 I eC ~ cs-- f I r~ V AS
J qd r1~ ~T Ii 4-)~f o~b ~t~i P )br ~ u~J~ ~~I ~t
~11A4)~ u~~1 ubiJ~ 4ol1 bull1~(S ~ J~ -)=~ uA )~ jl ~T~~
~)~ ~~~f IS ~JIJ T JJlt4 ~Tr ~r JI cs~~~JI
T=PxE (r-Y)
~ jl ~lgt- u~rSJ1 ~I~ dAcJTJ1 ~ J ~~ U ~JJJ~ ~ 4ebl 1 Ji ~~u-- JIJi ~Jt ~JA ~Tjl JJ~ ~I J~ IJ uJ=iJ)
~~f~~~r-tJ
~~J csA J0 ltljl d ~l- v~ 1uJJ lSJL J~J~ jl~SJ ~J) It
JJ) ~ ~I J) I) ulU J) 4ol ~~5J I ~~ J~r~J1 ~I~ ~Igtu-
(-~)J~ (r~d) ~ ~~
EP (field point) (a) P r-(z) ~E
~ - -- bull ~--- E ~r r I 1 -_~~ dlq
(~o)
~ - ----- --- - I-q~ I r) (y)
(a) (II)
d R~r--eos8 r
q ( ) V(r)= -- shyfl1fo R s
deos8 d Y
rlt -eosf8 f
~4J ~~ -L df
rlt ~ - COS 8 f
(r--o)
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
EP (field point) (a) P r-(z) ~E
~ - -- bull ~--- E ~r r I 1 -_~~ dlq
(~o)
~ - ----- --- - I-q~ I r) (y)
(a) (II)
d R~r--eos8 r
q ( ) V(r)= -- shyfl1fo R s
deos8 d Y
rlt -eosf8 f
~4J ~~ -L df
rlt ~ - COS 8 f
(r--o)
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
LiJ
av aVE=-VV=--8 ---aar r dB r
sqd cos 8 +qd sin 8= a ayen1TE r( r Y1Tpound rr fJ
o 0
(-v)
J5U- )~I 1) dl~ )~gt jl )J) Jpli)) ~SJ 1~J) 4 r~u ~
~gt- --i ~ li (r I ~ J )) b ~T~ J u-ki -) ~ l oj 11 I- cs~-~ (
~~ ~~~ ~iJ 0~ 41pJ) J- ~r~lgt ~J iJL 4 I)~-)I iJ ))
(P) ~) u
)gt j )J) AJpti)) iJ~ I tSl)b iJ~ ~~I ~t 4 cJJJ) J) u=
o~ ~~ -S~Jl~ Jf )~ ~J1 u1~ -~
~)) xsu j ~J) ~ ~ (15J-A oJ u~ki J~~ r~ ~ l$L~J~ bull))Jcs-- )~J~ ~SJI iJIJ-- (JsJ jl ()6-) )J) ~L)) ow ~ JsL
- tAJSJ )~gt ~~I 4 ~T)~J~ ~Jtgt ~Il-- 4 JIJ ~1 sect-o r iJl u~
(lA~J~I)cJ-- lStA)~ J (~) ~~)t~ ~)) cli JotgtiJ)tA l$1)b
t~ u-J ~I olo ~bi ~ ~lgt)) ~ ~)oJ UIJ) ~~li ~ rJl)r
)~~I J ogti tSlgtIJ ~~ lSI)) uiJ1r JW ~I~)) lAukiJ)~1 ~L
vi~ )) ~~ I X JJT~ )ftJ~ ~JgtJI )1 4JL )J s lS uo csgtshy
iJl~ ~)) jlo~ csbi siJT~~J~~ )$ )I ~Jl-gt- ~rltJ ~l-- ~
Jjy~ c~ ~ 4 ~JsL1 J~tgt l$1AiJl-4a ~J) J middotJJuo ~o 4$S~rltJ1
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
AJ~ ~ I ow ~ ($tAJftmiddot ~)ytJ )J ~u- ~ ~ ~Jl ~I~ J
r~ ~IJ )~ ~-~ n ($I)b oJ ~ ~Il j~IJ J~~ laquoJ1ycs P LAiJT
~ ~I rI J ~V cs~ r1 )J LA~kJJ JS~t- ~)gt--P ltJl)J w~
(-A)
r~)u u)r- ~I tlA~ jl j-f)~ ($~ojlil ($IJ~ IJ jL L ~ts 01 )J
lgtIJ )J cs~bJJ ~t iJ1y- I) (~n-Ij)~) ~t~ J~l 1iJ (~)L)
~ ~0rU ~
~middotIycs bull-IC r jA J Jsect1gtIJ ($IJIJ 4) ~llS ~ ~ uhi JbJ IJ P JIJJ
) ~~ P ~JJ ~ cj~ 4~ ($)bj~ laquoJl~4 ~)J4~ I~ P Jb~~ljl
I )J ~JJ ~I s J54-Y middot~4cs ~JJ J ~ ($ lAu1iJJ ($LAiJL ~TJ~
(JQ)~ ~) ~ )~ IJ LA)l~ li~1 bull1ilo~ J~I lAJ~- 1A J~ ~t)~
~bjU2il~l~ JJb 4k~1J laquoJT)l ~ J~ 4 cs~biJJ ~t ~($))zlA middotrJ~ u
cgt) 1 ~~ lil ~ 1iJ ~ ~IJ ~IJ ~ ($lA)~ J~ ~ jd P JbJ~ ~
~ 4 ))a 01 ($IJ ~)S)r-~ I) ~ ($LA)~ J P ~I~ ~I) ~1Jgtl5
f I ~ ~ ltS P = 0 J uk ($u b ~ cJ b)J U r-$(s )ai)~ IJ ~
LAJ ($LAJ~) O~ ($ki ~T($lAJJSJ ~J)I) cs~SJ1 ~I~ ~)J IJ~~I
~ Tlaquos JYJ~ ~SJ I J ~ J cs~ ~~ ~ J (j~cs 1~ jl laquoJ ~~ J )J
~ ~ ~ rAi ~~jl ~ lj J5~ lgt)J jljcs ~ c~~ p ~ ltlJ u hi
OJ ~~~)4)~ ~if)J- ~Igtcs J r-~cs p)J I) V ~gt ~ S ~
J~cs ~b 4 J ~Jlgt- ~ ~ jl
1 ~I~ JIoIJJ J JJ~cs uk lAJ~ JYff JlIJ cs~)lgt laquoJ1~iJ
4~ ~I u~~ ~~ S ~ Cb Jgtb ~ vaLgt)~ ~~ ($I J 1ir ~
