Engineering Electromagnetics Cheng Solutions Ch3 4 5

J

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,_\".

z -

" ^ ( T 1 ( 4 )

v r n o * erw,)

b) A+ flre s".e.n, (do)^or: D/z

L rnqsl be. z /-,"vx

_t

w r h e ^ r e '

L

_rlr4x

= * [ * + ^ - ; 2 h ) -

_{z I}

ew _{Vri r/}

c)E.ubl--

_{Wo* Ay dou-6/;* Lt, or}

./ouL/,'n34"-A node 4-ece /arotrn.t vo l4a-qe-,

.o

2 s - z

P ,

s - 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 l e D q l S i v e f a r c e s I oF thz triangrle.

r y :

- al 3 + d"4, e-p : ar4-d.z

At P

: E,

:

_ffiV

Gart+

d"4)

,

E,: Vfu (*+-v"3).

+ a ) b) N" y- cotnpanent : -3 Q, f y / o Z - C o r r r P o n e n t ' _{4 Q ,}

4 Q , - O + *=+

3 o - . : o + +=+.

' L q 2 4

l l

: fl' , D 3 - 4

3' 1"

g'

' 1 < t ' C ) + ( A ( ' ) - l ( _{Q . c )} : ' '

*-x -J

I

r ' ' - 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 ( c m ) - i + c a n b e p r o n e J Q , o n l Q 3 a r e o , / s o z e r o . A F

-..'. J -t _{)- rom}

Eg (s-4za-), f,: r;!:ffi

:

-*-E - a , * f

_{t - ( r + !;)-','l}

P - e i L ) r t . 4 \ 1

: 9tr I, - ( '- E* -?;"-....,)J

: d"#ryLie) -*er* j

t +=---(r-*4.:*.".\-l ,

' = + l

a " O ' \ t - V V

/ )

w h e , ' e , the {;.sf _{( e - r a is lhz po,'nt-<hcrrqe tern-t qncl (h<_} ( e s l . , e p r e s < n f t h < - e r r o r . 1 o n ' ' , ' - l e r i ' y o n / 7 lhe [i.s/ e.rror

+ i- 3

o.ol '--> z 2,8 b, ,,, s.(,4

b.

, 2 3 - 6

_{A t o n q r b i t r a r y}

r { t t -* nP t h . - ! _{/ h e . n e - l { a - . = s o n} Tefr-Yt : St'-i/or sifualion P ( o , o , : . ) o n f f > e a . x i s .

I, b 4d'

4 l r g ( z ' * L ' ) ' 4

fliiErl:

o

vtt he-n P is be /o,+t lhe /oott

ur-ffi,:4fu,.y.

Ep--

-v4:-o"* -a"ZfuE

q ) At point @,qh),

E ,

= Q # #

b) To {;n.l the /o<olion o$ r,dr.lEul _{, s.f}

xn*lEr,l=fu'

P 3 - *

ctE

.: - f!9'9 si,,-l',

- ' Y 4 f t € o b

/ : = ; F - - - n

t z

f r s i - a e d

4 -7 */ -'/ ttlob ro a

=-7rtffi

t e J

t

, p .

I - g

S p A " r l c c z l t f - - e l n y t

F : 4 r E ^

A 7 p / y Q . a u ' s l f , z - .

r) o < R<b. 4tn'fo,=

_{*{-f ,}

t1+r{an=

Te-#),

t'': **ry -#)

z ) A>R<R; 4rrR': *^{t(,-fl#rtR?e: W^t,

,u'iJH^

z ) R;< R< Ro. Eor:o-.

+)

_R>R".

_{tr; ffi.}

,P,

e-9 Cy[;n,t.;co.l sy,n^.*.y , E: drE, Apely 6:outri (aw

a ) E r = o , d o , r < ( t .

E^:

?9:

_{€ o l}

, fo. 4-1 r z b,

er=ft?,

{or r)b

b ) * : - _*

P 3-lo W.: - 7I E. ai :

-V/04x+xr/).

q) A long *he pora6o/e Y =2/'-, 4/ :4*-l*.

W " : -

l g x l o ' a ; / - t f z t + + z ' ) " { x

: ? x t d ' ( r )

-

? o ( 7 t ) .

i l A(ong 1y'.,s

_{s { r c t ; g h t ( i o e #=}

_{# - - 2 ,}

Y = - 2 x + 4 , 4 7 = - 2 / 7 .

W" =- ( E * t6t)l-' f{-zz,++)

dx - zx4xl : ?o (/t).

,o

v

{ 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 [uz

7tr€sJo .l x-.yz

P ^ r .

