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!"#$%$%&' ("')"*+&' ," *"$+#%$%,-, . /-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%! ,"& C$ &-/1-%#,% H5( *5 ,$#,5 F~E = qEˆi B525 E I&.(-@-25& *$ ,%#&-7# 1,-*-D$#05 &1& .52)5#%#,%& (%.,$#/1*$(%&F

T~ = −T sin(θ)ˆi + T cos(θ)ˆj

!"#$% &' ()'*' &)+%$ ,%& -)'$.%& (#$ /#+(#0'01' 2 %(,3/%$ ,% /#0*3/340 *' '5)3,36$3#7 8)'$.% '0 97 8)'$.% '0 :7 ;' 7

X

FX = qE − T sin(θ) = 0

(1)

X

FY = T cos(θ) − mg = 0

(2)

T = ?''+(,%.%0*# '0 < >7

qE −

mg cos(θ)

mg sin(θ) = 0 cos(θ)

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

q=

mg tan(θ) E

! ;', $'&),1%*# %01'$3#$ A'+#& 1'0'+#& 5)' q =

mg tan(θ) B E

C#+# θ ≪ 1 1'0'+#& 5)' tan(θ) ≈ θB D#$ ,# 1%01#@ (#*'+#& '&/$363$ ,% /%$E% '0 -)0/340 *', 13'+(# /#+#7

q(t) =

mgθ(t) E

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

dq mg dθ = dt E dt F)'E#@ /#+#

dq dt

= α@ *'&('G%0*#

dθ dt

$'&),1%7

αE dθ = dt 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 −2kqQx ˆ i F~ = F~1 + F~2 = l3

(1)

! "#$%& '(# )%*#$%& +#,-./%0-+ #, ,-+1% ℓ .%0 ,-& */&2-0./- d/2 #0 3(0./40 *#, 501(,% θ /0*/.-*% #0 ,- 61(+- -02#+/%+ )%+ ,- +#,-./40 2+/1%0%$72+/.-8

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

−16kqQx F~ = (2) d3 D%+ ,- )+/$#+- ,#@ *# 0#E2%0 &# 2/#0# '(# F~ = m~a> A0 #&2- &/2(-./40 ,- )-+2F.(,- &# $(#?# 2 &% I- 3+#.(#0./- -01(,-+ &#+5 #02%0.#&8 r kqQ w=4 md3 A, )#+/%*% *# %&./,-./40 &#+5 #02%0.#& =⇒

π 2π = T = w 2

s

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$ ˆj @ 9* 0+62# (*70.)&0# 2)#'-0&'# 2#) *+ 0+),+ 0($.)+* %#/)( (* 2-$.# P %()? %&62*(6($.(3

~ 1 = 2kq ˆj E r2 "#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() ($ *+ %&,-&($.( E,-)+3

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

E=

kq d2 + r 2

! "#$%&'( $' )&*+, -$. /0$ cos(θ) =

Ey = (

√ r 1 d2 +r2

d2

2$ $'34 %45$.4 3$5$%6' /0$

r kq kqr ·√ )ˆj = 2 2 2 2 +r (d + r2 )3/2 d +r

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

~y = − E

(d2

kqr ˆj + r2 )3/2

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

~ =E ~y + E ~y + E ~1 E ?$$%9,4@45#6 ,6' -4,6.$' A4 6:3$5+#6'

2kqr ~ = ( 2kq − )ˆj E 2 2 r (d + r2 )3/2

! 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' .$#0*+#4 *6%6E r ~ = 2qk( 1 − ) E r2 (r2 + d2 ) 32 ;'36 ,6 96#$%6' $'*.+:+. #$ )6.%4 $/0+-4,$53$ *6%6E

1 ~ = 2qk( 1 − E 2 r r2 (1 +

d2 32 ) r2

)

d2 − 3 2qk (1 − (1 + ) 2) r2 r2 ;5 8$5$.4,( ,4 )05*+>5 (1 + x)α '$ 90$#$ 49.6F+%4. 96. 34A,6.( *045#6 x ≈ 0 4E =

(1 + x)α = 1 + αx 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 2 ~ = 3qkd ˆj E r4

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

!

