01+%'2&pauli.fis.puc.cl/~rramirez/e_m/html/libro_electro... · !"# $%&'%#...
TRANSCRIPT
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~N ", 4-,(, ,"+0%#%0 -"'*(! "-" +!.4!*,*&," 0,+&'*1-$'0," +!.!?
~N = N cos(π/3)i + N sin(π/3)j
@' 7-,05'
~Fe ," $' ,A,0+%(' 4!0 $' #!$%&' (, $' (,0,+:' "!#0, $' #!$%&' (, $' %5;-%,0(' / ,"?
~Fe =kq2
R2
B
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∑
FX = N cos(π/3)− kq2
R2= 0 (1)
8'*$+% *. :7
∑
FY = N sin(π/3)−mg = 0 (2)
;* <=> ?*(#& 6'*
N =mg
sin(π/3)
@**(,1%+%.3# *&/% *A,$*&2#. *. <B>7
mg
sin(π/3)cos(π/3)− kq2
R2= 0
;*&,*C%.3# q
q = ±R
√
mg cot(π/3)
k
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#"%")'(A
mg = kq2
x02
.'% x0 ;'($.$?% &" "0,$-$</$'5
=⇒ x0 = q ·√
k
mg(1)
<= F'%($&"/")'( ,%* .''/&"%*&* 9 "% #'/%' -* ;'($.$?% &" "0,$-$</$' x05 G"%")'( ;'/ H&*
-": &" I"J#?%
F = md2x
dt2= −mg +
kq2
(x0 + x)2
K*/* '(.$-*.$'%"( ;"0,"@*(A |x| << |x0| =⇒ | xx0| << 1A ;'/ -' 0,"
F = md2x
dt2= −mg +
kq2
(x0 + x)2= −mg +
kq2
x02· (1 +
x
x0)−2 ∼= −mg +
kq2
x02· (1− 2
x
x0)
K"/' &" LM= #"%")'( 0," mg = kq2
x02
=⇒ md2x
dt2∼= −2kq2x
x03
=⇒ d2x
dt2+
2kq2
mx02· x
x0= 0
K"/' &" LM=A mx02 = kq2
g
=⇒ d2x
dt2+
2g
x0· x = 0
K'/ #*%#'A #"%")'( 0," -* 7/".,"%.$* &" '(.$-*.$'%"( ;"0,"@*( "(
w =
√
2g
x0
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~F1 = kq2
L2· (cos(π
3)x− sen(
π
3)y) = k
q2
L2· ( x
2−√
3
2y)
~F2 = kq2
L2x
32 A.,(B2 (,%.#)2"), ,%
~F = ~F1 + ~F2 =3
2k
q2
L2x−
√3
2k
q2
L2y =
√3 · k q2
L2· (√
3
2x− y
2) =
√3 · k q2
L2· (cos(π
6)x− sen(
π
6)y)
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61 : 61 -, =2%, #2 =2%, -,# )(*/"0.#$ ,1.*#/),($7 H%I;
r
sen(π6 )
=L
sen(2π3 )
=⇒ r =L√3
K$( )2")$; 2# *0.2#2( #$% <D-.#$% -, #2% A.,(B2% $=),",<$%
√3 · k q2
L2= k
( L√3)2
=⇒ Q =1√3· q
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#":&- -, -A- 73 !- -#+% 9"&5%> +-*-5"# 6)- ~r = R · (cos(θ)x + sen(θ)y)> ~r1 = az @ ~r2 = −az>2"*2- ~r -# ,% ("#.$.B* 2- ,% $%&'% 2- (&)-:% @ ~r1, ~r2 #"* ,%# ("#.$."*-# 2- ,%# $%&'%# ("#.+./%#
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2- (&)-:% -#
~F = ~F1 + ~F2 =kqQ
(R2 + a2)3
2
· (Rcos(θ)x + Rsen(θ)y − az + Rcos(θ)x + Rsen(θ)y + az)
=2kqQR
(R2 + a2)3
2
· (cos(θ)x + sen(θ)y) .
