!"#$%$%&' ("')"*+&' ,"

*"$+#%$%,-, . /-01"+%'2&

!"# $"%&' ("))*+' ,-

."#*/& 0"))*"1 2-

!"#$% &%!%'()

! "#$ %# &'()'*+ ,

-! ./0#123)#4 %# &'()'*+ 5

6! "#$ %# 73(44 65

8! 9'0#/:;3) 6

?! @;#)#:02;:'4 -

A! &'22;#/0# !"#$ %# E;'0FG3H320 ?,

!"#$ %# I*D#2# A

-!"#$ %# B323%3$F"#/C 5

6!./%(::;'/ 5?

JKLIM !" #$"%&'()!" *!)#+(, $-$,'%(.$,'$ '$,$# $##!#$" )$ '+*$!/ 0,'$ 1%(&2%+$# )%)(

$"1#+3+# (4 536(##+)7%1/1& ! ),(##+("7%1/1&

8

!"#$% &%!%'()

!"#$%&' (

!" #! $%&'%()

!"#$%&' (

!" #!$%&'" %()*&%+'" &%,*,* -*' .'"' m / +'01' q2 3-'*(! ", 4!*,* ,* -* &'56* ,"7)0%+!(, 0'(%! R +!* 4'0,(," *! +!*(-+&!0'" / "%* 70%++%6*8 $'" #!$%&'" ", .-,9,* :'"&' ;-, ,* $'4!"%+%6* (, ,;-%$%#0%! ,"&

!"#$%&' () &*+ ,* '%&'-.

!"#$% &'(%(#& )'*$+%& ,#$ -#(,#.*./* $* 1% 0#12/% 3* 1% 3*$*-"% 4 %,12-%(#& 1% -#.32-25.

3* *6'2120$2#7

8'*$+% *. 97 FX = N cos(/3) kq

2

R2= 0 (1)

8'*$+% *. :7 FY = N sin(/3)mg = 0 (2);* ?*(#& 6'*

N =mg

sin(/3)

@**(,1%+%.3# *&/% *A,$*&2#. *. 7

mg

sin(/3)cos(/3) kq

2

R2= 0

;*&,*C%.3# q

q = Rmg cot(/3)

k

!"#$%&' (

!" #$"%"% &'( )*(*( $+,*-"( ) &" )$()* .*/+* 0 ,%$&*( * #/*12( &" ,% ("+)"%#' 1"/#$.*- 4'5

6* )*(* $%7"/$'/ "(#8 4* *- "9#/")' $%7"/$'/ &"- ("+)"%#' : -* )*(* (,;"/$'/ (" ;,"&" )'1"/

-$

!"#$%&' () &*+ ,* '%&'-.

!"#$%&' (

!" #$% &'()*+,% -, ." )(*/"0.#$ ,1.*#/),($ -, #2-$ 3 425 )(,% +2(02% ",02)*&2% 617 8* %, 9$",

."2 +2(02 : ," ,# +,")($ -, 0(2&,-2- -,# )(*/"0.#$; ,"+.,")(, : )2# 1., ,# %*%),

!"#$%&' (

!"# $%&'%# ()*+)%,-# ("#.+./%# 0 -#+1* #-(%&%2%# ("& )*% 2.#+%*$.% 2a3 4"& -, ()*+" 5-2." 2-,#-'5-*+" 6)- ,%# )*- #- +&%7% )* (,%*" (-&(-*2.$),%& %, 5.#5"3 8, ,)'%& 2- ,"# ()*+"# -* 6)-

,% 9)-&7% #":&- )*% $%&'% 2- (&)-:% 6 -* -, (,%*" -# 51;.5% -# ("& #.5-+& ~r1 = az @ ~r2 = az>2"*2- ~r -# ,% ("#.$.B* 2- ,% $%&'% 2- (&)-:% @ ~r1, ~r2 #"* ,%# ("#.$."*-# 2- ,%# $%&'%# ("#.+./%#%*+-# 5-*$."*%2%#3 C2-51#> |~r ~r1| = |~r ~r2| =

R2 + a23 D% 9)-&7% -,E$+&.$% #":&- ,% $%&'%2- (&)-:% -#

~F = ~F1 + ~F2 =kqQ

(R2 + a2)3

2

(Rcos()x+Rsen()y az +Rcos()x+Rsen()y + az)

=2kqQR

(R2 + a2)3

2

(cos()x+ sen()y) .

!- -#+" /-5"# 6)- ,% 9)-&7% #":&- ,% $%&'% 2- (&)-:% -#+1 #":&- -, (,%*"3 F,%&%5-*+- ," 6)-

2-:-5"# =%$-& -# 5%;.5.7%& -, 5B2)," 2- -#+% 9)-&7%> @ -*$"*+&%& $"* -,," -, ,)'%& '-"5E+&.$"

2-#$&.+" ("& ,"# ()*+"# 2"*2- -, 5B2)," 2- -#+% 9)-&7% -# 51;.5%3 8# 2-$.&> 2-:-5"# 5%;.5.7%&

,% 9)*$.B*

|~F | = 2kqQR(R2 + a2)

3

2

= 2kqQ f(R)

= df(R)dR

=(R2 + a2)

3

2 3R2(R2 + a2) 12(R2 + a2)3

= 0

= (R2 + a2) 12 ((R2 + a2) 3R2)) = 0 = R = 12 a

G-5"# 6)- -, ,)'%& '-"5E+&.$" -# )*% $.&$)*9-&-*$.% @ #) &%2." -# R = 12 a

! !"#$%&' () &*+ ,* '%&'-.

!"#$%&' (

"#$ %&'%($ )*+&,-.$ .$(/$0$ ,-%#% 1#$ 2$&$ 2 3 &% .1%*/$ 0% 1# 4-*5 $-&*$#,% 0% *$(/5 *6 %# 1#$

(%/-7# 05#0% %8-&,% 1# .$2)5 %*9.,(-.5

~E :;%( /1($=> ?- *$ @5*-,$ )%(2$#%.% %# %A1-*-@(-5 %#1# +#/1*5 %#,(% *$ ;%(,-.$* 3 %* 4-*5>

:$= B$*.1*$( *$ .$(/$ A 0% *$ @5*-,$>

:@= ?1)5#/$ A1% *$ @5*-,$ )-%(0% .$(/$ $ 1#$ ,$&$ 0% (C seg1)> B$*.1*% *$ ;%*5.-0$0$#/1*$( A1% )(501.% %&,$ 0%&.$(/$ )$($ 1>

)"$*+,-./

'0 C5 )(-2%(5 A1% &% 0%@% 4$.%( %& 0%,%(2-#$( ,50$& *$& '1%(D$& A1% %&,+# $.,1$#05 &5@(% *$

@5*-,$ 3 *1%/5 1,-*-D$( *$ .5#0-.-7# 0% %A1-*-@(-5> C$ &-/1-%#,% /1($ 21%&,($ *$& '1%(D$& A1%

%&,+# $.,1$#05>

E5#0%

~FE %& *$ '1%(D$ )(501.-0$ )5( %* .$2)5 %*9.,(-.5 &5@(% *$ .$(/$ q 3 &% .$*.1*$ .525F

~FE = q ~E

B525

~E %& )$($*%*5 $* %G% X6 &% ,%#0(+ A1% ~E = Ei> H5( *5 ,$#,5 ~FE = qEi

I&.(-@-25& *$ ,%#&-7# 1,-*-D$#05 &1& .52)5#%#,%& (%.,$#/1*$(%&F

~T = T sin()i+ T cos()j

!"#$% &' ()'*' &)+%$ ,%& -)'$.%& (#$ /#+(#0'01' 2 %(,3/%$ ,% /#0*3/340 *' '5)3,36$3#7

8)'$.% '0 97 FX = qE T sin() = 0 (1)8)'$.% '0 :7

FY = T cos()mg = 0 (2);' 7

T =mg

cos()

?''+(,%.%0*# '0 < >7

qE mgcos()

sin() = 0

;' '&1#@ $'&),1% 5)'7

q =mg tan()

E

! ;', $'&),1%*# %01'$3#$ A'+#& 1'0'+#& 5)' q = mg tan()E B C#+# 1 1'0'+#& 5)' tan() B D#$ ,# 1%01#@ (#*'+#& '&/$363$ ,% /%$E% '0 -)0/340 *', 13'+(# /#+#7

q(t) =mg(t)

E

;'$3A%+#& /#0 $'&('/1# %, 13'+(# #61'0'+#&7

dq

dt=

mg

E

d

dt

F)'E#@ /#+#

dqdt = @ *'&('G%0*#

ddt $'&),1%7

d

dt=

E

mg

! !"#$%&' () &*+ ,* '%&'-.

!"#$%&' (

"#$ %&'(&$ +Q $) *&+,-)+)+ /&$ & 0+& 1-$,&+%-& d 1) $)2&'&%-3+4 5+& 2&',6%07& 1) %&'(&+)(&,-8& q 9 *&$& m $) $-,:& )+ )7 %)+,'# 1) )77&$ 9 70)(#; ,'&$ 0+ 2)&*-)+,#2)'2)+1-%07&' & 7& 76+)& & ')$07,&+,) $#?') 7& %&'(& q $)'FA

~F = ~F1 + ~F2 =2kqQx

l3i (1)

!

"#$%& '(# )%*#$%& +#,-./%0-+ #, ,-+1% .%0 ,-& */&2-0./- d/2 #0 3(0./40 *#, 501(,% /0*/.-*%#0 ,- 1(+- -02#+/%+ )%+ ,- +#,-./40 2+/1%0%$72+/.-8

d/2

= cos()

= = (d/2) cos()9%$% )+%*(.# (0 )#'(#:% *#&),-;-$/#02% *# ,- .-+1- *# )+(# A?-,(-0*% #0 B C 2/#0# '(#8

~F =16kqQx

d3(2)

D%+ ,- )+/$#+- ,#@ *# 0#E2%0 2/#0# '(#

~F = m~a> A0 #&2- &/2(-./40 ,- )-+2F.(,- $(#?#&% I- 3+#.(#0./- -01(,-+

+5 #02%0.#&8

w = 4

kqQ

md3

A, )#+/%*% *# %&./,-./40 +5 #02%0.#&

T =2

w=

2

md3

kqQ

! !"#$%&' () &*+ ,* '%&'-.

!"#$%&' (

"#$%&'()( *+ %&,-&($.( '&%.)&/-0&1$ '( 0+),+% 2-$.-+*(%3

') 4"-+$.# 5+*( (* 0+62# (*70.)&0# ($ (* 2-$.# P &$'&0+'#8#) 95+*-+) (* 0+62# (*70.)&0# ($ (* *:6&.( ($ ;-( P (%.+ 6-< +*(=+'# '(* %&%.(6+

*"$+,-./0

') >#) %&6(.):+ (* 0+62# (*70.)&0# (%.+)? ($ '&)(00&1$ j@ 9* 0+62# (*70.)&0# 2)#'-0&'# 2#) *+0+),+ 0($.)+* %#/)( (* 2-$.# P %()? %&62*(6($.(3

~E1 =2kq

r2j

"#6# (% A?0&* 5()B *+% '#% 0+),+% *+.()+*(% 2)#'-0&)?$ -$ 0+62# (*70.)&0# ($ (* 2-$.# P < *+%0#62#$($.(% ($ x %( +$-*+)+$ #) *# .+$.#B .($')(6#% ;-( 0+*0-*+) *+ 0#62#$($.( ($ y '( 0+'+ 0+62# < '(%2-7% %-6+)*#C2#) 2)&$0&2 '( %-2()2#%&0&1$D@ 9%.# %( 2-('( 5() ($ *+ %&,-&($.( ,-)+3

F+ 0#62#$($.( ($ y '(* 0+62# (*70.)&0# %()? Ey = E cos()@ 9* 6#'-*# '(* 0+62# (*70.)&0#BEB (% 2#) '($&0&1$3

E =kq

d2 + r2

!

"#$%&'( $' )&*+, -$. /0$ cos() = rd2+r21 2$ $'34 %45$.4 3$5$%6' /0$

Ey = (kq

d2 + r2 r

d2 + r2)j =

kqr

(d2 + r2)3/2

70$86( #4#6 /0$ ,4 *4.84 $' 5$843+-4( 96#$%6' $'*.+:+. -$*36.+4,%$53$ $'3$ *4%96 $,$*3.+*6

*6%6

~Ey = kqr(d2 + r2)3/2

j

;'3$ *4%96 '$.& $, 9.6#0*+#6 96. *4#4 *4.84 ,43$.4,1 5

~E = ~Ey + ~Ey + ~E1

?$$%9,4@45#6 ,6' -4,6.$' A4 6:3$5+#6'

~E = (2kq

r2 2kqr

(d2 + r2)3/2)j

! B045#6 P $'34 %0A 4,$C4#6 #$, '+'3$%4 '$ 3+$5$ /0$ d r1 ;, *4%96 $,=*3.+*6 $5*653.4#6$5 4D ,6 96#$%6' $'*.+:+. #$ )6.%4 %4' .$#0*+#4 *6%6E

~E = 2qk(1

r2 r

(r2 + d2)3

2

)

;'36 ,6 96#$%6' $'*.+:+. #$ )6.%4 $/0+-4,$53$ *6%6E

~E = 2qk(1

r2 1r2(1 + d

2

r2)

3

2

)

=2qk

r2(1 (1 + d

2

r2)

3

2 )

;5 8$5$.4,( ,4 )05*+>5 (1 + x) '$ 90$#$ 49.6F+%4. 96. 34A,6.( *045#6 x 0 4E

(1 + x) = 1 + x

B6%6 d r( '$ 3+$5$ /0$ dr 0 A 96. ,6 34536 (dr )2 01 5 6:3$5+#4 94.4 $, *4%96( '$ 3+$5$ /0$E

~E =3qkd2

r4j

! !"#$%&' () &*+ ,* '%&'-.