~I d= Jgtb lAgtT csA J4 l ~~)~ lp ~j~ p)J I) JLAJ~--~ ~l)~JIJ~I~JTjIJlgt~ bJ~ u~ S ~ Jgtb jtgtJ~J I laquoJ~
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
JuJ~~ tampi 0 1r ~ ))11 S ~Jgt) ~)4 )) (S)I u~ ~J~~jl
~ 4 ~)~ ~ O)J~ ciS p)) I) lU))I-i s ~ cSjr ylgt )) 45
~b jl ~)lraquo) u~T~ 4 ~)~ ~~ ~)) Ju~ ~ Jgtb u~T~
cJl y lt$ ~lgtw JJ)~ P )) I) ~s J~ ugtwb ~ )JoI- ~)) bull uL~ s ) ~ )J-Ai
~ J ~)lgt lraquo~ rA ~r ~jl )ap~I)) c~ ~ (Sjly P ))r J1
~s ~ LA U LA~J) jl (S)I1-i )j~ ~Jlj ~s ~ p)~ I u-J )~~ J-gtb
N )~ u-- )I h) j)) u~)~ J ~ W~ ss LA~T~)~ x~u
)yu-- ~)lgt 4 J ))IJ ~s ~ jl ~ lgt )~ ~
8 + A~~_I r--~~+ +
--1 I--~ - tl 1
I L __Y~+-
~ ~- ) -8+ ~ ~ - rgt sgt
~~~J)~~~ P)) ~J) ~ )4 JJgt I) lAJ~rgtr- 11
~)lgt ttT ~)~ -gt~ ~b- OI LSI) ~~u-- (i cos (J)~ (~ )) ~
)I1IU 11 ~)) (~~ o)lgt (J ~Jlj JI) w~u ~ jgt-b J) u~)t~ J ~JI
~s eb ylraquo) 451) ~6 ~ J~ ull n ~gt 1gtIJ J) ol~ ukiJ) cs~J-Sl ~ ~1 r1) cil ai)) l~~
8Q =nQd cos 8~s =nQdmiddot ~s
l J
~Q =pds
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
(_0)
ampmiddot-=fmiddotmiddot v41J=-fJ d~ ~
(Y- r)
bull~ ~ tJ~ -wn y tlmiddot~~middotrt~f~tilaquo-AA tJ ~I cfl ~~
~ )~ ~laquof ~ Jmiddotlt 1 J~~ b~ ~ ~I ~)~ Wiamp ~~
--t ~ t~ 4to- I
Pbull t-)t -)~ ~f1 ~ 4L4 ~J ~J ~Ju- tS~VJ4f bullIt~ uJ) I
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
bull 1middot d 1 I~J ) JlU-1 V ~gt~ ~ S ~
~p (R-R) P(R-R) lV(R) = Y7TeurooiR- RI == ~1TeurooIR-RI dv (Y-)
~jl J
) R-R (V lR-R1 =+IR-R I
(-If)vmiddot(fA) =VmiddotmiddotA+AVf
( ) ( P) pv IR-R =V R-RI - lR-R1 V p
~
(Y- r)
p=poa ~~J~ o~ ubi f~i JPLgt U~JSJ JI~ r-rJlo (u4cs-Rojta) -)JT~~~ 1) bull fO))
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
z
P= -v- -vmiddot(middotbull)=middot
-bullbullbullmiddotIimiddotmiddotCOlI
Imiddot middotmiddotmiddot~~~middotImiddot bullbull middotmiddotaLJ --~-- ~ --~ - _~ - - -- --
middotmiddotmiddotJ~ifmiddotmiddotbull JampiIIl iM~-ampV iimiddotmiddotiampwllmiddotmiddotmiddot~middotmiddoti~~middotWJr-Ymiddot-Y
middotA~~__ ~Wmiddot~ ~w~jJ f~ ~f~middotAimiddotmiddotjt~iiJ_tWmiddottit ~~
lyen-ilif~rJjll~~jJJ
Vmiddota=yen (--Y)
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
J ~ljT ($lA)~ Ja~ P 5 uJ~ ~ IJ) u~ r )ljTISlAJ ~ ($IJI ~ J I ~ -C ~~ ~ ~ tAJ~
~ J
(t- A)
~r J~) ~J-( uD J~l 4$ ~~-5A VmiddotD=p tlkl~ J ~I~ u~1 4-lA ~ r~)
~I ~ c 1 ~w ~k 4-tf-5b~ ~IJ rA J )ljT yenISIJ-~ rA j~ U1
~J) ~ -Jl4i 4IJ 01 jl oJ)1 bullJA~u- ~)~ r I) O~JU ~lojT~l cI )Jp )ttD J E
ltS~~) lIJ 0-1 ~gtfl LS~$ )1JA bull)JI) u~lJ o)lA1~ -5~~ ~IJ Lr- I lL siYlaquof
~IlA ~ - L Jc~r J ~ ()ok J b J ~TJ) -se--I -5~ l uk ~ l~
e-I Jagtd ~l~ ~JJ-D ~IJ~ J) P=XeEoE ~~~ 0 Jl1 u~J1 J o)lo ~ ~ J p J ~ ~ Jl)~ ~TJ) s~I ~l~ ~JJr-1 ~~
JS~ J~ ~TJIJA ~Il~ ~ ~ ~ Jlid ff JI- ~I~ 4~ ~~ 4A
~ ~J~Jj1 J~ -~$ ~J-PL1I~J) ~ ~l ~JIi ~~ 4A J) Xe ~
~T~) ~tl41W~ jl ~T~SJI ~~~ 0~ I u~~ ~j~ ~~
~~~lyJ~ JbJ~ E ~1~csS~101~~ cgtlfl ) ~J~~ bullx$~zU ~ l ss 0ribd- ~~ ~ J J4~ Irf- J) J~ u cgt-I~ ~~~bull)La 1 -A J)
~~ ~I sect u~ JJ JIX ~~ ~ IJ ~6- 01 AS)f~Y ~ ~
~JJJi1 J 0j~Jzgt- LAO~ fsl Jw J) middotw4 0rf J ~JiJT-1 J craquoshybullJA
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
0
~ Lb~ ~ l~ o~~ ~~ I)-JjJjIJ In- (S LA~ ~ fra ~t)UIJ
