Ejercicios Resueltos Electricidad y Magnetismo (Garrido - Narrias)

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@' 7-,05' *!0.'$ N~ ", 4-,(, ,"+0%#%0 -"'*(! "-" +!.4!*,*&," 0,+&'*1-$'0," +!.!?
~ = N cos(π/3)î + N sin(π/3)ĵ
N
@' 7-,05' F~e ," $' ,A,0+%(' 4!0 $' #!$%&' (, $' (,0,+:' "!#0, $' #!$%&' (, $' %5;-%,0(' / ,"?
kq 2
F~e = 2
R
B
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;* <=> ?*(#& 6'*
X
kq 2
=0
R2
(1)
FY = N sin(π/3) − mg = 0
(2)
FX = N cos(π/3) −
X
N=
mg
sin(π/3)
@**(,1%+%.3# *&/% *A,$*&2#. *. <B>7
mg
kq 2
cos(π/3) − 2 = 0
sin(π/3)
R
;*&,*C%.3# q
q = ±R
r
mg cot(π/3)
k
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#"%")'(A
q2
mg = k 2
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=⇒ x0 = q ·
s
k
mg
(1)
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-": &" I"J#?%
kq 2
d2 x
F = m 2 = −mg +
dt
(x0 + x)2
K*/* '(.$-*.$'%"( ;"0,"@*(A |x| << |x0 | =⇒ | xx0 | << 1A ;'/ -' 0,"
F =m
kq 2
kq 2
kq 2
x −2 ∼
x
d2 x
−mg
+
=
−mg
+
=
−mg
+
·
(1
+
)
· (1 − 2 )
=
dt2
(x0 + x)2
x0 2
x0
x0 2
x0
K"/' &" LM= #"%")'( 0," mg =
kq 2
x0 2
=⇒ m
=⇒
K"/' &" LM=A mx0 2 =
d2 x ∼ −2kq 2 x
=
dt2
x0 3
2kq 2 x
d2 x
+
·
=0
dt2
mx0 2 x0
kq 2
g
d2 x 2g
+
·x=0
dt2
x0
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√
2
2
π
π
q
x̂
3
q
ŷ)
F~1 = k 2 · (cos( )x̂ − sen( )ŷ) = k 2 · ( −
L
3
3
L
2
2
2
q
F~2 = k 2 x̂
L
32 A.,(B2 (,%.#)2"), ,%
√
√
2
2
2
√
√
π
q
q
ŷ
π
3
q
3
q2
3
k 2 ŷ = 3 · k 2 · (
x̂ − ) = 3 · k 2 · (cos( )x̂ − sen( )ŷ)
F~ = F~1 + F~2 = k 2 x̂ −
2 L
2 L
L
2
2
L
6
6
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r
L
L
√
=
2π =⇒ r =
sen( π6 )
sen( 3 )
3
K$( )2")$; 2# *0.2#2( #$% <D-.#$% -, #2% A.,(B2% $=),",<$%
√
3·k
qQ
q2
1
= k L 2 =⇒ Q = √ · q
2
L
( √3 )
3
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2"*2- ~r -# ,% ("#.$.B* 2- ,% $%&'% 2- (&)-:% @ r~1 ,√
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2- (&)-:% -#
kqQ
F~ = F~1 + F~2 =
(R2
=
3
+ a2 ) 2
2kqQR
3
(R2 + a2 ) 2
· (Rcos(θ)x̂ + Rsen(θ)ŷ − aẑ + Rcos(θ)x̂ + Rsen(θ)ŷ + aẑ)
· (cos(θ)x̂ + sen(θ)ŷ)
.
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,% 9)*$.B*
2kqQR
|F~ | =
3 = 2kqQ · f (R)
(R2 + a2 ) 2
3
=⇒
1
df (R)
(R2 + a2 ) 2 − 3R2 (R2 + a2 ) 2
=
=0
dR
(R2 + a2 )3
1
1
=⇒ (R2 + a2 ) 2 · ((R2 + a2 ) − 3R2 )) = 0 =⇒ R = √ · a
2
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~
F~E = q E
~ %& )$($*%*5 $* %G% X 6 &% ,%#0(+ A1% E
~ = E î> H5( *5 ,$#,5 F~E = qE î
B525 E
I&.(-@-25& *$ ,%#&-7# 1,-*-D$#05 &1& .52)5#%#,%& (%.,$#/1*$(%&F
T~ = −T sin(θ)î + T cos(θ)ĵ
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8)'$.% '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
$'&),1%7
αE
dθ
=
dt
mg
!
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0'$+*$" 1% F~1/
B& %&'(& <0) 2'#10%) 7& D0)'>& F~1 $) )+%0)+,'& )+ 7& 2#$-%-3+ ~r1 = d/2ĵ 9 7& 2&',6%07& 1)
2'0)?& )+ xî4 @#' 1).+-%-3+; 7& D0)'>& )+,') &*?&$ %&'(&$ )$A
k(−q)Q(xî + d/2ĵ)
F~1 =
l3
0'$+*$" 1% F~2/
G+ )$,) %&$#; 7& %&'(& <0) 2'#10%) F~2 $) )+%0)+,'& )+ 7& 2#$-%-3+ ~r2 = −d/2ĵ ; 2#' 7# ,&+,# 7&
D0)'>& 1) &,'&%%-3+ )+,') )$,& %&'(& 9 7& %&'(& 1) 2'0)?& $)'FA
k(−q)Q(xî − d/2ĵ)
F~2 =
l3
") )$,& *&+)'&; 7& D0)'>& ')$07,&+,) $#?') 7& %&'(& −q $)'FA
−2kqQx
î
F~ = F~1 + F~2 =
l3
(1)
!
"#$%& '(# )%*#$%& +#,-./%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 )#'(#:% *#&),-;-$/#02% *# ,- .-+1- *# )+(#<-= #, 501(,% θ #& 2-$</70
)#'(#:%8 θ ≪ 1> 9%0 #&2% ?#$%& '(# cos(θ) ≈ 1 @ )%+ ,% 2-02% ℓ ≈ (d/2)> A?-,(-0*% #0 B C &#
2/#0# '(#8
−16kqQx
F~ =
(2)
d3
D%+ ,- )+/$#+- ,#@ *# 0#E2%0 &# 2/#0# '(# F~ = m~a> A0 #&2- &/2(-./40 ,- )-+2F.(,- &# $(#?#
2
&%<+# #, #G# x @ )%+ ,% 2-02% m~a = m ddt2x î
H# #&2- $-0#+- &# 2#0*+5 '(#
−16kqQx
d2 x
=m 2
3
d
dt
d2 x 16kqQx
+
=0
md3
dt2
9%0 #&2% ?#+/6.-$%& '(# ,- .-+1- &/1(# (0 $%?/$/#02% -+$40/.% &/$),#> 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$ ĵ @ 9* 0+62# (*70.)&0# 2)#'-0&'# 2#) *+
0+),+ 0($.)+* %#/)( (* 2-$.# P %()? %&62*(6($.(3
~ 1 = 2kq ĵ
E
r2
"#6# (% A?0&* 5()B *+% '#% 0+),+% *+.()+*(% 2)#'-0&)?$ -$ 0+62# (*70.)&0# ($ (* 2-$.# P < *+%
0#62#$($.(% ($ x %( +$-*+)+$ <+ ;-( .&($($ &,-+* 0+),+ < (%.?$ + &,-+* '&%.+$0&+ '(* 2-$.#@
>#) *# .+$.#B .($')(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
·√
)ĵ = 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
ĵ
+ r2 )3/2
;'3$ *4%96 '$.& $, 9.6#0*+#6 96. *4#4 *4.84 ,43$.4,1 <6. ,6 34536 $, *4%96 $,=*3.+*6 .$'0,3453$
$5 $, 90536 P '$.& 96. 9.+5*+9+6 #$ '09$.96'+*+>5
~ =E
~y + E
~y + E
~1
E
?$$%9,4@45#6 ,6' -4,6.$' A4 6:3$5+#6'
2kqr
~ = ( 2kq −
)ĵ
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 <6. ,6 34536( $5 ,4 $F9.$'+>5 #$,
*4%96 96#$%6' G4*$. ,4 '+80+$53$ 49.6F+%4*+>5E
(1 +
d2 − 3
3 d2
2 ≈ (1 −
)
)
r2
2 r2
;-4,045#6 $'3$ .$'0,34#6 $5 ,4 $F9.$'+>5 6:3$5+#4 94.4 $, *4%96( '$ 3+$5$ /0$E
2
~ = 3qkd ĵ
E
r4
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!
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)2C
τ = −F a sin(θ) + −F a sin(θ) = −2F a sin(θ) = −2qE sin(θ)
<( 2%4#' #)4/,%>' 2) '7,%)#) / &/-,%- $) (/ -)4(/ $) (/ 0/#' $)-)+F/ ? 2) &.)$) +'0&-'7/- /(
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τ = Iα
−2qEa sin(θ) = Iα
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!
d2 θ
dt2
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−2qEaθ = I
d2 θ 2qEaθ
+
=0
I
dt2
51+ /1 '(%/ 6&21# 7(& /% 1.)&+$%')*+ %+8(/%. ,&/ ,)01/1 #&8().9 (+ 216)2)&+$1 %.2*+)'1
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r
2qEa
w=
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: 01. /1 $%+$1 /% -.&'(%+')% #&.94
1
w
=
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r
2qEa
I
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7 !$ )( 3&'#/#=$ ~r %!+#%& ( 0$( %#'",#+0/#=$ %! /(,-( Ω !'
F~ = q ·
~
dq(~r − r‘)
~ 3
4πǫ0 |~r − r‘|
Z
Ω
9&$-(*&' (*+&' ()(*+,!' !$ !) !8! <2 %! "() 1&,*( 70! 0$& /0+,( %! x = 0 ( x = L : !)
&",& %! x = L+d ( x = 2L+d4 !($ r~1 = x1 x̂2 r~2 = x2 x̂2 /&$ 0 ≤ x1 ≤ L : L+d ≤ x2 ≤ 2L+d2
)&' >!/"&,!' 3&'#/#=$ %! /(%( ()(*+,!4
?!$!*&' 70! )( 10!,6( 70! !8!,/! @ '&+,! 0$ !)!*!$"& %< %! A !'
dF~21 = dq2 ·
Z
Ω1
L
dq1 (r~2 − r~1 )
4πǫ0 |r~2 − r~1 |3
λ1 dx1 (x2 − x1 )x̂
4πǫ0 |x2 − x1 |3
0
Z L
dx1
λ1
= dq2 ·
· x̂ ·
2
4πǫ0
0 (x2 − x1 )
= dq2 ·
Z
L
λ1
1
· x̂ ·
4πǫ0
x2 − x1 0
1
1
λ1 x̂
· dq2 · (
− )
4πǫ0
x2 − L x2
1
1
λ1 λ2 x̂
· dx2 · (
− )
4πǫ0
x2 − L x2
= dq2 ·
=
=
@B
!"#$%&' () *+$,-.!&,/ 0, '%&'12
!
=⇒ F~21 =
=
=
Z 2L+d 1
1
λ1 λ2 x̂
dx2
·
−
4πǫ0
x2 − L x2
L+d
λ1 λ2 x̂
2L+d
· (ln(x2 − L) − ln(x2 ))|L+d
4πǫ0
λ1 λ2 x̂
x2 − L
· ln(
)
4πǫ0
x2
λ1 λ2
· ln
=⇒ F~21 =
4πǫ0
2L+d
L+d
(L + d)2
d(2L + d)
· x̂
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q
(-.$ %$2('$ 10)(%0- )(*23' -, )(2-3)$) )( 4$'+$ 40%0 λ1 = 2R
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(#(4.'340 A,( 1'0),4( -('<G
!"#$%&' () *+$,-.!&,/ 0, '%&'12
kdq
kλ1 dx
)î = (
)î
2
x
x2
~1 = (
dE
!"#$%&'"() ($*($ x = −3R +'*#' x = −R
~ 1 = k(λ1
E
Z
1
1
dx)î = (kλ1 ( − 2
x2
x
,) -.$ &$*./#'0
−R
))î
−3R
~ 1 = 2kλ1 î
E
3R
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2 5
λ2 =
2q
πR
λ3 = −
2q
πR
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(-%/(' .'" #&)! %!"3! &#"@ &/+.$#+#*)#5
dE =
kdq
R2
6'* !8-(! (# $! (#*&/(!( (# %!"3! λ2 .'(#+'& #&%"/:/" dq = λ2 ds7 ('*(# ds #& -* .#,-#?'
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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 î
E
R
9# %' $,:*,. ;%+ ./ *#&0#(%()% %( x $%. */&0# 0+#$2*,$# 0#+ %')% /+*# '%+< ,32/.
/. /()%+,#+4 $% %')/ &/(%+/7
!"# $%&
~ x3 = kλ2 î
E
R
~2 + E
~3 =
=', '% )%($+/ >2% E
'!#(# %)&
2kλ2
R î
?%/ q ′ ./ */+3/ >2% ),%(% %')% )+#@# $% /./&-+% )/. >2% /(2./ %. */&0# %.%*)+,*# 0$+#$2*,$#
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λ4 =
q′
R
12%3#4 0+#*%$,%($# $% ,32/. C#+&/ >2% 0/+/ %. )+#@# AB ;%&#' >2%
~ 4 = (kλ4
E
Z
2R
R
−kλ4
1
dx)î = (
)î
2
x
2R
?2&/($# )#$#' .#' */&0#' */.*2./$#' % ,32/./($# / *%+#7
~1 + E
~2 + E
~3 + E
~ 4 = 2kλ1 î + 2kλ2 î + ( −kλ4 )î = 0
E
3R
R
2R
!
!"#$%&' () *+$,-.!&,/ 0, '%&'12
"#$%#&'()* λ4 +
2
λ4 = (2λ1 + 6λ2 )
3
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7'18' 9:$7')'+
12
2
q ′ = q(1 + )
3
π
;5 7*($5)#1'-*$ π ≈ 3 +
q′ =
10
q
3
!
