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7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5
http://slidepdf.com/reader/full/engineering-electromagnetics-cheng-solutions-ch3-4-5 1/25
en"e:""Stotic E /ecf rie FietJs
P. z - t o) Mox. rol(ar. l/.._r. tui/l nto.Le 4) :h/, afz:,t
h : +[Vr^Vy \, V^-. ry o,t,_\".-"^ ( T1(4 ) vrno* e rw, )
b) A+ flre s".e.n, (do)^or: D/z L rnqsl be.z /-,"vx twrhe^ re '
L =* [ *+^ - ;2h ) -rlr4x z I ew Vr i r/
c)E.ubl-- Wo* Ay dou-6/;* Lt , or ./ouL/, 'n34"-A node 4-ece /arotrn.t vo l4a-qe-,
2 s-z
P, -t
/ t o,ti:
-'vr'; ( 2 / l o ' t ) t / -6,
-
ffif
(d*o.F dro'tse)
: 3 .6 (avo-5 + d .2o.3 .65) 1. r ) .
Er: 3. 6 (-dro.t + iy o-t( ,d) (nt)
E: 4, + Fzt= d,, 0.624 (N).
Similarly f" , f, orrdE ,\/l o're.
in the diTecfion a.way from thc canle.r
u \/ a
/ \/ \
r \
L----- \ -I O. l un ) a ia
t f
leDq lS i ve fa rcesI
oF thz triangrle.
ry : -al 3 +d"4, e-p ar4-d.z
At P E,ffiV Gart+"4)
E,:Vfu (*+-v"3).
+)
b)
N" y- cotnpanent : -3 Q,
fy/o Z- CorrrPonent ' 4Q,
4Q,-O+ *=+3o-.:o+ +=+.L q 2 4
l l
7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5
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: fl'
,D3 -4
3' 1" g'' 1 < t ' C ) + ( A ( ' ) - l ( Q . c )
: ' '*-x -J Ir ' ' - l
9 (crn)----*i
fo, zero {orre- on Q.1 :
Q , Q , , Q ' Q t:4 a i o x - 4 z r e o T t v '
,:1"8,:r/*:r cc,-).
\A / ; r h x : 3 (cm) - + can be p roneJ
Q, onl Q3 are o, /so zero.
A - F -..'. J -t )- rom Eg(s-4za-), f,: r;!:ffi : -*-
E - a,*f t -(r+ !;)- ' , ' lP - e i L ) r t . 4 \ 1: 9tr I, - ( - E* -?;"-....,)J
: d"#ryLie) -*er* jt +=---(r-*4.:*.".\- l=+l a"O'\ t-VV /)
whe, 'e, the {; .sf (e-ra is lhz po, 'nt-<hcrrqe tern-t qncl (h<_
(esl . ,epres<nf th<- error. 1on' ' , ' - ler i 'y on/7 lhe [ i .s/ e.rror
+ i- 3 o.ol '-->z 2,8 b, ,,,s.(,4 .
, 2 3 -6 At on qrb i t ra ry
r { t t -* nP
t h . - ! /he . ne- l {a - .=s on
Tefr-Yt :
St'-i/or sifualion
P (o,o, : . ) on f f>e a.x is .
I, b 4d'4 l rg ( z ' * L ' ) ' 4
fl i iErl:ovtt he-n P is be /o,+t lhe /oott
ur-ffi,:4fu,.y.Ep--v4:-o"* -a"ZfuE
q) At point@,qh), , Q##
b) To {;n.l th e /o<olion o$ r,dr.lEul , s. f
xn*lEr,l=fu'
/ 2
7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5
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P3-*ctE: - f!9 '9 si , ,- l ' , ' Y
4 f t € o b
/ : = ; F- - -n tz frs i -a ed4 -7 */ -' /tt lob ro
a=-7rtff ite J
t
,p.-g SpA"r lcczl t f - -e lnyt F:4rE^ A7p/yQ.au's l f ,z- .
r) o<R<b. 4tn'fo,=*{-f , t1+r{an=Te-#),
t'':**ry #)z) A>R<R; 4rrR': *^{t( ,- f l#rtR?e: W^t,
,u'iJH^z) R;< R< Ro.
Eor:o- .+)R>R".r; ffi.,P,-9 Cy[;n,t.;co.l y,n^.*.y , E: drE, Apely 6:outri (aw
a) Er=o, do, r<( t .
E^: ?9: , fo. 4-1 z b,€ o l
er=ft?, {orr)b
b)
* : - *
P3-lo W.: -7I E. ai : -V/04x+xr/).
q) A long *he pora6o/e Y =2/'-, 4/ :4*-l*.
W":- lgxlo 'a; / - t f z t++z')"{x :?xtd '( r ) - ?o (7t) .
i l A(ong 1y' . ,s{rct ;ght ( ioe#= #--2, Y=-2x+4,47=-2/7.