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,% 9)*$.B*
|~F | = 2kqQR
(R2 + a2)3
2
= 2kqQ · f(R)
=⇒ df(R)
dR=
(R2 + a2)3
2 − 3R2(R2 + a2)1
2
(R2 + a2)3= 0
=⇒ (R2 + a2)1
2 · ((R2 + a2)− 3R2)) = 0 =⇒ R =1√2· a
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~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*$(%&F
~T = −T sin(θ)i + T cos(θ)j
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8)'$.% '0 97
∑
FX = qE − T sin(θ) = 0 (1)
8)'$.% '0 :7
∑
FY = T cos(θ)−mg = 0 (2)
;' <=>7
T =mg
cos(θ)
?''+(,%.%0*# '0 < >7
qE − mg
cos(θ)sin(θ) = 0
;' '&1#@ $'&),1% 5)'7
q =mg tan(θ)
E
! ;', $'&),1%*# %01'$3#$ A'+#& 1'0'+#& 5)' q = mg tan(θ)E B 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
dt=
mg
E
dθ
dt
F)'E#@ /#+#
dqdt = α@ *'&('G%0*# dθ
dt $'&),1%7
dθ
dt=
αE
mg
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0'$+*$" 1%
~F1/
B& %&'(& <0) 2'#10%) 7& D0)'>&
~F1 $) )+%0)+,'& )+ 7& 2#$-%-3+ ~r1 = d/2j 9 7& 2&',6%07& 1)
2'0)?& )+ xi4 @#' 1).+-%-3+; 7& D0)'>& )+,') &*?&$ %&'(&$ )$A
~F1 =k(−q)Q(xi + d/2j)
l3
0'$+*$" 1%
~F2/
G+ )$,) %&$#; 7& %&'(& <0) 2'#10%)
~F2 $) )+%0)+,'& )+ 7& 2#$-%-3+ ~r2 = −d/2j; 2#' 7# ,&+,# 7&
D0)'>& 1) &,'&%%-3+ )+,') )$,& %&'(& 9 7& %&'(& 1) 2'0)?& $)'FA
~F2 =k(−q)Q(xi− d/2j)
l3
") )$,& *&+)'&; 7& D0)'>& ')$07,&+,) $#?') 7& %&'(& −q $)'FA
~F = ~F1 + ~F2 =−2kqQx
l3i (1)
!
"#$%& '(# )%*#$%& +#,-./%0-+ #, ,-+1% ℓ .%0 ,-& */&2-0./- d/2 #0 3(0./40 *#, 501(,% θ /0*/.-*%
#0 ,- 61(+- -02#+/%+ )%+ ,- +#,-./40 2+/1%0%$72+/.-8
d/2
ℓ= cos(θ)
=⇒ ℓ = (d/2) cos(θ)
9%$% &# )+%*(.# (0 )#'(#:% *#&),-;-$/#02% *# ,- .-+1- *# )+(#<-= #, 501(,% θ #& 2-$</70
)#'(#:%8 θ ≪ 1> 9%0 #&2% ?#$%& '(# cos(θ) ≈ 1 @ )%+ ,% 2-02% ℓ ≈ (d/2)> A?-,(-0*% #0 B C &#
2/#0# '(#8
~F =−16kqQx
d3(2)
D%+ ,- )+/$#+- ,#@ *# 0#E2%0 &# 2/#0# '(#
~F = m~a> A0 #&2- &/2(-./40 ,- )-+2F.(,- &# $(#?#
&%<+# #, #G# x @ )%+ ,% 2-02% m~a = md2xdt2
iH# #&2- $-0#+- &# 2#0*+5 '(#
−16kqQx
d3= m
d2x
dt2
=⇒ d2x
dt2+
16kqQx
md3= 0
9%0 #&2% ?#+/6.-$%& '(# ,- .-+1- &/1(# (0 $%?/$/#02% -+$40/.% &/$),#> I- 3+#.(#0./- -01(,-+
&#+5 #02%0.#&8
w = 4
√
kqQ
md3
A, )#+/%*% *# %&./,-./40 &#+5 #02%0.#&
T =2π
w=
π
2
√
md3
kqQ
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"#$%&'()( *+ %&,-&($.( '&%.)&/-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 %( +$-*+)+$ <+ ;-( .&($($ &,-+* 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
EB (% 2#) '(E$&0&1$3
E =kq
d2 + r2
!