!"#$%&' (

"# $%&'(' )(*+,-%+' )# .# +/0&' )(*+,-%+' .#%1'-0) 2) $)2&(/3/ (%4)-/0)#,) $) 2. &'2%+%5# $)

)6.%(%7-%'8 +'0' 2) 0.)2,-/ )# (/ 4.-/8 $'#$) )2 &)6.):';

!

2qEa = I d2

dt2

"#$% &'(%')*+ ,)-&.&+')%/ /% 01,&21# '.)3). '1214

d2

dt2+

2qEa

I= 0

51+ /1 '(%/ 6&21# 7(& /% 1.)&+$%')*+ %+8(/%. ,&/ ,)01/1 #&8().9 (+ 216)2)&+$1 %.2*+)'1

#)20/& '1+ -.&'(&+')% %+8(/%.4

w =

2qEa

I

: 01. /1 $%+$1 /% -.&'(%+')% #&.94

f =w

2=

1

2

2qEa

I

! !"#$%&' () &*+ ,* '%&'-.

!"#$%&' (

!"#$%&'#( )# *+,'+-.

!"#$%&' (

! "#!$!$ %&' ()(*+,!' %! )(,-& . /(,-(%&' /&$ %!$'#%(%!' )#$!()!' 0$#1&,*!' 12 22 '!3(,(%&'0$( %#'"($/#( %4 5()/0)! )( 10!,6( 70! '! !8!,/!$ (*+&' ()(*+,!'4

)"$*+,-./ 9&, )( )!: %! 5&0)&*+2 '(+!*&' 70! )( 10!,6( !);/",#/( 70! !

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

= ~F21 = 12x40

2L+dL+d

(1

x2 L 1

x2

)dx2

=12x

40 (ln(x2 L) ln(x2))|2L+dL+d

=12x

40 ln(x2 L

x2)

2L+d

L+d

= ~F21 = 1240

ln(

(L+ d)2

d(2L+ d)

) x

" #$%& #$ '& ()#*+& #',-%*.-& /)# $# #0#*-#1 &234$ &'&23*#$5 6)#$ $&3#24$ /)# '& '#7 8# 94)'423

$&%.$(&-# '& :*& '#7 8# ;#

!

!"#$%&' ()

"# $#$%&'( )( #$ +,'$ (-.$ /0'%$)0 10' ,2$ 1$'.( -(%343'4,#$' BCD5 )( '$)30 R6%75 8 10')0- '(4.3#92($- )( #02+3.,)(- AB = 2R 6%7 8 DE = R 6%7: ;0- $'40- BC 8 CD (-.

!"#$%&' () *+$,-.!&,/ 0, '%&'12

d ~E1 = (kdq

x2)i = (

k1dx

x2)i

!"#$%&'"() ($*($ x = 3R +'*#' x = R

~E1 = k(1

1

x2dx)i = (k1( 1

x2

R

3R))i

,) -.$ &$*./#'0

~E1 =2k13R

i

!"! #$ %!&' (# $'& !"%'& )#*("#+'& ,-# !$ &-.#".'*#"*$'& #$ %!+.' #$#%)"/%' "#&-$)!*)# #&)!"!

#* (/"#%%/'* hati0 1&)' ,-#(! /*+#(/!*)!+#*) %$!"' ! 2#" $! &/3-/#*)# 3-"!5

6'+' 2#+'&7 #$ !"%' BC )/#*# %!"3! .'&/)/2! #* #$ 8 .'" $' )!*)' %!(! #$#+#*)' (# %!"3!3#*#"! -* %!+.' &!$/#*)#7 +/#&*)"!& ,-# #$ !"%' CD )/#*# %!"3! *#3!)/2! 8 %!(! #$#+#*)'(# %!"3! 3#*#"! -* %!+.' #*)"!*)#0 9# #&)! +!*#"!7 $' ,-# (#:#+'& ;!%#" #& &/+.$#+#*)#

%!$%-$!" $!& %'+.'*#*)#& #* x (# %!(! %!+.' 8 (#&.-#& &-+!"$!&0

R2 5

2 =2q

R

3 = 2qR

!"# $%& 6'*&/(#"!+'& -*! .#,-#?! %!"3! dq (#$ !"%'7 #$ +'(-$' (#$ %!+.' #$#%)"/' ."'>(-%/(' .'" #&)! %!"3! "@ &/+.$#+#*)#5

dE =kdq

R2

6'* !8-(! (# $! (#*&/(!( (# %!"3! 2 .'(#+'& #&%"/:/" dq = 2ds7 ('*(# ds #& -* .#,-#?')"'A' (#$ !"%'0 1&)# )"'A' (# !"%' $' .'(#+'& #&%"/:/" #* =-*%/'* (#$ !*3-$' ,-# $' "#%'""#5

ds = Rd

!

"#$%' %()#(*%' %'*+,-,+ %. */&0# *#

dE2 =k2Rd

R2=

k2d

R

12%3#4 ./ *#&0#(%()% %( x $%. */&0# dE2 '%+/

dEx2 = dE2 cos() =k2 cos()d

R

5()%3+/($# $%'$% = 0 6/')/ = 2 7

Ex2 =k2R

pi2

0cos()d =

k2R

8%*)#+,/.&%()%

~Ex2 =k2R

i

!"# $%& 9# %' $,*,. ;%+ ./ *#&0#(%()% %( x $%. */&0# 0+#$2*,$# 0#+ %')% /+*# '%+< ,32/./. /()%+,#+4 $% %')/ &/(%+/7

~Ex3 =k2R

i

=', '% )%($+/ >2%

~E2 + ~E3 =2k2R i'!#(# %)&

?%/ q ./ */+3/ >2% ),%(% %')% )+#@# $% /./&-+% )/. >2% /(2./ %. */&0# %.%*)+,*# 0$+#$2*,$#0#+ )#$# %. /./&-+% %( OA B# ),%(% ./+3# R4 ./ $%(',$/$ $% */+3/ '%+2% 0/+/ %. )+#@# AB ;%' >2%

~E4 = (k4

2RR

1

x2dx)i = (

k42R

)i

?2&/($# )#$#' .#' */&0#' */.*2./$#' % ,32/./($# / *%+#7

~E1 + ~E2 + ~E3 + ~E4 =2k13R

i+2k2R

i+ (k42R

)i = 0

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

"#$%#&'()* 4+

4 =2

3(21 + 62)

,##-%.'/'()* .*$ 0'.*1#$ )# 12 2 3 4 )#(5)*$ '(6#15*1-#(6#2 #(7*(61'-*$ #. 0'.*1 )# .'7'18' 9:$7')'+

q =2

3q(1 +

12

)

;5 7*($5)#1'-*$ 3 +q =

10

3q

!

!"#$%&' ((

"#$%&$' '$ %#()* '$+%,-.%* )-*/&%./* )*- &0 )$#0* .00.,* /' /'02./#/ /' %#-3# &0.4*-(' 5

)"$*+,-./

"*02./'-'(*2 '$ )$#0* '0 %&'2,.60 '$ )$#0* 78 /' %**-/'0#/#2 %#-,'2.#0#25 9#:'(*2 ;&' '$

%#()* '$+%,-.%*

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

= ~E(~r) = 20

sign(z)z

"#$% &'% %( )*($# +% +,)-#.$,.',+*+ +%( /0+'(# +%( -*/1# %(2-$3,-# 4&'% %) .#3/*( *( 1(*.#5

%) + 0 6

!

!"#$%&' () "#$%&'()( *$ +,-$# &$$&/# '( '($%&'-' '( 0-)1- *$&2#)3( 4 0#$ *$ #)&00&)0*,-) '( )-' 56

-7 "-,0*,( (, 0-3+# (,80/)&0# ($ 0*-,9*&() +*$/# '(, (:( 9*( +-%- +#) (, 0($/)# '(, #)&0

+()+($'&0*,-)3($/( -, +,-$#6

;7 < ,# ,-)1# '(, (:( %( 0#,#0- *$ -,-3;)( =$# 0#$'*0/#)7 '( ,-)1# -4 0#$ '($%&'-' '(

0-)1- ,&$(-, *$&2#)3( > 0*>- '&%/-$0&- 3?% +)@A&3- -, +,-$# (% '6 B$0#$/)-) ,- 2*()C-(,80/)&0- 9*( (A+()&3($/- (, -,-3;)(6

*"$+,-./0

-7 D%-$'# (, +)&$0&+ '( %*+()+#%&0&@$4 +#'(3#% ($0#$/)-) ,# '(%(-'# 0#$%&'()-$'# ,-

%*3- E(0/#)&-, '(, 0-3+# (,80/)&0# 9*( 1($()- *$ +,-$# &$$&/# '( '($%&'-' '( 0-)1- %#;)( (, (:( > *$ '&%0# '( )-' 5 > '($%&'-' '( 0-)1- 6 F(- (, +,-$# >C 0##)'($-'#(, +,-$# ($ 0*(%/&@$ > (, (:( A (, (:( ($ 0*(%/&@$6

G( ,#% &/(3% -$/())(%4 /($(3#% 9*( (, 0-3+# (,80/)&0# 1($()-'# +#) (, +,-$# (%

~E1(~r) =

20 sign(x)x

> (, 0-3+# (,80/)&0# 1($()-'# +#) (, '&%0# (%

~E2(~r) =20

x(sign(x) x

R2 + x2

)

6

= ~E = ~E1 + ~E2 = 20

(

xR2 + x2

)x

;7 H($(3#% 9*( ,- 2*()C- (,80/)&0- 9*( (A+()&3($/- *$ (,(3($/# dx '(, -,-3;)( '(;&'# -,

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

"#$%& '&% (# &)*'*& (,-. /$/$ "&) d~F = dq ~E0

= ~F = d+ad

dq ~E con dq = dx

=

d+ad

dx

20(

xR2 + x2

)x

=

20 d+ad

xdxR2 + x2

x

=

20(

R2 + x2)d+a

d x

=

20(

R2 + (d+ a)2 R2 + d2

) x

!

!"#$%&' ()

"# $%&'('%#)% *%+',%+'*-%$'&. #. &.#/0&).$ /% $1/'. a )'%#% 0#1 &1$21 ).)13 Q 0#'-.$+%+%#)%/'*)$'40'/1 %# *0 *0(%$&'% '#)%$'.$6 7#&0%#)$% %3 &1+(. %38&)$'&. %# %3 (0#). O6

*"$+,-./0

7*)% ($.43%+1 3. /%4%+.* $%*.39%$ 0)'3':1#/. 31 31; /% &.03.+4 (1$1 /'*)$'40&'.#%* *0(%$.+. ;1 ,%+.* 9'*).? %3 %3%+%#). /% &1$21 3. (./%+.* %*&$'4'$ &.+. dq = dA? /.#/% %*31 /%#*'/1/ /% &1$21 *0(%$&'13 %# %3 '#)%$'.$ /%3 $%&'('%#)% @&.#*)1#)%A6 B%4%+.* %*&$'4'$ %3

%3%+%#). /% C$%1 dA /% )13 +1#%$1 =0% (./1+.* $%&.$$%$ 31 *0(%$&'% ; (1$1 %*). ,1$%+.* 3.*'20'%#)%D >.#*'/%$1$% 0#1 &1*&1$1 %*-8$'&1 &.# 31 *'20'%#)% /'*(.*'&'E# (1$1 31* 91$'143%* ;

F3 ,1&%$ 0#1 (%=0%G1 91$'1&'E# /%3 C#203. ? 9%+.* =0% *% $%&.$$% 0# (%=0%G. 1$&. /% 3.#2')0/ds1 = R sin()6 H.$ .)$. 31/. 13 ,1&%$ 0#1 (%=0%G1 91$'1&'E# /%3 C#203. *% $%&.$$% 0# 1$&./% 3.#2')0/ ds2 = Rd6 B% %*)1 +1#%$1 %3 %3%+%#). /% C$%1 /% 31 &'$&0#-%$%#&'1 3. (./%+.*%*&$'4'$ &.+.

dA = ds1 ds2 = R2 sin()ddB% %*)1 +1#%$1 (./%+.* %*&$'4'$ %3 %3%+%#). /% &1$21 &.+.

dq = R2 sin()dd

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

"#$ #%$# &'(#) (*+*,#- (*%*$,./'$ - 0*1%#$*-

~r 2 ~r3 4&'$',*/%* ~r = ~0 2 5#$ #%$# &'6(#7,.$'/(# &' 9:$'; 0*$*,#- ?1.& 0*$

!