!"#$%&' (

"# $%&'(' )(*+,-%+' )# .# +/0&' )(*+,-%+' .#%1'-0) 2) $)2&(/3/ (%4)-/0)#,) $) 2. &'2%+%5# $) )6.%(%7-%'8 +'0' 2) 0.)2,-/ )# (/ 94.-/8 $'#$) θ )2 &)6.):';

(-%/(' .'" #&)! %!"3! &#"@ &/+.$#+#*)#5

dE =

kdq R2

6'* !8-(! (# $! (#*&/(!( (# %!"3! λ2 .'(#+'& #&%"/:/" dq = λ2 ds7 ('*(# ds #& -* .#,-#?' )"'A' (#$ !"%'0 1&)# )"'A' (# !"%' $' .'(#+'& #&%"/:/" #* =-*%/'* (#$ !*3-$' ,-# $' "#%'""#5 ds = Rdθ

!

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

kλ2 Rdθ kλ2 dθ = R2 R

dE2 =

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

dEx2 = dE2 cos(θ) =

kλ2 cos(θ)dθ R

5()%3+/($# $%'$% θ = 0 6/')/ θ = π2 7

Ex2

kλ2 = R

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

Z

π 2

cos(θ)dθ =

0

kλ2 R

~ x2 = kλ2 ˆi E R 9# %' $,:*,. ;%+ ./ *#&0#(%()% %( x $%. */&0# 0+#$2*,$# 0#+ %')% /+*# '%+< ,32/. /. /()%+,#+4 $% %')/ &/(%+/7

!"# $%&

~ x3 = kλ2 ˆi E R ~2 + E ~3 = =', '% )%($+/ >2% E

'!#(# %)&

2kλ2 ˆ R i

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

~ 4 = (kλ4 E

Z

2R

R

−kλ4 ˆ 1 dx)ˆi = ( )i 2 x 2R

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

~1 + E ~2 + E ~3 + E ~ 4 = 2kλ1 ˆi + 2kλ2 ˆi + ( −kλ4 )ˆi = 0 E 3R R 2R

!

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

"#$%#&'()* λ4 +

2 λ4 = (2λ1 + 6λ2 ) 3 ,##-%.'/'()* .*$ 0'.*1#$ )# λ1 2 λ2 3 λ4 )#4(5)*$ '(6#15*1-#(6#2 #(7*(61'-*$ #. 0'.*1 )# .' 7'18' 9:$7')'+ 12 2 q ′ = q(1 + ) 3 π ;5 7*($5)#1'-*$ π ≈ 3 +

q′ =

10 q 3

!

!"#$%&' ((

"#$%&$' '$ %#()* '$+%,-.%* )-*/&%./* )*- &0 )$#0* .010.,* /' /'02./#/ /' %#-3# &0.4*-(' σ 5

)"$*+,-./

"*02./'-'(*2 '$ )$#0* '0 %&'2,.60 '$ )$#0* 78 /' %**-/'0#/#2 %#-,'2.#0#25 9#:'(*2 ;&' '$ %#()* '$+%,-.%* '($%&'-' '( 0-)1- −σ 6 F(- (, +,-$# >C 0##)'($-'# (, +,-$# ($ 0*(%/&@$ > (, (:( A (, (:( ($ 0*(%/&@$6 G( ,#% &/(3% -$/()&#)(%4 /($(3#% 9*( (, 0-3+# (,80/)&0# 1($()-'# +#) (, +,-$# (%

σ E~1 (~r) = · sign(x)ˆ x 2ǫ0 > (, 0-3+# (,80/)&0# 1($()-'# +#) (, '&%0# (%

  −σ x E~2 (~r) = ·x ˆ sign(x) − √ 2ǫ0 R2 + x2 6

~ = E~1 + E~2 = σ · =⇒ E 2ǫ0



x √ 2 R + x2



x ˆ

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

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

!