W" =- ( E t6t)l-' f{-zz,++) x - zx4xl : ?o (/t).
o
{
l
t3
7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5
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P z-t r- l
D J
l \ r t ^P 1 ' ! /
-L/,
a
- -/ - -
t ) r : -
F,3' is .+)
P 3-/4
-n [uz7tr€sJo .l x-.yz
P^ r . , - (-#"1Lnvtr)if'*rJu (
)- LrY j '
E : dny- drx , E ' dI : Y1*-*4/ '
We: -7.-f
-^(z; - 4*') dl : toqri) '
W. = -7 J
-ir'zz ++ +2r )JN
: -6o (1'' ) ''
fU e qiven E d;tH is fio4c44sarvq-/-ive'
t / ' c / 2 o ; ' 'a-) r/ = 21 {u '*
y-)t)
I,neo/*
I n ra'a*r ^ - - - ,
: wt E
l:'o''\ -
| \,q
l\
f f , :F ' zn : Po* 6n 621/s) r
9fn: - 7'P - -3Po
o.. : ( ( t ' ) f7r :sPot; an
T"+*t bou,.d chc.r1e : 4,
c'l-x4?,i- b) froe Ca,l.^b7 /o-,,
a,=z[qLFvvd*. , : ;
l l , f%11/ ty- "Jo uy *'rr€uR' ny 2n<o./ lJ1tJ"f"y, i
FV 3ire-s 1lr- toonz ansve-tr a-s n b).
b)
{o. . , 4 th* cuLs-
: ( Lt ) fp; -3Pot3
'- e.r :O,
b)
F =arPo.
f f t : F ' d" : Po , ' r16cot { .
fpn= -0 .p : o.
d, = n b'srn 'ecosfcry gno to
: O.
t+
7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5
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a' -
P s-ts p : I (a-*3x+ar4y).
o) fp_:-V.F:- 74.
fs ta/ , rof , ,ne ch,-"ge Qn: - IPorr{1 ' ' r r ' )/)er unir /engfh
o; ' j .o f , , ",
' ^=2- .f t ' r " :V 'd , =
Q(a; rocos l+474r in / ) 'a ,
: Po o (3.os"* * 4 3in '{)
- - Po 'o 3 + t ; " '41 .
tlX:L""., 'u^:=! u, ffr;=- Po, (3 + s;n'41.
b) r21r n z l27lTolat Qro d"lrro ro f
: Po,' I '13 + ti n'#) di : 7r r Po1.,
. Pe ' rur t i l / e . t3 t6 .
lot- l Qr, = '7rrP"r; ', 2er oni/ /zagt/ '7
T o f o - t A o u n J c h o r q e : Q n * Q r o * Q r ; : O .
l,p.3- /L
( / ) R>Ro.
( t ) Ri<R< R".
(z ) R<R;f =-a3
Vt=
t r : -a @tR,:
6<rR.' V, : 747'
A F F ADo,: oEr,:fu, Po,: .
t r . ^ -q '-Rz+'lf€oerR: u4,
+i'E'
P*= Do, €oEn"=t -+,)*7
.
vz: J
R'
eo,anr[R
E,"rL=J,,-+,tA,<^
o#F, DAr:#, Pp)-0.
I rLvtlo=r,' )o , Eardq'
d . f r t - t r J - - / t - l - \ r rd .+n f tL \ ' € r tRo \ ' €? le t ' A )
tg
Spher i .o l syrnmelny 'Apply Qo-ut t ' s low' . E-aEt ,6=QDo.
7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5
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; < J - / . /
- ,
P:- ts
P. 3 ;9
Ll;e subscript
Vr: Et "4" :
Vt: Erydp:
a fo, air, p {or plexiglass,aoJ bior 'breakclown
(s^ to ' ) * (so x o- t ) = /s o x c?(v) : /9o ( tv) '
2 O x o :y' Ooo (*V)
Vb = Eu- i^ , EF{O: Eu( so-dr ) + FO. i ,
Alo*' D^= D, - €oE^: {o€rfEf - [^-=trfEp , Ef '
W : 8,.^(o-/o) ' + "/o:3 (zo-{vo) : /3o LD'
€r^ 1 f tt*Br"okJ"rtn occurt rh c,)r re3)ort {i''st
b
ya.
ba
bP 2qtoh fr.-sr,
3J
,4t the z=0 plnne ; E,: d-:7 - d, 3v + dt 5 .
1(z -O 1 = f r r ( z :O ) = 4r ' 2 ' 7 - a , : > '
D^G:c) : D-., (z: o) --- ZE,,(z=o) : 3 Err(z=o)z- 21 - l ot1
E,(z=o) -4r2y - drsx * d"f
O. tz=o)= ( 'q*2y- vy'32. 1# ) l r"
f , r r ,^e Er=d.Er-- i*E.(
/ lo.,r7glory conJ; 't ;on: dn s{
= dnx Er .