!"#$%&' ()

"# $% '()%* +$ %$%,-)+ AB ./+#+ $0#'/.(1 2L 2 +3 4+)4+#1/5($%) %$ %$%,-)+ CD* 1+ $0#'/.(1L6 7%1% (#0 1+ +$$03 ./+#+ $% ,/3,% 5%)'% Q* 1/3.)/-(/1% (#/80),+,+#.+6 7%$5($+ $% ,%'#/.(12 1/)+55/9# 1+ $% 8(+):% ;(+ (#0 1+ +$$03 +

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

"# #$%& '&(#)&

dEy =k1ddx

(x2 + d2)3

2

*(%#+)&(,- ,#$,# x = L .&$%& x = L/

Ey = k1d

LL

1

(x2 + d2)3

2

dx = k1d(x

(d2x2 + d2

L

L) =

2k1L

dL2 + d2

"# #$%& '&(#)& 0& 12#)3& 42# #5#)6# #0 &0&'7)# .-)83-(%&0 $-7)# 2(& 9#42#:& 6&)+& dq ,#0&0&'7)# ;#)%86&0 42# #$%& & 2(& &0%2)& y $#) 4/(/3/4" &- y 6 &+ #" 2/+2" &- $%"#*%/&( 8%-15 4 "#"27(&:A5( 1"-159 -5+ 7"+1" E&( #" '%&()" 85( %-/4"4 4& #"(35 &- BFG 4 "#"27(&:

H&"- ~r2 = ~09 ~r1 = xx+ yy9 $5- x + 6 d y b+ d9 #" 85+/$/=- 4 &2&-15 4B&- BFG 4 "#"27(& 6 #" 85+/$/=- 4& #" .%/-$."9 (&+8&$1/E"2&-1&: I+D9 1&-&25+ *%& #" '%&()"

+57(& 1"# &2&-15 &+

~dF21 = dq2

dq1 ~r140|~r1|3 , con dq1 = dxdy

= dq2 +

b+dd

dxdy xx40(x2 + y2)

3

2

dq2 +

b+dd

dxdy yy40(x2 + y2)

3

2

= dq2y40

+

b+dd

dxdy y(x2 + y2)

3

2

=dq2y

40

+

(1

x2 + y2

)b+d

d

dx

=dq2y

40

+

(1

x2 + (b+ d)2 1

x2 + d2

) dx, pero dq2 = dx2

=~dF21dx2

=y

40

+

(1

x2 + (b+ d)2 1

x2 + d2

) dx

=y

20 ln

(x+

x2 + (b+ d)2

x+x2 + d2

)+

0

=y

20 lm

x+ln

1 +

1 + ( b+dx )

2

1 +

1 + ( dx)

2

ln(b+ d

d

)=

y

20 ln(

d

b+ d

)

=~dF21dx2

=y

20 ln(

d

b+ d

)

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

!"#$%&' ()

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dq *.&/+'+ ($ (, ;*$0# ~r = xi '(, +,+-.)( %#.)( (, ;*$0# '( ~r = dj %()7B

d ~E =kdq(~r ~r)|~r ~r|3

=kdq(xi+ dj)(x2 + d2)

3

2

6?#)+D %( 0&($( >*( dq = dx A ;#) ,# 0+$0#B

d ~E =kdx(xi+ dj)

(x2 + d2)3

2

E$0(4)+$'# '(%'( + +B

~E =

+

kdx(xi+ dj)(x2 + d2)

3

2

= k(( +

x

(x2 + d2)3

2

)i+ (

+

d

(x2 + d2)3

2

)j)

!

"# $%&'()& f(x) = x(x2+d2)

3

2

*+ (,-#. / -0. 10 2#&20 1# (&2*3.#1 4* *+2# $%&'()& *& %& (&2*.5#10

-#. 20,# 5#10. 67 80. 02.0 1#409 1# $%&'()& g(x) = d(x2+d2)

3

2

*+ -#. / 1# (&2*3.#1 4* *+2# $%&'()&

4*+4* # + +*.: *1 40;1* 4* 1# (&2*3.#1 4* 3 4*+4* 6 # +? +

g(x)dx = 2

+0

g(x)dx

80. 10 2#&209 1# *=-.*+()& -#.# *1 '#,-0 +* .*4%'* #

~E = 2kd

+0

1

(x2 + d2)3

2

j

@* 2(*&* A%*

1

(x2+d2)3

2

= xd2(x2+d2)

1

2

80. 10 2#&20

~E = 2kd(x

d2(x2 + d2)3

2

0

)j =2k

dj

BC0.# A%* +#;*,0+ '0,0 *+ *1 '#,-0 *1D'2.('0 3*&*.#40 -0. %& #1#,;.* (&&(20 -04*,0+ '#1F

'%1#. 1# $%*.G# +0;.* *1 #1#,;.* +*,('(.'%1#.7 8#.# '#1'%1#. *+2# $%*.G# '0&+(4*.#.*,0+ -*A%*H0+

2.0G0+ 4* '#;1* '0& '#.3# dq *& *1 '%#1 1# $%*.G# #-1('#4# +0;.* *1 +*.: -0. 4*&('()&?dF = dqE940&4* E *+ *1 '#,-0 *1D'2.('0 -.04%'(40 -0. *1 #1#,;.* (&&(20 # 1# #12%.# *& A%* +* *&'%*&2.#1# '#.3# dq7 I*'0..*.D 10+ *1*,*&20+ 4* '#.3# 4*1 #1#,;.* *& $%&'()& 4* %& :&3%10 A%* .*'0..#*1 #1#,;.# +*,('(.'%1#.7 J0&+(4*.#.* 1# +(3%(*&2* 3%.#?

K1 '#,-0 *1D'2.('0 -.04%'(40 -0. *1 #1#,;.* (&&(20 4*-*&4* 4* 1# 4(+2#&'(# # 1# '%#1 +* *+2#

4* *19 '0,0 1# #12%.# 4* '#4# *1*,*&20 dq 4* '#.3# 5#.(#9 1# $%*.G# -.04%'(4# +0;.* '#4# %4* *+2#+ '#.3#+ 2#,;(D& 10 C#.:7 "# #12%.# 4* '#4# *1*,*&20 dq *& $%&'()& 4*1 :&3%10 A%*4#*=-.*+#4# '0,0 d() = R+R sin()7 B4*,:+9 *1 *1*,*&20 4* '#.3# dq *+ -0. 4*&('()& (3%#1# ds9 40&4* ds *+ %& -*A%*H0 2.0G0 4* #1#,;.* '(.'%1#. *1 '%#1 -04*,0+ *+'.(;(. *& $%&'()&4*1 :&3%10 '0,0 ds = Rd7 L* *+2# ,#&*.# 1# $%*.G# +0;.* % '#.3# dq +*.:?

dF = dqE = Rd 2kR+R sin()

=2k2d

1 + sin()

M&2*3.#&40 4*+4* = 0 # = 0;2*&*,0+ 1# $%*.G# 202#1 +0;.* *1 #1#,;.* +*,('(.'%1#.7 8#.#*+20 %2(1(G#,0+ *1 C*'C0 4* A%*?

! !"#$%&' () *+$,-.!&,/ 0, '%&'12

1

1 + sin()=

sin() 1cos()

"#$ %&'# '%$%(#& )*% +, -*%./, .%&*+',$'% %&

F = 4k2

!"#$%&' (

!" #! $%&''

!"#$%&' ()

!" #"$%" &'!('") q > 0 *+(" $,-*"-" &,$ '!" +'&*$#/* #*$$"-" 0,$1"-" &,$ '! 1"!(, #2!/#,-* $"-/, R 3 ")('$" H4 3 '!" +'&*$#/* +*1/*+05$/#" #,!#5!($/#" #,! )" #"$%"4 +*%6! +* ,7+*$8"*! )" %'$"9 :")#')* *) ''* ))"1"1,+ DE +*$B /%'") " )" +'1" -*

),+ ''* &"+*! &,$ )" +*1/*+0*$" C>'* ))"1"1,+ SeE 3 ),+ ''* &"+*! &,$ *) 1"!(,#2!/#,C>'* ))"1"1,+ ScE9 F+(, +* &'*-* *+#$/7/$ #,1,@ = e + c

= c = + e (2)G* *+(" 1"!*$"4 *!#,!($"!-, *) 8"),$ -* e $*+,)8*1,+ *) &$,7)*1" /!1*-/"("1*!(*9 e +*#")#')" #,1,@

e =

Se

~E d~S

F) #"1&, *)5#($/#, &$,-'#/-, &,$ )" #"$%" q *+ E = 140qR2r9 G,!-* r *+ *) 8*#(,$ '!/("$/,

*! -/$*##/2! $"-/")9 F) 8*#(,$ d~S ("17/5! *+(" *! -/$*##/2! $"-/")4 3" >'* d~S = ndS 3 n *+&"$")*), " r #,1, 1'*+($" )" %'$"@

HI

! !"#$%&' () &*+ ,* -!%..

"# #$%&' $# %#()*+ ,-#

~E d~S = 140

q

R2r ndS = 1

40

q

R2dS

.(%#/*0()& $# &1%2#(# #3 -5& 60*0 30 $#72#$8#*09

e =

Se

~E d~S =Se

1

40

q

R2dS =

1

40

q

R2

Se

dS

:0 2(%#/*03

SedS ;&**#$6&()# 03 +*#0 )# 30 $#72#$8#*0< =(%&(;#$

SedS = 2R2< "# #$%070(#*0 #3 -5& $#*+9

e =1

40

q

R2 2R2 = q

20(3)

"# >?@'>A@ B > @ $# %2#(# (037#(%# ,-#9

c =q

20

!"#$%&' ()

D&($2)#*# -(0 ;+$;0*0 ;23E()*2;0' $2( #$6#$&*' )# *0)2& F B 30*/& 2((2%&' ;&( )#($2)0) )# ;0*/0

$-6#*;203 -(28&*7# >G#* /-*0@< =(;-#(%*# #3 ;076& #3H;%*2;& #( %&)& #3 #$60;2& >-$# 3#B )#I0-$$@72#(%*0$ #$%0 ;0(%2)0) $#0 30 72$70@ B ,-#

#$ )2*#;%07#(%# 6*&6&*;2&(03 0 30 ;0*/0 #(;#**0)0' ;&( ;&($%0(%# )# 6*&6&*;2&(032)0)

10< =$

)#;2*'

!

~E ndS = Qint0

!" # Qint #$ %& '&()& #"'#((& & *!( %& $+*#('-# . /$0& 1!(2& -"0#)(&% # %& 3#4 #5&+$$ $# *+# # #6*(#$&( #" 1!(2& -1#(#"'-&% +$&" ! #% 0#!(#2& # %& -7#()#"'-& 8%! '+&% $#

#9& &% %#'0!(:; !+# #% '&2*! #%?'0(-'! $#& "+%! #"0(! #% '-%-" (!.

+#

~E ndS = E(r)(2r)h = (2R)h0

= E(r) = 0

(R

r

)

A!( 0&"0!;

~E(r) =

{~0 r < R

0

(Rr

) R rJ!0# >+# #% $&%0! # -$'!"0-"+- & #% 2= +%! #% '&2*! #%?'0(-'! 8>+# #$ "!(2&% & %& $+*#(E

'-#: #$ + 0 .

! !"#$%&' () &*+ ,* -!%..

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"# $%# '&( )%*$+%,'-%.& #*/0+%-( )# -(+1( )# +()%2 3 4 )#&*%)() 5(+%(,6# = r 7(+(r < R8 9(6-'6# 6( #+1:( 72$#&-%(6 (-';'6()( 72+ 6( )%*$+%,'-%.&8

*"$+,-./0

"(,#;2* R r 92&*%)#+#;2* &'#5(;#&$# -2;2 *'7#+-%# )# %&$#1+(-%.& '&( -?*-(+( #*/0+%-( )#+()%2 + -2& ;%*;2 -#&$+2

!

"#$ %&'%#( %)')*#+ ,-)

~E(r) =

{40

r2r r < R40

(Rr

)2R2r R r

./#$& ,-) 0#'#0)*#+ )1 0&*2# )130%$40# )' %#5# )1 )+2&04#( 2#5)*#+ 0&10-1&$ 1& )')$67&

2#%)'04&1 &0-*-1&5& )' 1& 54+%$48-049' 5) 0&$6& :

U =02

R3

~E2d3x

=02

20

0

+0

E(r)2r2sen()drdd

=402

+0

E(r)2r2dr

=402

( R0

2

1620r6dr +

+R

2

1620

R8

r2dr

)

=2

80(r7

7

R

0

R8 1r

+

R

)

=2

80(R7

7+R7

)

=2

70R7

! !"#$%&' () &*+ ,* -!%..