~J 01J~ ~
(f-Yo)
J rA is ~ l~ e 4 J r-- t (SA CS-i LS~ SH laquoS ~ jA ~ 4 I) ( +x) o~t~ ~ ~ Ugtvpll ~l-gt -~ I) e
Ogt)~ J
D=eeE=eEo
J~~~ ~~ (fi~ ~) ~ ~T-~jAi -~ e
IJ ~I J o~ cP ~14lbA ~~)l- ~I)r-~ jl bull lA1 ~
~J J I~I ~)J~ 01Ji
v-D=p 4 PD-clS=Q (r-Y r)
v-p= -Pl p p-cls= -Q
~ J~ ~l J ~~Yu-W PI ljTJ~~IJ U f -Jir- j ~~ OIjlsf 41
~~ 0~ P ~ IJ J ~ 4JoJ J ~ltf p b
(agtb) b J a ~tAtW~ jr~ ~LA ~J jl ~Jr-Jjlgt 4 --fJ U~~l~ J ~I oJ ~ OEo u~~1 ~i~jJ4 -4 ~~ ~ J oc J~
p J p P D E1Lgt JbJIi b ~ ~JJ ( -Q) J~ J a cb- ~JJ
IJ ~llAJl ii=
iJ~0y li jl 0141 4 ~
g D-cls=JJf1lr~ =Q
Q QD = -- a E = ---------shyfllr rOllE r
o
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
Qp= (e-e )E= ar
o lJlTr
d ( Q)- _ bull -- - r T -0Ph - V P- r dr lJlTrshy
d ( T Q)p=t7D=- - r =0 V r d r flTr
)br 45~) D) oc j-P 0J~ J~ Lr- ~ )l~ J ~ 4is ~~l ~i~ J ~~ LSJ -=- ) bJ~ ~ 11) ~ c= ~~ r-) I) ~ J u~0
I5)J~ 441Ja )) JlaAil ~~l u~ jt~ 4~~j r)~JgtsJlAJt )) iJ~=)li
J ~ uk==-s ~~ JLA)) Jbull~A J~) l51 0) -~cseb ~ TlS J) ~~SJ1 iJll~
ck lSJ) di6- iJ~)) bull lilo~ ~~i ~ 0)1- ~ 1 l~If u-)r ~I)
~ J=~ J) ~)~f )) ~ J bullco~ u- JaA- UIS) h~ iJ110 ri ~O) L ~ lr t-)~ I) ~IJ) ~I rlllgtU1$~I )Ir lJ~c~ wbJ) (5ltiJl~ j)lia
o~)JT
pO)) I) )~ ~ (~U~J1 ~~4agt ~) ~ ~J) JU~ i
~~ 0))1) )J~J p J~~ U~ ~~I)~ ~iJTlJ~f)) 45rJ$ if )tJ~ ~ jli J r-is ~ D J D ~ ~J~ I) b~J~)) ~Jl (sJU
0l-S- U A ~~ ~sect ~~~ ttAr)1 ~ ~=-5 Jgt ~ ~k ~
PDmiddotds=Q
~ ~U 4j P))~kcs-~J I)~ 4~1~s ~~t tA)I~~
olilui~ 0~lc5h lSJ) J c JI~ ~~ ~~J) ~o)~ ~ c JI)~ii ~)if ~Lj D ~ I) (y) 4tgtli ~)~~~ () 4~l 1 ckJ~ ampAS- ~IJ )I)
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
(r-Y)
($1 ~~ CSJjJ- Wy u~ ~ ~JJ P cJ~~ lljT ~)~ 1 ~
~~r ~ ljT J~ uJ~ ttjlil~ J ttlJi ~ ~WfJJ J u~J1 JS shy ~~
($)ljTJ4 ~~J)lr-jfJJ Jf 1euroy J E Jel)~ ~J)l cb-- -tgt ~bshy
bullt-A J-lgt~~ cb-- j)-J~J Y bull) ~ li 1~ J I j T)~ ~~ 0)Ix )I
tJ~ ltiIJ) ~
DtoDH D Ony (r-TV)
E(E- D If =eE oD ( 0
Loli I_~iJl~1~lA 4) ribcs- cS) LA - ~ 1 l~ cb--y tgt JIgt
cSJJ 1i1cs- JljTJ~J ~I IjlJ~ cSlJI J cSloA JJ ~I ~~l cjll-shy
oJ ~ lJr-- jr cb-shy
P =Doo
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
I
I
~
~
I f
t~fJI ~f-cSj s ABCDEF JJ--aJ AJtLtu- lmiddotmiddotj~)I4 J Ja--~ T- ~~ ira
gttJt Mt I ~u ~ Jt~ Ji)~ 1Jt-J~ j tSJJ P ~ J Jjl JtII J- ~gtJI
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
JE dl+J E~middotdl+ JE~dl+ J Eydl+J Edl+
+JE middotdb
E nlAB+E~ nBC+E~ (-t)CD+E(e (-ol)DE
+E middot ( -D()EF+E middot tAD =0
AF=CD BC=DE AB=EF
(-7)
~ J
(r-v)
bullc ~~ E( E J~ j J Ule ~ lAw -dJ~ ~ E ~
iJl~ ))Ili iJ~~ 45~) lA ~ )) iJ~ ~)LA _ ~~ lr ~ -gt ~ lgt
oIfI iJ1l--Ul- 4iI- ~~I ~)~ ~ ~J) )L-~~ ~I ~ ~J1
)cSl~ iJtA ~I ~)~ Wy ~I)I) IW ~I~ cS)LA ~ cSJ) ~ bullc~u
r-)jJL1 ~T~ ~
~~ ~ J~) s~~ ~4MJo sj~ J~
J lagt-aJ)J~ J cSJ) P ~ J ~P ~~ ~~ ~J jAi)) l
Ji li ~t- J Y
(-YA)
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
~ -1 ~~~ ~~ ~ jl bullt ~)Lgt ~llo ~-li ~I ~t~ s~
liJ
pCmiddotas-p ~S= - P~S
~ J
bulll~u- Y ~~ ~ jl ~J) l ~ )~ ~IJ )brl D( )I) AS
)La JeL Xlto ~L J fo ~~~ ~~ bull )l J-L Xgto ~li yr-6 Jt1o J~ ~ ~JI )~ 1)b yz ~JI l~u-- be ~rP ~) ~u
~ rll~ xgto 4gt1 J) ~Jt tJl~ J l~ P= -Yfo