~0 "#$%& '&% (# &)*+'*& (,-. /$/$ "&) dF~ = dq E =⇒ F~

=

Z

= = =

~ dq E

con dq = λdx   Z d+a σ x λdx x ˆ · √ 2ǫ0 R2 + x2 d Z d+a λσ xdx √ · ·x ˆ 2ǫ0 d R2 + x2  d+a λσ p 2 ·x ˆ R + x2 · 2ǫ0 d  p λσ p 2 · ˆ R + (d + a)2 − R2 + d2 · x 2ǫ0 d

=

d+a

!

!"#$%&' ()

"# $%&'('%#)% *%+',%+'*-%$'&. #. &.#/0&).$ /% $1/'. a )'%#% 0#1 &1$21 ).)13 Q 0#'-.$+%+%#)% /'*)$'40'/1 %# *0 *0(%$5&'% '#)%$'.$6 7#&0%#)$% %3 &1+(. %38&)$'&. %# %3 (0#). O6

*"$+,-./0

7*)% ($.43%+1 3. /%4%+.* $%*.39%$ 0)'3':1#/. 31 31; /% &.03.+4 (1$1 /'*)$'40&'.#%* *0(%$5< &'13%* /% &1$216 7*)1 3%; #.* /'&% (1$1 %*)% &1*. =0%

~ ~0) = E(

Z

kdq(~r − r~′ ) k~r − r~′ k3

>.+. ;1 ,%+.* 9'*).? %3 %3%+%#). /% &1$21 3. (./%+.* %*&$'4'$ &.+. dq = σdA? /.#/% σ %* 31 /%#*'/1/ /% &1$21 *0(%$5&'13 %# %3 '#)%$'.$ /%3 $%&'('%#)% @&.#*)1#)%A6 B%4%+.* %*&$'4'$ %3 %3%+%#). /% C$%1 dA /% )13 +1#%$1 =0% (./1+.* $%&.$$%$ 31 *0(%$5&'% ; (1$1 %*). ,1$%+.* 3. *'20'%#)%D >.#*'/%$1$% 0#1 &1*&1$1 %*-8$'&1 &.# 31 *'20'%#)% /'*(.*'&'E# (1$1 31* 91$'143%* θ ; γ

F3 ,1&%$ 0#1 (%=0%G1 91$'1&'E# /%3 C#203. θ? 9%+.* =0% *% $%&.$$% 0# (%=0%G. 1$&. /% 3.#2')0/ ds1 = R sin(γ)θ6 H.$ .)$. 31/. 13 ,1&%$ 0#1 (%=0%G1 91$'1&'E# /%3 C#203. γ *% $%&.$$% 0# 1$&. /% 3.#2')0/ ds2 = Rdγ 6 B% %*)1 +1#%$1 %3 %3%+%#). /% C$%1 /% 31 &'$&0#-%$%#&'1 3. (./%+.* %*&$'4'$ &.+. dA = ds1 · ds2 = R2 sin(γ)dγdθ B% %*)1 +1#%$1 (./%+.* %*&$'4'$ %3 %3%+%#). /% &1$21 &.+.

dq = σR2 sin(γ)dγdθ

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

!