!"#$%&' ()

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~E ndS = Qint0

E(r)4r2 =4R3

30

= ~E(~r) = R3

30r2r

>#) /,$/#0 /($(*#% ;4( (3 -,*=# (3.-/)&-# 1($(),'# =#) 3, (%+(), *,-&B, (%

~E(r) =

{r30 r r < R

R330r2

r R r

>#$1,*#% (3 #)&1($ '( -##)'($,',% ($ (3 -($/)# '( 3, (%+(), '( '($%&',' '( -,)1, 7 G%C0/($(*#% ;4( 3#% -,*=#% (3.-/)&-#% ($ 3#% &$/())(% '( -,', (%+(), (%/:$ ','#% =#)

~E1(r) =

30 ~r para |~r| < R

~E2(r) = 30

(~r ~d) para |~r ~d| < R

8#% 2(-/#)(% ($ 3, &$/()%(--&?$ '( ,*9,% (%+(),% %,/&%+,-($ %&*43/:$(,*($/(

|~r ~d| < R 6 |~r| < R7 G%C0 =#) (3 =)&$-&= '( %4=()=#%&-&?$0 (3 -,*=# (3.-/)&-# ($ 3,&$/()%(--&?$ '( ,*9,% (%+(),% (%

~E(~r) = ~E1(~r) + ~E2(~r) =

30 ~r

30 (~r ~d) =

30 ~d

(3 -4,3 (% -#$%/,$/(0 /,3 -#*# %( ;4()C, =)#9,)7

!

!"#$%&' ()

"#$ %&'()&*+,&-# %. ,$)/$ 012+34()&,$ 6$ %. %.#'&%$% +#&71)3. > 0 1,+8$ +# 012+3.#.'74)&,1 %. )$%&1 9:

$; %. ?$+'';:

*; @. ,121,$ +#$ ,$)/$ 8+#(+$2 Q < 0 .# .2 ,.#()1 %. 2$ .'7.)$A > '. 2$ %.6$ 2&*).: BC+.%$1 #1 .# .D+&2&*)&1 2$ ,$)/$ .# .2 ,.#()1E @& .' $)3$(&0$ 2$ ).'8+.'($A B.' .'($*2. 1

.'($*2.E

@& 2$ ,$)/$ (&.#. +#$ 3$'$m > '. %.'82$F$ %.2 ,.#()1 %. 2$ .'7.)$A 8.)1 '&.38). D+.%$#%1%.#()1 %. .22$A ,$2,+2. 2$ 7+.)F$ D+. 2$ ,$)/$ .G8.)&3.#($ > 8)+.*. D+. '+ 310&3&.#(1

.' +# 310&3&.#(1 $)3-#&,1 '&382.: H#,+.#(). '+ 8.)I1%1 %. 1',&2$,&-#:

,; @+81#/$31' D+. 2$ ,$)/$ '-21 8+.%. 310.)'. .# .2 .6. G: @& $%.3J' '. $82&,$ +# ,$381

.24,()&,1 .G(.)#1

~E0 = E0xA E0 > 0A ,$2,+2. 2$' #+.0$' 81'&,&1#.' %. .D+&2&*)&1 %. 2$,$)/$ > %.(.)3. ,+J2.' %. .22$' '1# .'($*2.' 1 .'($*2.': K&',+($ D+. ,1#%&,&1#.' %.*.

,+382&) E0 8$)$ 2$ .G&'(.#,&$ %. ($2.' 81'&,&1#.' %. .D+&2&*)&1 > '+ #L3.)1:

*"$+,-./0

$; M1#/$31' .2 1)&/.# ,11)%.#$%1 .# .2 ,.#()1 %. 2$ .'7.)$: K.2 &(43 $#(.)&1)A $2 +'$) 2$ 2.>

%. ?$+''A '$*.31' D+. .2 ,$381 .24,()&,1 /.#.)$%1 81) 2$ .'7.)$ .'

~E(r) =

{r30 r r < R

R330r2

r R r

*; N.#.31' D+. .2 ,$381 .# .2 ,.#()1 %. 2$ .'7.)$ .'

~E(~0) = ~0A 81) 21 ,+$2 .2 ,.#()1 .'+# 8+#(1 %. .D+&2&*)&1: @& 310.31' +# 81D+&(1 2$ ,$)/$A .'($ .G8.)&3.#($ +#$ 7+.)F$

~F = Q r30 rA 2$ ,+$2 $8+#($ O$,&$ .2 ,.#()1 %. 2$ .'7.)$ =' &381)($) .# D+4 %&).,,&-#'. O$>$ 310&%1 2$ ,$)/$;A 81) 21 ,+$2 .' +# 8+#(1 %. .D+&2&*)&1 .'($*2.:

@& 310.31' 2$ ,$)/$ D+.%$#%1 '&.38). %.#()1 %. 2$ .'7.)$A (.#.31' D+. '+ .,+$,&-# %.

310&3&.#(1 .'

m~a = ~F = Q r30

r

~a + Qr

3m0 r = ~0

= r + Q 3m0

r = 0r + w2 r = 0M1) ($#(1A .2 310&3&.#(1 %. 2$ ,$)/$ .' +# 310&3&.#(1 $)3-#&,1 '&382. > '+ 8.)I1%1 %.

1',&2$,&-# .' T = 2w = 2

3m0Q

,; M1) .2 8),&8&1 %. '+8.)81'&,&-#A .2 ,$381 .24,()&,1 ).'+2($#(. .'

~E(r) =

{r30 r + E0x r < R

R330r2

r + E0x R r

!"#$%&' () &*+ ,* -!%..

!""#$%&'&"()* "+ $,-.* "+/$0%&$* , +* +,%1* )"+ "2" 34 %"#5+0,

~E(r) =

( x30 + E0) x R < x < R( R

330x2

+ E0) x R x(E0 R

330x2

) x x R

6($*(0%"-*# +,# (5"7,# .*#&$&*("# )" "85&+&'%&*9

&: R < x < R ;5"%"-*# 85"x

30+ E0 = 0 = x1 = 30E0

,? .*#&$&=( )" "85&+&'%&* .,%, R x9

!

"""# x R $%&'&()* +%&

E0 R3

30x2= 0 = x3 = R

R

30E0

,-'- +%& x1 &."*/&0 *& 1&2& 3%(45"' +%&

x3 = R

R

30E0< R =

R

30E0> 1 = E0 < R

30

+%& &* 5- ("*(- 3)61"3"76 +%& &63)6/'-()* -6/&'")'(&6/& 4-'- +%& 8%2"&'- &+%"9

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! !"#$%&' () &*+ ,* -!%..

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~E ndS +manto

~E ndS = qinterior0

(1)

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tapas

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8, 5(+2#) n3 $#)-/, /, -/$2#3 %(): 4/)/,(,# / r . 4#) ,# 2/$2# r n = 1 ,# 1*( &-4,&+/ 1*(@manto

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dS

C/ &$2(0)/,

manto dS +#))(%4#$'( /, :)(/ '(, -/$2#@

manto dS = 2rL7 A#) ,# 2/$2#@

manto

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D((-4,/E/$'# FGH . FIH ($ FJH )(%*,2/ 1*(@

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D&&0+)(E(57# FGH &5 FIH@

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R)

= dq = 2L 0(r r2

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r0

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= q = 2L0 (r2

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3

3R) = 2L0 (3Rr

2 2r36R

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= q = L03R

(3Rr2 2r3) (7)K& F H 4 F!H@

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(3Rr2 2r3)

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R2

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0 R

2

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~E = ~E =a

0i, si a < x < a+ 2R

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0+

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(a0 +R2

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20x

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40

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1

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q

(a x)2 x

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(

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1

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(a x)2) x

G&'! :+ ?(&

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z= 0 Vy = 0

V

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(

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(a x)2)

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20+

1

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JK

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eV (a/2) = mv2

2 eV (0)

= K = mv2

2= e (V (0) V (a/2))

= e

( 140

q

a+ C +

1

40

2q

a+

a

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>!&-

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= (~r)A0

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~E(~r) ndS +carasup

~E1(~r) n1dS +carainf

~E2(~r) n2dS

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0

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= E(~r) = +(~r)0

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0= ~E(r) = q4

40

r

r2

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r 1b

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a 1b

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a 1b

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q040

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bV0r b r

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= q1 = 0 = q2 = q0

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'(F2!-0' V (+) = 0"7 H2 'A'&,-+ V (b) = b+ ~E dr 6= 0+ 3/'0 ~E 6= 0 : 2- I%:01.',(D% */' 3/!1'(% %2/$%( $% 12,')(%$7 B% !1A'('2&1% !' 3-,'2&1%$ '2,(' %(.%!/(%0

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.%!/(%0 2- I% &%.>1%!-+ 3-( $- */' 0/ &%3%&1,%2&1% 2- I% &%.>1%!-+ : ,%.3-&-

I% &%.>1%!- $% &%()% '2 '$$%0"7

!

!"#$%&' ()

"#$%&'()( *$+ ,+)&--+ '(-.+'+ '( '($%&'+' -&$(+- *$&/#)0( 1 -+).# L2 3$4*($5)( %* 6#5($4&+-(-745)&4# ($ 5#'# (- (%6+4 8*( -+ )#'(+2

*"$+,-./0

9+:(0#% 8*( (- 6#5($4&+- (%5; '+'# 6#)

V (~r) =

dq

40|~r ~r1|3$ (%5( 4+%#< %& ~r = xx+ yy + zz 1 ~r1 = x1x< 4#$ 0 x1 L< %#$ -+% 6#%&4$(% '(- 6*$5# 1'( -+ ,+)&--+ )(%6(45&,+0($5(< 5($(0#% 8*(

V (~r) =

L0

dx1

40(x1 x)2 + y2 + z2

3$4#$5)(0#%

dxx2+12 9& x = tan()< ($5#$4(%

dxx2 + 1

=

sec2()dtan2() + 1

=

sec()d = ln(sec()+tan()) = ln(x+

x2 + 1) = arccosh(x)

3=5($'&($'# (%5# +

dx1

(x1x)2+y2+z2)(%*-5+ >,()?8*(-#A

dx1

(x1 x)2 + y2 + z2= ln(x1 x+

((x1 x)2 + y2 + z2)

B%?< (- 6#5($4&+- (%

V (x, y, z) =

L0

dx1

40(x1 x)2 + y2 + z2

=

40

L0

dx1(x1 x)2 + y2 + z2

=

40 ln(x1 x+

(x1 x)2 + y2 + z2)

L0

=

40 ln

(L x+

(L x)2 + y2 + z2

x+x2 + y2 + z2

)

!"#$%&' 12

"#$%&'()( (- '&6#-# (-745)&4# '( -+ .*)+2

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8*( 2d< -+ %(6+)+4&C$ ($5)( 4+).+%AE

+A 3- 6#5($4&+- (-745)&4# ($ 5#'# (- (%6+4 > !"# E F%( 4##)'($+'+% 6#-+)(%< 1 *%( -+ -(1 '(

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:A 3- 4+06# (-745)&4# ($ 5#'# (- (%6+4

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

"# $% &'&()*% +,-&'".%/ 0&/ 0.+,/, 1&2-,3 +%(% 4' 0.+,/, &' )&'&(%/#5

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%# 6,( &/ +(.'".+., 0& 24+&(+,2.".7'3 &/ +,-&'".%/ &/8"-(.", &' &/ +4'-, 6 &2-9 0%0, +,(

V =q

40

(1

r+ 1r

)

6,( /% /&: 0&/ ",2&',3 -&'&;,2

!

r2 = r2 + d2 2rd cos

(2+

)= r2 + d2 + 2rd sen ()

" #$ %&'() )*+,&-) ) ,& )*.$'/&'

1

r= 1

r(1 1

2

(d

r

)2 dr sen()

)

0&' ,& 12$ $, 3&.$*4/), 3)') r >> d '$52,.)

V (r, ) =q

40

(1

r+ 1r

)

= q40r

((

1 12

(d

r

)2+d

r sen()

)(1 1

2

(d

r

)2 dr sen()

))

=q

40r(2d

r sen()

)

=2qd

40 sen()

r2

=p

40 sen()

r2

V (r, ) =1

40 ~p rr2

#&*#$ ~p $5 $, (&($*.& #/3&,)' $,64.'/4& #$, #/3&,&7

89 :)8$(&5 12$

~E = V = Vr

r 1r

V

;5&*5/#$'$(&5 ,) 5/-2/$*.$ -2')7 @)#& 12$ ,) '$-/A* &423)#) 3&' $, #/3&,& $5 )4&.)#)=

3$(&5 .&()' V (+) = 0 4&(& 3&.$*4/), #$ '$%$'$*4/)7;, .')$' ,)5 4)'-)5 #$5#$ $, /**/.& " %&'()' $, #/3&,&= ,) $*$'-

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

U = qV (x+ dx, y + dy, z + dz) qV (x, y, z)= q (V (x+ dx, y + dy, z + dz) V (x, y, z))= qdV

= q

(V

xdx+

V

ydy +

V

zdz

)

=V

xpx +

V

ypy +

V

zpz

= ~p ~VU = ~p ~E

!