!"#$%&' ((
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)"$*+,-./
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%#()* '$+%,-.%* <.'0' /#/* )*Z Z
dq(~r − r~1 )
~
E(~r) =
r − r~1 |3
Ω 4πǫ0 |~
%*0 r~1 '$ <'%,*- )*2.%.60 /' $# /.2,-.:&%.60 /' %#-3# Ω ;&' 3'0'-# '$ %#()* '$+%,-.%* '0 ~r5
='0'(*2 ;&' ~r = xx̂ + y ŷ + z ẑ > r~1 = x1 x̂ + y1 ŷ 8 |~r − r~1 |2 = (x − x1 )2 + (y − y1 )2 + z 2 5
Z +∞ Z +∞
σdx1 dy1 ((x − x1 )x̂ + (y − y1 )ŷ + z ẑ)
~
=⇒ E(~r) =
3
−∞
−∞
4πǫ0 ((x − x1 )2 + (y − y1 )2 + z 2 ) 2
?#3#(*2 %#(:.* /' <#-.#:$'25 w = x − x1 > v = y − y1 =⇒ x1 = x − w,
A#%*:.#0* /' $# ,-#024*-(#%.60 '2
(x1 )w (x1 )v
(y1 )w (y1 )v
J=
~ r) =
=⇒ E(~
Z
+∞ Z +∞
−∞
=
=
σ
4πǫ0
Z
−1 0
0 −1
=1
σdwdv(wx̂ + v ŷ + z ẑ)
3
−∞
4πǫ0 (w2
+∞ Z +∞
−∞
y1 = y − v 5 @$
+ v2 + z2) 2
dwdv · (z ẑ)
−∞
3
(w2 + v 2 + z 2 ) 2
B&'<#('0,' C#3#(*2 %#(:.* /' <#-.#:$'25
w = rcos(θ)
v = rsen(θ)
J=
~ r) =
=⇒ E(~
=
=
=
=
=
σ
4πǫ0
Z
0
2π
wr wθ
vr vθ
Z
+∞
=
cos(θ) −rsen(θ)
sen(θ) rcos(θ)
=r
rdrdθ · (z ẑ)
3
(r2 + z 2 ) 2
Z +∞
rdr
0
σ
· 2πz ẑ
3
4πǫ0
0
(r2 + z 2 ) 2
Z +∞
du
σ
· z ẑ
3
4ǫ0
z2
(u) 2
z2
1
σ
· z ẑ 2u− 2
4ǫ0
+∞
σ
z
·
ẑ
2ǫ0 |z|
σ
· sign(z)ẑ
2ǫ0
pero
u = r2 + z 2 → du = 2rdr
!
!"#$%&' () *+$,-.!&,/ 0, '%&'12
~ r) = σ · sign(z)ẑ
=⇒ E(~
2ǫ0
"#$% &'% %( )*($# +% +,)-#.$,.',+*+ +%( /0+'(# +%( -*/1# %(2-$3,-# 4&'% %) .#3/*( *( 1(*.#5
%) + ǫσ0 6
!
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0&)0*,-) '( )-'&# 56
-7 "-,0*,( (, 0-3+# (,80/)&0# ($ 0*-,9*&() +*$/# '(, (:( 9*( +-%- +#) (, 0($/)# '(, #)&.0&#4
+()+($'&0*,-)3($/( -, +,-$#6
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0-)1- ,&$(-, *$&2#)3( λ > 0*>- '&%/-$0&- 3?% +)@A&3- -, +,-$# (% '6 B$0#$/)-) ,- 2*()C(,80/)&0- 9*( (A+()&3($/- (, -,-3;)(6
*"$+,-./0
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%#;)( (, (:( > *$ '&%0# '( )-'&# 5 > '($%&'-' '( 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)x̂
2ǫ0
> (, 0-3+# (,80/)&0# 1($()-'# +#) (, '&%0# (%
−σ
x
E~2 (~r) =
· x̂ sign(x) − √
2ǫ0
R2 + x2
6
~ = E~1 + E~2 = σ ·
=⇒ E
2ǫ0
x
√
2
R + x2
x̂
;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
x̂
· √
2ǫ0
R2 + x2
d
Z d+a
λσ
xdx
√
·
· x̂
2ǫ0 d
R2 + x2
d+a
λσ p 2
· x̂
R + x2
·
2ǫ0
d
p
λσ p 2
·
R + (d + a)2 − R2 + d2 · x̂
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
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γ
F3 ,1&%$ 0#1 (%=0%G1 91$'1&'E# /%3 C#203. θ? 9%+.* =0% *% $%&.$$% 0# (%=0%G. 1$&. /% 3.#2')0/
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dS
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Z
Z
Z
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R
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2
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q
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~ 1+#(& # %& -"0#)(&%. A!( #%%!; '!"$- #(#2!$ +"& $+*#(,'-# '-%@" (-'& # (& -! ( '!" #9# #
|E|
$-2#0(@& #% #9# B 84 #% #9# #% '-%-" (! '&()& !:. A! #2!$ -$0-")+-( !$ '&$!$F
&: G- r < R.
/" #$0# '&$! 0#"#2!$ >+# "! H&4 '&()& #"'#((& & *!( #% '-%-" (!; *!( %! '+&%
~ · n̂dS = Qint = 0
E
ǫ0
Ω
I
C #2I$;
I
Ω
~ · n̂dS =
E
Z
manto
= E(r)
E(r)r̂ · n̂dS +
Z
Z
tapas
E(r)r̂ · n̂dS
dS
manto
= E(r)(2πr)h
~ = ~0
=⇒ E(r)(2πr)h = 0 =⇒ E(r) = 0 =⇒ E
/$ -"0#(#$&"0# #% H#'H! >+# #% '&2*! #%?'0(-'! $#& "+%! #"0(! #% '-%-" (!.
<: G- R ≤ r.
/" #$0# '&$! 0#"#2!$ >+#
~ · n̂dS = E(r)(2πr)h = (2πR)hσ =⇒ E(r) = σ
E
ǫ0
ǫ0
Ω
A!( 0&"0!;
I
~
E(r)
=
J!0# >+# #% $&%0! # -$'!"0-"+- &
,'-#: #$ + ǫσ0 .
σ
ǫ0
~0
R
r
R
r
r<R
· ρ̂ R ≤ r
#% 2= +%! #% '&2*! #%?'0(-'! 8>+# #$ "!(2&% & %& $+*#(E
!"#$%&' () &*+ ,* -!%..
!
!"#$%&' ()
"# $%#&# '&( )%*$+%,'-%.& #*/0+%-( )# -(+1( Θ )# +()%2 3 4 )#&*%)() 5(+%(,6# ρ = αr 7(+(
r < R8 9(6-'6# 6( #&#+1:( 72$#&-%(6 (-';'6()( 72+ 6( )%*$+%,'-%.&8
*"$+,-./0
"(,#;2* <'# 6( #&#+1:( 72$#&-%(6 (*2-%()( ( '&( )%*$+%,'-%.& )# -(+1( =2 (6 -(;72 #60-$+%-2
<'# 0*$( 7+2)'-#> #*$? )()( 72+
Z
ǫ0
~ 2 d3 x
U=
E
2 R3
)2&)# 6( %&$#1+(6 #* *2,+# $2)2 #6 #*7(-%28 @2+ $(&$2A -2&2-%#&)2 #6 -(;72 #& $2)2 #6 #*7(-%2A
72)#;2* -(6-'6(+ 6( #&#+1:( 72$#&-%(6 <'# <'#+#;2*8
B()2 <'# 6( )%*$+%,'-%.& )# -(+1( #* *%;0$+%-( #*/0+%-(;#&$# 4 $(;,%0& 6( )#&*%)() )# -(+1(
$%#&# *%;#$+:( #*/0+%-(A ( 7+%2+% *(,#;2* <'# #6 -(;72 #60-$+%-2 $%#&# *%;#$+:( #*/0+%-(A #* )#-%+A
~ r) = E(r)r̂A )2&)# r #* 6( )%*$(&-%( )#6 -#&$+2 )# 6( #*/#+( (6 7'&$2 #& -'#*$%.& 4 r̂ #* #6
E(~
5#-$2+ '&%$(+%2 )%+%1%)2 )#6 -#&$+2 )# 6( #*/#+( (6 7'&$28
@2)#;2* )%5%)%+ #6 7+2,6#;( #& )2* +#1%2&#*
(> r < R 92&*%)#+#;2* -2;2 *'7#+C-%# )# %&$#1+(-%.& '&( -?*-(+( #*/0+%-( )# +()%2 + -2&
;%*;2 -#&$+2 <'# 6( )%*$+%,'-%.& )# -(+1( Θ8 @2+ 6( 6#4 )# D('** $#&#;2*
I
Ω
I
Ω
~ · n̂dS =
E
E(r)r̂ · n̂dS =
E(r)4πr2 =
E(r)4πr2 =
E(r)4πr2 =
=⇒
Qint
ǫ0
Z
1
ρd3 x
ǫ0 Θ
Z Z
Z
1 2π π r 3
αr sen(ϕ)drdϕdθ
ǫ0 0
Z r 0 0
4π
αr3 dr
ǫ0 0
παr4
ǫ0
α 2
E(r) =
r
4ǫ0
,> R ≤ r 92&*%)#+#;2* &'#5(;#&$# -2;2 *'7#+C-%# )# %&$#1+(-%.& '&( -?*-(+( #*/0+%-( )#
+()%2 + -2& ;%*;2 -#&$+2 <'# 6( )%*$+%,'-%.& )# -(+1( Θ8 @2+ 6( 6#4 )# D('** $#&#;2*
Qint
ǫ0
Ω
Z
4π R 3
2
αr dr
E(r)4πr =
ǫ0 0
παR4
E(r)4πr2 =
ǫ0
α R 2 2
R
=⇒
E(r) =
4ǫ0 r
I
~ · n̂dS =
E
!
"#$ %&'%#( %)')*#+ ,-)
~
E(r)
=
(
α 2
4ǫ0 r r̂
R 2 2
α
R r̂
4ǫ0 r
r<R
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&$6& Θ:
U
=
=
=
=
=
=
=
Z
ǫ0
~ 2 d3 x
E
2 R3
Z Z
Z
ǫ0 2π π +∞
E(r)2 r2 sen(ϕ)drdϕdθ
2 0
0
0
Z
4πǫ0 +∞
E(r)2 r2 dr
2
0
Z R 2
Z +∞ 2 8 α R
4πǫ0
α 6
r dr +
dr
2
2
16ǫ20 r2
R
0 16ǫ0
!
"
R
+∞
r7
1
πα2
·
− R8
8ǫ0
7 0
r R
7
2
πα
R
·
+ R7
8ǫ0
7
πα2 7
R
7ǫ0
!"#$%&' () &*+ ,* -!%..
!
!"#$%&' ()
"#$%&'()(*#% '#% (%+(),% $# -#$-.$/)&-,% '( ),'&# R0 -#$ '($%&','(% '( -,)1, 2#34*./)&5
-,% ρ 6 −ρ 4$&+#)*(%7 8#% -($/)#% '( ,*9,% (%+(),% (%/:$ , 4$, '&%/,$-&, *($#) ;4(
2R7 <(, d~ (3 2(-/#) ;4( 2, '(3 -($/)# '( 3, (%+(), =#%&/&2, ,3 -($/)# '( 3, (%+(), $(1,/&2,7
>)4(9( ;4( (3 -,*=# (3.-/)&-# ($ 3, &$/()%(--&?$ '( 3,% (%+(),% (% -#$%/,$/( 6 ($-4($/)(
%4 2,3#)7
*"$+,-./0
@(3 =)#93(*, !A0 %,9(*#% ;4( (3 -,*=# (3.-/)&-# ($ (3 &$/()&#) '( 4$, (%+(), *,-&B, '(
ρr
~
'($%&',' '( -,)1, 4$&+#)*( ρ (% E(r)
= 3ǫ
· r̂7
0
@( 3, %&*(/)C, '( 3, '&%/)&94-&?$0 , =)&#)& %,9(*#% ;4( ,'(*:% =,), R ≤ r (3 -,*=#
~ r) = E(r)r̂7 <& 9&($ $# (% $(-(%,)&# =,), 3, )(%#34-&?$ '(3 =)#93(*, -#$#-() (3
-4*=3( E(~
-,*=# (3.-/)&-# +4(), '( 3, (%+(),0 ($-#$/).*#%3# 4%,$'# 3, 3(6 '( D,4%%7
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(*#% ;4(
I
~ · n̂dS = Qint
E
ǫ0
Ω
4πR3 ρ
E(r)4πr2 =
3ǫ0
3
~ r) = R ρ r̂
=⇒
E(~
3ǫ0 r2
>#) /,$/#0 /($(*#% ;4( (3 -,*=# (3.-/)&-# 1($(),'# =#) 3, (%+(), *,-&B, (%
( ρr
3ǫ0 · r̂ r < R
~
E(r)
=
R3 ρ
r̂ R ≤ r
3ǫ0 r 2
>#$1,*#% (3 #)&1($ '( -##)'($,',% ($ (3 -($/)# '( 3, (%+(), '( '($%&',' '( -,)1, ρ7 G%C0
/($(*#% ;4( 3#% -,*=#% (3.-/)&-#% ($ 3#% &$/()&#)(% '( -,', (%+(), (%/:$ ','#% =#)
ρ
· ~r para |~r| < R
3ǫ0
ρ
~ para |~r − d|
~ <R
E~2 (r) = −
· (~r − d)
3ǫ0
E~1 (r) =
8#% 2(-/#)(% ($ 3, &$/()%(--&?$ '( ,*9,% (%+(),% %,/&%+,-($ %&*43/:$(,*($/(
~ < R 6 |~r| < R7 G%C0 =#) (3 =)&$-&=&# '( %4=()=#%&-&?$0 (3 -,*=# (3.-/)&-# ($ 3,
|~r − d|
&$/()%(--&?$ '( ,*9,% (%+(),% (%
~ = ρ · d~
~ r) = E~1 (~r) + E~2 (~r) = ρ · ~r − ρ · (~r − d)
E(~
3ǫ0
3ǫ0
3ǫ0
(3 -4,3 (% -#$%/,$/(0 /,3 -#*# %( ;4()C, =)#9,)7
!
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"#$ %&'()&*+,&-# %. ,$)/$ 012+34()&,$ 56$ %. %.#'&%$% +#&71)3. ρ > 0 1,+8$ +# 012+3.#
.'74)&,1 %. )$%&1 9:
$; <$2,+2. .2 ,$381 .24,()&,1 .# (1%1 .2 .'8$,&1 =+'$#%1 2.> %. ?$+'';:
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&#.'($*2.E
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%. ?$+''A '$*.31' D+. .2 ,$381 .24,()&,1 /.#.)$%1 81) 2$ .'7.)$ .'
( ρr
3ǫ0 · r̂ r < R
~
E(r)
=
R3 ρ
r̂ R ≤ r
3ǫ0 r 2
~ ~0) = ~0A 81) 21 ,+$2 .2 ,.#()1 .'
*; N.#.31' D+. .2 ,$381 .# .2 ,.#()1 %. 2$ .'7.)$ .' E(
+# 8+#(1 %. .D+&2&*)&1: @& 310.31' +# 81D+&(1 2$ ,$)/$A .'($ .G8.)&3.#($ +#$ 7+.)F$
ρr
F~ = −Q 3ǫ
· r̂A 2$ ,+$2 $8+#($ O$,&$ .2 ,.#()1 %. 2$ .'7.)$ ='&# &381)($) .# D+4 %&).,,&-#
0
'. O$>$ 310&%1 2$ ,$)/$;A 81) 21 ,+$2 .' +# 8+#(1 %. .D+&2&*)&1 .'($*2.:
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310&3&.#(1 .'
ρr
· r̂
3ǫ0
ρr
· r̂ = ~0
~a + Q
3mǫ0
ρ
=⇒ r̈ + Q
·r =0
3mǫ0
r̈ + w2 · r = 0
m~a = F~
= −Q
M1) ($#(1A .2 310&3&.#(1 %.
q 2$ ,$)/$ .' +# 310&3&.#(1 $)3-#&,1 '&382. > '+ 8.)I1%1 %.