E=as+ya+ra
1) xlt 0 ~li J) D J E tJl~ 4 l- ~~
~c--IrlIF xlt 0 ~tJ) ~JI cJl~ Ji5~J ~
E= Ez8s+Ea+E8
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
r- ~ C-- j~A ~JkJ)J) CS1 ~SJI iJ1~~u 4JYJ ~J~ UIJ) bullt=~
v-i JJ Ir
Ea+E8=V8+r
E==Y E~=r
J5(S r-U u~ J~ bull j I~ 1 ~rgtJ I ltsIi ul~ (s)Jw 4CJ j) ~ jj I ~
JY iJtU u~ T~ xgto ~~ ~ xlto ~JI ~
a 1iJ
E = ~
E=az+ya+ra
D =Of (a~ + Y8+a)
u4u-- ($)lAJ) J~ I jl ~JA ~ 4 jl ~~ ~ iJj~ 4Jj~ ~N
4$ ~ I ~4Jj~ ~J- ilou I~ ~ jl u~~ amp lA~ ~ ~ ~J~ ~
bull)Jb iJ j ~ J) bull t bull~ i U~SJ I ($)J 1 J IJA- ~ rA- - lj
Jlou4Jjrtl rjJu~~ Jy~ 4~J~ QI liLJ ~) (Y-r) ~~J)
J~J ~Tjl c-lSJJ (+Q) ~5J1 ~ r~JJ V jlJJ ~ tSgt~ ~~ Jfl lA(S)l (SJ LAJ~ VI ($)~~ ~br jt ~ ~)YU4 c-e- 4S~~ tSJ ( -Q)
~ 4 ~J)alA J )~U ~j~ LA(S)Ut ~lgt ~L (S JJ Q j~ bullliL~ ~~
~ ~J~ 1J) ~ )U ~~ r-~ ~ ($)lA A r-)J) ~ lA~)lA )ysJ)
~ )cr Vo ~rSJl J-i~ ~~I ($IJI ~) ~JlA ~ ~ 4SJlA ~ jl ~
~ l~ -~ I ~ rA- - li Q J~ J)JuIJ ~rsJ_I iJ1~ tl ~lAJ) ~
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
j -QI
r I ~---- + 1-----1
~t )JbLllT~~ Ia~~ ~($)lA u ~ J~~ ~ 4S1J~L -rP 0)J1) V
r)u JL C 4 ~ rAIu-- Jj[gt
CA~ (_0)- V
6 lti1~ ~Y ~ J -U d 1A($)lA ~~I ~b )JyJ) ~T~ ~y ~ lSj-i1 tSJJJ~~ JI ~J) 4Su--41 ~~ ~ ~ ~J~~ ~j6J1
~ 6tS)IAJ) ~J~4 IJ lA~jlgt 1- 1iJ JIifU--dl ~jlgtJ) oJ o~~
sj U-- l Lt-~ IJ tS~)
- -- -
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
hDdS claquo (1_11)-~JEdl
4~ )~ JJ) I) lAtS)lA jl ~ 6 ~ clw ~ eraquo r5V)- JI~I ~
jl 4bA ~~ ($llA ~ 4Ai ~jl I-~b ~ ~ tS-) r5~ JI~I s )~
bull)~4S 4--- be ~) cS) la cSji1 ~I) jl cJlyU cJjLgt) bullu ~~ ~JI cSjr1 ~b ($I~
u~ V s V LAtS)A ~l ~I middotf bull)tA1 tU ~I)I f-) ~)) 0 ~JI
~vJ~ ~I)) V=V -Vy ~J~
W =-QV - -QV = -QVerr r (-11)
=~CV = ~ Q y C
c= We = Q V yW
e
bullu ~ ~jl d 41oPli~ )r ulgt eb-- ($)tA ~J) jl cJjlgt ~ Y-P JIici ~~Lgt I) cJjlgt ~I -i)i ~I
~)I~~ clw ($-1 I) (-Q))~- ~4 clwcS-) I) (+Q))~ ~
c~ A ~LA J-~ Jf I SA r=-j ~I~)k ($t clw cS-) )~~I
IJ~ cJot jllA1 ~ U P=QIA ($)IAeb- )~ u~ v)-~IJ
~~l- IJ eb-J) ~ hJ I cJl~ vu cJ1iu-- cJ )l~
~ds=Q
pAs=eEds
E=RAi
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
z
I
- - -
+ + + +
l l pound
- - - - -
f lt QJd QV= - EmiddotdI=- dZ=----d Ae Aeuro0
Q AC= -=e-shy
V d
J b J a ($btW~ r 4$Jf~LA 4t J~ j( tSJ)lJjf ~ Y-VJL
~ l-gt JJ Tv ~f oUr ~ e ~ lo
J1i b t ~ J~ tSJJ I) (-Q) J a t ~~ ofISJ) ) lt+Q) J~ j
~ ~1Jr oJJ ~ ~SJ f ~1Je ~jJPUI)) ~J~
E=-___
--~-_ __ -_
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
V=- EmiddottD= _ - Q ( )Jbull fT D b
c= ~= 1fTt V I 1
-6
=j u~ bullu A Ilj JJ~t e J ~ t- J~ ~ l J ~tJj I yen-A JL (-It ~- cltltb yen-t J f Qltrltc~) yen I) ~j~
~)gtIJ be-- ffJ) tJ (-Q) J~J a~middot dJ) I) (+0) JJ t ~
~ -t JtIJt~Sjl I~ tJJ- ciC 1)) QD=
ffTr
J~ r =c j c)amp I)~ J -1 ~- ~J~ J -J JJ j amp$1J ~1 iJ~ J
Upound rot)11 J-
hi m- middotIF _ _ -e
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
middotL
L J
JiJ
- -Qshy
shyraquo ---shy
I
I I I I I I
--_shy
v= R[_ (~-~)+~(~-~)J f1f pound~ c b pound a c
- Q -----------shyC--_ v
I
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
V bullbullbullJ~ ~ ~St 1 ~f~
- I I- I lt bull 1_ _r - c ltlt2_) u=~ ~~ C J r =C J r =a ~p ~ f5 J r ~)~ ~~ C 4S