"#$ #%$# &'(#) (*+*,#- (*%*$,./'$ &#- 0*1%#$*- r~′ 2 ~r3 4&'$',*/%* r~′ = ~0 2 5#$ #%$# &'6 (#7,.$'/(# &' 89:$'; 0*$*,#- ?1.& 0*$ - "275+ 57,&15+ &# 8#"-5 B6: C& #" +/2&1(D" 4&# 8(57#&2"9 E&25+ *%& #" '%&()" *%& &,&($& #" .%/-$." +57(& %&#&2&-15 4B 4&# "#"27(& &+1> 4/(/3/4" &- ±ˆ y 6 &+ #" 2/+2" &- $%"#*%/&( 8%-15 4&# "#"27(&: A5( 1"-159 -5+ 7"+1" E&( #" '%&()" 85( %-/4"4 4& #"(35 &- BFG 4&# "#"27(&: H&"- r~2 = ~09 r~1 = xˆ x + y yˆ9 $5- −∞ ≤ x ≤ +∞ 6 d ≤ y ≤ b + d9 #" 85+/$/=- 4&# &#&2&-15 4B &- BFG 4&# "#"27(& 6 #" 85+/$/=- 4& #" .%/-$."9 (&+8&$1/E"2&-1&: I+D9 1&-&25+ *%& #" '%&()" +57(& 1"# &#&2&-15 &+ Z Z dq1 · −r~1 ~ dF21 = dq2 , con dq1 = σdxdy 3 Ω 4πǫ0 |r~1 | Z +∞ Z b+d Z +∞ Z b+d σdxdy · xˆ x σdxdy · y yˆ = −dq2 3 − dq2 3 d d −∞ −∞ 4πǫ0 (x2 + y 2 ) 2 4πǫ0 (x2 + y 2 ) 2 Z Z dq2 σ yˆ +∞ b+d dxdy · y = − 3 4πǫ0 −∞ d (x2 + y 2 ) 2 ! b+d Z 1 dq2 σ yˆ +∞ p · dx = 2 2 4πǫ0 −∞ x + y d ! Z 1 dq2 σ yˆ +∞ 1 p = · dx, pero dq2 = λdx2 −√ 4πǫ0 −∞ x2 + d2 x2 + (b + d)2

dF~21 =⇒ dx2

=

=

=

=

! 1 p −√ · dx x2 + d2 x2 + (b + d)2 −∞ ! +∞ p x + x2 + (b + d)2 λσ yˆ √ · ln 2πǫ0 x + x2 + d2 0 q     2   ) 1 + 1 + ( b+d λσ yˆ  b + d x   − ln q l´ım ln  · x→+∞ 2πǫ0 d d 2 1 + 1 + (x)   d λσ yˆ · ln 2πǫ0 b+d   dF~21 λσ yˆ d =⇒ = · ln dx2 2πǫ0 b+d λσ yˆ 4πǫ0

Z

+∞

1

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

!

!"#$%&' ()

"#$%&'()( *$ +,+-.)( )(/0&,1$(# &$2$&0# + *$+ '&%0+$/&+ R '( *$ +,+-.)( %(-&/&)/*,+) '( )+'&# R /#-# %( 3( ($ ,+ 24*)+5 6-.#% (%07$ *$&8#)-(-($0( /+)4+'#% /#$ '($%&'+' ,&$(+, '( /+)4+ λ5 9$/*($0)( ,+ 8*():+ '( )(;*,%&*(C+ /+)4+ dq *.&/+'+ ($ (, ;*$0# r~′ = xˆi '(, +,+-.)( %#.)( (, ;*$0# '( ~r = dˆj %()7B

~ = dE

kdq(~r − r~′ ) kdq(−xˆi + dˆj) = 3 |~r − r~′ |3 (x2 + d2 ) 2

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

~ = dE

kλdx(−xˆi + dˆj) 3

(x2 + d2 ) 2

E$0(4)+$'# '(%'( −∞ + +∞B

~ = E

Z

+∞

−∞

kλdx(−xˆi + dˆj) (x2

+

3

d2 ) 2

= kλ((−

Z

+∞

−∞

x (x2

+

Z ˆ ) i + ( 3

d2 ) 2

+∞

−∞

d (x2

3

+ d2 ) 2

)ˆj)