!"#$%&' ()

"#$%&'()( *$+ )(,&-$ (%./)&0+1 '( )+' b1 2*( 3&($( *$+ '&%3)&4*0&-$ '( 0+),+ *$&.#)5( (r) =0 6+)+ 7+ )(,&-$ '(3()5&$+'+ 6#) a < r < b 8 '($%&'+' $*7+ 6+)+ r < a9 :(3()5&$( (76#3($0&+7 (7(03)#%3;3&0# ($ 3#'# (7 (%6+0

*"$+,-./0

'1

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

=

S

~E ndS = E4r2

"# $#% '()*%'+% qint ,* -.*/* $#0$.0#% 12$'03*()* 4# 5.* 0# /*(,'/#/ /* $#% *, $+(,)#()*67* )*(/%2 5.* qint = 0V 6 V *, *0 8+0.3*( *()%* 0# *,1*%# /* %#/'+ r 4 0# *,1*%# /* %#/'+ a $+(0+ 5.* %*,.0)#9

qint = 04

3(r3 a3):-0'$#(/+ 0# 0*4 /* .,,9

E4r2 = 04

30(r3 a3)

;+( 0+ 5.* ,* +

!

"#$%&'#()*+ #, -.&/* #,0-12'-* #( 1*)* #, #$/.-'* 2#$%,1.

~E =

0 r < a0(r3a3)

30r2r a < r < b

0(b3a3)30r2

r r > b

34*2. 5%# 1#(#&*$ #, -.&/* #,0-12'-* #( 1*)* #, #$/.-'* /*)#&*$ -.,-%,.2 #, /*1#(-'., #,0-12'-*6

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~E d~ = Er d~ = Edr6 7# #$1. &.(#2. #, /*1#(-'., #,0-12'-* $#2BC

V (r) = r

~E d~ = rEdr =

r

0(b3 a3)

30r2)dr

D(1#92.()* $# *81'#(# 5%#C

V (r) =0(b

3 a3)30r

, si r > b

34*2. 5%#2#&*$ -.,-%,.2 #, /*1#(-'., . %(. )'$1.(-'. r+ )*()# a < r < b6 E*2 )#('-'

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

"# #$%& '&(#)&* #+ ,-%#(./&+ #( r )#$0+%& $#)

V (r) =030

(b3 a3

b+b2 r2

2+ a3

(r b)rb

), si a < x < b

1&)& +& )#2/3( r < a 4#$.-',-(#'-$ +& %)&5#.%-)/& 4#$4# #+ /((/%- 7&$%& ) #( 8 %)&5#.%-)/&$9:#(4)#'-$ ;0# #+ ,-%#(./&+ #$

V (r) = r

~E d~ = b~E d~+

ab

~E d~r + ra

~E d~

4# #$%& '&(#)& #+ ,-%#(./&+ )#$0+%& $#)?

V (r) =030

(b3 a3

b+b2 a2

2+ a3

(a b)ab

), si r < a

!

!"#$%&' ()

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+"$,-./01

'* C-)- #

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

"##$%&'(')*+ ,-. / ,0. #) ,1.2 +34#)#$+5 67#

E =a

r0(4)

89'&7')*+ #) r = d 4#)#$+5 67#

E =a

d0

:;+4+ %=*+ %+< >'*' >=&=)*=+)'$+5 #& >'$%+ #&?>4+ #5 4+#54'=B) i / 4#)*=='& #)4=&=)*'&>7&'< #& >'$%+

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!"#" $% &' ()*"+ ', -%#." ',/-01)-" '2 ', )20'1)"1 (', -),)2(1" &'13 24,"+ ."1 ," 0%20"

~E1 = 0, si 0 < x < a

54'1% (', -),)2(1" ', -%#." ',/-01)-" &' -%,-4,%1% (' )64%, #%2'1% -"#" ," 7)-)#"& '2 ,% .%10'

%8+ (' '&0% #%2'1% 0'2'#"& 94':

~E1 =a

x0i, si x > a

!"#$ %&'()*+($ #*$,-(+,$ #$* %& (+&+.,*$ ,%*%(/$

;2 ', )20'1)"1 (', -),)2(1" ', -%#." .1"(4-)(" ."1 /, &'13 24,":

~E2 = 0, si d a < x < d'1'#"& 94' % .%1% 42 -)'10" x94' -4#.,% 0 < x < d a+ ,% ()&0%2-)% (', -'201" (', -),)2(1" ('1'-7" % x &'13 d x+ (' '&0%#%2'1% ', -%#." ',/-01)-" 1'&4,0% &'1:

~E2 =a

(d x)0 i

?' '&0% #%2'1%+ %, &4.'1."2'1 ,"& -%#."& ',/-01)-"& "@0'2)("&+ >'#"& 94' ', -%#."& 1'&A,0%0'

'&:

~E =

a(dx)0 i 0 < x < a

( ax0 +a

(dx)0 )i a < x < d aax0

i d a < x < dB7"1% .%1% -%,-4,%1 ,% ()C'1'2-)% (' ."0'2-)%, V '201' ,"& -),)2(1"& ('@'#"& -%,-4,%1 ,% )20'61%,$ .%1% '&0" 40),)D%#"& ,% 01%$'-0"1)% #%& "@>)%: ', -%#)2" &"@1' ', '*' x:

V = d0

~E d~

= a0

~E d~+ daa

~E d~+ dda

~E d~

=

a0

a

(d x)0dx+ daa

(a

x0+

a

(d x)0 )dx+ dda

a

x0i

= (a0

( ln(d x)|a0)) + (a

0( ln(x) ln(d x)|daa )) + (

a

0(( ln(x)|dda)))

= (a0

ln(d ad

)) + (2a

0ln(

d aa

)) + (a0

ln(d ad

))

=2a

0(ln(

d aa

) ln(d ad

))

?' ("2(' &' "@0)'2'

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

V =2a

0ln(

d

a)

!

!"#$%&' ((

"# $%&'( '%)'*#+) $, )+$%( R $, #+ .*)+ ,&/0 '+).+$( '(1 *1+ $,1&%$+$ = 0(1 Rr )2 '(10 > 02 &%,1$( r #+ $%&/+1'%+ $, *1 3*1/( '*+#4*%,)+ $,# $%&'( + &* ',1/)( O5') "1'*,1/), ,# 3(/,1'%+# ,1 ,# 3*1/( P &(6), &* ,7,8OP = R95

#) :;+'%+ $(1$, &,

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

! "!#$! dq %& '! ()*&+)% &%"#,-,# ")+)

dq = (r) dA.)+) '! *&/%,*!* *& "!#$! *&(&/*& *& '! *,%0!/",! r 0&/*#&+)% 12& "!*! !/,'') *&' *,%") 12&")/%,*&+)% 0&/*#3 ,$2!' *&/%,*!* 4! 12& "!*! &'&+&/0) *& 5' &%0! ! ,$2!' *,%0!/",! r *&'"&/0#) *&' *,%")6 7& &%0! +!/! ")/%,*!#& ")+) dA &' 3#&! *& 2/ !'!+-#& *& !/"8) dr9

dA = 2rdr

7& &%0! +!/! )-0&/&+)%

dq = 20(r R)dr:' ()0&/",!' *& (#)*2",*) ()# &%0& !/,'') % &/0)/"&%9

dV =2k0(r R)dr

r2 +R2

;/0&$#!/*) *&%*& r = 0 8!%0! r = R9

V = 2k0

R0

(r R)drr2 +R2

= 2k0(

R0

rr2 +R2

dr R R0

1r2 +R2

dr)

= 2k0((r2 +R2

R0) ( ln(

r2 +R2 + r)

R0))

= 2k0(R2RR ln(R

2 +R) +R ln(R))

= 2k0(R2RR ln(R(

2 + 1)) +R ln(R))

= 2k0(R2RR ln(R)R ln(

2 + 1)) +R ln(R))

= 2k0(R2RR ln(

2 + 1))

= 2k0R(2 1 ln(

2 + 1))

!

V kQR

"#$# %&$#'( &) *#+&,-./) &,-#,+0/1# &' .23/) /) *#+&,-./) 43& *0#13-& 3,/ -/02/ *3,+3/) /

3,/ 1.'+/,-./ R 1& &))/5

! "/1/ &)&$&,+# 1& -/02/ dq &' ,&2/+.%/( )3&2# &) -/$*# &)6-+0.-# '&07 /+0/8&,+&5 9#0 '.$&+0:/(&) -/$*# &)6-+0.-# /*3,+/0/ &, 1.0&--.;, 1&) &/ &' *#'.+.%/( &'+/ '& $#%&07 +/$>.6, =/-./ &) -&,+0# 1&) 1.'-#5

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!"#$%&' ()

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!

!"#$%&' ()

"#$%& '() *)+) ,&-.&,/0%.*) 1& %)1.# R ,& 2.&(& '() 1.,2%.$'*.3( 1& *)%4) '(./#%-& 5 6)7*'7&8'* 97 +#2&(*.)7 &7&*2%#,2:2.*# ) 7# 7)%4# 1&7 &;& Z< +)%) z > 05#* 97 *)-+# &70*2%.*# ) 7# 7)%4# 1&7 &;& Z< +)%) z > 05+* 97 *)-+# &70*2%.*# &( &7 +'(2# O5

,"$-+./01

'* 6#(,.1&%)-#, &7 #%.4&( &( O5 =)%) *)7*'7)% &7 +#2&(*.)7 >'& +%#1'*& 2#1) 7) ,'+&%*.& ,#$%&'( +'(2# zk 2#-)%&-#, '( &7&-&(2# 1& *)%4) dq 1& &77) @ *)7*'7)%&-#, &7 +#2&(*.)7 +%#1'*.1#*#-#8

dV =kdq

r

A#(1& r &, 7) 1.,2)(*.) 1&,1& dq B),2) zk5=#,2&%.#%-&(2&< +)%) *)7*'7)% &7 +#2&(*.)7 2#2)7< .(2&4%)-#, dV ,#$%& 2#1) 7) ,'+&%*.&56#-# C.-#, &( 7) )@'1)(2.) D< 7) *)%4) dq 7) +#1&-#, &E+%&,)% *#-# &7 +%#1'*2# &(2%& 7)1&(,.1)1 ,'+&%*.)7 1& *)%4) +#% &7 )%&) >'& #*'+) &,) *)%4) >'& 77)-)-#, dA @ C.-#, >'&

dA = R2sin()dd

F) 1.,2)(*.) r &( &,2& *),# 7) #$2&(&-#, '2.7.G)(1# &7 2%&-) 1&7 *#,&(#8

r2 = R2 + z2 2zR cos()H,I< &7 +#2&(*.)7 +%#1'*.1# +#% dq &,

dV =kR2 sin()ddR2 + z2 2zR cos()F'&4#< .(2&4%)-#, &,2) &E+%&,.#( &7.4.&(1# 7#, 7.-.2&, 1& .(2&4%)*.#( 1& 2)7 /#%-) 1& %&*#%%&%

7) ,'+&%*.&8

V =

20

pi2

kR2 sin()ddR2 + z2 2zR cos()9,2) .(2&4%)7 ,& +'&1& %&1'*.% )

V = 2kR2 pi2

sin()R2 + z2 2zR cos()

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

"#$# $%&'()%$ %&*'+ ,%-%.'& -%$ /#(/0(#$ (# 12*%3$#( ,% (# 4'$.#5sin

A+B cos()d

6#/%.'& %( /#.-1' ,% )#$1#-(% u2 = A+B cos()7 8% %&*# .#2%$# d = 2uduB sin() 7 9 #&1sin

A+B cos()d = 2

B

1du = 2

Bu

:'.' u =A+B cos()5

sin A+B cos()

d = 2B

A+B cos()

"#$# %( /#(/0(' ,% ;'*%2/1#( *%2%.'& %%.;(?#2,' %2 (#12*%3$#(5

sin R2 + z2 2zR cos()d =

1

zR

R2 + z2 2Rz cos()

@A'$#+ ;',%.'& /#(/0(#$ %( 12*%3$#( &12 ,1/0(*#,%&5

V (z) =2kR2

zR(R2 + z2 2Rz cos()

pi2

) =2kR

z(z +R

R2 + z2)

! @A'$#

!

~E(z) = 2kR2(

R2 + z2 Rz2R2 + z2

)k

! "#$%& '(# )* #+,-#&.%/ */0#-.%- #&0* ./1#/.1* #/ z = 03 4./ #$5*-6%7 ,%1#$%& *8#-8*-/%& *O 0*/0% 8%$% '(#-*$%& 9 8*)8()*- #) :*)%- 1#) 8*$,% #);80-.8%3CDE,.0*) ,*-* 8*)8()*-)%=

2kR2 lmz0

(1 R

z2+R2

z2) = 2kR2 lm

z0((1 R

z2+R2)

(z2))

= 2kR2 lmz0

(R(z2 +R2)

3

2

2z)

= 2kR2 12R2

= k

F# #&0* $*/#-*

~E(0) = kk

! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '

!"#$%&' (

!"#$%&!'()

!"#$%&' ()

! "#!$!$ %&' !'()*+' ,&$%-,"&*+' %! *+%#&' r1, r2 . ,+*/+' q1, q2 '!0+*+%+' 0&* -$+ /*+$ %#'1"+$,#+ d >> r1, r22 # +34+' '! ,&$!,"+$ + "*+5)' %! -$ ,+46! ,&$%-,"&* 7%!'0*!,#+46!8 9-!'#*5! ':6& 0+*+ "*+$'0&*"+* ,+*/+ %! -$+ + &"*+;8 !$,-!$"*! 6+' %!$'#%+%!' %! ,+*/+ '-0!*,#+6!'