, - (

--#"1 Lnvtr)if'*rJ

u ( )

- LrY j'

E : d n y - d r x ,

E ' d I : Y 1 * - * 4 / '

We: -7. -f

-^

(z; - 4*') dl : toqri)

'

W. = - 7 J

-

ir'zz ++ ) + 2r )JN

: -6o (1''

i )'

'

fUe qiven E d;tH is fio4c44sarvq-/-ive'

t / ' c / 2 o ; ' '

a-) r/ = 21

{u "'*

y -)t) I,neo/*

I

n ra'a*

r ^ - - - , : w t E

l:'o''

t \ -| \,q

l\

f f , : F ' z n : P o * 6 n 6 2 1 / s ) r

9fn:

- 7'P - -3Po

o . . : ( ( t ' ) f 7 r : s P o t ;

a n

T"+*t bou,.d c hc.r1e : 4,

c'l-x4

?,i-

_{b) froe Ca,l.^b7 /o-,,}

a , = z [ q L

F v v d * . , : ;

l l , f % 1 1

^/ ty-

_"Jo

uy

*'rr€uR'

ny 2n<o./

_{lJ1tJ"f"y, i}

F V 3 i r e - s 1 l r - t o o n z a n s v e - t r a - s i n b ) .

b)

{ o . . , 4 t h * c u L s

-: ( Lt ) fp; -3Pot3

' - e.r:O,

b)

F = a r P o .

f f t : F ' d " : P o t, ' r 1 6 c o t { .

f p n = - 0 . p : o .

d , =

_{n b ' s r n ' e c o s f c r y}

4 g

no to

: O.

t+

*a'2-P s - t s p : I (a-*3x+ ar4y).

o ) fp_:-V.F:-

7 4 .

f s t a / , r o f

_{. , , n e c h , - " g e Qn: -IPorr{1'' rr') /)er unir /engfh}

o ; ' j . o f , , "

,

' ^ = 2 - .

f t ' r " : V ' d , = Q (a; rocosl+474r tin/)'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 l27l

Tolat Qro

:d"

_{lrro ro i f}

: Po

r,'I ' 13 + ti n'#) di : 7rr Po1.

_,

. P e ' r u r t i l / e . t 3 t 6 .

l o t - l Q r , = ' 7 r r P " 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 w ' . '

l,

t L e .

p. 3- /L

( / )

_{R > R o .}

( t ) Ri<R< R".

( z ) R < R ;

f = -a3

V t =

t r :

_{t R , :}

- a

@

6 < r R .

'

V , : 747'

A F F A

Do,: ,o Er,: fu,

Po,:

c.

t r . ^ - q ' -Rz

+'lf€oerR: u4, +i'E'

P*= Do,

- €oEn"=

_{(t - +,) *7 .}

v z :

_J

R'

_eo,an

-r[R

E,"rL=

*J,,-+,t A

& ,<^

o

#F,

DAr:#, Pp)-0.

I

r L

vtlo=r,' )o, Eardq'

d . frt-trJ--/t-l-\

r rd.

+ n f t L \ ' € r t R o \ ' € ? l e t ' A )

tg

; < J - / . / - ,

P : - t s

P . ,3

- ; 9

Ll;e subscript

V r : E t " 4 " :

V t : E r y d p :

a fo, air, p {or plexiglass,aoJ bior 'breakclown ( s ^ t o ' ) * (so x /o-t) = /s o x /c?(v): /9o (tv)'

2 O x lo :y' Ooo (*V)

V b = E u - i ^ , E F { O : E u ( s o - d r ) + F O . i ,

Alo*' D^= D, -

€oE^: {o€rf Ef -

[^-=trfEp , Ef '

W : 8,.^(

ro-/o)' +

_"

/o:3 (zo- {vo) : /3o lLD'

€r^ 1 f tt *Br"okJ"rtn occurt rh c,)r re3)ort {i''st

b ya. ba

bP

2qtoh fr.-sr, 3 J

,4t the z=0 plnne ; E,: d-:7 - d, 3 v + dt 5 . 1 ( z - O 1 = f r r ( z : O ) = 4 r ' 2 ' 7 - a , : > '

D ^ G : c ) : D -

D., (z: o) --- ZE,,(z=o) : 3 Err(z=o)

,

z- 21 - l o l t 1

E, (z=o) -- 4r2y - drsx * d"f

O . t z = o ) = ( ' q * 2 y - v y ' 3 2 .

_{+ 1# ) lr"}

f , r r , ^ e E r = d . E r - - i * E . ( /lo.,r7glory conJ;'t;on: dns { = 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;

riia,b,n

_{, Zn' >, -- 4'\}

-- ' _{€r Err=€t ErrT* €oS =€o€rr3 '}

+ _{4 r r =}

{ : r e r 7

P . 3 - z o t :

_{T l + € , .}

,/ss,t*e Q- on Ptq'fe' o-f Y-J

.P

s-zr

r/=d _{A-d ln (<,/<,)}

V : -J

_{- Y - o}

E.,iZ : ---=4.L.