2π
0
1',&2$,&-# .' T = w = 2π 3mǫ
Qρ
,; M1) .2 8)&#,&8&1 %. '+8.)81'&,&-#A .2 ,$381 .24,()&,1 ).'+2($#(. .'
( ρr
3ǫ0 · r̂ + E0 x̂ r < R
~
E(r)
=
R3 ρ
r̂ + E0 x̂ R ≤ r
3ǫ0 r 2
!"#$%&' () &*+ ,* -!%..
!""#$%&'&"()* "+ $,-.* "+/$0%&$* , +* +,%1* )"+ "2"
 ρx

 ( 3ǫ30 + E0 ) · x̂
~
( 3ǫR0 xρ2 + E0 ) · x̂
E(r) =

3

(E0 − 3ǫR0 xρ2 ) · x̂
34 %"#5+0,
−R < x < R
R≤x
x ≤ −R
6($*(0%"-*# +,# (5"7,# .*#&$&*("# )" "85&+&'%&*9
&: −R < x < R ;5"%"-*# 85"
ρx
−3ǫ0 E0
+ E0 = 0 =⇒ x1 =
3ǫ0
ρ
<,%, 85" x1 "3&#0"4 #" )"'" $5-.+&% 85"
x1 =
Rρ
−3ǫ0 E0
> −R =⇒ E0 <
ρ
3ǫ0
6#0, $*()&$&=( )"'" $5-.+&% E0 .,%, 85" >,?, 5( .5(0* )" "85&+&'%&* )"(0%* )" +,
"#@"%,9
A,'"-*# 85" #& "+ +,.+,$&,(* )" +, "("%1B, .*0"($&,+ "+/$0%&$, ∇2 U "( "+ .5(0*
)" "85&+&'%&* "# -,?*% 85" $"%*4 "(0*($"# 0,+ .5(0* "# "#0,'+"C #& "# -"(*% 85"
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.*0"($&,+ "+/$0%&$*9 E#B4 0"("-*# 85"
Z
~
E(x)
· x̂dx + c
Z ρx
= −
+ E0 dx + c
3ǫ0
ρx2
+c
= −E0 x −
6ǫ0
V (x) = −
U (x) = −QV (x)
= QE0 x + Q
=⇒ ∇2 U =
d2 U
dx2
=
ρx2
− Qc
6ǫ0
Qρ
>0
3ǫ0
<*% +* 0,(0*4 "# .5(0* )" "85&+&'%&* "#0,'+"9
&&: R ≤ x
;5"%"-*# 85"
R3 ρ
+ E0 = 0 =⇒ x2
3ǫ0 x2
imaginario
.*% +* 85" (* >,? .*#&$&=( )" "85&+&'%&* .,%, R ≤ x9
!
"""# x ≤ −R
$%&'&()* +%&
R3 ρ
E0 −
= 0 =⇒ x3 = −R
3ǫ0 x2
r
ρR
3ǫ0 E0
,-'- +%& x1 &."*/&0 *& 1&2& 3%(45"' +%&
r
r
Rρ
ρR
ρR
< −R =⇒
> 1 =⇒ E0 <
x3 = −R
3ǫ0 E0
3ǫ0 E0
3ǫ0
+%& &* 5- ("*(- 3)61"3"76 +%& &63)6/'-()* -6/&'")'(&6/& 4-'- +%& 8%2"&'- &+%"9
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Rρ
?- 4'%&2- 1& 5) *"@%"&6/& *& 1&<- -5 5&3/)': A&()* ="*/) +%& 4-'- E0 < 3ǫ
&."*/&6 1)* 4%6/)*
0
1& &+%"5"2'")0 %6) &*/-25& B &5 )/') "6&*/-25&: ,-'- E0 = 0 &."*/& %6- *)5- 4)*"3"76 1& &+%"5"2'")0
Rρ
&."*/& %6- *)5- 4)*"3"76 1& &+%"5"2'")0 &*
&6 &5 3&6/') 1& 5- &*C&'- B &* &*/-25&: ,-'- E0 = 3ǫ
0
Rρ
&6 x = −R B &* "6&*/-25&: D 4-'- E0 > 3ǫ0 6) 8-B 4)*"3")6&* 1& &+%"5"2'"): E)1) &*/) *& 4%&1&
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ρ = ρ0 (1 − Rr )3 '#$'( ρ0 (% *$/ +#$%2/$2( 4#%&2&5/3 %&($'# r ,/ '&%2/$+&/ -('&'/ '(%'( (, (6(
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~ = E r̂3 '#$'( r̂ (% (, 5(+2#) *$&2/)&#
A#'(-#% ($2#$+(% (%+)&B&) (, +/-4# (,9+2)&+# +#-# E
)/'&/, ($ +##)'($/'/% +&,>$')&+/%7 A#) ,/ ,(. '( 0/*%% %( 2($'): 1*(@
Z
Z
Z
~ · n̂dS = qinterior
~
~
E
(1)
E · n̂dS +
E · n̂dS =
ǫ0
manto
tapas
S
8, 5(+2#) $#)-/, n̂ '( ,/% 2/4/% (% 4()4($'&+*,/) / r̂7 "#$ (%2# %( 2($'): 1*( r̂ · n̂ = 0 . 4#)
,# 2/$2#
Z
Z
~ · n̂dS =
E r̂ · n̂dS = 0
(2)
E
tapas
tapas
8, 5(+2#) n̂3 $#)-/, /, -/$2#3 %(): 4/)/,(,# / r̂ . 4#) ,# 2/$2# r̂ · n̂ = 1 ,# 1*( &-4,&+/ 1*(@
Z
Z
Z
~ · n̂dS =
E
E r̂ · n̂dS = E ·
dS
manto
manto
manto
R
C/ &$2(0)/, manto dS +#))(%4#$'( /, :)(/ '(, -/$2#@ manto dS = 2πrL7 A#) ,# 2/$2#@
Z
~ · n̂dS = E · 2πrL
E
(3)
R
manto
D((-4,/E/$'# FGH . FIH ($ FJH )(%*,2/ 1*(@
qinterior
E · 2πrL =
ǫ0
(4)
!
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dq = ρ(r′ ) · dV
(5)
A(,( 1()1%)(, dV 1#5-67&,(0#- &) .#)%0&5 &5*,& 7#- 16)657,#- 7& )( )(,2# " 4 06-0# &B& $%&
&) 16)657,# 7&) +,#/)&0(C 1#5 ,(76#- r′ 4 r′ + dr′ 1#0# 0%&-*,( )( 92%,(@
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dV = 2πr′ dr′ L
D&&0+)(E(57# FGH &5 FIH@
(6)
dq = 2πr′ · dr′ · L · p(r′ )
dq = 2πr′ · dr′ · L · ρ0 (1 −
=⇒ dq = 2πL · ρ0 (r′ −
r′
)
R
r′ 2
)dr′
R
J5*&2,(57#@
q = 2πLρ0
Z
0
r
Z r ′2
Z r
r′ 2
r
′ ′
′
(r −
r dr −
)dr = 2πLρ0 · (
dr′ )
R
0 R
0
′
r3
3Rr2 − 2r3
r2
−
) = 2πLρ0 · (
)
2
3R
6R
πLρ0
(3Rr2 − 2r3 )
(7)
=⇒ q =
3R
=⇒ q = 2πLρ0 · (
K& F H 4 F!H@
E · 2πrL =
πLρ0
(3Rr2 − 2r3 )
3Rǫ0
K&-+&B(57# E @
E=
ρ0
(3Rr − 2r2 )
6Rǫ0
(8)
!"#$%&' () &*+ ,* -!%..
!
"#$%$&'( ()*$% +)%) ,#$ -).'% /$ r $. 0)&+' $.102%30' $( &)43&'5 6)%) $(2' /$%3-)&'( E
%$(+$02' ) %7
ρ0
dE
=
(3R − 4r)
dr
6Rǫ0
89#).):/' ) ; ($ '*23$:$ ,#$ $. -).'% /$ r *#(0)/' $(7
r=
3R
4
<$ $(2) &):$%)
|E|max =
3ρ0 R
16ǫ0
!
!"#$%&' ()
"#$%& '() *+,#() -#.#-$() *,%,+&+() /0& )& &#10&#$%,# , 0#, '-)$,#1-, 2a &+ 0#( '&+ ($%( )&
$-&#& 0#, '-)$%-201-3# 4(5(67#&, '& 1,%6, 1(# '&#)-',' ρ8 9,#6&#$& ,+ *+,#( '& +, '&%&14,
4,: 0# 1,)1,%3# 1,%6,'(; '& %,'-( R : '&#)-',' )0*&%.1-,+ σ <=&% .60%,>8 ?,+10+,%; 1(# &+
(%-6&# &# &+ *0#$( 5&'-( '& +, %&1$, /0& 0#& +() *+,#(); &+ 1,5*( &+&1$%-1( *,%, $('( x > 08
*"$+,-./0
@, 1(#.60%,1-3# '&+ *%(2+&5, &) +, )-60-&#$&A
B,%, &#1(#$%,% &+ 1,5*( &+71$%-1( )(2%& &+ &C& x<x > 0> 1,+10+,%&5() &+ 1,5*( &+71$%-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<x<a
"+/1.0('#'( %+)+ 1&*('6%.( 2#&11.#/# &/# %#7# 0( 8'(# $#-('#$ A 4 $#0+ 2x %+)+ )&(1-'# $#
1.2&.(/-( 62&'#5
9#0# $# 0.'(%%.:/ ;&( -.(/( ($ %#)*+ ($,%-'.%+ 1( -(/0'8 ;&( *+' $#1 -#*#1 <'+/-#$= *+1-('.+'=
./<('.+' 4 1&*('.+' ($ >&7+ 1('8 %('+= 4# ;&( ($ ?(%-+' /+')#$ # (1-#1 %#'#1 (1 *('*(/0.%&$#' #$
~ · n̂ = 0 $+ ;&( .)*$.%# ;&(
%#)*+ 4 0( (1-+ E
Z
~ · n̂dS = 0
E
@./ ()A#'2+= 1. B#A'8 >&7+ *+' $#1 %#'#1 $#-('#$(1 4 1( -(/0'8 ;&( ($ %#)*+ ($,%-'.%+ (1 *#'#$($+
# $+1 ?(%-+'(1 /+')#$(1 0( (1-#1 %#'#1 %+/ $+ %&#$ 1( %&)*$.'8 ~(E) · n̂ = E $+ ;&( .)*$.%# ;&(
Z
Z
~
dS
E · n̂dS = E
S
S
9( (1-# )#/('#= #*$.%#/0+ $# $(4 0( 2#&11= 1( -(/0'8 ;&( ($ >&7+ -+-#$ 1+A'( $# %#7# 1('8 ($
>&7+ # -'#?,1 0( $#1 %#'#1 $#-('#$(15
Z
Z
qint
(1)
dS + E
dS = EA + EA = 2EA =
φ=E
ǫ0
S2
S1
9+/0( S1 4 S2 1+/ $#1 1&*('6%.(1 $#-('#$(1 ./0.%#0#1 #/-('.+')(/-(3
C+0()+1 %#$%&$#' $# %#'2# ./-('.+' qint 4# ;&( 1#A()+1 $# 0.)(/1.+/(1 0( $# %#7# 4 $# 0(/1.0#0
?+$&),-'.%# 0( %#'2# ρ3 D# %#'2# ./-('.+' 1('8 1.)*$()(/-(
qint = ρV = ρ2xA
E(()*$#F#/0+ (1-( ?#$+' (/ GHI +A-(/()+15
ρ2xA
ǫ0
ρx
E=
ǫ0
2EA =
J 0#0# $# 1.)(-'K# ?()+1 ;&( ($ %#)*+ (1-# (/ 0.'(%%.:/ î= *+' $+ -#/-+ 1( -.(/( ;&(5
~ ρ = ρx î, si 0 < x < a
E
ǫ0
!
• x>a
"# $%&'#(# (# )*+,- .&%/, 0+# $,%, -, %#*)12 ,23#%)&% 4, 0+# -, 5)/#3%6, $%#5#23# #5 -, /)5/,
#2 #- ',/$& #-7'3%)'& 3,- '&/& 5# /+#53%, #2 -, 5)*+)#23# 8*+%,9
:- )*+,- 0+# #- ',5& ,23#%)&% #;)53)%< =+>& 5&-& #2 -,5 ',%,5 -,3#%,-#5 4 5#%<9φ = 2EA?
@, ',%*, )23#%)&% 5#%< -, ',%*, 0+# #53, (#5(# −a < x < aA $&% -& 3,23&9 qint = ρ2aA? :$-)',2(&
-, -#4 (# *,+559
ρ2aA
ǫ0
ρa
E=
ǫ0
φ = 2EA =
B- ',/$& #53, #2 ()%#'')12 î $&% -& 3,23&9
Eρ =
ρa
î, si x > a
ǫ0
C&3,% 0+# #- ',/$& #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<R
K,(& -, 5)/#3%6, #5.7%)', 4 -, (#25)(,( 5+$#%8'),- σ #5 '&253,23#A 5# 3#2(%< 0+# #- ',/$&
#-7'3%)'& #53,%< #2 ()%#'')12 %,(),-? L&25)(#%,%# '&/& 5+$#%8')# *,+55),2, +2, ',5',%, #5.7%)',
(# %,()& r < RE5+$#%8')# "H 3,- '&/& /+#53%, -, 8*+%,9
!"#$%&' () &*+ ,* -!%..
!