* ,rl: -3
Fo, E, , ct,tc/ hence E , *o 6e po-rol/el
/ -, : f he x--*t)s , Ez/=-
Err.E"=3'
8ou,"Jo,y c ct; i ia,b,n Zn' >, -4'\
-- ' €r Err=€t ErrT* €o S =€o€rr3'
+ 4r r={ : r -e r7 -
l6
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P.3-zo t : Tl+€, .,/ss,t*e Q- on Ptq'fe' o- f Y- J
.P -zr
r/=d A-d ln (<,/<,)V : -J E.,iZ : ---=4.L.
- Y - o J \ s : - e , )
4 -A- ' l ( € - -< ' )e V cJ bGr/e,)
Le_+ Pt b< the-/ite-u_/ q$c{-i3e. 4ensi*y ct-trbe-)'trle-r'
c-ondc,chor-E: E.
#;1 . 7 ' e - P , , r 6 \ _ € _ 2 t € Vt 'o- - -J6 E'crF :Vf th(at - t . t - TBn)
f f A : d , % ='
a_ n(b/o)
Fo, o f i 'x.a b , fhe {un</bn io be ft ')n)mtze./ is: (x=L/o).
{ t4 : #^*. S . r r in3
W: t y te tJs Lr ,x - - / ,
ee: -dy? a: f f i '
' \ - T - Y * c , )
o .
t
ot')
b)
Q r V : * - - e : 2 . 7 / g .
C) rn in. E(a. l : e q /b.
'1) ': +:fu^l= 2tr€ r/^)
@ E =u,# E,=d,r#, r ;1..b;
Er=&r# ' b< r ' - r o
ConJucfars) res?e<iivel7. E = drE*
v = L'*8.axdR--# (*"Eo
P.3-23 Atrr , , - , , ef ia,rges+Q-an. l 'Q on fh 'e inner a.n.J ot i far
= ? i = Qz .ftr€oqz
- - / \ .4/
t ' - _ L -v y
t7
( r/a, - t/tz")
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Ps-24 T*-,f C-afa-e.t'fony a-cfos s Luhuy fe,.,nino/",
cr: t Qfl 'sfor+..1<).-f i< <n ery/ W, =i cr(rzo)'= .t(^w).
We ;,', 2-{7uF)capct iior = ! tz'6i' sL)z:6-+(,nw).
\Ug ,'r, lV O cctl>ct.'7:f.,,- ! (r o'-"2 49':o-y q^1n).
Wo in ){7y) c:.pctci/tt-= 4 Sxfo'c)xorz :2.g1nW).
E:F; r6rs in f+e/ rcos{ ;
4f = dr 4r^+
a4rd{ rdr4z,i-',\t= *a-f t . d,E= - Tsrro-'"lli4;n6f'ar r +3 / !; fJ;1.
" 1 L I : j r / J _ i
r - 2, , $ {.o^ U/z'/o -'tr/a; f h.-n 6- -T/t-t f }rorn Llo4:
W.: -(r^ ro-to) lr"1" in4i--"* ( s;,(-n5l*- lL , T / 3 , ^ , L l . L J
: - ( rx t6 t " ) L- ts -3GJ 27x/ ;q ( t ) :z f (nT) .
9 =tT,/s, r {rom 2Y4 ;
.fhen r = 4, Q Srem?rrb ro -.tr/z :
L(-=(r* /,t,") le;ng)i i ' ,- 'LfL-
: - ( 9 x (o - to )| r r - f z ) : z?x r c l l ( t ) - ZT GI ) .
--.Sea4e dr.i:?{oin 1:n.t, a,). jxl:6-,E;rronr.Tyqlfve
P ,_?5 P.
c+) Fi rst
P'3'24 A tturT)e- the inrl"r qnJ ou.fer ra-dr ' t ' fu be- a- a.ne
a. + r respu.-liwe//.Subs/H.+,-1 tr1rs*F7) t.9 Cs-a7)
a.rt s/ ct s i n3 €7 ( s - // 52 *o [r...u"_
'"::sh:? i +LlizL(46
--4;<Lb ' 'J"'::-!:;;:'rJ,,^d,La).
{s
t
Ps
I"
.=---*'-'
ulr:Ils1,lL
/2-_
rfir1,
' ,7) t1 ytr ,l 'J { , /
= looi
Jf ,j)../'A I :'
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ES-zz .fwitchConsta.nt .
P E-zg
a i r
oPen
a\-
-VW":
1 Cho rJes
c16 ,
71,*a'arT
on the- p/ole-s, a.?^e
.1 Ly'Y-:fr+€o(-"t
e / t \
4\T)rt z -
LIs. subscripls d aoJ a,-fa Jena4e Jie/eclr ic and
regrions re.sp'cch'vely. €'V:0 in both regions.
Vd : Cr/ I Cr, E" - '9c, , D4 = -rfolrer.
Vn: czy1 c+., Eu= dyl, , D.: -dr€oc, .
B.C.: Af ,/ :O , Ya= OI al y=4, V*: Vo
af y :O."d t Vl=V , 6a=D; .S.,/n;n"i",: , c^:Ot =fu' c+:ff i
r;:ffi' E":-a/&T^:WV,Eo=-dyffi(f)y=a=-(D)y=a=
ffi(f)=o: (D)y=s=-ffi
a-)
b)
c)
P g -zq poi . r ,or i .7. VrV = -*
.4
So/q \ ian : Tf : ' '+ , + c , ln r+cL-
&C. , fA t r= a - , V : ' *o "c , laa tc , .
