en"e:"": Stotic
J
P. z - t
o) Mox.
E /ecf rie FietJs
l/.._r. tui/l nto.Le 4) :h/,
rol(ar.
h : +[Vr^Vy z-
afz:,t
\, V^-. v r n o *: ry e r[o,t,_\". w,) T 1 ( 4 ) "^( b) A+ flre s".e.n, (do)^or: D/z L rnqsl be. z /-,"vx t wrhe^re' L rlr4x = *z[ * I+ ^ - ; 2 h ) ew
c)E.ubl-A node
.o
Wo* Ay dou-6/;* 4-ece /arotrn.t
/t o
2 s-z
,ti:
u\
leDq I
P,s-t
a
ia
tf
ry
Similarly f", f, orrdE
triangrle.
: - al 3 + d"4,
At P: E,: ffiV
b)
N"
y-
fy/o Z-
e-p : ar4-d.z
d"4), Gart+
E,:Vfu a)
,\/l o're.
diTecfion a.way from thc canle.r
in the
lSi ve farces
oF thz
= d,, 0.624 (N). 4, + Fzt
E:
r\
O.l un)
/(d*o.F * dro'tse)
: 3 . 6 ( a v o - 5 + d . 2 o . 3 . 6 5 )( 1 . r ) . Er: 3. 6 (-dro.t + iy o-t(,d) (nt)
/a
L-----\-
vo l4a-qe-,
;
/\ /\ I
Lt, or ./ouL/,'n34"-
(2/lo't)t 6, ffif
-'vr'
Vri r/
cotnpanent : CorrrPonent'
ll
(*+-v"3). -3 Q, + 4Q,
4Q,-O+ *=+ 3o-.:o+ 'Lq24 +=+.
:
fl'
, D3 - 4
zero {orre-
fo,
g'
3' 1"
'1
:'
+ (A(')
'
*-x -J r''-l
Q,Q,
Q'Qt
4zreoTt
v '
,:1"8,:r/*:r
I
cc,-).
9 (crn)----*i
3 (cm)- i+ can be proneJ
\A/;rh x: th.-! A-F..'. J -t
-,:
4aio x-
- l( Q.c)
on Q.1 :
/he.
)- rom
ne-l {a-.=s
on Q,
onl
o,/so zero.
r;!:ffi
Eg (s-4za-), f,:
E - a,*f
Q3 are
: -*-
t -(r+ !;)-','l
: 9tr I, - ( '- E* -?;"-....,)J P-
e
i
L)
r
t.4
\1
: d"#ryLie) -*er* '=+l
whe,'e, (esl Tefr-Yt
,2 3-6
t +=---(r-*4.:*.".\-l, /) a"O'\t-VV
the {;.sf
.,epres
j
(e-ra is lhz po,'nt-
'--> z 2,8 b, ,,,s.(,4b. + i- 3 o.ol
:
At
on qrbitrary
P (o,o,:.) on ff>e a.xis .
I, b 4d'
*r { t tnP -
4lrg(z'*
L')'4
ur-ffi,:4fu,.y. -v4:-o"* Ep--
-a"ZfuE
q) At point@,qh),E,=Q## b)
To {;n.l
the /o
o$ r,dr.lEul
xn*lEr,l=fu'
fliiErl:o St'-i/or
sifualion
vtt he-n
P is
/2
, s.f
be /o,+t lhe
/oott
P3-*
ctE - ' Y .: f!9'9 4ft€ob ; */F 4/ : = -7
si,,-l', tz
- - -'/ n
ttlob
frsi-a ro
ed
=-7rtffi a
teJ
,pI-g .
S p A " rl cczl tf- - elnyt
F:4r E^
A7p/yQ.au' slf,z- .
t1+r{an= r) o < R
t'':**ry-#)
z) A>R
,u'iJH^
z) R ;< R < R o .
t
Eor :o- .
+)R>R". tr; ffi.
e-9 Cy[;n,t.;co.lsy,n^.*.y , E: drE, ,P, a) Er=o, do, r<(t. , fo. 4-1 r z b, E^: ?9: €ol
er=ft?, b)
*:-
Apely 6:outri (aw
{or r)b
*
P 3-lo W.: - 7I E. ai : -V/04x+xr/). q) A long *he pora6o/e
Y =2/'-,
zt++z' ) "{ x i l A (o n g 1 y'.,s s { r ct;ght ( ioe #=
,o v
W" :-
l g xl o' a;/- tf
4/ :4*-l*. :?xtd' ( r ) #- - 2,
- ?o ( 7t). Y= - 2x+4, 47=- 2/7.