Jp- r1lo~ 4$- ~gtP~ ~jtgt- ~ 01 )~lS- ~~ 6~ J~U--
-tS) I) cJ ~ jljU Ul5 ) -~ I~~
IJS~ oJgtamp- cl~ eb- 4 jl wLj gt01)) 6 uS~I)~ )1Ji jIJ~ J
degr~lju ~J~ I ~J
1= dQ dr
ul) d JJoJ1 dJ-gt ltgt1)1 ~1 ~ r~4)J Jlji4$~)~
C-)Jgt J 1 ~ 14 4 Sigtlr Sgt l~Tr ~ioJ J)iAi )J t~ (~ 1)
~llj S Jlb~JtA)J oJ) u )Ir S~5J1 JI~ 4 JL~I r )) LA I
s 4AiJb1) oJL ~d6-)) J~~ LS~I JljTf5~~~J(]IJ~1 JlAi1 Ja j~ll
~ y ~ juS~ )ltJ I ~l1e ~r I1~rltJ1 J~ r- 0 Ir ~ e ~ u ~
N J I bulllAJU--J~ I -Jle 4 jl ~~ -i ~ ~~Jf )2) 11 (1- A) I$
gt4J ~1 )) )L 1lie JoJ-0IJ u~ ~gt LgtlJ J) )ljTf5lb~JJ5Ji J ~
l~ -- Ir Ir
LiQ=~VN=Nedsdl
aQ1=-=NelJ dsdt e
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
shy pound
eft tf = bullu~ cs- E ~J~S1 iJll-e JI JJ lA~J~ I Jo -- v cJ TJJ W
~~ut ~l J ~ I$ J~U--~ ~I- J~ ~4~ J ~ Neve
J=Nevbull (r-Yv)
j ~- ~IU-- ~r ~~ J~ JbH ~ ~J~A Igt cJ-~ rJ ~
JJ~ ~ I-~
J=-Nevbull
bull~~u ~ IJJ) )ljT~J1 ltslA-~ J~ f1J Ne gtIJA
JJ ~~gt~ $ ~ amp- 4]1 ~J~ YiJ~~ c)1J ~
~~f ~- ~ftJA ~ r~ 1Y ~J ~I Ju~UA ~~I iJ1l-- ~I J) ~JJI ~IJJ-- 45 r~~~ -~
~1 iJlAre ~ ~ - 1
lbull =_ItrE
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
)) I) bull)Le ~~ ~ c ~ tu-- cJJSJ IJ~ --rP IJ Pe - li --~ 1iJ XSd J~ r-J) JiJu ~L cJJ~JIJ) jJlir~ 1 )IjTJ~ a)~ cJ bull J
(r-f )
J=oE (Y-YY)
J-Is~~ j~ J ~f f ~l )f ~Ir J )~j cS)LA ~I- cSl~ a ~IJA ~
u~u-- (J ~ 00 J-l)~)lA LSr~ J a
)Z)~ ~lt~ lb ) a ~fLA ~~~ ~ jl 4llLi 4 ~SJI ~JlA
S4 J-i~ ~ ~ jJ cJ~~ ~ )IJir cJT cSlfi1 J~~ Vbullbull ~~ 4-S~ -shy
Ult~I cJ 11 IJ~ ~ ~ljTLAJ~ ~~u- c~b 8 J~lr--~il J ~)I ~U oJ lAT~r r~ ~ T ltS LA r J ~ ~)P-r I)) bullJ)is d ~ ~ t J
uJI J~ cSjil~ ~J~J cSj~Jjl cS)l-ll IiJ J)Jf fS la--sr~ ($1)1)
tJP )1) 1 ~5J1 cJ~~ ~ ri-- -1 Vbullbull ~~ ~~ )~ ~
bull~~ cf ~SJI ~Jli JJ ~lJ
j C E dl jtlEdlV 6 6
R = T=J J middotds =-rshya -E shybullbull-
bull
0
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
)~ ~SJI I~ ~b ~ c~U-- Sbull J S ~J~ ~ ~JI ~bullJIL R
~)JT ~4~ IJ ~fiJ I ~JtA Icsshy
S eb jl 4bAi4 jl ofJ) ~~ ($JJ JJ~ JJ~I y~ 4~ J ))
~cb-~jl 4S~SJI ~ft J)~r- JI~IJ ~~cs- S ~ j1-bAi
~I Y~~ 4 tSJJ JI~i1 1lJ J c~u-- x5If)
11~rgtJ I ~JA ~~ )aiJ) I) a ~1lA ~~~ cgt)1A ~ 4 _~ Jlo
(~l a) yent
1-4---d__-I~ ~I
--- J~
---~ E
v -----+~ 111--------
)IJi~ E=VId rT ~I~ c)~J u=JJ V ~ 4 ~rJ)~J ~
~ ))f Ushy
fEmiddot Ed I d R- - --_-f JmiddotlIs -aEA -a A
~ ~ (altb) b s a ~1At~~ JlS~)LA ~J~~JJ) ~ 4lPli _ 0 Jlo JbJ a ~J) ~ u--ltrgtJI ~JlL ~Ioc A 0 ~11A ~rP~ j )lA
bullLS~ Lgt I- ~~~it_16)u )I~ 1 0 iJ~J~ ))fJ~J~J)~ JJZ~ J ~I~
~ bull)~ 4S ))1 tb ~J) (d iJJ~~u~) ~~I~ )~
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
-
Y~ bullbullbull JtJ f ~
Y
- t Jb flmiddot JCl
-I)~
lIJ
- ~ ~b -r ~yen~~Jt~_JW7~l ~ it ~-A ItampiJflUf-t J~ utJa J--t u ~~ I
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
middotu ~t ~It (r-yy)~ lt~ ubrua T~~ ~ ~~ u-(J
Jgtb ~ slAJL ~~~ ~ ~~ iJ~~ U laU-- iJL J 4amp ~ ~ ~I)fl)f ~ jl bullu~Jt tLJ~~ bulls~ )f1J S ~
1 ~ - ~ Jjf~ ~ - C ~~middot~middotltrlllft eJwt ~ UJf
middot1I
(r-ff)
i
(I
I
f
II I I bullbull=--- It I
y
1- cJ ~ ~ fW )f ~IJ citl
~nImiddot 4middotf tJpound
II
1
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