J \ s : - e , ) 4 - A - ' l ( € - - < ' ) e _{V cJ bGr/e,)}

Le_+ Pt b< the-/ite-u_/ q$c{-i3e. 4ensi*y ct-t rbe-)'trle-r' c - o n d c , c h o r - E : E . # ; 1 .

_{t ' o - - - J 6 E ' c r F : V f t h ( a t -}

7 ' e - P , , r 6 \ _ € _ 2 t € V

t . t - TBn)

f f A : d , _' % = a_ ln(b/o)

F o , o f i ' x . a b , fhe {un</bn i o be ft')n)mtze./ is: (x=L/o).

{ t 4 :

_{# ^ *}

.

S . r r i n 3

W

: t y t e t J s L r , x - - / ,

e

e : - d y ?

: f f i '

a

' \ - T - Y * c , ) + :) l o . 3

t

ot')

b )

Q r _{V : * - - e : 2 . 7 / g .} C ) r n i n . E(a.l: e q /b.

'1)

_{c': +:fu^l= 2tr€ (r/^)}

@

E = u , #

E , = d , r # ,

r ; 1 . . b ;

E r = & r # '

b < r ' - r o

ConJucfars) res?e<iivel7. E = drE*

v = -L'*8. axdR--

_#

(*

"Eo

P . 3 - 2 3 Atrr,,-,, efia,rges +Q- an.l'Q

o n f h ' e i n n e r a . n . J o t i f a r

= ? i = Q z .ftr€oqz - - / \ .

4 /

t ' _ L -v y

t 7

( r/a, - t/tz")

P s-24 T*-,f

C-afa-e.t'fony

a-cfos s Luhuy

fe,.,nino/",

cr: t Qfl'

sfor+..1

_{<).-f ri< <n ery/ W, =i cr(rzo)'= f.t(^w).}

We ;,', 2-{7uF) capct

_{ciior = ! tz'6i'}

sL)z:6-+(,nw).

\Ug ,'r, lV O cctl>ct.'7:f.,,- -

! (r o'-"2 x 49':o-y q^1n). Wo in ){7y) c:.pctci/tt-= 4 Sxfo'c)xorz :2.g1nW).

E : F ; r 6 r s i n f + e / r c o s { ;

4f = dr 4r

^+

a4rd{ rdr4z,

i-',\t = *a-f t . d,E = - Tsr ro-'"lli4

s;n6f'ar

_{Jr + 3 r/ !; 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 ^ r o - t o ) l

3 r " 1 "

s i n 4 i - - " * ( s ; , ( - n 5 l * - l

L , T / 3 , ^ , L l . L J

: - ( r x t6t") L- ts -3GJ :27x/;q (t):zf (nT).

9 =tT,/s, r {rom 2 Y4 ; .fhen r = 4, Q Srem?rrb ro -.tr/z :

L(-=

-(r* /,t,")le

,;ng)ii'

,-'Lf

L -: - ( 9 x ( o - t o ) | r r - f z ) -: z ? x r c l l ( t ) - Z T G I ) . --.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:?

_lizL(46

--

_{4;<Lb ' 'J"'::-!:;;:'rJ,,^d,La).}

i +L

{s

t

Ps

I"

.=---*'-'

ulr:Il s

_{1,lL / 2-_ rfir1,}

' , 7 ) t 1 ytr,

l ' J { , /

= l o o i

Jf ,j)../'A I :'

ES-zz .fwitch

C o n s t a . n t .

P E - z g

a i r

o P e n

a

\-- V W " :

1 C ho rJes

c16 ,

71,*

a'

- a r

T

on the- p/ole-s, a.?^e

.1 L

y'Y-:

_fr

+€o(

r-"t

',tiD. .W).

*w)

e / t \

4 \ T )

tr

t z

-l

LIs. subscripls d aoJ a,-fa Jena4e Jie/eclric and regrions re.sp'cch'vely. €'V:0 in both regions.

Vd : Cr/ I Cr, E" - '9c, , D4 = -rfolrer.

Vn: czy 1 c+., Eu= - dy l, , D.: -dr€oc, .

B.C.: Af ,/:O , Ya = O I

al y=4, V*: Vo ;

a f y : O . " d t V l = V , 6 a = D ; .

S.,/n;n"i

_",:

, c^:Ot

_Z=fu'

c+:ffi

r;:ffi'

E":-a/&

T^:WV,Eo=-dyffi

(f)y=a=-(D)y=a=

ffi

(f)=o: (D)y=s=-ffi

a-)

b)

c) live.

P g - z q

_{p o i . r , o r i .7. VrV = -}

*

. 4

S o / q \ i a n : T f : ' ' + , + c , l n r + c L

-& C . , f A t r = a - , V : ' * o " c , l a a t c , .