W" =- ( E* t6t)l-' f{-zz,++)dx - zx4xl : ?o (/t). { l
t3
P z-t r -l
DJ
E' dI : Y1*-*4/' E : dny- drx, -^ (z; - 4*') dl : toqri) ' We: -7. -f : -6o (1''i )' W. = - 7 J ir'zz ++) + 2r )JN ' fUe
l\ P
r 1'!
t^
qiven
is fio4c44sarvq-/-ive'
E d;tH
/
t
y
l:'o'' t\-
| \,q
t)
r
c'l-x4
?,i-
a,=z[qL ^/ ty-
F,3'is
"Jo
-
:
FV
3ire-s
.+) f f , : F '
t
u(
l\
a
I,neo/*
:w
r^---,
/----
;''
{u"'*
n [uz ra'a* -n 7tr€sJo .l x-.yz P^ r. ,-( --#"1 Lnvtr)if'*rJ
I
-L/,
o
/'c/2
a-) r/ = -)t) 21
b) froe
uy
E
- L r Y j) '
Ca,l.^b7 /o-,,
Fvvd*.,:; *'rr€uR'
1lr-
zn : Po*
toonz
ll ,f%11 ny 2n
ansve-tr
6 n 6 2 1 / s ) r { o..,
a-s in
4
b).
th* cuLs-
- -3Po 9fn: - 7'P a n : ( Lt ) fp; -3Pot3 b) o . . : ( ( t ' ) f 7 r : s P o t ; T"+*t bou,.d c hc.r1e : 4, ' - e.r :O, F =arPo.
P 3-/4 fft
: F' d":
P o t , ' r 1 6c o t { .
fpn= -0 .p : o. b) d , = n b ' s r n ' ec o s f c r y4 g no to : O.
t+
*a'2-
P s-ts
p : I ( a - * 3 x +a r 4 y ) .
o) fp_:-V.F:-
74.
fsta / ,ro f.,,ne ch,- "ge Qn: - IPor r { 1' ' r r ' ) /) er unir /en gfh o ; ' j . o f , , " ' ^ = 2 - . f t ' r " : V ' d , = ( a ; r o c o s l + 4 7 4 rt i n / ) ' a , Q , : Po,o (3.os"* * 4 3in'{)
- - P o ' o( 3 + t ; " ' 4 1 .
'u^:= ! u, ffr;= - Po', (3 + s;n'41.
tlX:L""., b)
n z l27l' r,'I Tolat Qro:d"r21r lrro roi f : Po 13 + ti n'#) di : 7rr Po1.,
Qr, = '7rrP"r;'
lot-l
l,
p. 3- /L (/)
Spheri.ol
Qn*
chorqe:
Tofo-t AounJ lw'.'
.
syrnmelny 'Apply
t rR: , : - a 6
R>Ro.
Do,:,oEr,: fu, (t) Ri
Qro * Qr;:
Qo-utt's low'. E-aEt,6=QDo.
Po,:c.
tr.^-q' -Rz +'lf€oerR:
u4,
+i'E'
P*= Do,- €oEn"= (t - +,) *7
R'
.
E,"rL= *J,,-+,t A
vz: J eo,an-r[R ,<^ & (z) R
o
#F, I
DAr:#,
Pp)-0.
rL
V t = vtlo=r,' )o, Eardq' r rd. d. frt-trJ--/t-l-\ +nftL\' €rtRo \' €?let' A)
tg
/e.t3t6.
O.
@ V,: 747'
'
AFFA
tLe.
Pe'rurtil
, 2er oni/ /zagt/'7
;< J-/./
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 lo :y' Ooo (*V)
Vb= Eu-i^ , EF{O: Eu( so-dr) + FO.i, €oE^: {o€rfEf Alo*' D^= D, *Br"okJ"rtn 1 f bP €r^ tt ya.
occurt
[^-=trfEp , Ef ' {i''st rh c,)r re3)ort 2qtoh fr.-sr,
: /3o lLD' ro-/o)' +J /o:3 (zo-{vo) Wb : 8,.^( 3 ba " P:-ts
- d, 3 v + dt 5 . ,4t the z=0 plnne ; E,: d-:7 (z-O 1 = frr(z:O) = 4r'2'7 - a, : >' 1
D - (z: o) --D ^ G : c ) : D., lt1
ZE,,(z=o) : 3 Err(z=o) , -lo 21 z-
E, (z=o)--4r2y - drsx * d"f O. tz=o )= ( ' q* 2y- vy' 32.+ 1# P . , 3- ; 9
) lr "
f ,rr,^e
Er=d.Er-- i*E.(
/lo.,r7glory
conJ;'t;on:
dns
{
* Fo,
E, , ct,tc/ hence /-,: f he x--*t)s ,
= dnx Er.