! "# $%&'()& f (x) =

x 3

(x2 +d2 ) 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) =

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

d 3

(x2 +d2 ) 2

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

Z

+∞

g(x)dx = 2

Z

+∞

g(x)dx

0

−∞

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

~ = 2kdλ E

Z

+∞

(x2

0

@* 2(*&* A%* 80. 10 2#&20

R

1 3

(x2 +d2 ) 2

=

1 +

3

d2 ) 2

ˆj

x 1

d2 (x2 +d2 ) 2

∞ ˆ 2kλ ˆ ~ = 2kdλ( )j = j E 3 d d2 (x2 + d2 ) 2 x

0

BC0.# A%* +#;*,0+ '0,0 *+ *1 '#,-0 *1D'2.('0 3*&*.#40 -0. %& #1#,;.* (&E&(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*E&('()&?dF = dqE 9 40&4* E *+ *1 '#,-0 *1D'2.('0 -.04%'(40 -0. *1 #1#,;.* (&E&(20 # 1# #12%.# *& A%* +* *&'%*&2.# 1# '#.3# dq 7 I*'0..*.D 10+ *1*,*&20+ 4* '#.3# 4*1 #1#,;.* *& $%&'()& 4* %& :&3%10 θ A%* .*'0..# *1 #1#,;.# +*,('(.'%1#.7 J0&+(4*.#.* 1# +(3%(*&2* E3%.#?

K1 '#,-0 *1D'2.('0 -.04%'(40 -0. *1 #1#,;.* (&E&(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 θ A%*4# *=-.*+#4# '0,0 d(θ) = R + R sin(θ)7 B4*,:+9 *1 *1*,*&20 4* '#.3# dq *+ -0. 4*E&('()& (3%#1 # λds9 40&4* ds *+ %& -*A%*H0 2.0G0 4* #1#,;.* '(.'%1#. *1 '%#1 -04*,0+ *+'.(;(. *& $%&'()& 4*1 :&3%10 θ '0,0 ds = Rdθ7 L* *+2# ,#&*.# 1# $%*.G# +0;.* %&# '#.3# dq +*.:?

dF = dqE = λRdθ ·

2kλ 2kλ2 dθ = R + R sin(θ) 1 + sin(θ)

M&2*3.#&40 4*+4* θ = 0 # θ = π 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

!

Z

1 sin(θ) − 1 = 1 + sin(θ) cos(θ)

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

F = 4kλ2

!"#$%&' ( !" #! $%&'' !"#$%&' () !" #"$%" &'!('") q > 0 *+(" $,-*"-" &,$ '!" +'&*$.#/* #*$$"-" 0,$1"-" &,$ '! 1"!(, #2!/#, -* $"-/, R 3 ")('$" H 4 3 '!" +'&*$.#/* +*1/*+05$/#" #,!#5!($/#" #,! )" #"$%"4 +*%6! +* ,7+*$8" *! )" .%'$"9 :")#')* *) ;''* ))"1"1,+ DE +*$B /%'") " )" +'1" -* ),+ ;''* &"+*! &,$ )" +*1/*+0*$" C>'* ))"1"1,+ Se E 3 ),+ ;''* &"+*! &,$ *) 1"!(, #2!/#,C>'* ))"1"1,+ Sc E9 F+(, +* &'*-* *+#$/7/$ #,1,@ φ = φe + φc =⇒ φc = φ + φe

(2)

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

Z

Se

~ · dS ~ E

F) #"1&, *)5#($/#, &$,-'#/-, &,$ )" #"$%" q *+ E = 4πǫ1 Rq rˆ9 G,!-* rˆ *+ *) 8*#(,$ '!/("$/, *! -/$*##/2! $"-/")9 F) 8*#(,$ dS~ ("17/5! *+(" *! -/$*##/2! $"-/")4 3" >'* dS~ = nˆ dS 3 nˆ *+ &"$")*), " rˆ #,1, 1'*+($" )" .%'$"@ 0

HI

2

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

!