\A+ ,= b, o

: -

*t * gln brc,.

1 a / 2 V \

T t-rTT/=-

1 r

A

. : {4b'o)-Jtr ,

vt - La6/a.) n = VoLnb+ lblnt-t^n, . -t La (b/a-)

/1

P s - s o

V V :

. ) o l - , l i o n : V = , , ( . n r + c , ,R .t ,, n d o| _{,u e-o,.r ef i.f r'o} y.t: :

A t r : c ) - , _{V = 6} = C , / n d \ + c L

A t r - : O ,

_{V = 0 :}

_{C , L n b . * c .}

!'Y : - , v o - , , . . - v o l n b | (n (6/c-) -r - _{ln ( b,/a)} ,/ = U ln ' ' 6 (b/,) F l,,r_ . L ; - G / A t L : - V f : a r " T ; C D 5 r , r " r u _{< | e r r s i l , e s : A f r : 4 t} E o : € u t , =

4 t r : b ,

l r , ,

_{f . : -}

i " I 6

b Ln (,;,--") C a t T ; c - c i / ' - . a . ' : a _{a ' - A - 2 T - ? l r ^} : _{? r € , < t / r i \} , : ) e r c t n r i , ' e n g l h < / : 1 i : " - . J n $ , / ^ ) \ w , t , t r 1 . , /

. P . 3 - s t

a)

I.o,,.,

E ,"r:i:t onr..u,

_{, -f f.(3-rzfl:}

_{* 0,"€ H)= 0}

S'olu tion:

#

:

#

---+ V(e) : C, ln (/o,

*j * C,

B.(. O VG) = V, : C, tn (n,^ !) * s. .

/ @ v ( 4 ) = i

//

. .L

t,: r, (n (foo

!)* c,

4c,:#6

_{+ v(e)=}

_n+*+?!

+ r?(,#)

:"

b )

_{E : - ; , r r v}

- - ;

v ,

L n f t a ' a G / z t )

"u eot

- -qo

nZ;W;U/rf

s,-,ie

'

c ) C:

_{n t'he <one e= ot.}

, t = <oE/a): _ V7ffin;,=;,;_

6n ,he. lJ rounJeci plet* . e =?r/+, _{t:_.$} ({)-: n#* 1 1 ; a . / J J - < . / C o n t ; j . " f / > e c o n d i l i o n s i n * h e n \ 1 / -

_{' P - 4 : ; € L q -}

A _{/ t . _ / _ t ! _ l _ t}

[ -

R ,

-

e )

4,= i {. -.( 1'+

( y -.t ),f'",

R t - l-'.x+4 )z + (y+ J )'f'" ,

2 0

x y - plane (z =9). w L . r u

R^: L(--.1)'* (y+4 i'l'",

Rr: l(z+4 )'+ (y-J f)".

r t

I

- o a J C I L_ ,..' R, / fn) ) ' 1 ' r ' 2 \L1 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

+ uY

4ne L-

-,rl '

,\t

- -nI - -

RT)

€ O w i / l A o . r e a Z - c o n t T o t a e n f r - [ ' / / ' r e 2 a ; o t P J o e s n u l / i e i n '( he zY- P /on. . D C n + h . c o n - l u e . l i n z r hnl*-plonrt , 9 , : D n : € E n . / t o n 3 4 l ' t e x : - a Y t s , y - = a I R , : ' f C - - " ! ) ' i d ' z f ' / 2 7 R ^ ,

F - n r - a i z *

! ' r : t ' * + J s ' + 4 ' - l ' / ' : R , '

- z - u ) - r - r*ln,r-

_{n ; )}

o / - \ A d ( t

)'uY=<t1- 2.t(

_t

r&:Tr:d,T-JT.-( O a - t x : o .

: 1 . | ft'to.*. , ol x: d

P l - l +

z t

I r

u'l':f;i"'

' l / t . . E ,

x

t_/_-- l / r , Y 3 l / ' - o {

" l

E , : E ,

S inr; l<, ly {"r -?, &: o) on fhe v27{,'.o./ C o n J u . h ' n g 7/o,r. hy chaliin? z l o f.

f ( x + 4 ) ' + _{" l ' \ t t '}

4 ,, ,,n e f, { s c nc /m) 1o be at /: e i^n=i z.j(,al a) Ve, (or {i-o,n f*t" 4 fu,+,3) : r, =614

l/e c ior /'ro,n -.12 la /i : 7- . al, + t 8"6 F = l p I . : f r d /

L * =

G T -

# T r 7 = 7 x t o s ( 6 , , o . 2 s )

= - P ^

E-:

*%

?

:-7xtos( dyc.c??+

Vro.us).