,rl:
-3
*o 6e po-rol/el E , Ez/= - Err.
E"=3' 8ou,"Jo, y c ct;riia,b,n, Zn' >, --4'\ --
+
l6
'
€r Err=€t
4rr=
{:
ErrT*
r-er7-
€oS =€o€rr3'
P.3-zo t:
Tl+€,. Q- on Ptq'fe'
,/ss,t*e
r/=d
(<,/<,)
:)
J \s:-e,
)
'l(€--<') cJ bGr/e,)
4 e - A -V
.Ps-zr
Le_+ Pt b< the-/ite-u_/ q$c{-i3e. 4ensi*y c-ondc,chorE: E. #; 1.
t ' o - - - J7 '6e -E ' c r F : V fPt h, ( a, t -r 6 \ _ € _
ot')
b)
ffA:d, Fo,
{t4:
lo. 3
t
Qr
ct-t rbe-)'trle-r'
2t€V
t.t- TBn)
%= a_ ln(b/o)
'
o fi'x.a
#^*
io be ft')n)mtze./ is: (x=L/o).
b , fhe {un
.
S.rrin3
:t
W
ytetJs Lr,x--/,
V:*--e:2.7/g.
C) rnin. E(a.l:
e q /b.
'1)c': +:fu^l= @
:ffi' '\-T-Y*c,)
A-d ln ---=4.L.
: V : -J - Y - o E.,iZ
+
a
e e: -dy?
o-f Y-J
E =u,#
2tr€(r/^) r;1..b;
E,=d,r#,
b
Er=&r#'
P.3 -2 3 A trr,,-,,
e fi a ,r ges +Q- an.l' Q
ConJucfars) res?e
=Q E = drE* = ?zi .ftr€oqz
v = -L'*8. axdR-# "Eo vt y' - _ L -
on fh'e inner
( r/a, - t/tz") t7
(*
-
-/
4/
\.
a.n.J
otifar
Ps-24 T*-,f
C-afa-e.t'fony
a-cfos s Luhuy
fe,.,nino/",
cr: t Qfl' sfor+..1<).-fri<
P ,_?5
(r o'-"2x 49':o-y O cctl>ct.'7:f.,,!
in ){7y)
P. i-
s;n6f'arJr + 3r/ !; fJ;1. t . d,E= - Tsr ro-'"lli4 LI:jr/J_i
"1
c+) Fi rst
q^1n).
c:.pctci/tt-= 4 Sxfo'c)xorz :2.g1nW).
E:F;r6rsinf+e/rcos{; a4rd{ rdr4z, 4f = dr 4r ^+ ',\t
= *a-f
sL)z:6-+(,nw).
r - 2,, $ {.o^
U/z'/o -'tr/a;
f h.-n 6- -T/t-t f }rorn
Llo4:
t
( s;,(-n5l*-l W.: -(r^ ro-to)l 3r"1"sin4i--"* ,^, L l.LJ ,T/3 L : - (rx t6t") L- ts -3GJ:27x/;q (t):zf
9 =tT,/s, r {rom 2
Y4 ;
(nT).
.fhen r = 4, Q Srem?rrb ro -.tr/z :
,-'Lf
L(-=-(r* /,t,")le,;ng)ii' L-
:
- (9 x (o-to) rr -fz): |
--.Sea4e
P' 3'24
a. + r a.rt s/
A
z?xrcll(t) - ZT GI).
dr.i:?{oin 1:n.t, a,). jxl:6-,E;rronr.Tyqlfve
tturT)e- the
inrl"r qnJ
respu.-liwe//.Subs/H.+,-1 tr1rs*F7) t.9 ct s i n3
'":
€7
( s - // 52 *o
:sh:? -lizL(46
Ps
ou.fer ra-dr't' fu be- a- a.ne
Cs-a7)
[r...u"_
I"
i +L 4;
{s
.=---*'-'
Jf ,j)../'A
I :'
ulr:Ils 1,lL / 2-_ rfir1, ',7)t1 ytr, l'J{,/
=looi
ES-zz
.fwitch o P en 1 C ho rJes
Consta.nt.
',tiD.
ttr z-
on the- p/ole-s,
y'Y-:fr
.1 L
a
c16 ,
\-
71,*+€o(r-"t
-VW":
.W).
a.?^e
- a r a' e / t \ T 4 \T)
*w) P E-zg
d aoJ LIs. subscripls re.sp'cch'vely. a i r regrions
l
a,-fa Jena4e Jie/eclric and €'V:0 in both regions.
Vd : Cr/ I Cr, E" - '9c, , D4 = -rfolrer. Vn: czy1 c+., Eu=- dyl, , D.: -dr€oc, . = OI al y=4, V*: Vo; B.C.: Af ,/:O , Ya af y:O."d t Vl=V , 6a=D;.