R
~ · n̂dS 5 0/*' /# +/2)'* n̂ /- 0(*(#/#' (# 2(67
"# $%&' ( )*(+,- ./ /-)( -%0/*123/ -/*4 φ = S E
R
~ · n̂ = E ; 0'* #' )(=)' φ = E dS = E4πr2 >
0'8(69'- *(.3(#/-: ; 0'* #' <%/ E
S
?'* ')*' #(.'5 #( 2(*@( 3=)/*3'* /- =%#( ;( <%/ -'#' A(; 2(*@( /= #( 2(-2(*(B qint = 0> ?'* #( #/;
./ @(%-- -/ )/=.*4 <%/
qint
φ = E4πr2 =
=0
ǫ0
E = 0, si r < R
• r>R
C'=-3./*(6'-5 (# 3@%(# <%/ /= /# 2(-' (=)/*3'*5 %=( 2(-2(*( /-D,*32( ./ *(.3' r > R 2'6'
-%0/*123/ @(%--3(=(> E(&' #'- 63-6'- (*@%6/=)'- ./ -36/)*F( (=)/*3'* +/6'- <%/ /# 2(60'
/#,2)*32' -/*4 *(.3(# ; 0'* #' )(=)'5 0(*(#/#' (# +/2)'* ='*6(# ; ./ /-)( 6(=/*(B
Z
Z
~
dS = E4πr2
E · n̂dS = E
φ=
S
S
G( 2(*@( 3=)/*=( qint -/*4 #( 2(*@( 2'=)/=3.( /= )'.( #( -%0/*123/5 #( 2%(# -/ '9)3/=/ 6%#)30#37
2(=.' /# 4*/( )')(# 0'* #( ./=-3.(. -%0/*123(# ./ 2(*@(B qint = σ4πR2 >
?'* #( #/; ./ @(%--5 -/ )/=.*4 <%/
φ = E4πr2 =
qint
σ4πR2
=
ǫ0
ǫ0
H/-0/&(=.' /# 2(60' /#,2)*32' -/ )3/=/ <%/
E=
σR2
ǫ0 r2
"-)/ 2(60' /-)(*4 /= .3*/223I= *(.3(#5 0'* #' )(=)'B
2
~ σ = σR r̂, si r > R
E
ǫ0 r2
H'=./ r̂ /- /# +/2)'* %=3)(*3' /= .3*/223I= *(.3(#>
JA'*( <%/ /=2'=)*(6'- #'- .'- 2(60'- /#,2)*32'- 0'./6'- /=2'=)*(* /# 2(60' */-%#)(=)/
-%0/*0'=3,=.'#'-> ?(*( /-)'5 2'=-3./*(*/ K */@3'=/-B
L: 0 < x < a
LL: a < x < a + 2R
LLL: x > a + 2R
~ σ = σR22 r̂> JA'*( )/=/6'~ ρ = ρx î ; E
L: 0 < x < aB "= /-)( */@3I= #'- 2(60'- /#,2)*32'- -'=B E
ǫ0
ǫ0 r
<%/ /M0*/-(* r ; r̂ ./ D'*6( 2'=+/=3/=)/ 0(*( 0'./* -%6(* (69'- 2(60'-> C'=-3./*/6'- #(
-3@%3/=)/ 1@%*(B
N/6'- <%/ -/ 2%60#3*4 <%/ a + R = x + r 2'= #' <%/ */-%#)( r = a + R − x> J./64-5 +/6'<%/ r̂ = −î ; 0'* #' )(=)' 0'./6'- /-2*393* /# 2(60' /#,2)*32' 2'6'B
~σ =
E
−σR2
î
ǫ0 (a + R − x)2
!
~ =E
~ρ + E
~ σ * #$ 4#,0)
"# #$%& '&(#)&* #+ ,&'-. #+/,%)0,. )#$1+%&(%# #( #$%& )#203( #$ E
σR2
~ = ( ρx −
)î
E
ǫ0
ǫ0 (a + R − x)2
556 a < x < a + 2R7 8( #$%& )#203( #+ ,&'-. +. &-.)%& $.+. Eρ 9& :1# #+ ,&'-. -).41,04. -.)
+& ,&$,&& #$;#)0,& #$ (1+. #( #+ 0(%#)0.) #++&< =.) +. %&(%.7
~ =E
~ ρ = ρa î, si a < x < a + 2R
E
ǫ0
5556 x > a + 2R 8( #$%& )#20.* &'>.$ ,&'-.$ #$%&)?( #( 40)#,,03( 9 $#(%04. î< =&)& .>%#(#)
~ σ #( ;1(,03( 4# @* ,.($04#)&'.$ +& $0210#(%# A21)&7
+& #@-)#$03( 4# E
B#'.$ :1# $# ,1'-+# +& )#+&,03( x = a + R + r* ,.( +. :1# )#$1+%& :1# r = x − a − R< C4#'?$*
r̂ = î< "# #$%& ;.)'&* %#(4)#'.$ :1#7
~σ =
E
σR2
î
ǫ0 (x − a − R)2
D1#2.* -.) -)0(,0-0. 4# $1-#)-.$0,03(* #+ ,&'-. #+/,%)0,. #( #$%& )#203( $#)?7
σR2
~ = ( ρa +
)î, si x > a + 2R
E
ǫ0
ǫ0 (x − a − R)2
E0(&+'#(%# #+ ,&'-. #+/,%)0,. )#$1+%&(%# -.) &'>&$ 40$%)0>1,0.(#$ 4# ,&)2& $#)?7
!"#$%&' () &*+ ,* -!%..
!
~ =
E
 ρx

 ( ǫ0 −

 ( ρa +
ǫ0
σR2
)î
ǫ0 (a+R−x)2
ρa
ǫ0 î
σR2
)î
ǫ0 (x−a−R)2
0<x<a
a < x < a + 2R
x > a + 2R
!"#$%&' ()
!"#$%&'(')"$ &"$ '$*'(+$ #" ,"#,-#.(%,+$ &' (+&%" R/ ,"# &'#$%&+&'$ &' ,+(0+ 1"23)-.(%,+$
ρ 4 −ρ 3#%*"()'$5 6"$ ,'#.("$ &' +)7+$ '$*'(+$ '$.8# + 3#+ &%$.+#,%+ )'#"( 93' 2R5 :'+ d~
'2 1',."( 93' 1+ &'2 ,'#.(" &' 2+ '$*'(+ ;"$%.%1+ +2 ,'#.(" &' 2+ '$*'(+ #'0+.%1+5 <(3'7' 93' '2
,+);" '2-,.(%," '# 2+ %#.'($',,%=# &' 2+$ '$*'(+$ '$ ,"#$.+#.' 4 '#,3'#.(' $3 1+2"(5
*"$+,-./0
>'2 ;("72')+ ?@/ $+7')"$ 93' '2 ,+);" '2-,.(%," '# '2 %#.'(%"( &' 3#+ '$*'(+ )+,%A+ &' &'#$%&+&
ρr
~
&' ,+(0+ 3#%*"()' ρ '$ E(r)
= 3ǫ
· r̂5
0
>' 2+ $%)'.(B+ &' 2+ &%$.(%73,%=#/ + ;(%"(% $+7')"$ 93' +&')8$ ;+(+ R ≤ r '2 ,+);" ,3);2'
~ r) = E(r)r̂5 :% 7%'# #" '$ #','$+(%" ;+(+ 2+ ('$"23,%=# &'2 ;("72')+ ,"#",'( '2 ,+);"
E(~
'2-,.(%," *3'(+ &' 2+ '$*'(+/ '#,"#.(-)"$2" 3$+#&" 2+ 2'4 &' C+3$$5
!"#$%&'(')"$ ,")" $3;'(D,%' &' %#.'0(+,%=# 3#+ ,8$,+(+ '$*-(%,+ &' (+&%" r > R5 E'#')"$
93'
I
~ · n̂dS = Qint
E
ǫ0
Ω
4πR3 ρ
E(r)4πr2 =
3ǫ0
3
~ r) = R ρ r̂
=⇒
E(~
3ǫ0 r2
<"( .+#."/ .'#')"$ 93' '2 ,+);" '2-,.(%," 0'#'(+&" ;"( 2+ '$*'(+ )+,%A+ '$
( ρr
3ǫ0 · r̂ r < R
~
E(r) =
R3 ρ
r̂ R ≤ r
3ǫ0 r2
<"#0+)"$ '2 "(%0'# &' ,""(&'#+&+$ '# '2 ,'#.(" &' 2+ '$*'(+ &' &'#$%&+& &' ,+(0+ ρ5 F$B/
.'#')"$ 93' 2"$ ,+);"$ '2-,.(%,"$ '# 2"$ %#.'(%"('$ &' ,+&+ '$*'(+ '$.8# &+&"$ ;"(
ρ
· ~r para |~r| < R
3ǫ0
ρ
~ para |~r − d|
~ <R
E~2 (r) = −
· (~r − d)
3ǫ0
E~1 (r) =
6"$ 1',."('$ '# 2+ %#.'($',,%=# &' +)7+$ '$*'(+$ $+.%$*+,'# $%)32.8#'+)'#.'
~ < R 4 |~r| < R5 F$B/ ;"( '2 ;(%#,%;%" &' $3;'(;"$%,%=#/ '2 ,+);" '2-,.(%," '# 2+ %#.'($',,%=#
|~r −d|
&' +)7+$ '$*'(+$ '$
~ = ρ · d~
~ r) = E~1 (~r) + E~2 (~r) = ρ · ~r − ρ · (~r − d)
E(~
3ǫ0
3ǫ0
3ǫ0
'2 ,3+2 '$ ,"#$.+#.'/ .+2 ,")" $' 93'(B+ ;("7+(5
!
!"#$%&' () &*+ ,* -!%..
!"#$%&' (
!"#$%&'( )(#%"*!#+"'"&%!
!"#$%&' ()
!"#$%&'& (" )*+"! $","$-! %& %&"#$%+% %& .+'/+ #()&',.$+* ("$0!'1& σ > 0 "!'1+* +* &2& 3
%& &.(+.$4" x = 05 6" ax̂ #& &".(&"-'+ ("+ .+'/+ )("-(+* −q < 05
+7 6".(&"-'& &* )!-&".$+* &*8.-'$.! #!9'& &* &2& 3 : &"-'& *+ .+'/+ −q < 0 : &* !'$/&"
.!!'%&"+%! ;5
97 <"+ )+'-=.(*+ %& 1+#+ m : .+'/+ −e < 0 #& (9$.+ &" &* )("-! 1&%$! &"-'& −q : ; : #& %&2+
*$9'&5 > !" ?(8 &"&'/=+ .$"8-$.+ **&/+ *+ .+'/+ +* )*+"!@ A$/"!'& &0&.-!# /'+B$-+.$!"+*&#75
*"$+,-./0
+7 !"!.&1!# &* .+1)! &*8.-'$.! /&"&'+%! )!' &* )*+"! : *+ .+'/+ −q #!9'& &* &2& 3 ?(& *!#
("&5 C+*&# .+1)!# #!"
σ
E~1 =
E~2 =
2ǫ0
x̂
−q
1
q
1
· −x̂ =
· x̂
2
4πǫ0 (a − x)
4πǫ0 (a − x)2
D#=E &* .+1)! &*8.-'$.! &"-'& ; : −q A?(& &#-FE %$/+1!#E &" &* )("-! D7 &#
~ = E~1 + E~2 =
E
σ
1
q
+
2ǫ0 4πǫ0 (a − x)2
· x̂
~ E &"-!".&# &" Ω A*+ '&/$4" )&%$%+7
G&'! :+ ?(& E~ = −∇V
∂V
q
1
σ
=0 −
=
+
∂x
2ǫ0 4πǫ0 (a − x)2
q
1
σx
+C
+
=⇒ V (x) = −
2ǫ0 4πǫ0 (a − x)
∂V
=0
∂z
∂V
∂y
97 6* .+1)! &*&.-'!#-F-$.! &# .!"#&'B+-$B!E +#= ?(& )!%&1!# (#+' .!"#&'B+.$4" %& &"&'/=+
&"-'& &"&'/=+ )!-&".$+* : .$"8-$.+5 H+ .+'/+ −e )+'-& %&* '&)!#! %&#%& a/2x̂ I+.$+ &*
)*+"!5 G!' .!"#&'B+.$4" %& &"&'/=+E -&"&1!# ?(&
JK
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
−eV (a/2) =
=⇒ K =
mv 2
2
mv 2
− eV (0)
2
= e (V (0) − V (a/2))
1 q
1 2q
σa
= e −
+C +
+
−C
4πǫ0 a
4πǫ0 a
4ǫ0
σa
1 q
= e
+
4πǫ0 a 4ǫ0
!
!"#$%&' ()
"# $%#&#& '() #)*+,-) .(&'/.$(,-) '# ,-'%() r1 , r2 0 .-,1-) q1 , q2 )#2-,-'-) 2(, /&- 1,-& '%)3
$-&.%- d >> r1 , r2 4 "% -56-) )# .(&#.$-& - $,-7+) '# /& .-68# .(&'/.$(, 9'#)2,#.%-68#: ;/#
)%,7# )<8( 2-,- $,-&)2(,$-, .-,1- '# /&- - ($,-=: #&./#&$,# 8-) '#&)%'-'#) '# .-,1- )/2#,>.%-8#)
'# .-'- /&- 9#& */&.%<& '# 8-) 7-,%-68#) .(&(.%'-)= /&- 7#? ;/# #8 )%)$#5- -8.-&?- #8 #;/%8%6,%(4
*"$+,-./0
@-'( ;/# #8 )%)$#5- #)$A #& /&- ,#1%<& -.($-'- '#8 #)2-.%(: 2('#5() $(5-, .(5( 2/&$( '#
,#*#,#&.%- '#8 2($#&.%-8 #8 %&>&%$( # %1/-8-, #8 2($#&.%-8 - .#,( -88B: #) '#.%,: V (+∞) = 04 C)B:
#8 2($#&.%-8 )(6,# 8-) )/2#,>.%#) '# 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-.%<& '# .-,1-:
)% (q1 )f , (q2 )f )(& 8-) .-,1-) #& .-'- #)*#,- /&- 7#? -8.-&?-'( #8 #;/%8%6,%(: $#&#5() ;/#
q1 + q2 = (q1 )f + (q2 )f
C8 -8.-&?-, #8 #;/%8%6,%(: #) '#.%,: ./-&'( '#F- '# D-6#, $,-&)*#,#&.%- '# .-,1-) #&$,# 8-) #)*#,-):
$#&#5() ;/# 8- '%*#,#&.%- '# 2($#&.%-8 #8+.$,%.( #&$,# -56-) #) &/8(: #) '#.%,:
V1
(q1 )f
k
r1
2
4π(r1 ) (σ1 )f
r1
(σ2 )f
=⇒
(σ1 )f
= V2
(q2 )f
= k
r2
4π(r2 )2 (σ2 )f
=
r2
r1
=
r2
@# #)$( 7#5() ;/# #& 1#&#,-8: 8-) ,#1%(&#) #& 8- )/2#,>.%# '# /& .(&'/.$(, .(& 5#&(,
,-'%( '# ./,7-$/,- 92/&$-)= .(&.#&$,-& /&- 5-0(, '#&)%'-' )/2#,>.%-8 '# .-,1-: 2(, 8( ./-8 #8
.-52( #8+.$,%.( .#,.- '# #88-) 9#& )/ #G$#,%(,= #) 5A) */#,$# ;/# #& ,#1%(&#) .(& 5#&(, ,-'%(
'# ./,7-$/,-4
H)-&'( 8- ,#8-.%<& #&.(&$,-'- #& 8- #./-.%<& '# .(&)#,7-.%<& '# .-,1-: (6$#&#5()
(q1 )f + (q2 )f
2
2
4π(r1 ) (σ1 )f + 4π(r2 ) (σ2 )f
= q1 + q2
= q1 + q2
4πr1 (σ1 )f (r1 + r2 ) = q1 + q2
1 q1 + q2
=⇒ (σ1 )f =
4πr1 r1 + r2
1 q1 + q2
(σ2 )f =
4πr2 r1 + r2
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
!"#$%&' ()
!"#!$%&! '#! () *+$,-.%+.#+*)* !. |E~⊥ (~r)| )( /)$)& *! #.) &!0+1. 2),3) ) -%&) 2),3) /-& #.)