"#$%&'( $' )&*+, -$. /0$ cos(θ) = r√d2+r2
1 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)3/2j
;'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 = ~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− 1
r2(1 + d2
r2 )3
2
)
=2qk
r2(1− (1 +
d2
r2)−
3
2 )
;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 (d
r )2 ≈ 01 <6. ,6 34536( $5 ,4 $F9.$'+>5 #$,
*4%96 96#$%6' G4*$. ,4 '+80+$53$ 49.6F+%4*+>5E
(1 +d2
r2)−
3
2 ≈ (1− 3
2
d2
r2)
;-4,045#6 $'3$ .$'0,34#6 $5 ,4 $F9.$'+>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,-/ )# (/ 94.-/8 $'#$) θ )2 &)6.):'; <( 0'0)#,' $) %#)-+%/ $)( $%&'('
)2 I; =% )( $%&'(' 2) (%7)-/ $)2$) (/ &'2%+%5#8 $)0.)2,-) 6.) 2. '-%)#,/+%5# /#4.(/- &-)2)#,/
0'>%0%)#,' /-05#%+' 2%0&() ? )#+.)#,-) (/ 1-)+.)#+%/ $) '2+%(/+%5#;
)"$*+,-./
@/2 1.)-3/2 )(*+,-%+/2 6.) /+,A/# 2'7-) (/2 +/-4/2 2'# %4./()2 )# 0/4#%,.$ ? )# $%-)++%5#8 &)-'
)2,B# )# 2)#,%$'2 '&.)2,'2; <2,/2 1.)-3/2 2'# 4)#)-/$/2 &'- )( +/0&' )(*+,-%+'
~E ? ,%)#)#
0/4#%,.$ F = qE; <2,' 2) &.)$) /&-)+%/- )# (/ 2%4.%)#,) 94.-/C
<( ,'-6.) 2'7-) )( $%&'(' 2)-B )#,'#+)2 (/ 2.0/ $) ,'-6.)2 &-'$.+%$' &'- +/$/ 1.)-3/8 +'#D
2%$)-/#$' )( )E) $) -',/+%5# 2'7-) )( &.#,' O; <#,'#+)2 2) ,)#$-B 6.) )( ,'-6.) 2'7-) )( $%&'('
)2C
τ = −Fa sin(θ) +−Fa sin(θ) = −2Fa sin(θ) = −2qE sin(θ)
<( 2%4#' #)4/,%>' 2) '7,%)#) / &/-,%- $) (/ -)4(/ $) (/ 0/#' $)-)+F/ ? 2) &.)$) +'0&-'7/- /(
F/+)- (/ '&)-/+%'# $) 0'0)#,'2 $) 1'-0/ >)+,'-%/(;
=/7)0'2 6.) )( ,'-6.) 2'7-) .# 2%2,)0/ 2)-B %4./( /( 0'0)#,' $) %#)-+%/ I &'- (/ /+)()-/+%5#
/#4.(/- αG/07'2 -)2&)+,' /( )E) 2'7-) OH8 )2 $)+%-
τ = Iα
−2qEa sin(θ) = Iα
I'0' 2) &-'$.E' 2'(' .# &)6.):' $)2&(/3/0%)#,' $)( $%&'(' 2) +.0&(/ 6.) θ ≪ 1 ? &'- ('
,/#,' &'$)0'2 F/+)- (/ /&-'J%0/+%5# sin(θ) ≈ θ; K$)0B2 #',/0'2 6.) α = d2θdt2
; L))0&(/3/#$'
)2,'2 >/('-)2 )# (/ )+./+%5# /#,)-%'- 2) ,%)#) 6.)C
!
−2qEaθ = Id2θ
dt2
"#$% &'(%')*+ ,)-&.&+')%/ /% 01,&21# &#'.)3). '1214
d2θ
dt2+
2qEaθ
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
2π=
1
2π
√
2qEa
I
! !"#$%&' () &*+ ,* '%&'-.
!"#$%&' (
!"#$%&'#( )# *+,'+-.
!"#$%&' (
! "#!$!$ %&' ()(*+,!' %! )(,-& . /(,-(%&' /&$ %!$'#%(%!' )#$!()!' 0$#1&,*!' λ12 λ22 '!3(,(%&'
0$( %#'"($/#( %4 5()/0)! )( 10!,6( 70! '! !8!,/!$ (*+&' ()(*+,!'4
)"$*+,-./ 9&, )( )!: %! 5&0)&*+2 '(+!*&' 70! )( 10!,6( !);/",#/( 70! !<3!,#*!$"( 0$( /(,-(
7 !$ )( 3&'#/#=$ ~r %!+#%& ( 0$( %#'",#+0/#=$ %! /(,-( Ω !'