, c^:OtZ=fu'
S.,/n;n"i ",: a-)
b) c)
r;:ffi'
c+:ffi
E":-a/&
T^:WV,Eo=-dyffi (f)y=a=-(D)y=a= ffi (f)=o: (D)y=s=-ffi
live.
P g -zq
.4 Vr V = - * Tf: ''+, + c,lnr+cL-
p o i .r,o ri
So/q\ian :
.7.
& C . , fA t r = a - , V : ' * o " c , l a a t c , . : \A+ ,= b, o *t * gln brc,.
/1
1
a / 2V\
T t-rTT/=-
.vt :-
A 1r
{4b'o)-Jtr, La6/a.)
n = VoLnb+lblnt-t^n, -t La (b/a-)
.
Ps-so
V
+ r?(,#):"
V:
.)o l-, lion:V
= ,, (.n r +
c,
,R.t ,, n d o| ,u e-o,.ref i.f r'oy.t: : r:c)-, At V=6
At
r-:O,
Y:-
,/ ' = U '6 5r,r"ru
,
(n (6/c-)
|
ln (b/,) L;-G/A
volnb
.-r. --
( b,/a)
ln
l,,r_
LF: - V f : a r " T ; C D
t
<|errsil,es: Af
r:4t
a ' - :1i A
.
Eo:€ut,=
r:b,
4t CatT;c-ci/'-.a.':a ,:)er ctnri,'englh
C, Lnb.* c.
V=0:
,vo-,
!'
= C,/nd\+cL
lr,,
f.:-
2T-?lr^
:-
:
i"I6
b Ln (,;,--")
-.Jn ?r€,
/ri \ \ w,t,tr1.,/
"
.P.3-st a)
E ,"r:i:t onr..u, , -f f.(3-rzfl: * 0,"€H)= 0 I.o,,., ---+ S'olu tion: : : #
V(e) C,ln (/o, *j * C,
#
B.(. O VG) = V, : C,tn (n,^ !) * s. . /@ v(4)= it,: r, (n (foo . .L c, // 4c,:#6
b)
+
!)*
v, - - -qo ; E:-;,rrv "u eot nZ;W;U/rf
n+*+?! v(e)= Lnfta'aG/zt)
s,-,ie
'
c ) C:n t'he
11 /J
; a. J-<./
. e =?r/+, t:_.$
Cont;j." f/>e condilions in *he n \ 1' P / - - 4 :A; € L/ qt -. _ / _ t ! _ l _-t
R, e) [4,= i {. -.( 1'+( y -.t ),f'",
R t - l - ' . x + 4 ) z+ ( y + J ) ' f ' "
20
,
({)-:
n#*
xy- plane (z =9). w L.ru
R^: L(--.1)'*(y+4 i'l'", Rr: l(z+4 )'+ (y-Jf)".
rt I
-o a
I
L_ ,..' R, J
C
Er=-avr=-a.y, Fr# q -= -"2+tre
y-d
I
*
t-.J
/
-
x+ci
*
R: u 4, Y-d ' vtd Y./ - -nI +, =uY A f -,rl ,\t 4ne L;
€O wi/l Ao.re a Z-contTotaenf '( he zYP /on. . / fn)
r-['//'re2a;otPJoesnul
/ie in
o/
(
Ad
-\
!'r:t'*+Js'+4'-l'/':R,'
n; ) t
)'uY=
S inr; l<, ly ConJu.h'ng
Pl-l+
4 ,, ,,n e f, { s c nc /m) 1o be at /: e i^n=iz.j(,al a) Ve, (or {i-o,n f*t" 4 fu,+,3) : r, =614
u'l':f;i"' / t..
d
-?, &: o) on fhe v27{,'.o./ zlo f. hy chaliin? 7/o,r.
zt 'l
x:
f(x+4 )'+ "l'\tt'
{"r
Ir
E,
x t_/_/r, -l 3l / '
Y
l/e c ior /'ro,n -.12la /i : 7- . al, + t 8"6 d/ fr =7xtos(6,,o.2s) LF =* = l p I . : =
"l E,:
G- PT^ -
#Tr7
:-7xtos( dyc.c??+Vro.us). *% ? E, + r- : 7 y bt (ay o.r7s- a"o.r7) (.vIn) at l. E-:
-o{
\L1 y'z )_l
R; J - -v - d l RT)
D Cn +h. con-lu e.linzr hnl*-plonrt , 9, : Dn:€ En . / 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 ^ ,
-Fz --nu ) -rr - - r *aiz* ln,r-
)'1'r'2
z+d I
2l
G
ul
't.
b)
I
At
(the. grourt.J
41t\
\
tl
tl
,Z (o,4,o) on fhe ry-pla-ne
.
tr-
fa /o ,0. i t 4: ar+-12, l, o,n - fxf" P, lt i = dr++43. ay+-dri _ t = lJ_ U:4
\/9q4or f-rn
P,
71-/ "?r i ,C-'t"E,
Ve .io.