VV Jl ~iSJ J~
J-N iJ L j ~ (S~ JsJ IJ~ t J c ~ r4=-- ~ JgtJ I iJ ~ fi 4 iJ 4~ ~J~J~
~JJP ltJ IJ~ J~~
vmiddotJ=o (UJI t-fY)
(~ t-fv)
P==VI=Rlf (r-fA)
I C middot I G
----y~--
flp == flvfll (r-f~)
iiiflv == E fll==EmiddotJ J
fll == J middot fls == J ~s
~
flp=EmiddotJflv (r-~ 0)
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
i I
Jl EmiddotJ=aEC=shy (-6)
a
~$
i~ I f ~ ~ -r
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
IiJ J eo ~ tt ~~ ~ jl 01 ~J6 ~ lgt 0~ft
J middot~S +JAs e
u ~ r r j J UJ) J) ~~JSJl cJ ~~ ($~~ tliJ ~ ~ sJ J T~) 4~ 0 1J ~~ ~ rgtJ I ~ J-- Ul- bJ sJ J j I I) U L 4A jJ
L J1J (altb) -b raquo a lt5~t ~4 Jotgt lt$~lA ~1y1 ~J)~ clQj _ Jlic
~ oc ~ ~ J) ~ ~j uJ~~
YEo1dego ~~~~ ~ oltpltw ~gt(
Eo J rOo uLagt ~ ~ ~ 1TltfJJltrlt r ~L
yl- s-Jgt~ ~ v J 4 IJ -101 J I bull-1 ~ ~ ltSJI ubull
os ~~ Ju bi J braquo sJlts)i os ~ s fiJ I-= U (J I
~~ JJjt J) cJl~
10)MYiJ~ft (~
4ImiddotIJJ v~ ~J I ~JlA (~
AJJJ )s~JIr u~Lt~J) J+ LJlr-~ ~JA~JLA~ ~Ji~( ~ I ~
bullw~ ~l~ dli jl~1i lil-~ ~ ~
J J(r)a
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
b~bull bull
d - -(rJ)==or dr
J JJs-- ~ F
1iJ
VaC--_Q hshy0
Inshya
~ rAIJ ~ ~-~
~~
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
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 Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
- -
E = E~ =-V_O~b- a rInshy
a
~~~ L Jk Jt)o1ily1 4 l5~~ 2- c)fi Lbi (~
I = f J middotds=f Jd+ f JdS=JlTrL+JwrL
_ YOoV 0
1TLb
Inshya (C
bIn-shy
V o aR-_ shy
I () 1TL
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
1 Heyt W H [r laquoEngineering Electromagneticraquo McGraw-Hill N Y 1981
2 Della Torre E and C L Longo laquoThe Electromagnetic Fieldraquo Allyn and Bacon Inc Boston 1969
3 Schelkunoff S A Electromagnetic Fields Blaisdell Publishing Company NewYork 1963
4 Plonsey R and R E Collin laquoPrinciples and Applications of Elecshy
tromagnetic Fieldsraquo McGraw-Hill Book Company NewYork 1961
5 Plonus M A laquoApplied Elecromagneticsraquo McGraw-Hill Book COmshy
pany NewYork 1978
6 Skilling H H laquoFundamentals of Electric Wavesraquo 2d ed John Wiley A SODS Inc NewYork 1978
7 Attwood S S laquoElectric and Magnetic Fieldsraquo sd ed John Wiley amp Sons Inc NewYork 1949 There are a large number of weIlshy
drawn field maps of various charge distributions includtng the
dipole field Vector analysis is not used
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
u~U er=~ ~ ~lP J-U zgto ~t J Er=r ~ ~tp Jou zlto-middot~li iJl~ Zgto ~lj))JI
JSfj t ut
lJ z= 0)) ~~)~ tJ~ P D E zlto ~t ))
~~I J E ~SJI igt1~ ~u ))b )IJ et~ 11--)~ q (514LWJ~ r ~)T~~~ ) Jatl- ($If I) r=t J r= e-J) ~ ~~
middotu4 t 4 ~W J-u zgto ~t J 4 ~tp Jau zlto ~I (~1
(fSJf) bullu~ IT ~ ~lJJ-U rgt ylimiddott ~ J-U rltY 4t-li (= (lA ~JJt tA~ t-)
~~ (oltb) b Q ($-t~~ Jt) ) laquoSI~I-I ~lS c$2LJJ T JS~ t ni tJ -- cstJ ) ~ ~ ca)uo- ~jgt
bullu~ I ~ )) iJjb ~~ P (~I
middot~4 1fltlt ampSJ IT 0ltlt lSlJf e (-
cltrltb slf elf altrltc fSl pound (Go
~ ~iflP)lf- ~ J ~u 4)j ($jl ~cPlA ~J) ~ ~l
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