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

~ · dS ~= E

1 q 1 q rˆ · n ˆ dS = dS 2 4πǫ0 R 4πǫ0 R2

.(%#/*0()& $# &1%2#(# #3 4-5& 60*0 30 $#72#$8#*09 Z Z Z 1 q 1 q ~ ~ dS φe = E · dS = dS = 2 4πǫ0 R2 Se Se Se 4πǫ0 R R R :0 2(%#/*03 Se dS ;&**#$6&()# 03 +*#0 )# 30 $#72#$8#*0< =(%&(;#$ Se dS = 2πR2 < "# #$%0 70(#*0 #3 4-5& $#*+9

φe =

1 q q · 2πR2 = 2 4πǫ0 R 2ǫ0

(3)

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

φc =

q 2ǫ0

!"#$%&' ()

D&($2)#*# -(0 ;+$;0*0 ;23E()*2;0' $2( #$6#$&*' )# *0)2& F B 30*/& 2(C(2%&' ;&( )#($2)0) )# ;0*/0 $-6#*C;203 σ -(28&*7# >G#* C/-*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' ;&( ;&($%0(%# )# 6*&6&*;2&(032)0) ǫ10 < =$ )#;2*'

!

Qint ~ ·n E ˆ dS = ǫ0 Ω

I

!" # Qint #$ %& '&()& #"'#((& & *!( %& $+*#(,'-# Ω. /$0& 1!(2& -"0#)(&% # %& 3#4 # 5&+$$ $# *+# # #6*(#$&( #" 1!(2& -1#(#"'-&% +$&" ! #% 0#!(#2& # %& -7#()#"'-& 8%! '+&% $# #9& &% %#'0!(:; !+# #% '&2*! #%?'0(-'! $#& "+%! #"0(! #% '-%-" (!. +# (2πR)hσ σ ~ ·n E ˆ dS = E(r)(2πr)h = =⇒ E(r) = ǫ ǫ 0 0 Ω

A!( 0&"0!;

I

~ E(r) =

J!0# >+# #% $&%0! # -$'!"0-"+- & ,'-#: #$ + ǫσ0 .



σ ǫ0

~0


R r

  R r

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

! """# x ≤ −R

$%&'&()* +%&

R3 ρ E0 − = 0 =⇒ x3 = −R 3ǫ0 x2

r

ρR 3ǫ0 E0

,-'- +%& x1 &."*/&0 *& 1&2& 3%(45"' +%& r r Rρ ρR ρR < −R =⇒ > 1 =⇒ E0 < x3 = −R 3ǫ0 E0 3ǫ0 E0 3ǫ0 +%& &* 5- ("*(- 3)61"3"76 +%& &63)6/'-()* -6/&'")'(&6/& 4-'- +%& 8%2"&'- &+%"9 5"2'"): ;*/& 4%6/) 1& &+%"5"2'") &* "6&*/-25&0 *& 1& 08

*"$+,-./0

@, 1(#.60%,1-3# '&+ *%(2+&5, &) +, )-60-&#$&A

B,%,
(#$%,% &+ 1,5*( &+71$%-1( )(2%& &+ &C& x 0> 1,+10+,%&5() &+ 1,5*( &+71$%-1( *%(D '01-'( *(% +, %&6-3# &#$%& +() *+,#() -#.#-$() '& '&#)-',' ρ : &+ 1,)1,%(# &)E7%-1( '& '&#)-',' σ : *()$&%-(%5&#$& 0$-+-F,%&5() &+ *%-#1-*-( '& )0*&%*()-1-3#8

1'&2" %$3,4!-," 2!"5+,-5" 2"! $' !%6-./ %/4!% $"7 2$'/"7 -/8/-4"7 9Eρ:0 G,', +,

)-5&$%H, /0& *%&)&#$, &)$, %&6-3# %&)*&1$( ,+ &C& y ; &+ 1,5*( &+71$%-1( Eρ *%&)&#$,%, $,52-7# &)$, )-5&$%H, 1(5( 50&)$%, +, )-60-&#$& .60%,A

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

!