EL
_try
):
.T
: J-
4t * 3,
lnlo
I
- F-i
f'
+F '- =
/-t!l
a/rq
( I
:
- -dzz.re'to'(v/") 7 .. /€ G
P" o.24): (fr. -ao Ezz - - . ? ff"50.a4) ) d
,23'3E a-)
Qiuen
(c^) ,
: - '.7 | \ tc.s(c/-'\
:_.7t(pC/m.).
c-= 0.3 (.rn).
,F,-c,r't[". (3- lt=),
"J f (p*,,ffi,):f
lz+Jra6r3yJ=,,Js4(c,-\
ct:D-4-
O . 0 4 6 ( c r a ) - O ,4 6 ( m - ) '
o,iz:ffi: c)
D:2
4r1 i,,/
2 - /.95 4 :
21 1frxrdq)t /oo = 2.71 x 161 ( nt- )
The eTuiva.lent l;ni
-lcf,(v)
t ?+9, foofu) L"'n)' -'t
E
La ( r.?s+/o. t) cha-ges
4re
:2.9L (^F/-) sega.raf
d'-d-di:/,?54-o.a46 : f.?og (cm). P' ll 7 l tl:: G@x2:ltt-1
by
(v/^),
p/a.ne- Cort#t)ning th<-'wires
22-
,
9-@'t-",3 gz, '22 -
a)R: +-
P +-t
(s/-). L . * o##=3.trxro' ( x/a-, (v/^)
E: +:
b)
/ (w). The3iv.-r,
c) p : t l l :
(v/,t
Cttrrent s
S te oJy El"clric
r-a Z
a ) 9"= - #../
th.-f
e/e.lron o,abitity /.+x to-3 (m'.y/)
aF o3oo4
'-r:I tl: l+l:ly.el:
/vn- )
)-
:
g.4 , lo-'
cortJuclor.
/o'') ,.otro'x(4,
(m/r),
R,= P.esislapcepzr unif t"*+* * .-e-
P.4-z
is
=
+
7fr-'
Rr: Rr-is1y-.nce-pe.r ur'rtf /ength o{ coah'ng: #Sr' L e+ b - f /1i - k 6 ess * ra 4, g. ---+ S, : TG+ 6)'- rr4': T((24b+L), 1)a-:2.32a-. D : p,/\r-. + h - ( / Tu ^t
r)
v /
1\ ?-T b) It: Lz:- J 7'
T Jnt : ;fr-, ,
r Jz:
L
I--
q:
&-
I
m;
E,:*:#,8,-#:# Thus,
Jt:
loTu
o'n4
E,:
E-'
P.+-s a)
1:4W
Rtb
€l=Q-#
2. >b:
b>R - < b : R->b
4-
nrr-i*7.s
:a^#
|k/
: -n-#x/o6( r/:) x /o/op
= -A
"
2]
'?'12'/0'7
( A/-').
P 4-4
a-)
b) TAii
.
u::.:!jj:::,:r,o W;, - f : b/dtlz : a.ot / : lc-1 J L'* Tilo=
c ) E iu -t,n'sleh'c e-ner4/ tar : ? grcrec o.tlsiae fhe sph.re ru" z.
P.4-s
f,7" D
I , = O . l ( A ), P o , = 3 . 3 3 G W ) ; Iz= o. o t ll (n\, Por-5-s t (^tlt) ;
4tzR JR = _
= +s (kt)
#
caaifont.
z":o.oz (A), Por--8'Oo(.W); Ia=o.ost3 (a), P"f s.st 6W);
Is=. o.o667 (rt), Por:4a. E (nW),
Q' vl) . E Po^: VoA,: 70 7o ia f va 5i5l,ances e e n by ih e s o u r c e : 7 U \ ) .
P.4 -,:,
b!-
a-)
d.*zo- a"so (v/'-) Ert: E,,= 2o ' f,z,',:J,nErnttE-,
91":
TE,n
(so) *n,^- = # -75.