~F = q ·∫
Ω
dq(~r − ~r‘)
4πǫ0|~r − ~r‘|3
9&$-(*&' (*+&' ()(*+,!' !$ !) !8! <2 %! "() 1&,*( 70! 0$& /0+,( %! x = 0 ( x = L : !)
&",& %! x = L+d ( x = 2L+d4 !($ ~r1 = x1x2 ~r2 = x2x2 /&$ 0 ≤ x1 ≤ L : L+d ≤ x2 ≤ 2L+d2)&' >!/"&,!' 3&'#/#=$ %! /(%( ()(*+,!4
?!$!*&' 70! )( 10!,6( 70! !8!,/! @ '&+,! 0$ !)!*!$"& %< %! A !'
~dF21 = dq2 ·∫
Ω1
dq1(~r2 − ~r1)
4πǫ0|~r2 − ~r1|3
= dq2 ·∫ L
0
λ1dx1(x2 − x1)x
4πǫ0|x2 − x1|3
= dq2 ·λ1
4πǫ0· x ·
∫ L
0
dx1
(x2 − x1)2
= dq2 ·λ1
4πǫ0· x · 1
x2 − x1
∣
∣
∣
∣
L
0
=λ1x
4πǫ0· dq2 · (
1
x2 − L− 1
x2)
=λ1λ2x
4πǫ0· dx2 · (
1
x2 − L− 1
x2)
@B
! !"#$%&' () *+$,-.!&,/ 0, '%&'12
=⇒ ~F21 =λ1λ2x
4πǫ0·∫ 2L+d
L+d
(
1
x2 − L− 1
x2
)
dx2
=λ1λ2x
4πǫ0· (ln(x2 − L)− ln(x2))|2L+d
L+d
=λ1λ2x
4πǫ0· ln(
x2 − L
x2)
∣
∣
∣
∣
2L+d
L+d
=⇒ ~F21 =λ1λ2
4πǫ0· ln
(
(L + d)2
d(2L + d)
)
· x
" #$%& #$ '& ()#*+& #',-%*.-& /)# $# #0#*-#1 &234$ &'&23*#$5 6)#$ $&3#24$ /)# '& '#7 8# 94)'423
$&%.$(&-# '& :*& '#7 8# ;#<%=1 8# (4*2& ()#*%#>
!
!"#$%&' ()
"# $#$%&'( )( #$ *+,'$ (-.$ /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 (-.<2 ,23=
/0'%(%(2.( 4$'+$)0- 402 4$'+$- q>? 8 −q5 '(-1(4.3@$%(2.(5 %3(2.'$- A,( -0&'( AB .$%&3B2
-( )3-.'3&,8( ,23/0'%(%(2.( ,2$ 4$'+$ q: C,( 4$2.3)$) )( 4$'+$ )(&( '(1$'.3'-( 402 )(2-3)$)
402-.$2.( -0&'( (# .'$D0 DE5 1$'$ A,( (# 4$%10 (#B4.'340 -($ 2,#0 (2 (# 4(2.'0 O:
*"$+,-"./
;0 A,( E$'(%0- -('< )3@3)3' (# $#$#%&'( 40%1#(.0 (2 F 1$'.(- .$# 40%0 %,(-.'$ #$ *+,'$G
H$#4,#$'(%0- (# 4$%10 (#(4.'340 1'0),43)0 10' #$- F '(+302(- (2 O 8 #,(+0 #0- -,1('102)'(%0-
( 3%102)'(%0- #$ 402)34302 )( A,( (# 4$%10 (#(4.'340 (2 (-.$ '(+302 -($ 2,#0:
0!"1" 23/
"# $#%$&'( (- '(4.3#32(0 8 )( #$'+0 2R 8 .3(2( (2 (# ,2$ 4$'+$ q ,23/0'%(%(2.( )3-.'3&,3)$: I(
(-.$ %$2('$ 10)(%0- )(*23' -, )(2-3)$) )( 4$'+$ 40%0 λ1 = q2R : I( (-.$ %$2('$5 -3 .0%$%0-
,2$ 1(A,(J$ 4$'+$ dq )(# $#$%&'( -$&(%0- A,( -( 4,%1#( #$ '(#$4302 dq = λ1dx5 )02)( dx (-
,2 1(A,(J0 .'0D0 )(# $#$%$&'(
"# 4$%10 (#(4.'340 A,( 1'0),4( 4$)$ (#(%(2.0 dq (-.$'$ (2 )3'(44302 i5 10' #0 .$2.0 (# 4$%10
(#(4.'340 A,( 1'0),4( -('<G
!"#$%&' () *+$,-.!&,/ 0, '%&'12
d ~E1 = (kdq
x2)i = (
kλ1dx
x2)i
!"#$%&'"() ($*($ x = −3R +'*#' x = −R
~E1 = k(λ1
∫
1
x2dx)i = (kλ1(−
1
x2
∣
∣
∣
∣
−R
−3R
))i
,) -.$ &$*./#'0
~E1 =2kλ1
3Ri
!"! #$ %!&' (# $'& !"%'& )#*("#+'& ,-# !$ &-.#".'*#"*$'& #$ %!+.' #$#%)"/%' "#&-$)!*)# #&)!"!