$#/!&4,+! ,)&0)*) *! *!.$+*)* $#/!&4,+)( σ(~r)5 ,-. ~r !. () $#/!&4,+!5 !$
σ(~r)
ǫ0
6-.$+*!&).*- !$%-5 !.,#!.%&! !( ,)"/- !(7,%&+,- !. () $#/!&4,+! *! #. ,-.*#,%-& !. !'#+(+8&+!(!,%&-$%9%+,-:
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0 < A << 15 %)( '#! () $#/!&4,+! (- ,&#=! /-& () "+%)*: >-& () (!; *! 0)#$$5 %!.!"-$
~ r) · n̂dS = Qint
E(~
ǫ0
Ω
I
>!&-
Qint ∼ σ(~r)A
Qint ∼
=
= σ(~r)A =⇒
ǫ0
ǫ0
?!.!"-$ )*!"9$
I
Ω
~ r) · n̂dS =
E(~
Z
manto
~ r) · n̂dS +
E(~
Z
carasup
E~1 (~r) · nˆ1 dS +
Z
carainf
E~2 (~r) · nˆ2 dS
>-*!"-$ !@/&!$)& !( ,)"/- !(7,%&+,- ,-"- () $#") *! $# ,-"/-.!.%! %).0!.,+)( ; .-&")(
) () $#/!&4,+!5 !$ *!,+&5 E~ = E~⊥ + E~// : A*!"9$5 ,-"- !( 9&!) !$ "#; /!'#!<)5 !( "1*#(*!( ,)"/- !(7,%&+,- $-8&! 7( $! ").%+!.! ,)$+ ,-.$%).%!5 /-& (- '#! (- /-*!"-$ )/&-@+")& /-&
~ r)| !. %-*) !( 9&!): ?)"8+7.5 nˆ1 = −nˆ2
|E(~
6-. !$%- !. "!.%!5 %!.!"-$ '#!
Z
carasup
E~1 (~r) · nˆ1 dS +
Z
carainf
E~2 (~r) · nˆ2 dS = (E~1 )⊥ A − (E~2 )⊥ A
= ((E~1 )⊥ − (E~2 )⊥ )A
!@/&!$+1. '#! .- *!/!.*! *! h B)( +0#)( '#! σ(~ǫr)A C: ?)"8+7. %!.!"-$ '#!
0
Z
manto
~ r) · n̂dS → 0
E(~
cuando
h→0
!
"#$% &'(')*# +(,-)'(&'
σ(~r)A
ǫ0
σ(~r)A
△E⊥ A
=
ǫ0
σ(~r)
=⇒ △E⊥ (~r) = +
ǫ0
((E~1 )⊥ − (E~2 )⊥ )A
=
.,/')*# 01' 1( 2*(312&*4 '( '015-5/45* '-62&4*#&7&52* (* 84'#'(&, 2,49, '( #1 5(&'45*4% '2,)8* '-62&452* '( #1 5(&'45*4 '# (1-* : &*3, #1 2,49, '#&7 35#&45/153, '( -, #18'4+25' 3' &,;*4), 01' '- 2,)8* '-62&452* '( -, #18'4+25' '# (*4),- , -, )5#), <81'# #5 &1=5'4, 2*)8*('(&'
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'#
~ r) = + σ(~r) n̂
E(~
ǫ0
2*( n̂ '- ='2&*4 (*4),- 'A&'45*4 , -, #18'4+25' '( ~r?
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
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5 6 768*91 5 '# #):#)(+ '#):+#0%*63# *-&;-# <&%$(= >* *+,*'-+* %&$#+&* )# 0*+?* 0(& -&* 0*+?*
q0 > 0= "# )-:(&# ;-# 3* $%#++* 7;-# )# $(,* 0(,( (+%?#& '# :($#&0%*3#)9 #)$2 %&<&%$*,#&$#
*3#@*'*=
A= >* *+,*'-+* #B$#+&* )# 0(&#0$* * $%#++* * $+*C/) '# -&* 6*$#+D* 0-5* '%.#+#&0%* '#
:($#&0%*3 #&$+# )-) 6(+&#) #) V0 =
*9 E*30-3# 3* 0*+?* ;-# )# %&'-0# #& 3*) )-:#+<0%#) %&$#+%(+ 5 #B$#+%(+ '# 0*'* -&* '#
3*) *+,*'-+*)=
69 E*30-3# #3 0*,:( #3/0$+%0( #& $('() 3() :-&$() '#3 #):*0%(=
09 E*30-3# #3 :($#&0%*3 #3/0$+%0( #& $('() 3() :-&$() '#3 #):*0%(1 5 3* '%.#+#&0%* '#
:($#&0%*3 #&$+# 3*) *+,*'-+*)=
AA= "# 0(+$(0%+0-%$* 3* 6*$#+D* 70(&#B%F& '%+#0$* * $%#++*9= G#:%$* 3() 0*30-3() *&$#+%(+#)
%&$#3%?#&$#,#&$#=
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( &(I
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JB:3%;-# 6+#C#,#&$# )-) +#):-#)$*)=
*"$+,-./0
A= J3 :+(63#,* $%#&# )%,#$+D* #)./+%0* 0(& 0#&$+( '# 0((+'#&*'*) #3 0#&$+( '# 3*) 02)0*+*)
#)./+%0*)1 '#6%'( * ;-# 3* $%#++* #)$2 %&<&%$*,#&$# 3#@*&* 7+#0(+'*+ ;-# 3* $%#++* )# :%#&)*
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3* '%)$*&0%* *3 0#&$+( '# 3*) 02)0*+*) 5 #3 0*,:( *:-&$*+2 #& #3 )#&$%'( '# r̂1 #) '#0%+1
~ r) = E(r)r̂=
V = V (r) 5 E(~
*9 "#*& q1 , q2 , q3 , q4 3*) 0*+?*) '# 3*) )-:#+<0%#) %&$#+%(+ 5 #B$#+%(+ '# 3*) *+,*'-+*)
%&$#+&* 5 #B$#+&*1 +#):#0$%C*,#&$#= K#&#,() ;-# q0 = q1 + q2 =
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)-:#+<0%#) '# 3* :+%,#+* *+,*'-+*1 $#&#,()
I
~ · n̂dS = 0 = q1 =⇒ q1 = 0 =⇒ q2 = q0
E
ǫ0
~ = ~0 :-#) #3 0*,:( #3/0$+%0( #& #3 %&$#+%(+ '# -& 0(&'-0$(+ #& #;-%3%6+%(
'(&'# E
#3#0$+()$2$%0( #) &-3(=
!
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I
~ · n̂dS = 0 = q1 + q2 + q3 =⇒ q1 + q2 + q3 = 0 =⇒ q3 = −q2 = −q0
E
ǫ0
"#$%&'()* )- (1-9* -$ .-*,-/' )- 0'122 &*( 1(' 21#-,3&%- -245,%&' &*(&5(.,%&' ),')%* r > b 6 :1- &*(.-(7' $' 2-71()' ',/')1,'8 .-(-/*2
I
q4 r̂
~
~ · n̂dS = E(r)4πr2 = q1 + q2 + q3 + q4 =⇒ E(r)
=
E
ǫ0
4πǫ0 r2
;*, *.,' #',.-8 2'<-/*2 :1- $' )%4-,-(&%' -(.,- $' ',/')1,' -=.-,(' 6 $' .%-,,' >-(
-$ %(3(%.*? -2 V0 8 -2.* -2
Z b
~ · d~r
E
V (b) − V (+∞) = V (b) = V0 = −
+∞
b
V0 = −
V0
=⇒ q4
Z
+∞
q4 dr
4πǫ0 r2
q4 1
=
4πǫ0 b
= 4πǫ0 bV0
)*()- .*/'/*2 1( &'/%(* ,')%'$ 2*<,- $' %(.-7,'$ )- $@(-'A
<? B%9%)'/*2 -$ -2#'&%* -( 9',%'2 ,-7%*(-2A C*)'2 $'2 21#-,3&%-2 )- %(.-7,'&%D( :1.*/',-/*2 2-,E( -245,%&'2 6 &*(&5(.,%&'2 ' $'2 ',/')1,'2A
%? r ≤ a >,-7%D( %(.-,%*, 6 /-.'$ )- $' ',/')1,' %(.-,('?
F% -$ ,')%* )- $' 21#-,3&%- )- %(.-7,'&%D( -2 r ≤ a
I
~ · n̂dS = E(r)4πr2 = q1 = 0 =⇒ E
~ = ~0
E
ǫ0
%%? a < r < b >,-7%D( -(.,- ',/')1,'2?
G2 -9%)-(.- :1- -=%2.- &'/#* -$5&.,%&* -( -2.' ,-7%D(8 #1-2 +'6 1(' )%4-,-(&%'
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q0 r̂
~ · n̂dS = E(r)4πr2 = q0 =⇒ E(r)
~
E
=
ǫ0
4πǫ0 r2
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~
G( -2.- #1(.*8 E(b)
= ~08 #1-2 -$ &'/#* -$5&.,%&* -( -$ %(.-,%*, )- 1( /-.'$
-2 2%-/#,- (1$*A
%9? r > b >,-7%D( -=.-,%*, ' $'2 ',/')1,'2?
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bV0
~
~ · n̂dS = E(r)4πr2 = q4 = 4πǫ0 bV0 =⇒ E(r)
= 2 · r̂
E
ǫ0
ǫ0
r
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
"#$%&'#()*+ #, -.&/* #,0-12'-* #$




~
E(r)
=



~0
q0
4πǫ0
~0
bV0
r2
·
r̂
r2
· r̂
r≤a
a<r<b
r=b
b≤r
-3 4.&5'0( )'6').&*$ #, #$/.-'* #( 6.2'.$ 2#7'*(#$8
'3 a ≤ r ≤ b
4#(#&*$
V (r) = −
~ · dr̂ + C
E
q0 dr
+C
4πǫ0 r2
q0
1
· +C
4πǫ0 r
= −
=
Z
Z
)*()# #, -.&'(* )# '(1#72.-'9( %$.)* :%# 2.)'.,8
;.2. )#1#2&'(.2 < '&/*(#&*$ ,. -*()'-'9( )# 5*2)#8 =.5#&*$ >%# V (b) = V0 +
q0
/*2 1.(1* C = V0 − 4πǫ
· 1b + #(1*(-#$
0
1 1
q0
+ V0
−
V (r) =
4πǫ0 r
b
;*2 1.(1*+ 1#(#&*$ >%# ,. )':#2#(-'. )# /*1#(-'., #(12# ,.$ .2&.)%2.$ #$
1 1
q0
−
V (a) − V (b) =
4πǫ0 a b
''3 r ≤ a
?, /*1#(-'., #( r = a @.2&.)%2. '(1#2(.3 #$
q0
1 1
V (a) =
+ V0
−
4πǫ0 a b
<*&* #, -.&/* #,0-12'-* #( #$1. 2#7'9( #$ (%,*+ #(1*(-#$ 1*). ,. 2#7'9( )#5#
1#(#2 #, &'$&* /*1#(-'., @)#5# $#2 #>%'/*1#(-'.,3+ /*2 ,* >%#
q0
1 1
+ V0
V (r ≤ a) =
−
4πǫ0 a b
'''3 b ≤ r
V (r) = −
=
~ · dr̂ + C‘
E
q4 dr
+ C‘
4πǫ0 r2
q4
1
· + C‘
4πǫ0 r
bV0
+ C‘
r
= −
=
Z
Z
!
"#$% V (b) = V0 =⇒ C‘ = 0
=⇒ V (r) =
bV0
r
&#'()*#+,%- #. /%0#+1*2. #.310$*1% #'
V (r) =



q0
4πǫ0
q0
4πǫ0
1
a
1
r
− 1b + V0 r ≤ a
− 1b + V0 a < r < b
bV0
b≤r
r
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~ 5
/(#' '@.% #:*'0# 12)/% 1(2+,% 92A (+2 ,*>#$#+1*2 ,# /%0#+1*2.5 &#1(#$,# <(# E
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,*19% 2+0#$*%$)#+0#5 C# #'02 >%$)2- 0#+#)%'
12$?2D
q1 = 0
q2 = q0
q3 = −q0
q4 = 0
12)/%D
~ = ~0
r≤a : E
~ = q0 · r̂
a<r<b : E
4πǫ0 r2
~ = ~0
b≤r : E
/%0#+1*2.D
1 1
q0
−
r ≤ a : V (r) =
4πǫ0 a b
q0
1 1
a < r < b : V (r) =
−
4πǫ0 r
b
b ≤ r : V (r) = 0
E .2 ,*>#$#+1*2 ,# /%0#+1*2. #+0$# 2$)2,($2' #' V (a) − V (b) =
)*')2 <(# #+ #. 12'% 2+0#$*%$5
q0
4πǫ0
1
a
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1
b
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ǫ0
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ǫ0
ǫ0
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V (~r) =
Z
Ω
dq
4πǫ0 |~r − r~1 |
3$ (%5( 4+%#< %& ~r = xx̂ + y ŷ + z ẑ 1 r~1 = x1 x̂< 4#$ 0 ≤ x1 ≤ L< %#$ -+% 6#%&4&#$(% '(- 6*$5# 1
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V (~r) =
Z
L
0
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R
√ dx 2
x2 +1
9& x = tan(θ)< ($5#$4(%
sec2 (θ)dθ
p
=
tan2 (θ) + 1
R
dx1
3=5($'&($'# (%5# + √
2
2
Z
√
dx
=
x2 + 1
Z
Z
(x1 −x) +y +z 2
Z
p
sec(θ)dθ = ln(sec(θ)+tan(θ)) = ln(x+ x2 + 1) = arccosh(x)
)(%*-5+ >,()?@8*(-#A
dx1
p
λdx1
p
4πǫ0 (x1 − x)2 + y 2 + z 2
(x1 − x)2 + y 2 + z 2
= ln(x1 − x +
B%?< (- 6#5($4&+- (%
V (x, y, z) =
=
=
=
!"#$%&' 12
p
((x1 − x)2 + y 2 + z 2 )
L
λdx1
p
2
2
2
0 4πǫ0 (x1 − x) + y + z
Z L
λ
dx1
p
4πǫ0 0
(x1 − x)2 + y 2 + z 2
p
L
λ
· ln(x1 − x + (x1 − x)2 + y 2 + z 2 )
4πǫ0
0
"
!
p
2
2
2
L − x + (L − x) + y + z
λ
p
· ln
4πǫ0
−x + x2 + y 2 + z 2
Z
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q
1
1
V =
−
4πǫ0 r+ r−
6,( /% /&: 0&/ ",2&',3 -&'&;,2 <4&
2
r+
= r2 + d2 − 2rd · cos
=⇒
1
r+
π
−θ
2
= r2 + d2 − 2rd · sen(θ)
1
= p
2
2
r + d − 2rd · sen (θ)
1
1
·q
=
r
1 + ( d )2 − 2 d · sen (θ)
r
r
6&(, &2-%;,2 "%/"4/%'0, &/ +,-&'".%/ +%(% r >> d3 +,( /, <4&
1
r+
=
∼
=
=
= 24 >&?3 -&'&;,2 +%(% r−
d
r
<< 1 :
1
1
·q
r
1 + ( dr )2 − 2 dr · sen (θ)
!
""
! d
1
d 2
1
− 2 · sen(θ)
· 1−
r
2
r
r
!