: 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 ' \ - Qo'ts (Ai*'\ ir: arEr: lox/o3 (Ax2o'i"zs):6*o'l 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: { ' l o s ( n C / n ' ) '
24
P.4-7
f(y)-
4 + (r,-n)*. a) Neg I ecling.f,. ;1rqing ql{e-.t
v+ l,
-td/
T*
,s
d.tS.r,r1 in7
iI c{y)
_r-
a n -l
4e-tsi'ty
:
F:-dr#
urur={ U=-Io€ nffi:#rfr ,Q=+= #-- &rV(,',+ 1o(c,- oi)vo
-
b) (fr)-: €oEyu)T
tl
i:-trro-E=
TUo
A
oo uPfer Pta'E'
A;6 {o{a"-'!)V,o, r,4 LnCrJq)
(e-). -0-/*'(d--S&='ss'.I_=-1^E rt
) ),
o
on fon.-
7l-t-.
P 4-g
rx) Corth'nuit/ .{ 215. /torna,l Compcnenf ol -- atss ures thu sqrre
vo:(R,t {-)r.= (#. #) t |/o lCVg , = J -+: 5 G.l, +{4" 1d,/o;)+(4,/r,)
))
,)
b)
T*o_-7't+/ions
for
orre- ne.edeJ
the
Vo= EJ, +8,4" iEr: eE..
o$ E, and e: an4 S.,/r;ng t ule houe
1:
#
an4fr: ffi c) E?,;uo)ent l?-c circuit betwe.zrt /errnina/r q
q,n4
Je.lera;nq*Zn
c
'a',:#' Rr:*
6:
r - €-r S-J7'
"t
cr:ff. 25
P 4-q
c ' - )S a m e
€7c,i,,a.le.nf R-C
circuif
srt t/t^f
tn
vul*h
Probfen, P4-{
Qr:#{r(+)
/1,: #t"(*), t
^
u!
27f€'L
:
b) f:%q=1r
'o R,a 4.
7 4Tt :- lul -z - Z a ' -
p1-ro
(Fr+-/4)
, Q - e s i s l a n c eo - # 8e*"rreen
P +- rt
6 Ln (6/e)+ [ Ln(+/a) qr'V '
-
{fo^oSeneoLr7 toV ar-d !D :
rna4eria/ {laf 6a#o^ 4h clr (b'- *)
LG e Lap la.ce's e?sqfia
v,v:o + Solu{ion: flo. nJory
I/rr;:c,
2t<"i l" (a/c)
/' -z -
'
Ln Ce/a)
,,vi+6 a u n iform {a
ces.
":
lnr } cz'
con4ifions: Vto)= % ;
' VQ):v;ffi E-(r):-Ar#:rrfu
Vf 4:
g.
i(r.):rEcr)' tr chV t : J -- 5 j . a s - f n ' " j . ( a , h r d 4 ) - 2 ln (b/a)
26
E 1b'- a2).
a in c2r/inclrical uor/,nq4zs.
+#(,#):o
Q:V-2tn(6/d, rrah I
c fo ss ss cltb n.
P 4- lL
d- po/.e.r*i*( 4;rqe.ren*-W Attrrr" spAenes . ;nne-r a-nJ oifer
be-/--etn
+r\e-
+ V: k-.- Eo= V"Tf: 0+ fr fr(e^u)=g. # -_f:, = E^"R- - u{o,iq* : k (*,- t) W nY+' -> k:# i*:(Er' Tt-4
p2-s ine/e4g : t - ul'"f'l^ otoRRtR,
-Tn\T-E/'/ \ A:1=Lez
25.
27
?y{
! I l
I : :
Nl"Sneric Fi eid'
S foti,
g
ixgt:
F : e (8,
p.-/
Er 3y, 7.8.)
- - d,- uox ( ar ^*+ E - - z,Lx B /: uo(drBr'-tzg/).
E=doE/:44*
P s-z
B^Js: / '-g r6: Js :
*io-,Y'lh lr 44 ^e
.
O
l/-
n
-E
)a
a.t r= 33L "/
/i!t fo 4r a, 2n Ja-
errcr:
is ttssd, E':#('#) _/
#xloo :l--zt;-a)---1ix/oo, | < b +^ ) L n ( b / a . )
For
b
:s,,1,
u) U-re
t- -.
I
(4c" /ow).
d"dz', E=d,r-drz'' ardz' x ( i. r - drz") : d4 r'{z''
Er:aE+ifu
I
Z'= t' fan 4,
Let
$
- 17. L '/"
(f-.?z<). df': aZ x [:
",'
- t i xloo,
is l## "r.sr or
r. >'J
I
-% t3,#I*'o, : a+ D)
#
For an inf,ni(ely ll
DP
<14'
nJa.
Csin a1r- sin <,2. bg
b e comzs a( #
zg
Jz' =r;ec2a
vyire:
o(r+Q2o aad cl,+-?oi
' &s i,t e7' G' 3t) ' I I
t
i
D (-4
J
Llse F7. (s -3 s); a,d By ciE_ - P . . :* / / a-.lB- a ' : d.xGS4 sin€ + dr{as4:o,'",
l.