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

u l

_{b )}

I

41-t \ t l \ t l . t r - P ,

"?- 71-/

_{r i ,C-'t"E,}

_try

EL

I

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 _ lnlo _{4t * 3,} - F _{+ F =} f ' _{/ - t ! l} -i '- _a/rq ( 4r1 i,,/ I

:

7 .. /€ G <ra.:4)

- -dzz.re

'to'(v/")

(- o.24)

:

_ff"50.a4)

: - '.7 | \ tc.s

(c /-'\

: _ . 7 t ( p C / m . ) .

.T

fr. -ao Ezz

, 2 3 ' 3 E _{Q i u e n} D : 2 ( c ^ ) , c - = 0 . 3 ( . r n ) . a-) ,F,-c,r't [". (3- lt=),

P"

- - . ? ) d t ? + 9 , L " ' n ) ' f o o f u ) t '

-E

" J

-

f ( p * , , f f i , ) : f

l z + J r a 6 r 3 y J = , , J s 4 (c , - \

c t : D 4

-o, iz:ffi:

c) _{The eTuiva.lent} 2 - / . 9 5 4 : O . 0 4 6 ( c r a ) - O , 4 6 ( m - ) '

21 1frxrdq)t /oo

_{= 2.71 x 161 ( nt-)}

: 2 . 9 L ( ^ F / - )

La ( r.?s+/o. t)

l;ni

cha-ges 4re sega.raf<al by

d ' - d - d i : / , ? 5 4 - o . a 4 6

: f . ? o g ( c m ) .

l 7 l :

_{l l t :}

P '

G @ x 2 : l t t - 1

( v / ^ ) ,

p/a.ne- Cort#t)ning th<-'wires ,

22--lcf,(v)

G

9-@'t-",3

S t e o J y E l " c l r i c C t t r r e n t s r - a Z

gz,

'22 -

L .

* o -

_##=3.trxro'

(s/-).

(v/^)

(v/,t

/vn- ) )

-P + - t

P.+-s

a)

b>

b )

c)

a )

a) R:

+-E: +: ( x/a-,

p : t l l : / ( w ) .

9"= - #.. The3iv.-r, e/e.lron o,abitity

/ - _th.-f aF o3oo4 cortJuclor.

/.+x to-3 (m'.y/) is

'-r: I tl: l+l:ly.el: ,.ot ro'x(4,

/o'')

: g.4 , lo-' (m/r),

R, = P.esislapce-

pzr unif t"*+* * .-e-

₊

=

_7fr-'

Rr: Rr-is1y-.nce- pe.r ur'rtf /ength o{ coah'ng: _#Sr'

L e+ b - f /1 i - k 6 es s * ra 4, g. ---+ S, : TG+ 6)'- rr4': T( (24b+L), r ) D : p + h - ( / T - 1 ) a - : 2 . 3 2 a - . v / ^t ,/\r-. u 1 \ ? - T - J n T r L I - - I

b ) I t : L z : 7'

_{J t : ;fr-, , Jz:}

_{& - q :}

m ;

E,:*:#,8,-#:#

T h u s ,

J t : l o T u

o ' n 4

E , : E - '

R t b

2 . > 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/-').

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)rf e-sb'fz.

. W;, - f : b/dtlz : a. o t / : lc-1 u::.:!jj:::,:r,o

_{T i l o = L ' *}

_J

c ) E iu -t,n'sleh'c e-ner4/ tar : ?

f,7"

4tzR JR =

#

= +s (kt)

g r c r e c o . t l s i a e f h e s p h . r e r u " z . _D

_ c a a i f o n t .

I , = O.l (A) , Po,= 3.3 3 GW) ; Iz= o. o t ll (n\, Por- 5-s t (^tlt) ; Is=. o. o667 (rt), Por: 4a. E (nW),

7o ia f va 5i5l,ance s e e n by ih e

z":o.oz (A), Por-- 8'Oo

(.W);

Ia=o.ost3 (a), P"f s.st 6W);

E P o ^ :

V o A , : 7 0 Q ' v l ) .

s o u r c e : 7 U \ ) .

d.*zo - a"so (v/'-)

Ert: E,, = 2o '

f,z,',: J,n-

_{91": TE,n}

-

_{E r n -}

* n , ^ - # ( s o )

_{= - 7 5 .}

: 4*2o-427, (vl^\

b ) i : I E,: /s

x / o - t i A - 2 o - d o r a ) ' = d r o . ; ' d r o . z s Q | " n ' \

i r :

a r E r : l o x / o 3 ( A x 2 o ' i " z s ) : 6 * o ' l

-

Q o ' t s ( A i * ' \

c ) d,:tan4 (#u) -- 6s. 2o,

1t- ton' (#) : 1s. /l

C ) Drn- D,n = ls . ---- €, Ern- €, E,r: fs

'

f , = € o ? S ' ' 7 5 + 2 x s o ) = ' / 2 5 € o :