"
2
d
1
1 d
+ · sen(θ)
· 1−
r
2 r
r
!
2
r−
= r2 + d2 − 2rd · cos
π
2
+ θ = r2 + d2 + 2rd · sen (θ)
" #$ %&'() )*+,&-) ) ,& )*.$'/&'
1 ∼1
= ·
r−
r
!
1
1−
2
"
2
d
d
− · sen(θ)
r
r
0&' ,& 12$ $, 3&.$*4/), 3)') r >> d '$52,.)
V (r, θ) =
∼
=
=
=
=
V (r, θ) =
1
1
−
r + r−
!!
" !
""
q
1 d 2 d
1 d 2 d
+ · sen(θ) − 1 −
− · sen(θ)
·
1−
4πǫ0 r
2 r
r
2 r
r
2d
q
·
· sen(θ)
4πǫ0 r
r
2qd sen(θ)
·
4πǫ0
r2
p
sen(θ)
·
4πǫ0
r2
p~ · r̂
1
·
4πǫ0 r2
q
4πǫ0
#&*#$ p~ $5 $, (&($*.& #/3&,)' $,64.'/4& #$, #/3&,&7
89 :)8$(&5 12$
~ = −∇V
ˆ = − ∂V r̂ − 1 ∂V θ̂
E
∂r
r ∂θ
;5<= .$*$(&5 12$
∂V
1 psen(θ)
=−
∂r
2πǫ0 r3
1 pcos(θ)
1 ∂V
=
r ∂θ
4πǫ0 r3
~ θ) =
=⇒ E(r,
p 2sen(θ)r̂ − cos(θ)θ̂
4πǫ0 r3
49 >&*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&,&= ,) $*$'-<) )5&4/)#) ), #/3&,& $5
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
U
= qV (x + dx, y + dy, z + dz) − qV (x, y, z)
= q (V (x + dx, y + dy, z + dz) − V (x, y, z))
U
= qdV
∂V
∂V
∂V
= q
dx +
dy +
dz
∂x
∂y
∂z
∂V
∂V
∂V
px +
py +
pz
=
∂x
∂y
∂z
~
= p~ · ∇V
~
= −~
p·E
!
!"#$%&' ()
"#$%&'()( *$+ )(,&-$ (%./)&0+1 '( )+'&# b1 2*( 3&($( *$+ '&%3)&4*0&-$ '( 0+),+ *$&.#)5( ρ(r) =
ρ0 6+)+ 7+ )(,&-$ '(3()5&$+'+ 6#) a < r < b 8 '($%&'+' $*7+ 6+)+ r < a9 :(3()5&$( (7
6#3($0&+7 (7(03)#%3;3&0# ($ 3#'# (7 (%6+0&#9
*"$+,-./0
'1 <# 2*( =+)(5#% 6)&5()# %(); 0+70*7+) (7 0+56# (7/03)&0# 6)#'*0&'# 6#) (%3+ '&%3)&4*0&-$
'( 0+),+ ($ 3#'# (7 (%6+0&# 8 0#$ (%3( )(%*73+'# #43($')(5#% (7 6#3($0&+7 *3&7&>+$'#?
Z p
~ · d~ℓ
E
Vp = −
∞
"#$%&'()+$'# (7 6#3($0&+7 &,*+7 + 0()# ($ (7 &$@$&3#9
~ = E r̂1 '#$'( r̂ (% (7 C(03#)
A7 6)#47(5+ 3&($( %&5(3)B+ (%./)&0+ 8 6#) 7# 3+$3# %( 3($'); 2*( E
*$&3+)&# ($ '&)(00&-$ )+'&+79
"#$%&'()+)(5#% D )(,&#$(%?
EF r < a
EEF a < r < b
EEEF r > b
• 21 r < a0
"#$%&'()# 0#5# %*6()@0&( ,+*%%&+$+ *$ 0+%0+)#$ (%./)&0# '( )+'&# r < aG%*6()@0&( HF9 H( 3&($(
2*( (7 C(03#) $#)5+7 + (%3+ %*6()@0&( (% 6+)+7(7# + r̂9 :( (%3+ 5+$()+ %( 3($'); 2*( (7 0+56#
~ (% 6+)+7(7# +7 C(03#) $#)5+7 n̂ '( 7+ %*6()@0&( H 6#) 7# 2*( E
~ · n̂ = E 8 6#) 7# 3+$3#?
E
Z
Z
~
dS = E4πr2
E · n̂dS = E
φ=
S
S
I#) #3)# 7+'# 7+ 0+),+ ($ (7 &$3()&#) '( 7+ %*6()@0&( (% $*7+ 8+ 2*( (%3+ 0#$3($&'+ ($ (7
0#$'*03#)? qint = 09 J67&0+$'# 7+ <(8 '( K+*%% %( 3($'); 2*(?
E4πr2 = 0
L 6#) 7# 3+$3# E = 01 %& )M+9
• 221 a < r < b0
"#$%&'()# 0#5# %*6()@0&( ,+*%%&+$+ *$+ 0+%0+)+ (%./)&0+ '( )+'&# )G%*6()@0&( HF 0#5# 5*(%3)+
7+ @,*)+?
~ · n̂ = E 8 6#) 7# 3+$3#?
J7 &,*+7 2*( ($ (7 0+%# +$3()&#) %( 3($'); 2*( E
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
φ=
Z
S
~ · n̂dS = E4πr2
E
"# $#%&# '()*%'+% qint ,* -.*/* $#0$.0#% 12$'03*()* 4# 5.* 0# /*(,'/#/ /* $#%&# *, $+(,)#()*6
7* )*(/%2 5.* qint = ρ0 V 6 V *, *0 8+0.3*( *()%* 0# *,1*%# /* %#/'+ r 4 0# *,1*%# /* %#/'+ a $+(
0+ 5.* %*,.0)#9
4
qint = ρ0 π(r3 − a3 )
3
:-0'$#(/+ 0# 0*4 /* &#.,,9
E4πr2 = ρ0
;+( 0+ 5.* ,* +<)'*(*9
E=
4
π(r3 − a3 )
3ǫ0
ρ0 (r3 − a3 )
3r2 ǫ0
;+3+ $+3*()#3+, #()*%'+%3*()* *,)* $#3-+ *,)# *( /'%*$$'=( %#/'#0 r̂> -+% 0+ )#()+9
3
3
~ = ρ0 (r − a ) r̂
E
3ǫ0 r2
•
! r > b"
?%+$*/'*(/+ /* '&.#0 3#(*%# 5.* *( 0# %*&'=( @@A 8*%*3+, 5.* 0# $#%&# #03#$*(#/# *( 0#
/',)%'<.$'=( 5.* %+/*# #0 $+(/.$)+% ,*%29
4
qint = πρ0 (b3 − a3 )
3
B* *,)# 3#(*%# #-0'$#(/+ 0# 0*4 /* &#.,, )*(/%*3+, 5.*9
E4πr2 =
E=
ρ0 (b3 − a3 )
3rǫ0
ρ0 (b3 − a3 )
3ǫ0 r2
C*$)+%'#03*()* 5.*/# *D-%*,#/+ $+3+9
ρ0 (b3 − a3 )
~
r̂
E(r)
=
3ǫ0 r2
!
"#$%&'#()*+ #, -.&/* #,0-12'-* #( 1*)* #, #$/.-'*

0


ρ0 (r 3 −a3 )
~
r̂
E=
3ǫ0 r 2

 ρ0 (b3 −a3 )
r̂
3ǫ0 r 2
2#$%,1.
r<a
a<r<b
r>b
34*2. 5%# 1#(#&*$ #, -.&/* #,0-12'-* #( 1*)* #, #$/.-'* /*)#&*$ -.,-%,.2 #, /*1#(-'., #,0-12'-*6
7#8#&*$ #,#9'2 %(. 12.:#-1*2'. /.2. 1#(#2 %(. #;/2#$'<( /.2. #, =#-1*2 d~ℓ6 >,#9'&*$ %(. ,?(#.
2#-1. 5%# =.:. )#$)# #, '(@('1* 4.$1. %(. )'$1.(-'. r > b
~ · d~ℓ = E r̂ · d~ℓ = Edr6 7# #$1. &.(#2. #, /*1#(-'., #,0-12'-* $#2BC
A*&* /*)#&*$ =#2 E
Z r
Z r
Z r
ρ0 (b3 − a3 )
~
~
Edr = −
E · dℓ = −
V (r) = −
)dr
3ǫ0 r2
∞
∞
∞
D(1#92.()* $# *81'#(# 5%#C
V (r) =
ρ0 (b3 − a3 )
,
3ǫ0 r
si r > b
34*2. 5%#2#&*$ -.,-%,.2 #, /*1#(-'., . %(. )'$1.(-'. r+ )*()# a < r < b6 E*2 )#@('-'<(
1#(#&*$ 5%#
Z r
~ · d~ℓ
E
V (r) = −
∞
7#$-*&/*(#&*$ #$1. '(1#92., )# ,?(#. $#/.2.()* ,. 12.:#-1*2'. )#$)# #, '(@('1* 4.$1. r+ #( ,.$
12.:#-1*2'.$ )#$)# '(@('1* 4.$1. b &.$ ,. 12.:#-1*2'. )#$)# b 4.$1. r6
V (r) = −
Z
b
∞
~ · d~ℓ + −
E
Z
b
r
~ · d~ℓ
E
F. /2'&#2. '(1#92., 2#$%,1. )# #=.,%.2 #, /*1#(-'., .(1#2'*2 #( r = bC
V (b) = −
b
3
3
~ · d~ℓ = ρ0 (b − a )
E
3ǫ0 b
∞
Z
E*2 *12* ,.)*+ /.2. ,. *12. '(1#92., 1#()2#&*$
−
Z
b
r
r
ρ0 (r3 − a3 )
dr
3ǫ0 r2
b
Z r
Z b
ρ0
1
3
= −
rdr − a
·(
dr)
2
3ǫ0
b r
∞
r
ρ0
r2
1 r
= −
· ((
) − a3 ( − ))
3ǫ0
2 b
r b
~ · d~ℓ = −
E
=
Z
b2 − r 2
(r − b)
ρ0
·(
+ a3
)
3ǫ0
2
rb
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
"# #$%& '&(#)&* #+ ,-%#(./&+ #( r )#$0+%& $#)
V (r) =
(r − b)
ρ0 b3 − a3 b2 − r2
(
+
+ a3
),
3ǫ0
b
2
rb
si a < x < b
1&)& +& )#2/3( r < a 4#$.-',-(#'-$ +& %)&5#.%-)/& 4#$4# #+ /(6(/%- 7&$%& ) #( 8 %)&5#.%-)/&$9
:#(4)#'-$ ;0# #+ ,-%#(./&+ #$
V (r) = −
Z
r
∞
~ · d~ℓ = −
E
Z
b
∞
~ · d~ℓ + −
E
Z
a
b
~ · d~r + −
E
Z
a
r
~ · d~ℓ
E
<+ $#) (0+- #+ .&',- #+=.%)/.- ,&)& r < a %#(4)#'-$ ;0#
Z r
~ · d~ℓ = 0
E
−
a
> 4# #$%& '&(#)& #+ ,-%#(./&+ )#$0+%& $#)?
V (r) =
(a − b)
ρ0 b3 − a3 b2 − a2
(
+
+ a3
),
3ǫ0
b
2
ab
si r < a
!
!"#$%&' ()
"#$%&'()( (* %&%+(,- '-'# ($ *- ./0)-1 2( +&($($ 3 4&*&$')#% ,05 *-)/#%6 70(4#%6 4-'- 0$# '(
)-'&# r 5 '($%&'-'(% '( 4-)/-% %08().4&-*(% 4#$%+-$+(% σ 5 −σ 9
'* :$40($+)( (* 4-,8# (*;4+)&4# %#<)( *- *=$(- AB 6 >0( (>0&'&%+- '( *#% 4&*&$')#% ($ 0$- '&%?