"#$%&$#'( ($ %#)*+ ($,%-'.%+ (/ 0+1 '(2.+/(13 "&#/0+ 0 < x < a 4 x > a5

• 0& 5&-& #2 -,5 ',%,5 -,3#%,-#5 4 5#% a ǫ0

C&3,% 0+# #- ',/$& #2 #53, %#*)12 #5 '&253,23#? !"#$% &'()*+,)% $+%-.),-% $%+ '" )"/)"+" &/0(+,)" 1Eσ 2 2 "#$,%,%#/&5 #- #5$,')& #2 D %#*)&2#5A #- )23#%)&% (# -, ',5',%, #5.7%)', E%FGH 4 #- #;3#%)&% E%IGH 4 ',-'+-,/&5 #- ',/$& #-7'3%)'& #2 ',(, +2& +3)-)J,2(& -, -#4 (# *,+559

• r ?'* ')*' #(.'5 #( 2(*@( 3=)/*3'* /- =%#( ;( ?'* #( #/; ./ @(%-- -/ )/=.*4 R C'=-3./*(6'-5 (# 3@%(# JA'*( a + 2R

~ σ = σR22 rˆ> JA'*( )/=/6'~ ρ = ρx ˆi ; E L: 0 < x < aB "= /-)( */@3I= #'- 2(60'- /#,2)*32'- -'=B E ǫ0 ǫ0 r &$ 40$%)0>1,0.(#$ 4# ,&)2& $#)?7

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

!

~ = E

 ρx   ( ǫ0 −   ( ρa + ǫ0

σR2 )ˆi ǫ0 (a+R−x)2 ρa ˆ ǫ0 i σR2 )ˆi ǫ0 (x−a−R)2

0' 2+ $%)'.(B+ &' 2+ &%$.(%73,%=#/ + ;(%"(% $+7')"$ 93' +&')8$ ;+(+ R ≤ r '2 ,+);" ,3);2' ~ r) = E(r)ˆ E(~ r5 :% 7%'# #" '$ #','$+(%" ;+(+ 2+ ('$"23,%=# &'2 ;("72')+ ,"#",'( '2 ,+);" '2-,.(%," *3'(+ &' 2+ '$*'(+/ '#,"#.(-)"$2" 3$+#&" 2+ 2'4 &' C+3$$5 !"#$%&'(')"$ ,")" $3;'(D,%' &' %#.'0(+,%=# 3#+ ,8$,+(+ '$*-(%,+ &' (+&%" r > R5 E'#')"$ 93' I Qint ~ ·n E ˆ dS = ǫ0 Ω 4πR3 ρ E(r)4πr2 = 3ǫ0 3 ~ r) = R ρ rˆ =⇒ E(~ 3ǫ0 r2 > r1 , r2 4 "% -56-) )# .(&#.$-& - $,-7+) '# /& .-68# .(&'/.$(, 9'#)2,#.%-68#: ;/# )%,7# ).%#) '# 8-) #)*#,-) .(&'/.$(,-) #)

V1 = k

q1 q2 , V2 = k r1 r2

'(&'# D#5() )/2/#)$( ;/# #8 2($#&.%-8 '# /&- #)*#,- &( #) -*#.$-'( 2(, #8 '# 8- ($,- 9( 5A) 6%#& #) -*#.$-'( '# *(,5- '#)2,#.%-68#=: '-'( ;/# #)$A& 5/0 5/0 )#2-,-'-: #) '#.%,: d >> r1 , r2 : 0 $-52(.( #8 .-52( '# /&- #)*#,- ,#'%)$,%6/0# 8- .-,1- '# 8- ($,-4 E(, .(&)#,7-.%.%-8 '# .-,1-: 2(, 8( ./-8 #8 .-52( #8+.$,%.( .#,.- '# #88-) 9#& )/ #G$#,%(,= #) 5A) */#,$# ;/# #& ,#1%(&#) .(& 5#&(, ,-'%( '# ./,7-$/,-4 H)-&'( 8- ,#8-.%* *+,*'-+* #B$#+&* )# 0(�$* * $%#++* * $+*C/) '# -&* 6*$#+D* 0-5* '%.#+#&0%* '# :($#&0%*3 #&$+# )-) 6(+&#) #) V0 = *9 E*30-3# 3* 0*+?* ;-# )# %&'-0# #& 3*) )-:#+