\A+ ,= b, o : -*t * gln brc,.
1 a / 2V \T t-rTT/=- A1 r
. : {4b'o)-Jtrt -
La6/a.)
n = VoLnb+ blnt-t^n, .-tLa (b/a-)
/1
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Ps-so V V :
. ) o l - , l i o n : V = , , ( . n r + c ,
,R t ,, n d o| ,u e-o,.ref .f r'o .t : :
A t r : c ) - , V = 6 = C , / n d \ + c L
At r - :O, V=0 : C, Lnb .* c .!'Y : - , v o - , , . . - v o l n b
| (n (6/c-)-r -
ln ( b,/a)
,/ = Uln (b/,)
F l,,r_ . ' 6L ; -G /A t L : -V f :a r "T ;CD
5r , r " r u < |e r r s i l , es : A f r : 4 t Eo :€u t ,=
4t r :b, l r , , f . : - i " I6b Ln (,;,--")
C a t T ; c - c i / ' - . a . ' : a a ' - A - 2 T - ? l r ^: ? r € , < t / r i \
, : )er ctnri , 'englh</ :1i :
"
-.Jn$,/^) \ w,t, tr1., /
.P.3-st
a) I.o,,.,E ,"r:i:t onr..u,-f f.(3-rzfl:0,"€H)=0
S'olu ion:#
:#
---+ V(e) C,n (/o, *j * C,
B.(.O VG) = V, C, n (n,^ ) * s./@ v(4)= i
//. .L t,: r, (n (foo
!)* c,4c,:#6 + v(e)=+*+?!
+r?(,#)"
b) E: - ; , r rv - - ; v ,Ln f ta 'aG/z t )
"u eot- -qo
nZ;W;U/rf s,-,ie'
c ) C: t'he<one e=ot. t =<oE/a):_V7ffin;,=;,;_
6n ,he. lJ rounJeci plet* . e =?r/+,t:_.$ ({)-:
n#*
1 1 ; a ./ J J - < . / Cont; j ." f />e condi l ions in *he
n \ 1 / - A / t . _ / _ t ! _ l _ tP-4:;€Lq-[ - R,
-e)
4,= {. -.(1'+ y-.t ),f '" ,Rt- l - ' .x+4 z+ (y+ J ) ' f ' " ,
20
xy- p lane (z =9) .
w L.ru
R^: L(-- .1) '* y+4 ' l ' " ,
Rr: l (z+4 )'+ (y-Jf)" .
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r tIo
a
J
C
IL_
, . . ' R,
fn)
) '1 'r '2
y'z_ l
Er=avr=a.y, Fr#/
-= q I y -d t - .J x+c i z+d I"2+tre; 4,
*R:
-u
*R; J
, = A f Y - d v t d Y . / v - d l+uY4ne L-
-,rl ',\t
- -nI - -RT)
€O wi/ l Ao.re a Z-contTotaenf r - [ ' / / ' re2a;otPJoesnul / ie in'( he zY- P /on. .
D Cn +h. con- lu e. l inzr hnl*-p lonr t , 9 , : Dn:€ En .
/ ton34l ' te x: -aYts,y-=a I R, : ' fC- - " ! ) ' i d 'z f ' /2 7 R^ ,
F -n r - aiz* ! ' r : t '*+Js'+4'- l ' / ' :R, 'z- u) - r - r * ln , r - n; )o / - \ Ad ( t)'uY=<t1-2.t( t r&:Tr:d,T-JT.-
( O a - t x :o .: 1 .
| ft'to.*. ol x: d
Pl- l+ z tI r
u'l':f;i"'' l
/ t . . E ,x t_/_-- l / r , Y3 l / '
- o {
" lE,: E,
S inr; l<, ly {" r -? ,&: o) on fhe v27{,'.o./
ConJu.h 'ng7/o, r .
hy chal i in? zlo f .
f ( x + 4 ) ' + " l ' \ t t '
4 ,, ,,n e f, { sc nc m) 1obe at /: e i^n=i . j( ,al
a) Ve, (o r {i-o,n *t" 4 fu,+,3) r, =614
l/e c ior /'ro,n -.12 a /i : 7- . al , + t 8"6
F = l p I . : f r d /L*=GT- #Tr7 =7xtos(6, ,o.2s)
= - P ^E-:*% ?
:-7xtos( dyc.c??+Vro.us).
+ r- : 7 y bt (ay o.r7s a"o.r7) (.vIn)at l.
2 l
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ul b)I
41-t \
t l \t l
.t r -
P ,
"?- 71-/r i ,C- ' t "E ,
_try ELI
At ,Z (o,4,o) on fhe ry-pla-ne( t h e . g r o u r t . J ) :
\/9q4or f-rn fa /o ,0. i t 4: ar+-12,
Ve io . l, o,n - fxf" P, lt i = dr++43.