#* (/"#%%/'* hati0 1&)' ,-#(! /*+#(/!*)!+#*) %$!"' ! 2#" $! &/3-/#*)# 43-"!5
6'+' 2#+'&7 #$ !"%' BC )/#*# %!"3! .'&/)/2! #* #$ 8 .'" $' )!*)' %!(! #$#+#*)' (# %!"3!
3#*#"! -* %!+.' &!$/#*)#7 +/#&*)"!& ,-# #$ !"%' CD )/#*# %!"3! *#3!)/2! 8 %!(! #$#+#*)'
(# %!"3! 3#*#"! -* %!+.' #*)"!*)#0 9# #&)! +!*#"!7 $' ,-# (#:#+'& ;!%#" #& &/+.$#+#*)#
%!$%-$!" $!& %'+.'*#*)#& #* x (# %!(! %!+.' 8 (#&.-#& &-+!"$!&0
<!& (#*&/(!(#& (# %!"3! .!"! %!(! !"%' &# ':)/#*#* =!%/$+#*)# !$ &!:#" ,-# )/#*# (#.'&/)!(!
-*! %!"3! , 8 >, 8 )/#*#* -* $!"3'
πR2 5
λ2 =2q
πR
λ3 = − 2q
πR
!"# $%& 6'*&/(#"!+'& -*! .#,-#?! %!"3! dq (#$ !"%'7 #$ +'(-$' (#$ %!+.' #$#%)"/' ."'>(-%/(' .'" #&)! %!"3! &#"@ &/+.$#+#*)#5
dE =kdq
R2
6'* !8-(! (# $! (#*&/(!( (# %!"3! λ2 .'(#+'& #&%"/:/" dq = λ2ds7 ('*(# ds #& -* .#,-#?')"'A' (#$ !"%'0 1&)# )"'A' (# !"%' $' .'(#+'& #&%"/:/" #* =-*%/'* (#$ !*3-$' ,-# $' "#%'""#5
ds = Rdθ
!
"#$%&#' %()#(*%' %'*+,-,+ %. */&0# *#&#
dE2 =kλ2Rdθ
R2=
kλ2dθ
R
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
∫ π2
0cos(θ)dθ =
kλ2
R
8%*)#+,/.&%()%
~Ex2 =kλ2
Ri
!"# $%& 9# %' $,:*,. ;%+ ./ *#&0#(%()% %( x $%. */&0# 0+#$2*,$# 0#+ %')% /+*# '%+< ,32/.
