Engineering Electromagnetics Cheng Solutions Ch3 4 5

7/27/2019 Engineering Electromagnetics Cheng Solutions Ch3 4 5

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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)ôr: 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



: 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



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



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+



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.



; < 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 ,ê 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



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")



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 :'



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.?ê

.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



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)" .



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 Ê-:*% ?

:-7xtos( dyc.c??+Vro.us).

+ r- : 7 y bt (ay o.r7s a"o.r7) (.vIn)at l.

2 l



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.



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



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 - ,



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-). =-1Ê (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



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)

{foôSeneoLr7 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)



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(eû)=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



!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



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



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,



--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



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



- 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 )



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



---

:'.

P s-zt For .e in a Lrn i {orcn magne* ;c $ ;e- l J :

I+

t B=a..o.S/ L

F = I tÊ =-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

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Engineering Electromagnetics Cheng Solutions Ch3 4 5