: J- ay+-dri t = lJ_ U:4 _ln lo 4t * 3,
-F +F = f ' / - t ! li '- a/rq ( 4r1 i,,/I
:7 .. /€ G <ra.:4) - -dzz.reto'(v/")
(- o.24) ff"50.a4): - ' .7| \ tc.sc - '\:_ .7 t (pC/m. ) .
.T
fr . -ao Ezz
, 23 '3E Qiuen D:2 (c^ ) , c -= 0 .3 ( . rn ) .
a- ) ,F,-c,r't". (3 - lt=),
P"- - . ?
) d
t ?+9,
L" 'n) 'foofu)
t - ' -E
"J f (p*,, f f i , ) : f lz+Jra6r3yJ=,,Js4 c,-\ct :D-4-
o, z:ffi:
c) The eTuiva.lent
2 - / .95 4 : O.046 (c ra ) - O,46 (m-) '
21 1frxrdq)t /oo = 2.71x 161 nt-)
:2 .9L (^F / - )a ( r.?s+/o. t)
l;ni cha-ges 4r e sega.raf<al by
d ' -d -d i : / ,?54-o .a46: f .?og (cm) .
l7 l :P '
l l t : G@x2: l t t -1 (v /^ ) ,
p/a.ne- Cort#t)ning th<-'wires ,
22-
-lcf,(v)
G
't.
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9-@'t-",3
S te oJy E l "c l r i c C t t r r en t s- a Z
z,-
L . * o -##=3.trxro'
(s/-).
(v/^)
(v/,t
)
-
P +-t
P.+-sa)
b>
b)c)
a)
a)R:+-
E: +: ( x/a-,p : t l l : / (w ) .
9"= -#.. The3iv.-r, e/e.lron o,abit ity
/-
th.-f aF o3oo4 cortJuclor.
/.+x to-3 (m'.y/) is
' -r:I l : l+l: ly.el: ,.ot o'x(4,o' '): g. 4 , lo-' (m/r),
R,=P.esislapce-zr unif t"*+* * .-e- +=
7fr-'Rr: Rr-is1y-.nce-pe.r ur'rtf /ength o{ coah'ng:
#Sr'L e+ b - f /1 - k 6ess * ra 4, g. ---+ S, : TG+ 6)'-rr4': T( 24b+L),
r ) D :p +h - ( /T - 1 )a - :2 .32a - . / ^t ,/\r-. u
1 \ ? - T - J n T r L I - - Ib) It : Lz: 7' Jt :; f r - , , Jz:
&- q: m;
E,:*:#,8,-#:#Thus, J t : loTu o 'n4 E, : E - '
Rtb2. >b:
R-<b:R->b
"
2]
P.4-z
1:4W:a^# |k/<tt :ez.sx /o?RA''fz"o"lun)
€l=Q-# : -n-#x o6 r/:)
4- nrr-i*7.s x /o/op'?'12'/0'7
( A/-').= - A
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P 4-4
b) TAii
P.4-s
P.4 -,:,
a-) 4'<r/<)t
e
4'<r/<)t= +
: o.o .4 t :#8.:4.
sa, s-/tt"t-q.rrrro-2 0
71"..r; Jn' : zz!4'e-2G/4t : (hri)rfe-sb'fz.
. W;, - f : b/dtlz : a.o / : lc-1u::.:!jj:::,:r,o
Ti lo= L'* J
c ) E iu -t,n'sleh'c e-ner4/ tar ?f,7"
4tzRJR =#
= +s (kt)grcrec o . t ls iae fhe sph. re ru" z .
D_ caa i fon t .
I ,= O. l (A), Po,= 3.3 3 GW) ;
Iz= o.o l l (n\, Por- -s t (^tl t) ;
Is=. o.o667 (rt), Por: a.E (nW),
7o ia f va 5i5l,ances een by ih e
z":o.oz (A), Por--8'Oo.W);
Ia=o.ost3 (a),P" f s.st 6W);
E Po^: VoA,: 70 Q'vl) .
source 7 U\ ) .
d.*zo a"so (v/'-)
Ert: E,,= 2o
f,z,' ,:J,n- 91": TE,n
- Ern-*n ,^ - #(so)
= -75.