~~J P E D ul~ Vo ~J~ ~ ~L w)igtl 31 ~bullbullJ ~~~ JiS-l-gt-~L)~ fto )~ I) ~~ J ~ ~IA)~
~~L~ I) ~~ ~ ~ c~ ~J) ~b )ljTCSLA) ~
Ptp P PP V (z) J P D E
~ ~l-aJ Vo ~J~ ~ J-i~ w~1 ~I ~I~ ~jlgtV1 ($It I)
(~l ~l ~)Iupoundr) (~I u~ J JJJ J~Z) ~~ S
I- j rf4~ ~~ )~ ~ (altbJ b a csJbt~~ J-ts($)tA ~J~~ ltSLai ~
~~L~ bull~~ ~bflaquoSJJ (+Q) lts~JI)~ -1 J ~
~ )lR-fb)~ Pt Ph P7 E (~I
I) jl JS~) (~
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
I~J) -Yq ~ z=d J) +1 ~lAJ~jl ~I w ~ csbgt tJiJ~4 v -UoJ ~~ lilo JI~ z J~~~J sz= -d J) (+1) ~ ~~
~J~T ~)~ IJ J~) ~ ~Ii J) tJiJ~ ~ I jl ~b ~jSJ1~I~
(r~d)
J p= Po- ~I bullu ftJ~ ~l~ J)~~ rt) J~ ~1P 114 It )) 4i1~1 r1 jl JPL- ~5JI ~~ ~ ~) ~lt )JI)Jlj )ljT~L4i))
~ amp01 ~I~ z J~ ISJ) ~ ~
V(z)= -t-(ya+(z-~r-(z- )
- fa~+ (z+ ~) ~-( z+)
z
p= rraquo ~I bullt middotfl)~ Z)~ ~J)Jb)~ A tl4t cJtwf4 ~ ~l-- bull) middotf jJ )) middotf~ 1j J-L- ~$J I I~ u
liJu I~ r-j l JU ~u1y J Jt(J) ~ )~ ~f4JI-1 ep1A
JI ~)L-~II ($J -PLY )f~ J e1) ~II 4SJ) Pt(elm))~ )) IJY(a) ut~1 J=-i~ ~I u J-J ~j~ ltit- 4ift d~
~
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
UJ lAgtif )Igt ltl ~ rAj 1d ltl ~J A gt- L JoIgtlSgtlA ~Jgt 11 - ~ ltl ) 1TJ I-li W -1 lS~ ~ -rP lS iJI )Igt r
1
bulltSu Ji ET [j E jJ ukgt J~ ~~
~~) ~ l (uJ I
(bgta)l~ )OJgt IJ b rJIgt t l J a6b tl J-rA JIgtI H
4S~) ~ Li J
V Jo-l= 1 J (pound)a cIo lSJJ s~ r- -lt)b IJ YJI)
~I JJ~ (V) b a cIo J)
(~ a) u ~ 6blSgt lAt lJ b = cte lrJ 6lSgtlAt l1l
l J I-li ltl lS IJ gt~ r-=- J I-li pound ( u ~ J~ a J~ rIJgt
bullw~ ~J) V
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
~~t u~ d ~TJI ~ a4I1~1 t ~ ~ I
IJ~~~~(SlAJ~cJ~ (~I
F JJ ~Il-~JJ uJs~li jt J~I ~ ~ cSLA)~ ~J~JJ ~ (~
~)T~J~ IJ 4111
Jt~ zlt 0 4-li J~~ eo cmY uI ~~ u-~ J~ (SIJb xy ~ ~ul w~U VEo ~~ ~~~zgto yen-l ~ YEo ~~~ uL-lS
r1J zgto ~li JJ ampJI~
tJ) ~ Joe IS J Is JyJ ~ a s a cS IAt tJ~ Jets cSJ lA bullf ~ cent u li bull ~
~I bullu ~ Jj
YEo ~~ ~~ ~ ~ laquoltr-cslaquo ~l
-~ ~~ ~ ~ YaltltYa 4gt(
~~L cJ~ Qo ~SJJ~ J-lgt (S)lA ~lAfjI4A1
IJ yli J JJ P J~JJ)~ JIJJ E cJl~ tJ-U (Ull
IJ cJgt ~J~ (~
~JJ JAe)~ ~~ yaltrltra ~lJJ ~~ J~ J~(~
IJ= Ya
IJ altltYa ~lJJw middot~i cs~fiJl (SJJ (J
~L ~I bullu~ p ul~~ ~1~ ~jJi l~ ~J~ (SfJIJ Z Jpt ~ poundy --J~~ cS~J JetsciLs ygto ~( J E ~~ u1-ts JLPylto
s)oIl-- cL-
~r~
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
1)
JJ~~ d Jb1 b ~Jl~t bJ a u-lb t~ ~lP lts14i11 ~~4 V ~l-gt ~J oJ ojJ)l P=P 8 amp
0
IJ ~ ltsla)~ ul~ (~I
I) A(o~ o k) ~)~~sJ1 iJly~u ~
z
~ zgto ~li1 E ~~ ~ri1 0 ~Iu --~ fSo)t ~ zlto yli A cJ~~ J~)br zlto ~l)~J1 ~Iou ~ e (1 ht
yen1- r) z= 0 )) ~ yen )~ ) ~~ ~ J D Ey J y u~
s ~ ~ ~11 s ~~~ 1 r-- j l yd Jpti~ jlsltS~tA ~~1) cr- ltS~
Jl- ~~J jJJ VO ~ ~~ IJ ~ 1) ~I ~ ooU n ~
(~1~ ~ ~1lA R) rJ I 1 R ~1~ ~ J))
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
loS1) (Y -r z) 1bA)~I ~I loS~LA ~ r=V loS1-S ~ 0ribu-- ro )P~ cJl~ bullf jl
E= oaz+Ea-Ea
t~jf~bl)E )0 ~~~ (y) ~1E 10 J~~ () ~ r rIr jf lS1) () ~~r-~ 1 ~~ ~ ~~ft Jetgt ~J~Jl c~U
~ljT~)~ J~ c~ (l-ejr-r ~~ Wy Ja 1laquott wje J )
yenLgt I) l jr loSJ) )~Y
eU ~ oX) )P~ ~1) ~ ra 1 a loS tAt t-J~ JetslS~ LA bullJ~ U loSLai bullrr rEo r ~ ~ deglttpltIT ~l