- { ' lo s ( n C / n ' ) '

2 4

b ! -a-) t t E - ,

P . 4 - 7

b ) ( f r ) - : € o E y u ) -

T

-c) E?,;uo)ent l?-c circuit betwe.zrt /errnina/r q q , n 4 _{6 :} f ( y ) - 4 + ( r , - n ) * .

a) Neg I ecling.f,. ;1rq ing ql{e-.t a n -l

d.tS.r,r1 in7 o <urrenf 4e-tsi'ty :

i:-trro-E= F:-dr#

U=-Io€

urur={

_nffi:#rfr

,Q=+=

_{#-- &rV(,',+}

1 o ( c , - o i )vo

A;6

oo uPfer Pta'E'

v +

l ,

, s

- t d /

T*

i c{y)

TUo

I _ r -A ) ) , t l ( e - ) . = - 1 ^ E ( d - - S & = - { o { a " - ' ! ) V , o , _{o n f o n . - 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+i

vo [43e

/e.w'

vo: (R,

_{t {-) r. = (#. #) t}

J

-+:

₅

|/o , = lCVg

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:

V o

=

_{E J , + 8 , 4 "}

a n 4

_{i E r : e E . .}

S . , / r ; n g t ule houe

1 : #

an4

_{fr: ffi}

)) ,)

c

'a',:#'

R r : *

r - € r S

"t

- -J7'

cr:ff.

2 5

P 4 - q

_{c ' - ) S a m e € 7 c , i , , a . l e . n f}

P r o b f e n , P 4 - {

v u l * h

/1,: #t"(*),

t R - C c i r c u i f s r t t / t ^ f

Qr:#{r(+)

/ ' - 2 t < " i - z l " ( a / c ) t n ^ : 2 7 f € ' L ' u! _{Ln Ce/a)}

b ) f : % q = 1 r

_{'o R,a 4.}

T : l l - 7 - q r ' V 4 t - u z - Z a ' - '

p 1 - r o

_{, Q - e s i s l a n c e o - #}

_{( F r + - / 4 )}

{ f o ^ o S e n e o L r 7 r n a 4 e r i a / , , v i + 6 a 8e*"rreen toV ar-d 6a#o^ {laf {a ces.

u n iform c fo s s ss cltb n.

": E

1b'- a2).

D : 4 h !

clr (b'- *)

P + - r t

LG e Lap la.ce's e?sqfia a in c2r/inclrical uor/,nq4zs.

v,v:o + +#(,#):o

S o l u { i o n : I / r r ; : c , l n r } c z '

flo. nJory con4ifions: Vto)= % ; Vf 4: g.

' VQ):v;ffi

E-(r):-Ar#:rrfu

i ( r . ) : r E c r ) '

t : J - j . a s - f n ' " j . ( a , h r d 4 ) -

- 5 Q : V - 2 t n ( 6 / d , I rrah tr chV 2 ln (b/a)

2 6

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 fr(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^

_{u o t o R R t R ,}

p2-s

ine /e4g : ?y{

A:1=Le-

-

_z

-Tn\T-E/'

/ \

25.

! I l I : : p . - /

S f o t i , N l " S n e r i c F i e i d '

F : e ( 8 , i x g t :

g

E

- - z,L

x B - - d,-uox (ar^*+ Er3y, 7.8.)

/

-: u o ( d r B r ' - t z g / ) .

E=doE/:44*

6 : / B ^ J s : / i ! t f o 4 a ,

r _Js '-g _{2n Ja- r} : . _{*io-,Y'lh lr 4^ e 4} O ) a

a . t r = 3 3 L i s t t s s d , E ' : # ( ' # )

_ /

" / e r r c r :

_{# x l o o}

: l - - z t ; - a ) - - - 1 i x / o o ,

| < b + ^ ) L n ( b / a . ) I

: s , , 1 ,

" r . s r

i s

l # #

- t i xloo,

or - 17. L '/" (4c" /ow).

P s - z

l / - n - E For b

u) U-re

t- -. r . > ' J " , ' I I

$

D)

( f - . ? z < ) . d f ' : d " d z ' , E = d , r - d r z ' '

aZ x [: ardz' x ( i. r - drz") : d4 r'{z''

Er:aE+i fu

L 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

z g

an inf,ni(ely bg vyire: o(r+Q2o aad cl,+-?oi

b e c o m z s a ( #

' &s i,t e7'G'3t)'

I

I

i D ( - 4

-r-! - l Y , I t J

l.

,:'?