: 4*2o-427, (vl^\
b) i:
I E , : /s x /o - t iA -2o- dora ) '=d ro . ; ' d ro .zs Q| "n ' \
i r : a rEr : lox /o3 (Ax2o ' i " zs ) :6 *o ' l-
Qo ' ts (A i * ' \
c ) d,:tan4 (#u) --6s.2o, 1t- ton' (#) : 1s. l
C Drn- D,n= ls . ---- €, Ern- €, E,r: fs '
f , = €o ?S ' '75 + 2x so)= ' /25€o - { ' os (nC/n ' ) '
24
b ! -
a-)
t t E - ,
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P.4-7
b) ( f r ) - : €oEyu) -T
-
c) E?,;uo)ent l?-c circuit
betwe.zrt /errnina/rq q,n4 6 :
f ( y ) - 4 + ( r , - n ) * .
a) Neg Iecling.f,. ;1rqng ql{e-.t a n -ld.tS.r,r1 in7 o <urrenf 4e-tsi' ty :
i :-trro-E= F:-dr#
U=-Io€rur={nffi:#rfr,Q=+=#-- &rV(,',+
1o(c,- oi vo
A;6oo uPfer Pta'E'
v+l , , s
- t d /
T*i c{y) TUoI_ r -
A
)
) ,
l
(e-). =-1^E (d--S&=- {o{a"- ' ! )V,o, on fon.- 7l- t - .ss'.I_ -0-/*' rt r,4 LnCrJq)
P 4-g rx) Corth'nuit/ .{ 215. torna,l Compcnenf ol -- atss ures
thu sqrre <urren-f in bofh ,nec{ia, E7 ft;rqhhof+ivo [43e /e.w'
vo: R, {-) .=(#. #) tJ -+: |/o , = lCVg
5 1d,/o;)+(4,/r,) G.l, +{4"
b) T*o_-7't+/ions orre- ne.edeJ for the Je.lera;nq*Zn
o$ E, and e: Vo EJ, +8,4"
an4 iEr: eE..
S., / r ;ng t u le houe1:
#an4 r: ffi
c'a',:#'Rr : *
r - € r S"t
- -J7'
cr:ff.
25
7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5
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P 4-q c ' - )Same €7c, i , ,a . le .n f
Probfen, P4-{ vu l *h
/1,: #t"(*),
R-C c i rcu i f sr t t / t ^ f
Qr:#{r(+)/ ' - 2t<"i-z
l" (a/c)
t n
^ :2 7 f € ' L '
u! Ln Ce/a)
b) f :%q=1r 'o R,a4.
T : l l - 7 - q r 'V4 t - uz -Za ' - '
p1-ro ,Q-es is lanceo-# (Fr+- /4)
{fo^oSeneoLr7 rna4eria/ , ,v i+6 a
8e*"rreen toV ar-d 6a#o^ {laf {a ces.
u n iform c fo ssss cltb n.
": E 1b'- a2).D: 4 h!
clr (b'- *)
P +- rt LG e Lap la.ce's e?sqfia a in c2r/inclrical uor/,nq4zs.
v,v:o+ +#(,#):oSo lu { ion : I / r r ; : c , l n r } cz '
f lo. nJory con4if ions: Vto)= % ; Vf 4: g.
' VQ):v;ff iE-(r):-Ar#:rrfui ( r . ) : rEcr ) 't :J - j . as - fn ' " j . (a ,hrd4) -
- 5
Q : V - 2 t n ( 6 / d ,I rrah
tr chV2 ln (b/a)
26
6 Ln 6/e)+ [ Ln +/a)
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P 4- lL Attrrr" d- po/.e.r*i*( 4;rqe.ren*-W be-/--etn +r\e-
;nne-r a-nJ oifer spAenes .
V"Tf 0+fr r(e^u)=g. V: k-.- Eo=#
W=-_f:,E^"R -u{o,iq*: k (*,- t)
-> k:# i*:(Er-nY+'
' T t -4
t - l'"f'l^ p2-sne e4g : ?y{u o t o R R t R ,
A:1=Le- / \ z
-Tn\T-E/'
25 .
27
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!Il
I::
p . - /
S fot i , Nl"Sner ic Fi e id '
F : e (8, ixgt : g
E - - z,L B - - d,-uox (ar^*+ Er3y, 7.8.)/ -
: uo (d rBr ' - tzg / ) .
E=doE/:44*
6: / B^Js: / i ! t fo 4 a,r Js '-g 2n Ja- r
: *io-,Y' lh r 4 ^ e 4O ) a
a. t r= 33L is t tssd, E ' :#( '#)_ /
" / er rcr : #x loo
: l - -z t ; -a) - - -1 ix /oo,| <b+^) Ln(b /a . ) I
:s , ,1,"r .sr
is l##- t i x loo,
or - 17. L'/ " (4c" /ow).
P s-z
l / - n-E
Forb
u) U-re t- -.r . > ' J
" , 'I
I
$D)
( f - .?z<) . df ' : d"dz ' , E=d,r -drz ' '
aZ x [: ardz'x ( i. r - drz") : d4 r'{z''
Er:aE+i uL e t Z ' = t ' f a n 4 , J z ' = r ; e c 2 a < 1 4 '
t3,-%
#I*'o, nJa.
: a+# Csin a1r- sin <,2.
For
ll
DP
zg
an inf,ni(ely bg vyire: o(r+Q2o aa d cl,+-?oi
becomzs a( #' &s i , t e7 'G'3t) '
II
t
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i
D ( - 4
-r-!
- lY ,
It
J
l.