,:'?
-r- , , | \ !
1\
dBn-vvhereffi*,r*'
I
-l Y,
I t
i
,
"l
.rJ/ sin o = 6fi7rp'
--
cose: 6frry
F.-
Er,: 4, Bn + arB, , | Whe re
/oJd 6z = ET
and
rlr
I )o
q,tx , x,'+ d,
p, = /_i!-_ [- .i/i x"+d' 2rtJo
tan \T/,
#1"(t,T)
Dy
L
r5-)
tfzrl
AoI L
ffi
f-{o*t in*alhe PdPe'r ( in -Q 4irecfion)'
; = Foz4z' - qza;A4' t /bc-
E,:-a,# P.s-6 A t"I
ly AmpZre}
-d,# Ln (*T) : /:""'-ry circcrr'ta'l
assurvte
lhe
:
f E' ai :PoL'
: lI
z 4a-, E: d+#Fo" 6 r
I
II :J.<.
""''J
m e.4 i u n't to be non^^Snetfc
,sct ttta
lqv't ' E7'cf-/o)'
I r.!c: I t rL. I J
V_
.\ !
fo,
a-1 r
Fo,r
b a r*
Fo, :! ol,+-?Oi
3s).
21
f. >e
c, t
B=a{e*W, E=0.
D <-t
thqt
As*rt*"
f he cL!rrenf
rn
f he. coun(cre/ackvtrise
in
a
Fram
f riang/e
q
t -
a n4 and
2
4iree{ion
in a6e xy^p(a,ne.
lying
,t-34)
tf.
f iovws
noring
lloof
r=Pto-)c":-!!,
2..3
vve heve
z E: s' \ *(a
L./r
: Jl,
42--t
Ztr rJ 7'+ r, ,/
J L,+ ". : _z: \/3 3^toL --5--; g t t r z 'z - qz zri ur,/./T Z?tv'
d:d
P;-r
Us" Ef G-37). ,.P-? -
f-^
= /JZt' ,*,.. dJ ; E: az-VVi.*gy"(il r.',
--
{0,
E:v,+fs
_ ; .z , u p z f L-z LL l-v/e- z )\ 6,
zl ,Fon L*n, a. n cl E*
O"*g 4, 4{Z
'3;- q a-) 8:VxA:*€* d A, : d r
A ,:
:ar#rrfi!)rl,
aa{4*t, : d" ratnl .
: -9 3+ A-:-#lnr
,/toI 2rrr
O a -t 1 -: fo.- +
. --
A=d"A,:4#uC). 30
E
(rVochorse,*,+hz.) *c-. lr r o.
-i
Il@4,, t)
/, (an
e.
'la-f 7f =zE'
tv.
L(seE:f
A.aZ.
llorizonfal
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
]'r'-,
(ar :---'n, dn )x lo xo.L t
= 2.)4^to-{
(wv).
1 : 2.34 (fwr)
'{lP'',
Ps-to
4rJ so
cct rre-ot s he.t I n[;n;+t ; qnd -B ar#i-sTrnroefri=a-/
ind zp e-'t denf ol x *rJy.
o) /97r/7 AnPi-z'' (ow/ lop D
_t ;l
-;
{
th z.)
e(, b) Fo, z Ais
)o'
/2V(l:
e -aI: /or -2-B/-l;t,o4 B- : '/
t ",l
ot (o,o,z), [-,HoJ'Jz L *ruo4o/l *+ (qo,-z).
E:*jr^
FxA:E-ur$)
o{ X ildepenJenf 4Ar - - /-0T7'*
.,nd
y.
4z
A*: - ryz +c-, At r-zo, Ar:
o : - ryzo+c
,,. A: _qqz_z)4. 3t
--+ c - Mo"r.
P s-ll
m
Me-"/
, D .S - . 1 2
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).
Il vH^=4H^
t Fo=uo4
L) r
+,v; ) :/o lo* r(t0(H2
Br": BF-
i!:^", r ,L i- (t - r+). I oa r
8=\
M =,+o - H - = d - l l t - i. / n I . l z\Fo
a) J; *xil--o; P-EJJ-
Fr=do0/r- lrl).
J-F
kl xan=( 4"
^) j^:ix
t,
B- = a"Y"nT, M=0'.
;
,il(fr,-t)nr:a+(fr,-)*.
fr: O.
J^r= (e* c4s s - ar sln d M x dn : d+ Mosiae. t'
E7' g't7) rc q loolt a] raliu, trr',=i6 g
M- drMo
p,,(L,gJa)(Lt;-al' 2(L,)r,.
=a ^ 2_, 2! ? M o s i n r o .