, , | \ 1 \

I

i

" l --

F.-Llse F7. (s -3 s);

ciE_

- P . . * / /

: a-.lB- a a,d By

'

: d.x GS4 sin€ + dr{as4:o,'",

vvhere-

dBn--

_ffi*,r*'

. r J /

sin

_{o = 6fi7rp'}

_{cose: 6frry}

,

Er,: 4, Bn + arB, ,

rlr

,t ,

| - / o J d I q x W h e r e _{6 z = ET ) o} x , ' + d ,

and

_{D y}

p, = /_i!-_ [- .i/i

2 r t J o x " + d ' A o I L t f z r l

f f i

t a n \ T / ,

#1"(t,T)

L f - { o * t i n * a l h e P d P e ' r ( i n - Q 4 i r e c f i o n ) '

/ ;

= F o z 4 z '

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 .

2 1

r 5 - ) :J.<. :! ol,+-?Oi

3 s ) .

c, t

P . s - 6 A

t " I

: l

, s c t I t t t a I

I

I I r.!c: I t rL. I J

V_

. \!

E,:

- a, # /:""'-ry : - d,# Ln

(*T)

ly AmpZre}

F o "

fo,

Fo, r

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} r n f he. coun(cre/ackvtrise _{4 i r e e { i o n} i n a f r i a n g / e l y i n g _{i n a 6 e x y ^ p ( a , n e .}

Fram tf. ,t-34) a n4 noring lloof t - q _{a n d r = P t o - ) 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 'z 3 ^ t o L - - 5 - - ; g t t r z zri ur,/./T - qz _Z?tv'

Us" Ef G-37).

, F o n L * n ,

O " * g a a { 4 * t ,

a. n cl

_{E* 4, 4{Z}

_{: d"}

ratnl .

'3;- q a-)

_{8 : V x A : * € *}

d A, _{: ,/toI} d r 2 r r r J ; = / J Z t ' , * , . .

d E : az-VVi.-

_{*gy"(il r.',}

E:v,+fs

_ ;

_.z

, u p z f

L - z

LL l-v/e- z )\ 6,

z l

:ar#rrfi!)rl,

: -

9 3+

(rVo

chorse

,*,+h

z.)

A - : - # l n r

* c - .

A ,: O a-t 1-

: fo.-+

. - -

E

l r r o .

A=d"A,:4#uC).

3 0

4 2 - - t Z t r r J 7'+ r, ,/ f - ^

,.P-? -

-{0,

/, --i

I

l@4,,

]'r'-,

t)

_{L(se E:f A.aZ.}

l l o r i z o n f a l s i d e s h o w e - n o 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 ^ t o - { ( w v ) .

( a n e. 'la-f 7f

=zE'

t v .

1 : 2 . 3 4 ( f w r )

P s - t o

I n [ ; n ; + t c c t r r e - o t s h e . t 4rJ so _- ; -B ar#i-sTrnroefri=a-/ qnd

ind zp e-'t denf ol x *rJy.

'{lP'',

l > D t h z . )

_t

;l

t " , l

A i s

o) /97r/7 AnPi-z''

<it'cu)+o!

( o w / l o p < r h

/ 2 V ( l :

{

e -aI: /or -2-B/-l;t,o4

-; _{B- : [-,HoJ'Jz} ot (o,o,z), '/ _{L *ruo4o /l *+ (qo,-z).}

e ( ,

E : * j r ^

b) Fo, z

)o'

FxA:E-ur$)

i l d e p e n J e n f _{o { X . , n d y .} 4 A r - /07*

4 z

- - - T '

A*: - ryz +c-,

A t r - z o , A r : o : - ryzo+c --+ c - Mo"r.

,,. A: _qqz_z)4.

3 t

P s - l l

_{I H^=4H^}

l v

a-) G)ve.n 4--/,4

Err:Bo-F

_{Hr*FoHo-F": LHr:u"f ,o.}

5) Q i ve n E,--t'. G, + y).

'/

Br": B F- r(t0(H2 +,v ; ) :/o lo* Fr= do 0/r- lrl ).

+).

I oa r <b;

f=u, nt,

;

B - = a " Y " n T ,

n I . l

M = 0 ' .

,il(fr,-

t)nr:a+(fr,-)*.

m

_Me-"/

t Fo=uo4

, D . S - . 1 2 L ) r < a _ :

a ) J; *xil--o;

P-EJJ-

_t,

n - ! t r . > - t 4 Y - o

i!:^", r ,L i- (t - r

8 = \ u n I ) I '

M = + - H - = d - l l t - i

_{, o z \ F o} . / J - F k l x a n = ( 4"

^ ) j ^ : i x fr: O.

J^r= (e* c4s s - ar sln d M x dn

: d + Mosiae.

t'

trr',

E7' g't7)

rc q loolt a] raliu,

b = i 6 g < a . r r y t n q a - < c t r r e - n f $ . r b J e , / J t = p , , ( L , g J a ) ( L t ; - a l ' d B : o r # 2 ( L , ) r , .