,:'?, | \
1 \
Ii
" l
--F.-
Llse F7. (s -3 s);
ciE_ a-.lB- a a,d ByP . . * / /'
: d.xGS4 sin€ + dr{as4:o,'",
vvhere-Bn--ffi*,r*'. r J /sino=
6fi7rp'
cose:
6frry
,
Er,: 4, Bn + arB, ,rlr ,t ,
| - / o J d Iq x
Whe re 6z =ET )o x, '+ d,
and p, = /_i!-_ [- .i/iDy 2 r t J o x " + d '
A o I L t f z r l
f f i tan \T/ ,
#1"(t,T)
L f-{o*t in*alhe PdPe'r
( in -Q 4i rec f ion) '
/ ; = Foz4z 't bc- - qza;A4'
circcrr'ta'l lqv't ' E7'cf-/o)'"" ' 'J
assurvte lhe
m e.4 i u n' t to be non^^Snetfc :
f E' ai :PoL'
6 z r 4a-, E: d+#-
a-1 r <b, E:4{ #
B=a{e*W,E=0.
21
r 5 - )
J.<.
: ! ol,+-?Oi
3s) .
c,
t
P.s-6 A
t"I: l
, s c t I
t t t aI
III
r.!c:It rL. I J
V_. \!
E,:a,# /:""'-ry-d,# Ln*T)
ly AmpZre}
Fo"
fo,
Fo,
Fo,
b a r*
f . > e
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D < - t
P;-r
A s * r t * " t h q t f h e c L ! r r e n f f i o v w s
rn f he. coun(cre/ackvtr ise 4iree{ ion
in a f r iang/e ly ing in a6e xy^p(a,ne.
Fram tf. ,t-34) a n4 noring lloof
t - qand r=Pto - ) c " : - ! ! ,
2 2 . . 3
v v e h e v e
E: s (a\ * z
L . / r : J l , J L ,+" .
: _z :\/3
d :d 3^ toL - -5 - - ; g t t r z'zzri ur,/./T
- qzZ?tv'
Us" Ef G-37).
,Fon L*n , O"*g aa{4* t ,
a. n cl E* 4, 4{Z : d" ratnl .
'3;- q a-) 8:VxA:*€*d A, ,/toI:d r 2rrr
J ; = / J Z t ' , * , . .
dE az-VVi.-*gy"(il r.',E:v,+fs
_; ,upz f L -z.zLL l-v/e- z )\ 6,
z l
:ar#rrfi!)rl,
: -9 3+ (rVo horse*,+h.)
A-:-#lnr *c-.
A ,: O a-t 1- fo.-+ . - -E lr ro.
A=d"A,:4#uC).
30
4 2 - - t
Ztr rJ 7'+ r, , /
f - ^ - -,.P-? - {0,
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--i
Il@4,,]'r'-,
t) L(seE:f A.aZ.
l lor izonfal sides howe- no e-{{e.f .
S iie @: ,l^ A J,{=(# t,,fi)"o.e
s ice.@: J A . dZ: - (# t^ft),o.t,
9 n.dI :WL"H)x0.6( a r d n ) x l o x o . L t: - - - ' n ,
= 2.)4^ to- { (wv) .
(a n e.
7f
v .
1 : 2.34(fwr)
Ps-to I n[ ;n;+t cct r re-ot s he.t4r J so ;-
-B ar#i-sTrnroefri=a-/qnd
ind zp e-'tdenf ol x *rJy.
>
tl
" , l
Ais
o) /97r/7 AnPi-z'' <it'cu)+o!
(ow/ lop<rh /2V( l :
{e -aI: /or -2-B/-l;t,o4
-; B- : [-,HoJ'Jzot (o,o,z),
'/ L*ruo4o l *+ (qo,-z).
e(, E:* j r^
b) Fo, z )o'FxA:E-ur$)
i ldepenJenf o{ X . ,nd y.
4Ar - /07*4z
- - -T '
A*: - ryz +c-,
At r-zo,Ar: o : - ryzo+c --+ c - Mo"r.
,,. A: _qqz_z)4.3t
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P s - l l I H^=4H^l v
a-) G)ve.n 4--/,4Err:Bo-F Hr*FoHo-F": LHr:u"f ,o.
5) Q i ve n E,--t'.G, +y).'/
Br": BF- r(t0(H2,v;):/o lo* Fr=do0/r- lr l.
+). Ioa r <b; f=u, nt,; B-=a"Y"nT,
n I . l M= 0 ' .
,il(fr,-)nr:a+(fr,-)*.
me-"/
t Fo=uo4
,D.S- .12 L) r<a_:
a) J; *x i l - -o;
P-EJJ- t,
n - ! tr . > - t 4
Y- o
i ! :^", r ,L i- (t - r8 = \ u n I ) I '
M=+-H-=d- l l t - i, o z \ F o . /
J-F kl xan=( 4"
^) j^ : ix f r : O.
J^r= (e* c4ss- arsln d M x dn
: d+ Mos iae .
t' trr', E7 ' g't7) rc q loolt a] raliu,b =i6 g <a.rry t nq a- <ctrre-nf
$ . r b J e ,/ J
t= p , , ( L , g J a ) ( L t ; - a l 'd B : o r #
2 ( L , ) r , .