+-$4&- &/0-* - %0 %(8-)-4&@$ d9
#* A(+(),&$( *- '&B()($4&- '( 8#+($4&-* ($+)( *#% 4($+)#% '( *#% 4&*&$')#%9
+"$,-./01
'* C-)- #<+($() (%+( )(%0*+-'# '(<(,#% 4-*40*-) (* 4-,8# (*;4+)&4# '( 4-'- 4&*&$')# 70(4# 0$- '&%+-$4&- r '( ;*D0+&*&E-)(,#% (%+( )(%0*+-'# 8#%+()&#),($+(F 5 (G-*0-) ($ r = b9 A-'#
>0( *#% 4&*&$')#% %#$ ,05 *-)/#% 5 (%+H$ 4-)/-'#% 4#$ '($%&'-' %08().4&-* 0$&B#),( 8#'(,#%
'(%8)(4&-) *-% 4#$'&4&#$(% '( <#)'( 5 0+&*&E-) *#% -)/0,($+#% '( %&,(+)=-9 :* 4-,8# (*;4+)&4#
>0( 8)#'04( 4-'- 4&*&$')# (%+-)H ($ '&)(44&@$ )-'&-* 5 '( (%+- ,-$()- 0+&*&E-$'# 4#,# %08().4&(
/-0%%&-$- 0$ 4&*&$')# '( *-)/# L 5 )-'&# r
φ=
~ · n̂dS = qint
E
ǫ0
S
Z
(1)
A(%-))#**-,#% *- &$+(/)-* 4#,#
Z
Z
~
dS = E2πrL
E · n̂dS = E
S
(2)
S
"#,# *- '($%&'-' '( 4-)/- %08().4&-* (% σ 4#$%+-$+(D%( 7-4( '( &/0-* ,-$()- 8-)- 8-)- (*
4&*&$')# F6 *- 4-)/- '(* 4&*&$')# %()H %&,8*(,($+(
qint = σV = σ2πaL
(3)
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
"##$%&'(')*+ ,-. / ,0. #) ,1.2 +34#)#$+5 67#
E=
σa
rǫ0
(4)
89'&7')*+ #) r = d 4#)#$+5 67#
E=
σa
dǫ0
:;+<' 57%#<%+)=#)*+ #& >'$%+ #&?>4<=>+ %<+*7>=*+ %+< >'*' >=&=)*<+ 5+3<# #& %7)4+ %#*=*+ 5#
*#3# >+)5=*#<'< &' 5=@7=#)4# >+)A@7<'>=B)C
D+$+ /' $#)>=+)'$+5 #& >'$%+ #&?>4<=>+ #5 <'*='&2 / %+< &+ 4')4+ #54'<E) #) &' *=<#>>=B)
=)*=>'*' #) &' A@7<'F D+$+ &+5 $B*7&+5 *# &+5 >'$%+5 %<+*7>=*+5 %+< '$3+5 >=&=)*<+5 5+)
=@7'&#5 #) #& %7)4+ %#*=*+2 &'5 >+$%+)#)4#5 #) y 5# ')7&'<') / %+< &+ 4')4+ #& >'$%+ #&?>4<=>+
#54'<E #) *=<#>>=B) î / 4#)*<E $+*7&+ 2Ey = 2Ecos( π3 )2 #5 *#>=<
E=
σa
dǫ0
! G'<' >'&>7&'< &' *=H#<#)>=' *# %+4#)>='& #)4<# &+5 *+5 >=&=)*<+5 *#3#$+5 >'&>7&'< #& >'$%+
#&?>4<=>+ 5+3<# &' &I)#' 67# 7)# ' '$3+5 >=&=)*<+5F D+)5=*#<'<# #& +<=@#) #) #& >#)4<+ *#& >=&=)*<+
*# &' =(67=#<*'2 >+$+ $7#54<' &' A@7<'C
G'<' 0 < x < D #& >'$%+ #&?>4<=>+ 5=#$%<# #54'<E #) *=<#>>=B) îF D+$+ ;#$+5 9=54+ ')4#<=+<J
$#)4#2 #& ;#>;+ *# 67# >'*' >=&=)*<+ 4#)@' >'<@' )7&' #) 57 =)4#<=+< =$%&=>'<E 67# #& >'$%+
#&?>4<=>+ 67# %<+*7(>') #) 57 =)4#<=+< 5#' )7&+F :;+<'2 >'&>7&'<#$+5 #& >'$%+ #&?>4<=>+ <#57&J
4')4# 5+3<# #& #K# x 57%#<%+)=#)*+ &+5 >'$%+5 %<+*7>=*+5 %+< >'*' >=&=)*<+C
"#$%& '()*+,-*& %,&./*-.& %&, '( *-(-0.,& -12/-',.&
!"#" $% &' ()*"+ ', -%#." ',/-01)-" '2 ', )20'1)"1 (', -),)2(1" &'13 24,"+ ."1 ," 0%20"
~ 1 = 0,
E
si 0 < x < a
54'1% (', -),)2(1" ', -%#." ',/-01)-" &' -%,-4,%1% (' )64%, #%2'1% -"#" ," 7)-)#"& '2 ,% .%10'
%8+ (' '&0% #%2'1% 0'2'#"& 94':
~ 1 = σa î,
E
xǫ0
si x > a
!"#$ %&'()*+($ #*$,-(+,$ #$* %& (+&+.,*$ ,%*%(/$
;2 ', )20'1)"1 (', -),)2(1" ', -%#." .1"(4-)(" ."1 /, &'13 24,":
~ 2 = 0,
E
si d − a < x < d
<"1 "01" ,%("+ -"2&)('1%2(" ,% =641% #"&01%(% %20'1)"1#'20'+ >'1'#"& 94' % .%1% 42 -)'10" x
94' -4#.,% 0 < x < d − a+ ,% ()&0%2-)% (', -'201" (', -),)2(1" ('1'-7" % x &'13 d − x+ (' '&0%
#%2'1% ', -%#." ',/-01)-" 1'&4,0% &'1:
~2 =
E
σa
î
(d − x)ǫ0
?' '&0% #%2'1%+ %, &4.'1."2'1 ,"& -%#."& ',/-01)-"& "@0'2)("&+ >'#"& 94' ', -%#."& 1'&A,0%0'
'&:
~ =
E
σa
(d−x)ǫ0 î
σa
σa
( xǫ
+ (d−x)ǫ
)î
0
0
σa
xǫ0 î





0<x<a
a<x<d−a
d−a<x<d
B7"1% .%1% -%,-4,%1 ,% ()C'1'2-)% (' ."0'2-)%, δV '201' ,"& -),)2(1"& ('@'#"& -%,-4,%1 ,% )20'61%,
$ .%1% '&0" 40),)D%#"& ,% 01%$'-0"1)% #%& "@>)%: ', -%#)2" &"@1' ', '*' x:
∆V
= −
= −
=
=
=
=
Z
d
0
Z
0
a
a
~ · d~ℓ
E
~ · d~ℓ + −
E
Z
a
d−a
~ · d~ℓ + −
E
Z
d
d−a
~ · d~ℓ
E
Z d−a
Z d
σa
σa
σa
σa
(
dx +
+
)dx +
î
xǫ0 (d − x)ǫ0
0 (d − x)ǫ0
a
d−a xǫ0
σa
σa
σa
(− ( ln(d − x)|a0 )) + ( ( ln(x) − ln(d − x)|d−a
(( ln(x)|dd−a )))
a )) + (
ǫ0
ǫ0
ǫ0
−σa
2σa
−σa
d−a
d−a
d−a
(
)) + (
)) + (
))
ln(
ln(
ln(
ǫ0
d
ǫ0
a
ǫ0
d
d−a
d−a
2σa
(ln(
) − ln(
))
ǫ0
a
d
Z
?' ("2(' &' "@0)'2'
!
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
∆V =
d
2σa
ln( )
ǫ0
a
!
!"#$%&' ((
"# $%&'( '%)'*#+) $, )+$%( R $, #+ -.*)+ ,&/0 '+).+$( '(1 *1+ $,1&%$+$ σ = σ0 (1 −
σ0 > 02 &%,1$( r #+ $%&/+1'%+ $, *1 3*1/( '*+#4*%,)+ $,# $%&'( + &* ',1/)( O5
') "1'*,1/), ,# 3(/,1'%+# ,1 ,# 3*1/( P &(6), &* ,7,8OP = R95
R
r )2
'(1
#) :;+'%+ $(1$, &, <(=,)0 *1+ '+).+ 3*1/*+# q2 $,7+$+ ,1 ),3(&( ,1 P >
*"$+,-./0
') ?( 4*, @+),<(& &,)0 '(1&%$,)+) ,# 3(/,1'%+# 3)($*'%$( 3() *1 +1%##( $, )+$%( a A '+).+
/(/+# q &(6), &* ,7, + *1+ $%&/+1'%+ d A #*,.(2 */%#%B+1$( ,&/, ),&*#/+$(2 $%=%$%),<(& ,# $%&'(
,1 3,4*,C(& +1%##(& A &*<+),<(& #(& 3(/,1'%+#,& 3)($*'%$(& 3() '+$+ *1(5
"# 3(/,1'%+# $, *1 +1%##( #( 3($,<(& '+#'*#+) */%#%B+1$(
Z
kdq
V =
r
"# =+#() $, r #( (6/,1,<(& 3() 3%/+.()+&D
r=
p
a2 + d2
"&/+ $%&/+1'%+ &, <+1/%,1, '(1&/+1/, A 3() #( /+1/( 3*,$, &+#%) $, #+ %1/,.)+#
Z
kq
k
dq = √
V =√
a2 + d2
a2 + d2
E@()+2 '(1&%$,)+), +1%##(& $,# $%&'( '(1 '+).+ dq 5 F,# ),&*#/+$( +1/,)%() A '(1&%$,)+1$( d = R
A a = r &, /,1$)0 4*,
dV = √
kdq
r 2 + R2
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
! "!#$! dq %& '! ()*&+)% &%"#,-,# ")+)
dq = σ(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 = 2πrdr
7& &%0! +!/&#! )-0&/&+)%
dq = 2πσ0 (r − R)dr
:' ()0&/",!' *& (#)*2",*) ()# &%0& !/,'') %&#3 &/0)/"&%9
dV =
;/0&$#!/*) *&%*& r = 0 8!%0! r = R9
V
<, ")/%,*&#!+)%
2kπσ0 (r − R)dr
√
r 2 + R2
R
(r − R)dr
√
r 2 + R2
0
Z R
Z R
r
1
√
√
= 2kπσ0 (
dr − R
dr)
2
2
2
r +R
r + R2
0
0
p
p
R
R
= 2kπσ0 (( r2 + R2 ) − ( ln( r2 + R2 + r) ))
0
0
√
√
= 2kπσ0 (R 2 − R − R ln(R 2 + R) + R ln(R))
√
√
= 2kπσ0 (R 2 − R − R ln(R( 2 + 1)) + R ln(R))
√
√
= 2kπσ0 (R 2 − R − R ln(R) − R ln( 2 + 1)) + R ln(R))
√
√
= 2kπσ0 (R 2 − R − R ln( 2 + 1))
√
√
= 2kπσ0 R( 2 − 1 − ln( 2 + 1))
√
√
2 − 1 − ln( 2 + 1) ≈ 21 #&%2'0!9
= 2kπσ0
Z
V ≈ −kπσ0 R
=)*&+)% &%"#,-,# &%0& #&%2'0!*) &/ >2/",?/ *& '! "!#$! 0)0!' Q *&' *,%")6 :%0! "!#$! '! ()*&+)%
"!'"2'!# ,/0&$#!/*) %)-#& &' *,%") *& '! &@(#&%,?/ dq = 2πσ0 (r − R)dr
Q =
Z
0
R
2πσ0 (r − R)dr
R
r2
= 2πσ0 ( ( − Rr) )
2
0
7& &%0! +!/&#!
= −πσ0 R2
V ≈
Q
− kπσ0 R
−πσ0 R2
!
V ≈
kQ
R
"#$# %&$#'( &) *#+&,-./) &,-#,+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&) &<& 1&) 1.'-# 8 '&,+.1# =/-./ &) -&,+0# 8 -#$# )/
-/02/ 1& *03&>/ &' *#'.+.%/( &'+/ '& $#%&07 +/$>.6, =/-./ &) -&,+0# 1&) 1.'-#5
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
!"#$%&' ()
"#$%&'()( '#% *+),+% -.$/.+0(% '( 1+%+ m 2 *+),+ q +/+'+% -#) .$+ *.()'+ &'(+0 '( 0+),#
ℓ3 &$&*&+01($/( ($ )(-#%# ($ 0+ -#%&*&4$ I 56() 7,.)+89 :( %.(0/+$ 2 *+($ -#) 0+ +**&4$ '( 0+
,)+6('+' + 0+ -#%&*&4$ II 3 +0*+$;+$'# .$ <$,.0# θ *#$#*&'#9 =>+)+ ?.( 6+0#) '( 0+ *+),+ (0
%&%/(1+ +0*+$;+ (%/( <$,.0# θ@9 AB-)(%( %. )(%.0/+'# ($ /C)1&$#% '( ℓ3 m 2 θ9
*"$+,-./0
A0(,&1#% (0 *()# '( 0+ ($(),D+ ,)+6&/+*&#$+0 ($ 0+ -#%&*&4$ ℓ 1+% +E+F# '( 0+ -#%&*&4$ &$&*&+0
56() 7,.)+89 G+ ($(),D+ -#/($*&+0 ($ 0+ -#%&*&4$ H %()<9
UI = 2mgℓ +
kq 2
2ℓ
G+ ($(),D+ -#/($*&+0 ($ 0+ -#%&*&4$ HH
UII = 2mgℓ(1 − cos(θ)) +
kq 2
2ℓ sin(θ)
"#1# 0+ ($(),D+ %( *#$%()6+3 -#'(1#% I+*() UI = UII 2 -#) 0# /+$/#
2mgℓ +
kq 2
kq 2
= 2mgℓ(1 − cos(θ)) +
2ℓ
2ℓ sin(θ)
J(%-(F+$'# ?K
s
q = ±2ℓ
mg sin(θ)
k(sec θ − tan(θ))
!
!"#$%&' ()
"#$%& '() *)+) ,&-.&,/0%.*) 1& %)1.# R ,& 2.&(& '() 1.,2%.$'*.3( 1& *)%4) '(./#%-& σ 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# z k̂ 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) z k̂ 5
=#,2&%.#%-&(2&< +)%) *)7*'7)% &7 +#2&(*.)7 2#2)7< .(2&4%)-#, dV ,#$%& 2#1) 7) ,'+&%?*.&5
6#-# C.-#, &( 7) )@'1)(2.) D< 7) *)%4) dq 7) +#1&-#, &E+%&,)% *#-# &7 +%#1'*2# &(2%& 7)
1&(,.1)1 ,'+&%?*.)7 1& *)%4) σ +#% &7 )%&) >'& #*'+) &,) *)%4) >'& 77)-)-#, dA @ C.-#, >'&
dA = R2 sin(γ)dγdθ
F) 1.,2)(*.) r &( &,2& *),# 7) #$2&(&-#, '2.7.G)(1# &7 2&#%&-) 1&7 *#,&(#8
r2 = R2 + z 2 − 2zR cos(γ)
H,I< &7 +#2&(*.)7 +%#1'*.1# +#% dq &,
kσR2 sin(γ)dγdθ
dV = p
R2 + z 2 − 2zR cos(γ)
F'&4#< .(2&4%)-#, &,2) &E+%&,.#( &7.4.&(1# 7#, 7.-.2&, 1& .(2&4%)*.#( 1& 2)7 /#%-) 1& %&*#%%&%
7) ,'+&%?*.&8
V =
Z
0
2π
Z
π
π
2
9,2) .(2&4%)7 ,& +'&1& %&1'*.% )
V = 2kπσR2
Z
kσR2 sin(γ)dγdθ
p
π
π
2
R2 + z 2 − 2zR cos(γ)
sin(γ)
p
2
R + z 2 − 2zR cos(γ)
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!
"#$# $%&'()%$ %&*'+ ,%-%.'& &#-%$ /#(/0(#$ (# 12*%3$#( ,% (# 4'$.#5
Z
sin γ
p
dγ
A + B cos(γ)
6#/%.'& %( /#.-1' ,% )#$1#-(% u2 = A + B cos(γ)7 8% %&*# .#2%$# dγ =
Z
Z
2
2
sin γ
p
1du = − u
dγ = −
B
B
A + B cos(γ)
p
:'.' u = A + B cos(γ)5
Z
sin γ
2p
p
dγ = −
A + B cos(γ)
B
A + B cos(γ)
2udu
−B sin(γ) 7
9 #&1
"#$# %( /#(/0(' ,% ;'*%2/1#( *%2%.'& <0% A = R2 + z 2 = B = −2zR7 >%%.;(?#2,' %2 (#
12*%3$#(5
Z
sin γ
1 p 2
p
R + z 2 − 2Rz cos(γ)
dγ =
zR
R2 + z 2 − 2zR cos(γ)
@A'$#+ ;',%.'& /#(/0(#$ %( 12*%3$#( &12 ,1B/0(*#,%&5
V (z) =
!
2kπσR2 p 2
( R + z 2 − 2Rz cos(γ)
zR
π
π
2
)=
p
2kπσR
(z + R − R2 + z 2 )
z
@A'$# <0% *%2%.'& %( ;'*%2/1#( %(C/*$1/' %2 z k̂ + %( /#.;' %(C/*$1/' &% ;0%,% /#(/0(#$ /'.'5
~ = −∇V
E
D%.'& <0% %( ;'*%2/1#( %&*# &'(' %2 402/1'2 ,% (# )#$1#-(% z = ;'$ (' *#2*' *%2,$%.'& <0% %(
/#.;' %(%/*$1/' %&5
~ = − dV k̂
E
dz
8%$1)#.'& V (z)5
dV
dz
p
d 1
( (z + R − R2 + z 2 ))
dz z
p
−1
1
1
1
= 2kπσ( 2 (z + R − R2 + z 2 ) + (1 + √
2z))
2
z
z
2 R + z2
√
1
−1 R
R2 + z 2 1
− −
+ +√
= 2kπσ(
)
2
2
z
z
z
z
R + z2
√
R2 + z 2 − R
2
√
= −2kπσR (
)
z 2 R2 + z 2
= 2kπσR
8% %&*# .#2%$# %( /#.;' %(%/*$1/' %2 %( ;02*' z k̂ &%$E5
!