~ u-- jP1 u=J1 V ~~ rJ) ~l) bull1~ 1 YEoJ 0 ~~1Tlt9ltY1To
JAS 1-gt- I) ~ J)~)) P cJ-=Ii)~ )I~J (~I
1) 1) ~ ~JgtJ I ~JlUJ )11 ~cJ~ft (
~l bulll bull )1) loSJf~))) ~1 ifsinpyolr) ~SJ1 ~~ cJl~ rr ))))tj )IjTloSlQ J) lS)u 4 eb-lSJ) ( foO roO) 4A ~ril~ ~1
bullJ5 bull ~
IJ loS~t~ 4J)Lt (~ I
IJ~ loSJJ blAi rW )~~ J~ uI~ (
~IJA ~~ ~l bullu ~~ J~ J lb uIamiddot e-~ ~ loS)lA4~middot rf JL~ OJ ~ cs- a0 ~~
IJ ~ YI)J) E J (~I
IJ I cJ 4ft ( 1) ~ eJi (~
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
~Ir (~l( ~ ~ - 0) ~j~d ~lC1~ ~ ~)J~ ~ltft ~
JI ~pJ ~j J ~~ ~ ~gtJ It~ ~ ~I)l ~~) c)i
r~
p== -0 (ave-eva)middotvV bull E= -Vv
~j J) $)k I ~loH -ly u~tA ~ -J~ ~j J~I 4 )~I ~Ir
~j ~1JA~rPJ1 ~~ JI ~j~Ho~li csXAltS J1J
~ J) ~Jll o)b m~LAH tLtu4 ) X O-iVjm
bullJ5~Lgt I) ~j
~
~~ ~r)~ltftl ~))IJ ($~lA ~~)~~Il cSIJi 4SJ~J~
~lS ~ I) w~) c)~~~ 4J)l- jl bull)tA1 - cs-A ~ ~
~I rJ ~J~- ~ ~ Jl ~~~I~
middotrv
bullu ~ jd ~~ ~ ~ ~ Us cS jl Jots~) lA ~- fl ui bullfA lSet L bull~1
~~ lSlA)4J~J~ ~ )fjT~LA)~ J~ PJD EJ J IJ I ~~ 1j ~
z IJ----------+d
oL-shy
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
~ )~ ~jT~ ~SJI)~ li u~ )lM- e(z) )I~ (r-YA) 41)~ r bull amp L ~1i~
4 J ~I bullLt A ~J~ ~ Js Ulk tSJl J-t~~lA ~J~ ~ dl fo
~lgt ~~J J-J ~J~ ~ cJJ V ~ o
I) ~J~~)~ JJ DJ E (~I
1) c 4JStSJ) yen J ~ljT~)~ J ~ (
z
bullu A ~J~ ~ ~ ~llt A ~~ ($jl J-trcS~1 ~J~ ~ ~t bullf
ysL ~f u jPJ ~J~ ~ ~J Yo ~ ~J
I) ~ J~~)~ EJ J (Ull
I) ~~J )iT cSlaJ~ u~ (
I) z = d eIw cSJ) lJe J )ljTdlA)~ )~ (~
IJ ~)jJ ~ J) ($-1 J ciliJi ) (~
IJ ~J) ~ tJj t )i J ~SJ I ~Ju (A
bull Id
d
o
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
--rJ laquoJ(+z) A --r L j ~~ J bulltr J cr J llo J I -1 JIJJ1 ~ ~r- E ( +Z)
~ ~t- I ~ ~IjT ($~J~ u~ ~l~ (~~l c J ($)IiA U ~LI- ~)
O
LS lot ~ J L JJ Jotgtlts 10 LS Ie1-1 cb~Jr liI LS IJ I I 4 tf
~ T~~ 41$QgtI b a
lTr JTI ~ t- ~ JJ e II J r- J 4 I J ~ ltS
bullS~l~~I4iIljJ~ jl ~~ r) ~J(f jlJr
I) 4i I1 J~ ~ E J J (~J 1
I 4if1 J~ ~~ ~4 J ~ IjTcgt LA) ~ u~~ (=
11Il0T~ ui W 0 1 J ~IfJ~ V ow 0Jgt~ u~~l cgtJI (~
J~ cr-~ 0jlgt ~)J J~gtfJ~ ~I I J~ ~ ~jJ ~Jlt (~
lJ~ logtfJ)~ ~IJ
~rDJ U=uoar ~~ ~I)b ptA 4JmiddotIJ) iJ~ o~la ~)~ tf
~J- IJ cJL u~ YEOrJa ~~
~rP~ ~ ~ YaJ a ($Ltt~~ J-Igt ($)lA ($141-1 ampJ~ w~u t~ ~ uJJ V O ~ 4 I ~ I o~ ~ U ~1JJll~rPJ e=poundora ~~~ o
=1 t- ~ 4if 1JJ
oSJ tJJ JJ ~ JS-gt u A gt t LS]1s-JotgtLS lA ~Jj J ~ bullff ~=5- t- ou j-PJ ~~JJ U uJJ V 0 J~ ~ J ~ ow ~
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
JJ ~ )1A I) E J (uJ J
11 ~ )-1) ~gtJ~ )IjT~lAJ~ ~~ (-
11 ~) ~ ~SJ I cLA (c
z
d
o
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864
1 Planus M A laquoApplied electromegneticsraquo McGraw-Hill N Ybull 1978
2 Dekker A J laquoElectrical Engineering Materlalsraquo Prentice-Hall Incbull Englewood Cliffs NJ 1959
3 Fano R M L J Chu and R B Adler laquoElectromagnetic Fields Energy and Forcesraquo John Wiley amp Sons Iuc NewYork 1960
4 Fink D G and H W Beaty laquoStandard Handbook for Electrical Engineersraquo 11th ed McGraw-HiUBook Company NewYork 1Q18
5 Maisch L W cCapacitors Magnetic Circuita and Transformersraquo Prentice-Hall Incbull EDilewood Cliffs N J 1864