= a_ !?Mo sinro.

^ 2 , 2

E =fa Z - urryIrrin, e de : iziforo: ? f,r,

at the cente-r o.

a) B,: d* 2 - 6rto (^T),

Er= 4 Br* - u, 4l

u : B r , :!-r - 2 / | 2-t 9ooorUo ' 'lr _lo + _{B r x = l q o o o ( n T ) ,} 8ay: Erl -/o (twr).

Er: d.-/o,ooo- dvto (nz).

' Y

ta-n 4r:

#,tcn

o1,

di-a.04-". I :-904.0

3 2

M - d r M o

@ a ; .

F n n l -Br @ J r a n fn'5ooo

- N o , ,

,F'O

b ) f f 1:\rc*a,

( m T ) , E,:a-B,o+dzg,/.

H,*: #: Hr': ft - B/.=*,oo:#= o.ooz

E U E " y =

? .

. ' . E , : d * o . o o z + d r z

( m T ) .

4t': to;'fr

= 9y

: o' oo ! (ra-t) : o' o{?o

'!o-M). 7

P.

s- rs

P .

s - t s

g = n f 8 4 = a q # r

f : r , - g c 4 t e \ .

6 = , F I N I S a l z z r P a < 1 p , . , . - / - \

z - z'n lo Jo

_{\ -r*fr= foMZqt}

-lr; -d.

L : f =foMt(r.-ffa)

I { e > > 6 , i 3 r + #(canr/or*)..

E *g*s : Brftr6,)

- ,"2{tz -L !#

,o

frr I in Lhe- lor3 -s,-lr"..r'gLi

_{vvire, -S=4}

_lf

t

A,, ={ E.rE

_{-J tr} 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/oop}

t.inking with fhe. s6all laop by fha.f Jqe lo fwo in{,'nikty

/ o n g r ' v i r e s c a r r y l r t g . 7 . , o / ' a n . 7 o p p o s / f e < c t r r e . n f f .

A,,:#{*'(#-ft)*

_-ffr4,1ry#)

t _ A e _ L , ^ ' - T -

Sf t^

a!=a o4l

3 3

(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

3oo

:

"-*#*

Forre per ur,;/ /<ngth on wire 2 :

F;: - ayz

82

= -a/H:

: - d/ //sofo= -dy /.41 x 76-3

( U/il.

Forr:e-t on a-l/ l6ree uvirec q.. c>{eo,-rq.J r/'2a.9nriu4e qnJ Lw-r-/ fhe C-nler * fAc- rrr'a nqie.

Cu rrenl _{d f = *ly in an e/ernental}

C t ; / / S

,/

l " t 4 l : - * :

_2rrwJ't.fir

, r i Y .

Synrrn-/ry * _{H qf f he wire- ha-s} / , . / snly _{a >/- <onponenl.}

F : u, J (d H).

(*):dyr"{,

_{ffi

:"/#ton-,(#)

f': r x E : (- 4z)* Wrn: o=4#+^;,

(#) (u/^).

j; '" E(t * a1) '

o . [ : 9 d l

dF =Ia[x E

: -a,+#(,.

#r,

(,4 reif -gcrn _/>ro6/.-.). _ * . ,

T"

d

I*

x

l"

l-l

w r , I l Y l . L ?

PE:,!

' +

I

F:-+W[ii.*,),

=-d.ff t^ (* _t)

3+

l.--- o

---: '.

P s - z t

F o r . e i n a L r n i { o r c n m a g n e * ; c $ ; e - l J : I +

t

A B=a..o.S / L

F = I t ^ E

= - B x ( r L ) .

o). -Ex r Gn)

j(€h

r(fr)

- (d y o. s)t / oC Eo. t - {/ o.t), io ( eo. r t $ o.), / o 4 o. z forrct -azo.s -A"o.s a"t.o (N)

Torfue an looft:

T : 6 x B

- - ( a " Z 5 2 x E

: (dr/o, _{!x o. zxt.z)x(6.ro.s)=-dro.l} (N--). t / m , / . I " 7 Q a /

oF

B ful,0)

D t - z t _{B, a-f fhe cenler oF fhe lot3e-} circular fe'trn v.tire calr7i2g 4 currenl 2z is (b/ telfln3 - O i n E a . s ' s 7 ) : _ _{! o I ,}

B r : a n L E

of

, fa(

o

/ y) n ) .

TorTcre ofr fhe-srno/l c;r..t/or /u'n oS wire carryi2,r a - c q r r e - n * 1 1 i s

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.eclion {s6Jing to

2 f ,

a / 4 i f h e ' r n a g n e - l i e { / u ' ' e ' s p r o J z ' < e J b y T ' a n d 7 r .