%! ,+%+ -$+ 7!$ (-$,#:$ %! 6+' 5+*#+46!' ,&$&,#%+'; -$+ 5!= 9-! !6 '#'"!3+ +6,+$=+ !6 !9-#6#4*#&2

*"$+,-./0

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*!(!*!$,#+ %!6 0&"!$,#+6 !6 #$$#"& ! #/-+6+* !6 0&"!$,#+6 + ,!*& +66@8 !' %!,#*8 V (+) = 02 A'@8!6 0&"!$,#+6 '&4*! 6+' '-0!*,#!' %! 6+' !'(!*+' ,&$%-,"&*+' !'

V1 = kq1r1, V2 = k

q2r2

%&$%! B!3&' '-0-!'"& 9-! !6 0&"!$,#+6 %! -$+ !'(!*+ $& !' +(!,"+%& 0&* !6 %! 6+ &"*+ 7& 3?' 4#!$

!' +(!,"+%& %! (&*3+ %!'0*!,#+46!;8 %+%& 9-! !'"?$ 3-. 3-. '!0+*+%+8 !' %!,#*8 d >> r1, r28. "+30&,& !6 ,+30& %! -$+ !'(!*+ *!%#'"*#4-.! 6+ ,+*/+ %! 6+ &"*+2 C&* ,&$'!*5+,#:$ %! ,+*/+8

'# (q1)f , (q2)f '&$ 6+' ,+*/+' !$ ,+%+ !'(!*+ -$+ 5!= +6,+$=+%& !6 !9-#6#4*#&8 "!$!3&' 9-!

q1 + q2 = (q1)f + (q2)f

A6 +6,+$=+* !6 !9-#6#4*#&8 !' %!,#*8 ,-+$%& %!D+ %! B+4!* "*+$'(!*!$,#+ %! ,+*/+' !$"*! 6+' !'(!*+'8

"!$!3&' 9-! 6+ %#(!*!$,#+ %! 0&"!$,#+6 !6),"*#,& !$"*! +34+' !' $-6&8 !' %!,#*8

V1 = V2

k(q1)fr1

= k(q2)fr2

4(r1)2(1)fr1

=4(r2)

2(2)fr2

= (2)f(1)f

=r1r2

>! !'"& 5!3&' 9-! !$ /!$!*+68 6+' *!/#&$!' !$ 6+ '-0!*,#! %! -$ ,&$%-,"&* ,&$ 3!$&*

*+%#& %! ,-*5+"-*+ 70-$"+'; ,&$,!$"*+$ -$+ 3+.&* %!$'#%+% '-0!*,#+6 %! ,+*/+8 0&* 6& ,-+6 !6

EF

!"#$%&' () '*+% $',-.

!"#$% &'(!)*+!% !&*!" ,& &''"- .&/ -0 &1)&*+%*2 &- #3- 40&*)& 50& &/ *&6+%/&- !%/ #&/%* *",+%

,& !0*7")0*"8

9-"/,% '" *&'"!+:/ &/!%/)*"," &/ '" &!0"!+:/ ,& !%/-&*7"!+:/ ,& !"*6"; %

!

!"#$%&' ()

"#$ %&$%&'&$ ($)*'+%&$ %#,-.%/#'&$ %#,%*,/'+%&$ -( '&-+#$ a < b /+(,(, 0#/(,%+&1($ V1 2 V23'($0(%/+4&5(,/(6

'* 7&1%.1( 1& %&'8& -( %&-& ($)('&6

#* 9:#'& /'&(5#$ -($-( (1 +,,+/# .,& %&'8& Q -+$/'+,/# /'&

! !"#$%&' () '*+% $',-.

~E =Q1 +Q240r2

"#$ %&'#( )#*%+#& ,-,./-0 %/ )#'%$,1-/ %$ b2

V2 = b

Q1 +Q240r2

dr =Q1 +Q240b

3% %&'- +-$%0-( /- ,-04- *% ,-&,-0- %5'%01#0 &%062

Q2 = 40b(V1 + (V2 V1) aba b)

! 7-0- ,-/,./-0 %/ '0-8-9# '#'-/( ,#$&1*%0-0%+#& :.% /-& ,-04-& :.% ,#+)#$%$ /- ,-&,-0- *%

0-*1# c > b /-& %&'-+#& '0-;%$*# *%&*% %/ 1$$1'# =-&'- .$- *1&'-$1,- c> "/-0-+%$'%( %/ '0-8-9#$%,%&-01# )-0- /- ,-*- ,-04- dq ,#$'%$1*- %$ /- ,-&,-0- &%0- %/ +1&+#> ?&'% '0-8-9# &% ,-/,./-,#+#

dW = dqV (c)

3% %&'- +-$%0-( %/ '0-8-9# '#'-/ &%06 &1+)/%+%$'%

W = QV (c)

> @# :.% '%$%+#& :.% =-,%0 %& ,-/,./-0 %/ )#'%$,1-/ %$ r = c :.% %& *10%,'-+%$'%

V (c) =Q1 +Q240c

A&1(

W =Q(Q1 +Q2)

40

!

!"#$%&' ()

"#$% &'%&'#'% $%($#)&'% &*+,-&.*#'% &*+&$.#)&'% /-0 ,$12','% 3*%$$+ #',)*% a4 b 0 c4 #$%3$&.)56'/$+.$4 %)$+,* a < b < c7 8+)&)'1/$+.$ 1' &'%&'#' )+.$#)*# $%.' ,$%&'#2','4 1' ,$1 /$,)* 3*%$$-+' &'#2' .*.'1 +$2'.)6' Q 0 1' $9.$#+' ,$ &'#2' .*.'1 3*%).)6' +Q7

*'+ :+&-$+.#$ $1 3*.$+&)'1 $1$&.#)&* $+ &',' -+' ,$ 1'% &'%&'#'% &*+,-&.*#'%7

*#+ ;) 1'% &'%&'#'% )+.$#)*# 0 $9.$#)*# %*+ &*+$&.','% /$,)'+.$ -+ '1'/?-'1 $% 1' &'%&'#' $+ &',' -+' ,$ 1'% &'%&'#'%A

,"$-./"0

! !"#$%&' () '*+% $',-.

!"#$%&' ()

"#$ %$&%$'$ (&)*'+%$ %,#-.%/,'$ -( '$-+, a 0 (&1(&,' a %,#/+(#( %$'2$ #(/$Q3 4( -+&/'+5.0(.#$ %$'2$ q (# (6 7,6.8(# +#/('+,' -(6 %$&%$',# -( '$-+, a9.# $+&6$#/( (# 6$ 1$'/( +#/('+,' -(6%$&%$',# +81+-( :.( (&/$ -(#&+-$- -( %$'2$ &( 1$&( $6 %,#-.%/,';3

!

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q =

(r)dV

= 4

a0

60Kr3

a44r2dr

=240K

a4r6

6

a

0

)

= 4a20K

:) ),-& +&/)%&

K =q

40a2

",;' 7$()+$, ),1%.*.% 0& ()/,.(&( () 1&%6& )/ a

",. )0 7$-)/1.&0 7&%& 5/& %)6.$/ )/ 45) r > a ),9

V (r) =

~Eextd~r = Q+ q40

r

1

r2dr =

Q+ q

40r

"#$%&' 7&%& 0& %)6.$ ./-)%.$% Er < aF )0 7$-)/1.&0 0$ 1&0150&+$, 1$+$9

! !"#$%&' () '*+% $',-.

V =

~E d~r = b~Eext d~r +

rb

~Eint d~r, si r > a

V (r) =

~E d~r

= b~Eext d~r +

rb

~Eint d~r

=Q+ q

40a q

20a60(r5 a5)

"#$% #& '$&(& )*&

V (r) =Q+ q

40a q

20a60(r5 a5), si r < a

!

!"#$%&' ()

"#$%&$' '$ %#()* '$+%,-.%* )-*/&%./* )*- &0# /.1,-.2&%.30 /' %#-4# (r) ,#$ 5&' '$ )*,'0%.#$5&' '1,# /.1,-.2&%.30 )-*/&%' '1,# /#/* )*-

V (r) = qer

4r20

6&'4*7 '0%&'0,-' $# /.1,-.2&%.30 /' %#-4# (r)8

*"$+,-./0

9'0'(*1 '$ )*,'0%.#$ )-*/&%./* )*- $# /.1,-.2&%.30 /' %#-4# (r) : ;'(*1 # $# ;'< 5&' '1,')*,'0%.#$ '1,# '0 =&0%.30 /' r7 )*- $* 5&' '$ )*,'0%.#$ ,.'0' 1.(',-># '1=+-.%#8 ?@*-#7 &,.$.

! !"#$%&' () '*+% $',-.

!"#$%&' (

!"#$"%!'$%

!"#$%&' ()

! "#!$%"' "()*!$'("# $+ ',$(# a - ",'., Q +/ "#,0(,) "#! %! ",/",'#! "()*!$'("# $+ .'#/#'$+/1'+"(,2)+3 ',$(# b > a - ",'., Q4 5!"%+!&'+ ), ",1,"(&,!"(, $+ +/&+ ",1,"(' "()*!$'("# /(/% )#!.(&%$ +/ 4

*"$+,-./0

6+!+7#/ 8%+ ), ",1,"(&,!"(, /+ $+!+ "#7# C = QV 4 :,2+7#/ 8%+ ), ",'., $+ +/&+ ",1,"('+/ Q 1#' )# 8%+ 2,/&, +!"#!&',' ), $(;+'+!"(, $+ 1#&+!"(,) 1,', #2&+!+' )# 8%+ 2%/",7#/4

! !"#$%&' () '*+,*-!+'.,-

"# $%&'$#()$* $#(+,-.,+* %$ /-0.* 1&+ 2$3. +*($ *-0+(,)$ +4 &3. 6., 4$* ($6$* '+4 7-4-#',. +*

#&4.8 9., 4. ($#(.:

=

S

~E ndS =manto

~E ndS = E2r

;$ 7$,# *+,A #&4.8B$ 7$47&4$'. +4 7$06. +4>7(,-7. '+2+0.* +4+

!"#$%&' ((

!"#$%&'(' )# $%$*'+, -"(+,&" ."( &"$ $).'(0%'$ 0%12#&(%0,$ 0"#&)0*"(,$ &' '3'$ &' $%+'*(2,

.,(,1'1"$4 ,+5"$ &' (,&%" R4 1,(6" %##%*" 7 $'.,(,&"$ ."( )#, &%$*,#0%, &' 2d '#*(' $)$ '3'$40"# d >> R8 9% $"# 0,(6,&"$ , )#, &%-'('#0%, &' ."*'#0%,1 V04 0"# 0,(6,$ Q 7 Q4 '#0)'#*('1, 0,.,0%&,& ."( )#%&,& &' 1"#6%*)& ,.(":%+,&, .,(, '1 $%$*'+, &' 0%1%#&("$8

)"$*+,-./

;"&'+"$ '#0"#*(,( '1 ."*'#0%,1 )$,#&" '1 0,+." '1,&" ?)' '$*@# +)7 1'3"$4 '$ &'0%(4 d >> R4 ."&'+"$ 0"#$%&'(,( ?)' '10,+." &' )# 0%1%#&(" #" ('&%$*(%5)7' 1, 0,(6, '# 1, $).'(0%' &'1 "*(" A."( '11"4 ."&'+"$

0"#$%&'(,( '1 0,+." '1' '$*, -"(+,4 1, &%-'('#0%, &' ."*'#0%,1 '#*(' 1"$ .)#*"$ +'#0%"#,&"$ '$

V0 = V =

~E(~r) d~r

= d+RdR

(Rx

0(d+ x) R x0(d x)

) dxx

= R0

d+RdR

(1

d+ x+

1

d x)dx

= R0

ln

(x+ d

d x)

d+R

dR

=2R

0 ln

(2dRR

)

.'(" = Q(2R)L 4 ."( 1" ?)'

V = Q0L

ln(2dRR

)= |V | = Q

0L ln

(2dRR

)

;'("4 *'#'+"$ ?)' 1, 0,.,0%*,#0%, ."( )#%&,& &' 1"#6%*)& '$ C = Q|V |L 4

= C = 0ln(2dRR

) = 0ln(2dR

)

!! !"#$%&' () '*+,*-!+'.,-

"#$% d >> R& '()$*(% +$, -#$ ./ 0/"/012/301/ )$"$3)$ )$ ./ 4$(*$2,5/ )$. %1%2$*/ 6 )$.*$)1( $3 $. -#$ $%27&

!