√
R2 + z 2 − R
2
~
√
)k̂
E(z) = 2kπσR (
z 2 R2 + z 2
! "#$%& '(# )* #+,-#&.%/ */0#-.%- #&0* ./1#2/.1* #/ z = 03 4./ #$5*-6%7 ,%1#$%& *8#-8*-/%& *
O 0*/0% 8%$% '(#-*$%& 9 8*)8()*- #) :*)%- 1#) 8*$,% #);80-.8%3<-.$#-% :#$%& '(# )* #+,-#&.%/
1#) 8*$,% #)#80-.8% &# ,(#1# -##&8-.5.- 8%$%=
2
2kπσR (
1−
√ R
z 2 +R2
z2
)
>(#6%7 0%$*-#$%& #) ).$.0# 8(*/1% z → 0 1# )* ?(/8.%/ E(z)k̂ @=
2
lı́m E(z) = 2kπσR lı́m (
z→0
z→0
A%$% :#$%&7 #&0# ).$.0# #& 1# )* ?%-$*
2kπσR2 lı́m (
z→0
1−
1−
0
0
√ R
z 2 +R2
z2
√ R
z 2 +R2
z2
)
,%- )% '(# (0.).B*-#$%& >CDE,.0*) ,*-* 8*)8()*-)%=
) = 2kπσR2 lı́m (
z→0
(1 −
√ R
)′
z 2 +R2
)
(z 2 )′
3
R(z 2 + R2 )− 2
= 2kπσR lı́m (
)
z→0
2z
1
= 2kπσR2 ·
2R2
= kπσ
2
~
F# #&0* $*/#-* E(0)
= kπσ k̂
!
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!"#$%&' (
!"#$%&!'()
!"#$%&' ()
! "#!$!$ %&' !'()*+' ,&$%-,"&*+' %! *+%#&' r1 , r2 . ,+*/+' q1 , q2 '!0+*+%+' 0&* -$+ /*+$ %#'1
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*"$+,-./0
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!6 0&"!$,#+6 '&4*! 6+' '-0!*<,#!' %! 6+' !'(!*+' ,&$%-,"&*+' !'
V1 = k
q1
q2
, V2 = k
r1
r2
%&$%! B!3&' '-0-!'"& 9-! !6 0&"!$,#+6 %! -$+ !'(!*+ $& !' +(!,"+%& 0&* !6 %! 6+ &"*+ 7& 3?' 4#!$
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q1 + q2 = (q1 )f + (q2 )f
A6 +6,+$=+* !6 !9-#6#4*#&8 !' %!,#*8 ,-+$%& %!D+ %! B+4!* "*+$'(!*!$,#+ %! ,+*/+' !$"*! 6+' !'(!*+'8
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V1
(q1 )f
k
r1
2
4π(r1 ) (σ1 )f
r1
(σ2 )f
=⇒
(σ1 )f
= V2
(q2 )f
= k
r2
4π(r2 )2 (σ2 )f
=
r2
r1
=
r2
>! !'"& 5!3&' 9-! !$ /!$!*+68 6+' *!/#&$!' !$ 6+ '-0!*<,#! %! -$ ,&$%-,"&* ,&$ 3!$&*
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EF
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(q1 )f + (q2 )f
2
2
4π(r1 ) (σ1 )f + 4π(r2 ) (σ2 )f
= q1 + q2
= q1 + q2
4πr1 (σ1 )f (r1 + r2 ) = q1 + q2
1 q1 + q2
=⇒ (σ1 )f =
4πr1 r1 + r2
1 q1 + q2
(σ2 )f =
4πr2 r1 + r2
!
!"#$%&' ()
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%&$%&'& -( '&-+# a3 ($ -(%+'3 Q1 0#' 1# /&,/#C
Z
~ · n̂dS = Q1
E
ǫ0
S
R
~ · n̂dS = E4πr2 6 "( ($/& 5&,('& (1 %&50# (1(%/'+%#EF.( 2& :(5#$ 4+$/# F.( /+(,(
E
D('#
S
-+'(%%+#, '&-+&1G ($C
Q1
r̂
4πǫ0 r2
9:#'&3 1& -+)('(,%+& -( 0#/(,%+&1 (,/'( &5<&$ $.0('%+($ ($C
Z a
Q1 1 1
Q1
dr =
( − )
V2 − V1 = −
2
4πǫ0 b a
b 4πǫ0 r
7#, ($/#3 0#-(5#$ #</(,(' 1& %&'8& Q1 C
~ =
E
ab
b−a
9:#'&3 %&1%.1&'(5#$ (1 %&50# (1(%/'+%# & .,& -+$/&,%+& r > b ./+1+@&,-# 1(2 -( H&.$$6 I1(8+5#$
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F.( -(<(5#$ %#,$+-('&' (, ($/( %&$#3 ($ F.( 1& %&8& (,%(''&-& 0#' ($/& $.0(';%+( ($ Q1 + Q2 6
"( ($/& 5&,('& 4(5#$ F.(C
Q1 = 4πǫ0 (V2 − V1 )
E4πr2 =
Q1 + Q2
ǫ0
!"#$%&' () '*+% $',-.
!
~ = Q1 + Q2
E
4πǫ0 r2
"#$ %&'#( )#*%+#& ,-,./-0 %/ )#'%$,1-/ %$ b2
V2 = −
Z
b
∞
Q1 + Q2
Q1 + Q2
dr =
4πǫ0 r2
4πǫ0 b
3% %&'- +-$%0-( /- ,-04- *% ,-&,-0- %5'%01#0 &%062
Q2 = 4πǫ0 b(V1 + (V2 − V1 )
ab
)
a−b
! 7-0- ,-/,./-0 %/ '0-8-9# '#'-/( ,#$&1*%0-0%+#& :.% /-& ,-04-& :.% ,#+)#$%$ /- ,-&,-0- *%
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dW = dqV (c)
3% %&'- +-$%0-( %/ '0-8-9# '#'-/ &%06 &1+)/%+%$'%
W = QV (c)
> @# :.% '%$%+#& :.% =-,%0 %& ,-/,./-0 %/ )#'%$,1-/ %$ r = c :.% %& *10%,'-+%$'%
V (c) =
A&1(
W =
Q1 + Q2
4πǫ0 c
Q(Q1 + Q2 )
4πǫ0
!
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"#$ %$&%$'$ (&)*'+%$ %,#-.%/,'$ -( '$-+, a 0 (&1(&,' δ ≫ a %,#/+(#( %$'2$ #(/$ Q3 4( -+&/'+5.0(
.#$ %$'2$ q (# (6 7,6.8(# +#/('+,' -(6 %$&%$',# -( '$-+, a9.# $+&6$#/( (# 6$ 1$'/( +#/('+,' -(6
%$&%$',# +81+-( :.( (&/$ -(#&+-$- -( %$'2$ &( 1$&( $6 %,#-.%/,';3 <,& -+%(# :.( (6 %$81,
(6*%/'+%, (# (6 +#/('+,' -(6 %$&%$',# (&/$ -$-, 1,'
~ = K( r )4 r̂
E
a
=,#-( K (& .# %,#&/$#/( 1,' -(/('8+#$' 0 r̂ (& (6 7(%/,' .#+/$'+, '$-+$63 4( 1+-( (#%,#/'$'>
'* ?$ -(#&+-$- -( %$'2$ ρ(r) (# (6 7,6.8(# +#/('+,' -(6 %$&%$',#3
#* ?$& -(#&+-$-(& -( %$'2$& &.1('@%+$6(& $6 +#/('+,' 0 $6 (A/('+,' -(6 %$&%$',#3
+* B6 1,/(#%+$6 (6(%/',&/C/+%, (# /,-, (6 (&1$%+,3
,"$-+./01
'* ?$ 6(0 -( 2$.&& &( 1.(-( (&%'+5+' (# &. ),'8$ 8$& %,8.# %,8,>
φ=
~ · n̂dS = qint
E
ǫ0
S
Z
4+# (85$'2,D &( 1.(-( ./6+E$' /$85+(# 6$ ),'8$ -+)('(#%+$6 -( 6$ 6(0 -( F$.&& :.( (&/$56(%(>
~ ·E
~ = p(r)
∇
ǫ0
G,8, 7(8,&D (6 %$81, (6(%/'+%, (&/$ &,6, (# ).#%+,# -( rD -( (&/$ 8$#('$ /(#(8,&
~ ·E
~ =
∇
=
=
=
1 ∂ 2
(r E(r))
r2 ∂r
1 ∂ 2 r 4
(r K( ) )
r2 ∂r
a
6
1 ∂ Kr
)
(
r2 ∂r a4
6Kr3
a4
=( (&/$ 8$#('$D 6$ -(#&+-$- -( %$'2$ (# (6 +#/('+,' -(6 %$&%$',# &('C>
ρ(r) =
6ǫ0 Kr3
a4
!
"#$%&' ()*)+$, ()-)%+./&% 0& 1$/,-&/-) K 2 3&*)+$, 45) 0& 1&%6& -$-&0 &0+&1)/&(& )/ )0
./-)%.$% ()0 1&,1&%$/ ), q ' 7$% 0$ -&/-$ ,) 8& & 15+70.% 0& %)0&1.$/9
q =
Z
ρ(r)dV
Z a
6ǫ0 Kr3
= 4π
4πr2 dr
4
a
0
a
24ǫ0 K r6
)
=
a4
6 0
= 4πa2 ǫ0 K
:) ),-& +&/)%&
K=
q
4πǫ0 a2
",;' 7$()+$, ),1%.*.% 0& ()/,.(&( () 1&%6& )/ <5/1.$/ () 0& 1&%6& -$-&0 &0+&1)/&(&9
3 qr3
2 πa6
! 3&*)+$, 45) )/ )0 ./-)%.$% () 5/ 1&,1&%$/ 0& 1&%6& /)-& ), /50&2 ",. ,. 1$/,.()%&+$, 5/&
,57)%=1.) ),<)%.1& 1$/ %&(.$ r = a + αδ ' 1$/ 0 < α < 1' 8)%)+$, 45) 7&%& 45) ,) +&/-)/6&
/50& 0& 1&%6&' )0 1$/(51-$% ./(51) 5/& 1&%6& −q )/ )0 ./-)%.$% ()0 1$/(51-$%2 :) ),-& +&/)%&'
0& ()/,.(&( () 1&%6& )/ 0& ,57)%=1.) ./-)%.$% ()0 1$/(51-$% ),9
ρ(r) =
σint = −
q
4πa2
>$/ ),-$ ? 5-.0.@&/($ 0& 1$/,)%8&1.$/ () 0& 1&%6& 8)+$, 45) )/ 0& ,57)%=1.) )A-)%.$% ,) -)/(%&
5/& 1&%6& Q + q 2 >$/ ),-) %),50-&($ 7$()+$, 1&0150&% 0& ()/,.(&( () 1&%6& ,57)%=1.&0 )A-)%.$%9
σext =
Q+q
4πa2
:$/() #)+$, (),7%)1.&($ δ 2
"! B&%& ,&*)% 7$-)/1.&0 )/ -$($ )0 ),7&1.$ ()*)+$, ,&*)% )0 1&+7$ )0C1-%.1$2 B&%& 0& %)6.$/
./-)%.$% ?& -)/)+$, )0 1&+7$ )0)1-%.1$9
~ int =
E
q
r
( )4 r̂,
4πǫ0 a2 a
para r < a
D/ )0 )A-)%.$% ()0 1$/(51-$% -)/)+$, -)/)+$, 45) )0 1&+7$ )0)1-%.1$ ), ,.+70)+)/-)9
~ ext = Q + q r̂,
E
4πǫ0 r2
si r > a
",. )0 7$-)/1.&0 7&%& 5/& %)6.$/ )/ 45) r > a ),9
Z
Z
Q+q r 1
Q+q
~
V (r) = − Eext d~r = −
dr =
2
4πǫ0 ∞ r
4πǫ0 r
"#$%&' 7&%& 0& %)6.$ ./-)%.$% Er < aF )0 7$-)/1.&0 0$ 1&0150&+$, 1$+$9
!"#$%&' () '*+% $',-.
!
V =−
Z
~ · d~r = −
E
Z
∞
V (r) = −
= −
=
b
~ ext · d~r + −
E
Z
Z
Z
b
r
~ int · d~r,
E
si r > a
~ · d~r
E
b
∞
~ ext · d~r + −
E
Z
r
b
~ int · d~r
E
Q+q
q
−
(r5 − a5 )
4πǫ0 a 20πa6 ǫ0
"#$% #& '$&(& )*&
V (r) =
q
Q+q
(r5 − a5 ),
−
4πǫ0 a 20πa6 ǫ0
si r < a
!
!"#$%&' ()
"#$%&$' '$ %#()* '$+%,-.%* )-*/&%./* )*- &0# /.1,-.2&%.30 /' %#-4# ρ(r) ,#$ 5&' '$ )*,'0%.#$
5&' '1,# /.1,-.2&%.30 )-*/&%' '1,# /#/* )*-
V (r) = q
e−λr
4πr2 ǫ0
6&'4*7 '0%&'0,-' $# /.1,-.2&%.30 /' %#-4# ρ(r)8
*"$+,-./0
9'0'(*1 '$ )*,'0%.#$ )-*/&%./* )*- $# /.1,-.2&%.30 /' %#-4# ρ(r) : ;'(*1 # $# ;'< 5&' '1,'
)*,'0%.#$ '1,# '0 =&0%.30 /' r7 )*- $* 5&' '$ )*,'0%.#$ ,.'0' 1.(',-># '1=+-.%#8 ?@*-#7 &,.$.<#0/*
'$ @'%@* /' 5&'
~ = −∇V
E
A*/'(*1 '0%*0,-#- =B%.$('0,' '$ %#()* '$+%,-.%*C
q
~ = − ∂V r̂ =
e−λr (1 + λr)r̂
E
∂r
4πǫ0 r2
D1,' -'1&$,#/* 0*1 )'-(.,' '0%*0,-#- 1.0 /.E%&$,#/ $# /'01./#/ /' %#-4#8 D1,* 1' @#%' &,.$.<#0/*
$# =*-(# /.='-'0%.#$ /' $# $': /' F#&11C
~ ·E
~ = ρ
∇
ǫ0
~ = E r̂7 /' '1,# (#0'-#
9'0'(*1 5&' '$ %#()* '1 -#/.#$ : )*- $* ,#0,* E
1 ∂ 2
(r E(r))
r2 ∂r
1 q
=
(−λe−λr (1 + λr) + λe−λr )
4πǫ0 r2
1 q 2 −λr
λ e
= −
4πǫ0 r
~ ·E
~ =
∇
A*- $* ,#0,*
ρ(r) = −
1 q 2 −λr
λ e
4π r
!
!"#$%&' () '*+% $',-.