= a_ !?Mo s in ro .2 , 2
E=faZ- urryIrrin, ede iziforo: ? f,r,at the cente-r o.
a) B,: d* 2 - 6rto (^T),
Er=4 Br* u, 4lu : Br, : ! - r - 2/ |2-t 9ooorUo ' 'lr
lo+ Brx=lqooo (nT) ,
8ay: Erl -/o (twr).
Er: d.-/o,ooo-dvto (nz).Y
ta-n 4r:#,tcn
o1, di-a.04-".I
:-904.0
32
M- drMo
@a;.Fnnl
-Br
@Jr anfn'5ooo
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- N o , ,,F'O
b) f f 1: \ rc*a, (mT), E, :a-B,o+dzg, / .
H,*:#: Hr':ft
- B/.=*,oo:#= o.oozEU E"y=
?.. ' . E , :d*o .ooz+drz (mT) .
4t': to;'fr = 9y : o'oo (ra-t) o'o{?o!o-M).
7
P. - rs
P. -ts
g=nf 84=aq#r f : r , -gc4te\ .
6 = , F I N I S a l z z r P a < 1 p , . , . - / - \z -z'n oJo \ -r*fr= foMZqt lr ; -d.
L: f = foMt( r . - f fa)
I { e>>6, i3r+#(canr /or* ) . .
E *g*s : Brftr6,) ,"2{tz -L !#
ofrr I in Lhe- lor3 -s,-lr"..r'gLivire, -S=4 lf
t
A,,={E.rEJtr} c,-a4;#["t r' t4si:': #l-_?*ah(r*g+)1
L,*:+= #V-tt^(,.#)l
P.t'! Approxirnole {he rnagnel|c F lq;a due lo16e lang/oopt.inking with fhe. s6all laop by fha.f Jqe lo fwo in{,'nikty/ong r'v ires carrylrtg .7.,o/ 'an.7
oppos/fe <ctrre.nf f .
A,,:#{*'(#-ft)*_-ffr4,1ry#)
t _ Ae _L,^ '- T
- Sf t^
a!=ao4l
33
(14r+4)1!zr,+ )( 1 + v r + d )
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P S.-i";- /v{*gn..f;" f iot{ irtleasiiy a/ 4he uqire du.- /, fhe-
f t : I , : I r :2 S A) , C : 0 . S (n) .
Er:d*2 E,rcos oo"-*#*
Forre pe r ur,;/ /<ngth on wire 2 :
F;: -ayz82 -a/H:: - d/ //sofo= -dy /.41 x 76-3U/il.
Forr:e-t on a-l/ l6ree uvirec q.. c>{eo,-rq.Jr/'2a.9nriu4e qnJ Lw-r-/ fhe C-nler *
fAc- rrr'a nqie.
Cu rrenl d f =*l y
in an e/ernental
Ct; / /S,/
l " t4 l : -* : , r iY.2rrwJ't.fir
Synrrn-/ry * H qf f he wire- ha-s/ , . /snly a >/- <onponenl.
F u,J (dH).*):dyr"{,{ffi:"/#ton-,(#)
f': r xE (-4z)*Wrn:o=4#+^;,#) (u/^).
j; '" E(t * a1) o.[ :9dl
dF=Ia[x E: -a,+#(,.#r,
(,4 reif -gcrn/>ro6/.-.) .
_* . ,
T"d
I*x
l"l-l
w r, I
l Yl . L?
PE:,!
'+I
F:-+W[ii.*,),=-d.ff t^(* _t)
3+
l.--- o
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---
:'.
P s-zt For .e in a Lrn i {orcn magne* ;c $ ;e- l J :
I+
t B=a..o.S/ L
F = I t^E =-Bx(rL) .
o). Ex r Gn) j(€h r(fr)- (dyo. )t /oCEo. {/ o.t), o( eo. $ o.), / o4 o.
forrct -azo.s -A"o.s a"t.o (N)
Torfue an looft:
T:6xB -- (a"Z52xE
: (dr/o, !x o.zxt.z)x(6.ro.s)=-dro.l N--).
.
I
Bful,0)
D t - z t B, a-f fhe cenler oF fhe lot3e- circular fe'trn
v.tire calr7i2g 4 currenl 2z is (b/ telf ln3
- O in Ea. s 's7 ) :
_ !o I ,Br: anLE
of
,
a(
o
) .
TorTcre ofr fhe-srno/l c;r..t/or /u'n oS wire carryi2,r
a - cqr re -n* 11 is
7 - fr,x E, z (dn,r,,?rf)x"",#)
: (d-,Xdrr)ry'
vthi.h is a {or7ue houinS a. lnagniluJe-
Zrr|nr'2 sin 0 and a Ji.ecl ion {s6Jing to2 f ,
a/4i fhe' rnagne-l ie {/u' 'e 's proJz'<eJ by T'
a n d 7 r .
35