!"#$%&' ()

"#$%&$' $# (&')*# '+,)' $#- .$#%#- /' &+ %0+/'+-#/0) /' .$#%#- .#)#$'$#- /' 1)'# A 2 -'.#)#%34+d5 6#)# '-,07 #$'8' &+ .09&3,0 :3++3,'-3&3'+,'- %#-0-@

#= A$ %0+/&%,0) '-,1 #3-$#/0 %0+ %#)># Q

B= A$ %0+/'+-#/0) '-,1 %0+'%,#/0 # &+# B#,')C# /' /3(')'+%3# /' .0,'+%3#$ V05

*"$+,-./0

#= D'+'

!" !"#$%&' () '*+,*-!+'.,-

#$% &$ '() &* +()%,* )- &* ./-.* 01$.$ 2)3) -)%45

6) 2)7* *& &)18$% )91$98%*% )-8*- 2$- ):#%)-/$9)- 1$9-/2)%*92$ '() &* +()%,* #$% (9/2*2

2) ;%)* -$3%) &* -(#)%1/) 2) (9 1$92(18$% )-

~F/A = 2

20n

!"

!"#$%&' ()

#$%&'()*) +%, &+-)*/') /$%(+/0$*, )&12*'/, () *,('$ a /$%/2%0*'/, , $0*, () *,('$ b3 /$%a < b 4,56,& &+-)*/')& 1$*5,% +% /$%()%&,($*78 9% ): )&-,/'$ )%0*) ,56$& /$%(+/0$*)& &)'%0*$(+/) +% ,*5,; %)+0*$8 @' '%'/',:5)%0) :, &+-)*/') '%0)*'$* 0')%) /,*A, +Q > :, )?0)*'$* Q3('&/+0, B+2 $/+**) /$% :, /,-,/'0,%/', (): &'&0)5, :+)A$ B+) &) -$%) ): ,*5,; )%/+)%0*)

:, /,-,/'0,%/', '%'/',: > %,: (): &'&0)5,8

*"$+,-./0

C$()5$& )%/$%0*,* ): /,5-$ ):2/0*'/$ )%0*) :,& ($& &+-)*/')&3 ,%0)& B+) &) -$%A, ): ,*5,; () D,+&&8 E)%)5$& B+) ): /,5-$ 0')%) &'5)0*F, )&12*'/, > G, )% ): &)%0'($ ()

r3 )& ()/'*3 ~E(~r) = E(r)r8 E$5,%($ /$5$ &+-)*/') () '%0)A*,/' () D,+&&8 9% )1)/0$3 +&,%($ /$5$ &+-)*/') () '%0)A*,/' ()%0*$ (): ,*5,; b8 C,*, )%/$%0*,*): /,5-$ )% d < r < b > a < r < c +&,5$& ): 5'&5$ -*$/)('5')%0$ B+) )% ): /,&$ ,%0)*'$*3-$* 0,%0$3 $60)%)5$& B+) ): /,5-$ ):2/0*'/$ )&

~E(~r) =

{~0 b < r, a > r

Q40

rr2

c < r < b, a < r < c

!" !"#$%&' () '*+,*-!+'.,-

#$%&'( %) *'+%,-./) %,+0% )/1 1$*%0-.%1 %3+%0,/ % .,+%0,/ %1

V2 =

= ab

~E(~r) d~r = ba

~E(~r) d~r

=

ca

~E(~r) d~r + dc

~E(~r) d~r + bd

~E(~r) d~r

=

ca

~E(~r) d~r + bd

~E(~r) d~r

=Q

40

( ca

dr

r2+

bd

dr

r2

)

=Q

40

(1

a+1

d 1c 1b

)

4'0 +/,+'( )/ -/*/-.+/,-./ ,/) %1

Cf =Q

V2 =40(

1a +

1d 1c 1b

)4'5%6'1 7%0 8$% Ci < Cf 5% )' 1.&$.%,+%9 :%,%6'1 8$%

c < d = 1d 1c< 0 = 1

a 1b+1

d 1c1'%','1+ ()+ &'&.)01 &)*C

Ei =1

2C1V

2

@&,*'.1 )% D-%,'A% () Q1?

Ei =1

2

Q21C1

BF$*17 ,)**10$& )+ '%.)**-6.$* S2 01%.)%')%($ S1 ,)**1($4 81 ('D)*)%,'1 () 6$.)%,'1+ 61*110=$& ,$%()%&1($*)& &)*C +1 0'&01? V 4 G 1&>7 &'06+)0)%.) +1& ,1*31& &)*C%?

Q1 = V C1

Q2 = V C2

81 )%)*3>1 ()+ &'&.)01 &)*C +1 16$*.1(1 6$* C1 < C2? Ef =12C1V

2 + 12C2V2

B&>7 )+ ,10='$ () )%)*3>1 &)*C?

Ef Ei = 12C2V

2

#*

H' 1=*'0$& S1 1%.)& () ,)**1* S2 .)%(*)0$& /-) +1 ('D)*)%,'1 () 6$.)%,'1+ V

! !"#$%&' () '*+,*-!+'.,-

Q1C1

=Q2C2

"# $% '()#(*+, -)' ,'('.*%#/$. .' $0%*',' 1$& '2 1&*,(*1*$ 3' ($,.'&4#(*+, 3' (#&5#6 '2

($,3',.#3$& C1 -)' '.%# *,*(*#2/',%' (#&5#3$ ($, Q1 1$& 2$ %#,%$ 2# .)/# 3' 2#. ,)'4#.(#&5#. %',3&7, -)' .'& *5)#2 # '.%#8

Q1 +Q2 = Q1

9' '.%# /#,' ($/0*,#,3$ #/0#. '()#(*$,'. %','/$. -)'

Q1 =C1

C1 + C2Q1

Q2 =C2

C1 + C2Q1

9' '.%# /#,' 1$3'/$. '.(&*0*& 2# ','&5:# 3'2 .*.%'/# -)' .'&7 2# #1$&%#3# 1$& (#3# ($,;

3',.#3$&8

Ef =1

2

Q21C1

+1

2

Q22C2

=1

2(

C1C1 + C2

)2Q21C1

+1

2(

C2C1 + C2

)2Q21C2

= Q21(1

2

C1(C1 + C2)2

+1

2

C2(C1 + C2)2

)

!"#$%&' ()

!" #$"%"% #&"' ()%*"%'+*)&"' *" (+,+($*+*"' C1 = C0-C2 = 2C0 . C3 = 4C0- . '" ()%"(#+% +/%+ 0+#"&1+ 2/" "%#&"3+ /%+ *$4"&"%($+ *" ,)#"%($+5 V0- ()6) 6/"'#&+ 5+ 3/&+8'* 9+5(/5" 5+ (+&3+ 2/" +*2/$"&" (+*+ ()%*"%'+*)& 6$"%#&+' "5 $%#"&&/,#)& S "'#+ +0$")8#* 9+5(/5" 5+ (+&3+ 2/" +*2/$"&" (+*+ ()%*"%'+*)& (/+%*) "5 $%#"&&/,#)& S "'#+ ("&&+*)8+* :;)&+ '" *"'()%"(#+ 5+ 0+#"&1+ *"5)' 0)&%"' . < *" 5+ 3/&+ ()% "5 $%#"&&/,#)& S ("&&+*)8=9/>%#) ?+&1+% 5+' (+&3+' "% (+*+ ()%*"%'+*)&@

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?$("?"&'+8 9+5(/5" 5+ (+&3+ "% 5)' ()%*"%'+*)&"' 5/"3) *"5 %/"?) "2/$5$0&$) "5B(#&$()

-"$.+/012

'* !$ "5 $%#"&&/,#)& S "'#+ +0$")- ')5) '" (+&3+&+ "5 ()%*"%'+*)& C38 C/"3)- ()6) 5+ *$4"&"%($+*" ,)#"%($+5 "' V0 . '+0"6)' '/ (+,+($*+*- 5+ (+&3+ +56+("%+*+ '"&>D

Q3 = V0C3

9)6) C3 = 4C0- )0#"%"6)'Q3 = 4C0V0

#* :5 ("&&+& "5 $%#"&&/,#)& S '" (+&3+&+% #)*)' 5)' ()%*"%'+*)&"' ()6) 6/"'#&+ 5+ 3/&+

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! !"#$%&' () '*+,*-!+'.,-

"#$%&" "'&(" ) ! "% V0* +,"-. /.0. C1 ) C2 "%&1' "' %"($"2 34 %,04 5" 3.% 6.3&47"% 5" /454/.'5"'%45.( %"(1 V08

V1 + V2 = V0

9"(.2 :.( 5"'$/$" "%&4 04'"(4

Q1C1

+Q2C2

= V0

?5"01%2 "3 /.'5,/&.( =," &.04 34 :34/4 5"("/@4 5" C1 ) 34 :34/4 $A=,$"(54 5" C2 "%&4 '",&(.2 43/4(-4(%" %" &"'5(1 =," "%&" /.'5,/&.( %" :.34($A4 /.' /4(-4% Q1 ) Q22 :"(. :.( /.'%"(64/$

!

"#$# %&$#' (&)&$#' ! &*+,*-#)&' . ! %,/-,01&' 2#/ 1# (,)(# 2#3&$#' &)*#)(/,/ 1#' %,1#/&'

3& 1,' *,/4,' 3& *,3, *#)3&)',3#/5

6& 1, &*+,*-7) 8 9 '& (-&)&

Q2 = Q1 +Q2 Q16& 1, &*+,*-7) 8:9 '& (-&)&

Q3 = Q3 Q1 +Q1;(-1-

Q1 = Q2 =2

3C0V0

Q3 = C3V0 = 4C0V0

;(-1-

! !"#$%&' () '*+,*-!+'.,-

!"#$%&' (

!"#"$%&!$'(

!"#$%&' ()

! "#!$! %$ &'$(!$)*('+ (! ,-*&*) ,*+*-!-*) (! *+!* A . )!,*+*'$ d/ . )! &*+0* (! 1'(' "*-2%! )% ,-*&* )%,!+#'+ *(2%#!+! &*+0* Q/ . -* #$3!+#'+ Q4 5*$"!$#!$(' !- &'$(!$)*('+ *#)-*('/)! #$"+'(%&!$ 6-'2%!) (! 1*"!+#*- (#!-7&"+#&'/ (! ,!+1#"#8#(*(!) 1 = 20 . 2 = 40 9*)"* -*1#"*( (!- &'$(!$)*('+/ &'1' 1%!)"+* -* 0%+*4

'* ;!"!+1#$! -* $%!8* &*,*(*( (!- &'$(!$)*('+/ &'$ -') 6-'2%!) (#!-7&"+#&') !$ )% #$"!+#'+4

#* &'$(!$)*('+!) &'1' 1%!)"+* -*

0%+*?

@@A

! !"#$%&' () *+,&, $-+ '.

"#$%' ()*(+*), *)' ()-)(.$)$%' $% ()$) (#/$%/')$#, $% 0#,&) $.,%(1)2

C1 =A/2

2/3d1 =

3A

2d0

C2 =A/2

1/3d2 =

6A

2d0

C3 =A/2

d0 =

A

2d0

3+%4#5 -),) ()*(+*), *) ()-)(.$)$ $%* '.'1%&) 6)'1) +1.*.7), *)' ,%4*)' -),) (#&6./)(.#/%'

$% (#/$%/')$#,%'8 3#' (#/$%/')$#,%' C1 9 C2 %'1)/ %/ '%,.%5 $% %'1) &)/%,) *) ()-)(.$)$%:+.;)*%/1% -),) %'1#' < (#/$%')$#,%' (+&-*.,)2

1

C eq=

1

C1+

1

C1=

13A2d 0

+1

6Ad 0

=5

6

d

A0

=% *# :+% '% #61.%/% :+% C eq =65Ad 08>?#,)5 C eq !"# $ %#' '( )($ C3 * %(& '( "#$"( '# )#%#)+,#, -.+/#' $" ! '# !.0# , #01#! )#%#)+,#, !2

Ceq = Ceq + C3 =

6

5

A

d0 +

A

2d0 =

17

10

A

d0

! 3# %& ! $)+# , ' ,+ ' )"&+)( 4#&5 -. '# )#&6# Q * Q , %(!+"#,# $ '#! %'#)#! ! & ,7+"&+1.*#$ , .$# , " &0+$#,# 0#$ # , $+&9 : $+0(! q3 '# )#&6# -. #,-.+ & ' )($, $7!#,(& C3 * q '# )#&6# $ '(! )($, !#,(& ! C1 * C2; ! '# 0+!0# $ #01(! )($, !#,(& ! %(& !"#& $ ! &+ < )(0( 0. !" '# 6.

=(0( '# )#&6# ! )($! &/# ! " $,&5 '# ).#)+($2

q1 + q2 = Q (1)

>(& ("&( '#,(? ' )+&).+"( ! ) &,( * %(& '( "#$"(2

V3 V1 V2 = 0q3C1 qC2 qC3

= 0 (2)

: ;@

!

q( 1C3

+1

C1+

1

C2) +

Q

C3= 0

"##$%&'(')*+ &+, -'&+.#, *# &', /'%'/01'/0', #)/+)1.'*', #) &' %'.1# '23 .#,4&1' 54#6

q =12

17Q

q3 =5

17Q

! !"#$%&' () *+,&, $-+ '.