E=faZ - urryIrrin, ede: iziforo: ? f,r, at the cente-r o. n
-
r.>
a) B,: d* 2 - 6rto (^T),
!t -t4
Er= 4 Br*- u, 4l
@a;.
Br, :!-r - 2 : /u| 2-t 9ooorUo ' 'lr lo + Brx=lqooo (nT),
Fnnl
Y-o
-
Br
@Jran fn'5ooo
8ay: Erl
-/o
(twr).
'Y (nz). Er: d.-/o,ooo-dvto
ta-n 4r:
o1,:-904.0 #,tcn
di-a.04-". I
32
b) ff - No,, ,F'O
1:\rc*a,
(mT),
H,*:#:
Hr': ft
EU
E"y=
4t': to;'fr
'!o-M).
?.
E,:a-B,o+dzg,/.
B/.=*,oo:#= o.ooz (mT).
.'. E,:d*o.ooz+drz
= 9y
P.s- rs
: o'oo! (ra-t): o'o{?o
g=nf 84=aq#r f : r,-gc4te\. ,.,.-/ ISalzzr Pa<1p -\ z6 =-, F I Nz'n lo Jo \ -r*fr= foMZqt -lr; -d. L: f =foMt(r.-ffa)
7
e>>6, i3r+ #(canr/or*).. E *g*s : Brftr6,)- ,"2{tz -L !#
I{
,o
P .s - t s
frr
I
in Lhe- lor3
-s,-lr"..r'gLi vvire, -S=4
lf
t
-J tr} c,-a4;:#["t r' t4si:' A,,={E.rE
: #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
/ong
r'vires
carrylrtg
.7.,o/'an.7
oppos/fe
A,,:#{*'(#-ft)* _-ffr4,1ry#) t _ Ae_ L,^'- T
Sf t^(14r+4)1!zr,+/) (1+vr+d
a!=ao4l
33
)
I, : Ir:2
ft:
Er:d*2 E,rcos3oo: Forre
C : 0 ./ S ( n ) .
S A) ,
"-*#*
per ur,;/ /
F;: - ayz82= -a/H:
: - d/ //sofo= -dy /.41 x ( U/il. 76-3 a-l/ l6ree uvirec q.. c>{eo,-rq.J qnJ Lw-r-/ fhe C-nler * rrr'a nqie. fAcForr:e-t on r/'2a.9nriu4e
P S.-i";-
/v{*gn..f;"
f iot{ irtleasiiy a/ 4he uqire du.- /, fheCu rrenl
l"
Ct;/
,/
l-l wr
d f = *ly
l"t4l: -*:
Synrrn-/ry * /,./ snly a
,I
lY l.L ?
in an e/ernental
/S
, r iY.
2rrwJ't.fir
H qf f he wire- ha-s >/-
F : u,J (dH).(*):dyr"{,{ffi
l.--- o
:"/#ton-,(#)
f': r xE: (-4z)*Wrn: o=4#+^;,(#) PE:,! '+
j;
'" E(t
I
* a1) ' _*.,
T" I* x d
(,4 reif -gcrn />ro6/.-.).
3+
(u/^).
o.[:9dl dF =Ia[x E
: -a,+#(,. #r, F:-+W[ii.*,), =-d.ff t^ (* _t)
--: '.
P s-zt
F = It^E
A
t
B=a..o.S /L
forrct -azo.s I
T:6xB
B ful,0)
D t-zt fa(
of
/ y)
r(fr)
-A"o.s
a"t.o (N)
Torfue an looft:
"7Qa/
o
j(€h
- (dy o.s)t /oCEo.t- {/ o.t),io( eo.rt o.), / o o.z 4 $
t/m,/ .
oF
=-Bx(rL).
o).-Ex r Gn)
I
+
$;e-lJ:
in a Lrni{orcn magne*;c
For.e
,
--(a"Z52xE
: (dr/o, !x o.zxt.z)x(6.ro.s)=-dro.l(N--).
B, a-f
4 currenl
calr7i2g
v.tire
oF fhe
fhe cenler
- O in Ea. s's7):
circular
lot3e2z
is
(b/
fe'trn
telfln3
_
!oI, Br: anLE TorTcre
ofr fhe-srno/l
a- cqrre-n*
c;r..t/or
/u'n
oS wire
carryi2,r
11 is
7 - fr, x E, z (dn,r,,?rf)x("",#)
: (d-,Xdrr)ry'
n).
is a {or7ue houinS a. lnagniluJevthi.h Zrr|nr'2 sin 0 and a Ji.eclion {s6Jing to 2f,
a/4i and
fhe' rnagne-lie
{/u''e's
7r.
35
proJz'
by T'