/. /()%+,#+4 $% %')/ &/(%+/7
~Ex3 =kλ2
Ri
=', '% )%($+/ >2%
~E2 + ~E3 = 2kλ2
R i'!#(# %)&
?%/ q′ ./ */+3/ >2% ),%(% %')% )+#@# $% /./&-+% )/. >2% /(2./ %. */&0# %.%*)+,*# 0$+#$2*,$#
0#+ )#$# %. /./&-+% %( OA B#&# ),%(% ./+3# R4 ./ $%(',$/$ $% */+3/ '%+<
λ4 =q′
R
12%3#4 0+#*%$,%($# $% ,32/. C#+&/ >2% 0/+/ %. )+#@# AB ;%&#' >2%
~E4 = (kλ4
∫ 2R
R
1
x2dx)i = (
−kλ4
2R)i
?2&/($# )#$#' .#' */&0#' */.*2./$#' % ,32/./($# / *%+#7
~E1 + ~E2 + ~E3 + ~E4 =2kλ1
3Ri +
2kλ2
Ri + (
−kλ4
2R)i = 0
! !"#$%&' () *+$,-.!&,/ 0, '%&'12
"#$%#&'()* λ4+
λ4 =2
3(2λ1 + 6λ2)
,##-%.'/'()* .*$ 0'.*1#$ )# λ12 λ2 3 λ4 )#4(5)*$ '(6#15*1-#(6#2 #(7*(61'-*$ #. 0'.*1 )# .'
7'18' 9:$7')'+
q′ =2
3q(1 +
12
π)
;5 7*($5)#1'-*$ π ≈ 3 +
q′ =10
3q
!
!"#$%&' ((
"#$%&$' '$ %#()* '$+%,-.%* )-*/&%./* )*- &0 )$#0* .010.,* /' /'02./#/ /' %#-3# &0.4*-(' σ5
)"$*+,-./
"*02./'-'(*2 '$ )$#0* '0 %&'2,.60 '$ )$#0* 78 /' %**-/'0#/#2 %#-,'2.#0#25 9#:'(*2 ;&' '$
%#()* '$+%,-.%* <.'0' /#/* )*-
~E(~r) =
∫ ∫
Ω
dq(~r − ~r1)
4πǫ0|~r − ~r1|3
%*0 ~r1 '$ <'%,*- )*2.%.60 /' $# /.2,-.:&%.60 /' %#-3# Ω ;&' 3'0'-# '$ %#()* '$+%,-.%* '0 ~r5='0'(*2 ;&' ~r = xx + yy + zz> ~r1 = x1x + y1y 8 |~r − ~r1|2 = (x− x1)
2 + (y − y1)2 + z2
5
=⇒ ~E(~r) =
∫ +∞
−∞
∫ +∞
−∞
σdx1dy1((x− x1)x + (y − y1)y + zz)
4πǫ0((x− x1)2 + (y − y1)2 + z2)3
2
?#3#(*2 %#(:.* /' <#-.#:$'25 w = x − x1> v = y − y1 =⇒ x1 = x − w, y1 = y − v5 @$A#%*:.#0* /' $# ,-#024*-(#%.60 '2
J =
∣
∣
∣
∣
(x1)w (x1)v
(y1)w (y1)v
∣
∣
∣
∣
=
∣
∣
∣
∣
−1 00 −1
∣
∣
∣
∣
= 1
=⇒ ~E(~r) =
∫ +∞
−∞
∫ +∞
−∞
σdwdv(wx + vy + zz)
4πǫ0(w2 + v2 + z2)3
2
=σ
4πǫ0
∫ +∞
−∞
∫ +∞
−∞
dwdv · (zz)
(w2 + v2 + z2)3
2
B&'<#('0,' C#3#(*2 %#(:.* /' <#-.#:$'25
w = rcos(θ)
v = rsen(θ)
J =
∣
∣
∣
∣
wr wθ
vr vθ
∣
∣
∣
∣
=
∣
∣
∣
∣
cos(θ) −rsen(θ)sen(θ) rcos(θ)
∣
∣
∣
∣
= r
=⇒ ~E(~r) =σ
4πǫ0
∫ 2π
0
∫ +∞
0
rdrdθ · (zz)
(r2 + z2)3
2
=σ
4πǫ0· 2πzz
∫ +∞
0
rdr
(r2 + z2)3
2
pero u = r2 + z2 → du = 2rdr
=σ
4ǫ0· zz
∫ +∞
z2
du
(u)3
2
=σ
4ǫ0· zz
(
2u−1
2
)∣
∣
∣
z2
+∞
=σ
2ǫ0· z
|z| z
=σ
2ǫ0· sign(z)z
! !"#$%&' () *+$,-.!&,/ 0, '%&'12
=⇒ ~E(~r) =σ
2ǫ0· sign(z)z
"#$% &'% %( )*($# +% +,)-#.$,.',+*+ +%( /0+'(# +%( -*/1# %(2-$3,-# 4&'% %) .#3/*( *( 1(*.#5
%) + σǫ06
!