!"#$%&' ()

"# $%&$%'(# )&*+',$- .%$,/- 0,)1+$2',$-3 0) '%0,-& ,#2)'#- a 4 )52)'#- b3 )& &-.)2,0- 6%7- 8#$%.9- )52)'#- :8) 9'-08$) )# +1 8# ;)$2-' 9-1%',/%$,(#

~P (~r) =k

rr

0-#0) k )& 8#% $-#&2%#2) %'6,2'%',% 4 r 1% 0,&2%#$,% %1 $)#2'- 0)1 $%&$%'(#< =-2) :8) #- >%4$%'?% 1,6') )# )&2) 9'-61).%< @#$8)#2') )1 $%.9- )1+$2',$- 0) 9-1%',/%$,(# )# 2-0- )1 )&9%$,-> b, a3 1)-.4- 6- Q 7/- 8/-)- &'%$.'- $, 8*,&%&9- 1 /- $2%$.'- '$4-%&9-:; b

8*. /* ?($ /- +&E$.$')&- +$ 8*%$')&-/ $'%.$ 8/-)-, $,

V = Q20L

ba

dr

r=

Q

20Lln(b/a) = C = 20L

ln(a/b)

$, /- )-8-)&%-')&- +$/ )*'+$',-+*. )&/0'+.&)*;

I/ 6-. 9$.%&)-/A$'%$ $/ )*'+$',-+*.3 $/ >/*?($ +&$/@)%.&)* +$,)&$'+$ 8*. $/ 8$,* ?($+-'+* ('-

8-.%$ +$/ )*'+$',-+*. ,&' +&$/@)%.&)*3 8*. /* ?($ /- )-8-)&%-')&- +$/ ,&,%$A- )-A>&-; #$- x /-+&,%-')&- ?($ $/ >/*?($ +&$/@)%.&)* +$,)&$'+$3 A$+&+- +$,+$ /- 8-.%$ ,(8$.&*. +$/ )*'+$',-+*.

D-,%- /- 8-.%$ ,(8$.&*. +$/ +&$/@)%.&)*3 1 x $/ 9$)%*. ('&%-.&* $' $/ ,$'%&+* A$')&*'-+*; I/+$,)$'+$. x $/ +&$/@)%.&)*3 ,$ E*.A-' +*, )*'+$',-+*.$, $' 8-.-/$/* +$ )-8-)&%-')&-,

C1 =20x

ln(b/a)C2 =

2(L x)ln(b/a)

8*. /* ?($ /- )-8-)&%-')&- +$/ )*'+$',-+*. )*A8/$%* $,

C(x) = C1 + C2 =20x

ln(b/a)+2(L x)ln(b/a)

=2

ln(b/a)(L x( 0))

8-.- 0 < x < LJ- $'$.40- 8*%$')&-/ -/A-)$'-+- $' $/ )*'+$',-+*. $,

U(x) =1

2

Q2

C(x)

#->$A*, ?($ /- E($.B- ,*>.$ $/ +&$/@%.&)* $6$.)&+- 8*. $/ )*'+$',-+*.3 )(-'+* /- )-.4- $,

)*',%-'%$3 $,

~F = ~U 3 8*. /* ?($ /- E($.B- %*%-/ ,*>.$ $/ >/*?($ +&$/@)%.&)* $,

~F = dUdx

x+mgx = ~0

!"

#$%&'( )*+,&)&-+ ./0 10 ,020 )/'3$&4 0+ 0$ 0./&$&24&*5 6*4 %(+%*7

mg =dU

dx

= 12

(Q

C

)2 dCdx

=Q2ln(b/a)( 0)4 (L x( 0))2

= (L x( 0))2 = Q2ln(b/a)( 0)

4mg

|L x( 0)| = Qln(b/a)( 0)

4mg

(L x( 0)) = Qln(b/a)( 0)

4mg

= x = L( 0)

Q

( 0)

ln(b/a)( 0)

4mg

8020'*1 (3$&)(4 $( 401%4&))&-+ 0 < x < L ( ('2(1 1*$/)&*+015 90+0'*1 34&'04('0+%0 ./0

x+ =L

( 0) +Q

( 0)

ln(b/a)( 0)

4mg< L

= Q( 0)

ln(b/a)( 0)

4mg< L

(1

( 0))= L0

( 0) < 0

$* )/($ 01 /+( )*+%4(,&))&-+ 3/01

Q(0)

ln(b/a)(0)

4mg > 0

6*4 %(+%*7 $( #+&)( 3*1&)&-+ ,0 0./&$&24&* 3*4 $* 34*+%* 3*1&2$0 01

x0 =L

( 0) Q

( 0)

ln(b/a)( 0)

4mg

:3$&)(+,* $(1 401%4&))&*+017 %0+0'*1 ./0

x0 =L

( 0) Q

( 0)

ln(b/a)( 0)

4mg> 0

= L > Q

ln(b/a)( 0)

4mg

01 /+( )*+,&)&-+ ./0 ,020 )/'3$&4 0$ $(4;* ,0$ )*+,0+1(,*4 3(4( ./0 0 $( #$%&'( 401%4&))&-+ 01

!" !"#$%&' () *+,&, $-+ '.

x0 =L

( 0) Q

( 0)

ln(b/a)( 0)

4mg< L

= L < Q0

ln(b/a)( 0)

4mg

#$%&'( () $(*% +,*)&% -'./)-)0. $( -'&1%*)2,$ -'. ,% %.*$3)'34 5%3% 67$ $8)(*% 1'()-)0. /$

$67),)23)' /$2$ -7&1,)3($ ()&7,*9.$%&$.*$ 67$

Q

ln(b/a)( 0)

4mg< L

r < a1 B) $*+$,) /$ ,)/&. ) $* -.'/(-%.,) 0 2., 6. %)'%. '. F)0 -):2. $6$-%&,-. $' *( &'%$,&.,>~E = 0, si r < a

D$ $*%) :)'$,) $6 3$-%., /$*26)@):&$'%. *$,5>

~D = 0 ~E = 0, r < a

a < r < 2a1 ?' $*%$ -)*. (%&6&@):.* 6$0 /$ 1)(**>S

~E ndS = qenc0

G*H4 /)/) 6) *&:$%,H) $*+7,&-) 0 I($ 6) -),1) $' 6) $*+$,) -.'/(-%.,) $* q .;%$'$:.* $6 ,$*(6%)/./$ *&$:2,$>

~E =q

40r2r, si a < r < 2a

J),) $6 3$-%., /$*26)@):&$'%.>

~D = 0 ~E =q

4r2r, si a < r < 2a

2a < r < 2, 5a1 G26&-):.* 6$0 /$ 1)(** 2),) /&$67-%,&-.*>S

~D ndS = qenc

!" !"#$%&' () *+,&, $-+ '.

#$ %&'()* +&,-$./.01&2() &, -.*.$&$) .$ '.0-) &$3'(*1') 456& &, *.+1.$78 9& &,(. 0.2&*.

~D =q

4r2r

:;)*.< ,.=&0), 56&

~D = ~E< +)2+& &, $. -&*01(1%1+.+ +&$ +1&$3'(*1') 56& %.*1. &2 >62'1?2+& r8 9& &,(. 0.2&*.< &$ '.0-) &$3'(*1') ,&*@A

~E =q

40r2(1 +ra)r, si 2a < r < 2, 5a

r > 2, 5a B(1$1/.2+) $&C +& D.6,, )=(&2&0), +1*&'(.0&2(& 56&A

~E =q

40r2r

~D =q

4r2r

!" #$ %&'()* -)$.*1/.'1?2 &E1,(1*@ &%1+&2(&0&2(& ,)$) &2 $. *&D1?2 +)2+& ,& &2'6&2(*. &$ +1&$3'F

(*1')8 G10), &2 &$ *&,60&2 56& D = 0 ~E + ~P C D = ~E8 9& &,(. 0.2&*.

~P = ( 0) ~E

H)2 = 0(1 +ra) C &$ %.$)* +&$ '.0-) &$3'(*1') &2 &$ +1&$3'(*1') &2')2(*.+) &2 .7A

~P = (0(1 +r

a) 0) q

40(1 +ra)r

2r

I10-$1'.2+)A

~P =q

4ar(1 + ra)r

#" H)2)'1&2+) &$ %&'()* -)$.*1/.'1?2 -)+&0), '.$'6$.* +1*&'(.0&2(& $., +&2,1+.+&, +& '.*D.

,6-&*'1.$ C %)$603(*1'. +&$ +1&$3'(*1')A

$%&'()*) '+,%-#(*/ p G10), 56& $. +&2,1+.+ +& '.*D. ,6-&*'1.$ ,& '.$'6$. -)*A

p = ~P (n)K&2+*&0), 62. +&2,1+.+ ,6-&*'1.$ &2 &$ 12(&*1)* +&$ +1&$3'(*1') C &2 &$ &E(&*1)* +&$ +1&$3'(*1')8

L.*. &$ 12(&*1)* +&$ +1&$3'(*1') 4r = 2a7

pint =~P r = q

24a2

#2 &$ &E(&*1)* +&$ +1&$3'(*1') 4r = 2, 5a7A

pext =~P r = q

35a2

$%&'()*) 01/+234-(#* p M. +&2,1+.+ +& '.*D. %)$603(*1'. ,& '.$'6$. ')0)

p = ~ ~P

!"

#$%$ &'%$( )* +, -,./, ,01 )+ &)2/$. -$+,.'3,2'4* )( .,5',+ 6-,.,+)+$ ,+ 2,%-$ )+72/.'2$0 8 -$.

+$ /,*/$ +, 5'&).9)*2', 5) )(/) &)2/$. () 2,+2:+, 2$%$;

p = ~ ~P = 1r2

r(r2P )

!" !"#$%&' () *+,&, $-+ '.

!"#$%&' ()

#$ %&$'($)*'&+ '( ,-*%*) ,*+*-(-*). )(,*+*'*) /$* '0)1*$%0* d. 10($( /$ '0(-2%1+0%& ($ )/0$1(+0&+. */)($1( '( %*+3*) -04+(). %/5* ,(+601070'*' (-2%1+0%* () 5 '(,($'( '( -* '0)1*$%0* */$* '( -*) ,-*%*)8 9*-%/-*+ -* %*,*%0'*' '(- %&$'($)*'&+ )0 (x) 70($( '*'& ,&+:

(x) = 0(1

1 y23d2

)

*"$+,-".

;( ,0'( %*-%/-*+ -* %*,*%0'*' '(- )0)1(6* 5 )*4(6&) '(6*). -* %*+3* -04+( ($%(++*'* )(+D qenc = LSS

~D d~S = ~D1 ~S1 + ~D2 ~S2 = qenc = LS

E( -* 3/+* 7(6&) ?&+*. (7*-/*$'& -&) 7*-&+() '( L 5 G'*'& ($ (- ($/$%0*'&H8

E2 =Q

S0(1

1 y23d2

)

>?&+* ()1* (I,+()0&$ -* ,&'(6&) +(()%+040+ %&6&:

E2 =Q

3S0d2(3d2 y2)

!

"#$% &' ($)*+*,-$' (* ./0*,-$'& #* -'&-1&'+' -/2/

V =

d0Es(y)dy =

d0

Q

3S0d2(3d2 y2)

3* *#0/ #* /40$*,* 51*

V =8

9

Qd

S0

6' -'.'-$('( *# *,0/,-*#

C =Q

V=9

8

S0d

!" !"#$%&' () *+,&, $-+ '.

!"#$%&' ()

#$%&'( )$ *% +(%)$%,-)(' )$ ./-+-, .-'-/$/-,0 )$ ,$++12% A 3 $,.$,(' d0 1%&'()*+14(, *%)1$/5+&'1+( )$ .$'41&161)-) 6-'1-7/$0 ,1$%)( y /- )1'$++12% .$'.$%)1+*/-' - /-, ./-+-,8 #$,.'$9+1-%)( /(, $:$+&(, )$ 7(')$ 3 $% +-,( )$ %( $;1,&1' +-'/ +-4.( $/5+&'1+(0 $/ )$,./-?-41$%&( $/5+&'1+( 3 $/ 6$+&(' )$ .(/-'1?-+12%0 +*-%)( -./19

+-4(, *%- )1:$'$%+1- )$ .(&$%+1-/ V0 $%&'$ /-, ./-+-,8#* @-, )$%,1)-)$, )$ +-'

!!

~P = ~D 0 ~E = V00d ln(2)

y

y + d

! "#$# %&'&$#( &) *&+%#, -#).,/0.+/#'1 #2%&'&$#( ).( +.,3.( 4& -#).,/0.+/#' 4/,&+%.$&'%&

4& &+5.+/#' 6789

:. +.-.+/4.4 ). +.)+5).$#( +#$# (/&$-,&; C = QV0 9 < 4/=&,&'+/. 4&) &>&,+/+/# .'%&,/#,1 ?.%&'&$#( ). 4/=&,&'+/. 4& -#%&'+/.)9 @#, )# %.%'# %&'&$#( A5& *&, .)35'. $.'&,. 4& +.)+5)., ).

+.,3. Q &' =5'+/#' 4& V09 @.,. &(%#1 .-)/+.,&$#( ). :&? 4&