!"#$%&' () "#$%&'()( *$ +,-$# &$.$&/# '( '($%&'-' '( 0-)1- *$&2#)3( σ4 0#$ *$ #)&.0&#
0&)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( λ > 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- σ%#;)( (, (:( > *$ '&%0# '( )-'&# 5 > '($%&'-' '( 0-)1- −σ6 F(- (, +,-$# >C 0##)'($-'#
(, +,-$# ($ 0*(%/&@$ > (, (:( A (, (:( ($ 0*(%/&@$6
G( ,#% &/(3% -$/()&#)(%4 /($(3#% 9*( (, 0-3+# (,80/)&0# 1($()-'# +#) (, +,-$# (%
~E1(~r) =σ
2ǫ0· sign(x)x
> (, 0-3+# (,80/)&0# 1($()-'# +#) (, '&%0# (%
~E2(~r) =−σ
2ǫ0· x(
sign(x)− x√R2 + x2
)
6
=⇒ ~E = ~E1 + ~E2 =σ
2ǫ0·(
x√R2 + x2
)
x
;7 H($(3#% 9*( ,- 2*()C- (,80/)&0- 9*( (A+()&3($/- *$ (,(3($/# dx '(, -,-3;)( '(;&'# -,
! !"#$%&' () *+$,-.!&,/ 0, '%&'12
"#$%& '&% (# &)*+'*& (,-. /$/$ "&) d~F = dq ~E0
=⇒ ~F =
∫ d+a
ddq ~E con dq = λdx
=
∫ d+a
dλdx
σ
2ǫ0·(
x√R2 + x2
)
x
=λσ
2ǫ0·∫ d+a
d
xdx√R2 + x2
· x
=λσ
2ǫ0·(√
R2 + x2)∣
∣
∣
d+a
d· x
=λσ
2ǫ0·(
√
R2 + (d + a)2 −√
R2 + d2)
· x
!
!"#$%&' ()
"# $%&'('%#)% *%+',%+'*-%$'&. #. &.#/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%
~E(~0) =
∫
kdq(~r − ~r′)
‖~r − ~r′‖3
>.+. ;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 = Rdγ6 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:$'; 0*$*,#- <:*
~r = R sin(γ) cos(θ)i + R sin(γ) sin(θ)j + R cos(γ)k
=# *- (.>?1.& 0*$ <:* ‖~r − ~r′‖ = R@* *-%' ,'/*$' *-5*1.81'/(# &#- $*1#$$.(#- (* &'- 0'$.'+&*- 5'$' <:* $*1#$$'/ *& $*1.5.*/%* -*
%.*/* <:*
~E(~0) =
∫ 2π
0
∫ 3π2
π2
kσ(−R sin(γ) cos(θ)i−R sin(γ) sin(θ)j −R cos(γ)k)
R3R2 sin(γ)dγdθ
"#$ -.,*%$?' -'+*,#- <:* *& 1',5# *&A1%$.1# $*-:&%'/%* *-%'$B */ (.$*11.C/ k 2 5#$ &# %'/%#)
(* &' ./%*9$'& '/%*$.#$) /# 1#/-.(*$',#- &'- 1#,5#/*/%*- $*-%'/%*-D
~E(~0) =
∫ 2π
0
∫ π
π2
kσ(− cos(γ)k) sin(γ)dγdθ = kσ
∫ 2π
0dθ
∫ π
π2
(− cos(γ)k) sin(γ)dγ = πkσ
∫ π
π2
(− sin(2θ)k))dγ
@* *-%' ,'/*$'
~E(~0) = πkσ(1
2cos(2θ)
∣
∣
∣
∣
π
π2
)k = πkσ1
2(cos(2π)− cos(π))k = (πkσ)k
E./'&,*/%*) 5'$' *-1$.+.$ *-%* $*-:&%'(# */ >:/1.C/ (* Q) (*+*,#- (*%*$,./'$ *& 0'&#$ (* σ3F-%# *- >B1.& -'+.*/(# <:* &' 1'$9' *-%' :/.>#$,*,*/%* (.-%$.+:.(' */ &' -:5*$81.* ./%*$.#$)
%*/($*,#- <:*
σ =Q
2πR2
@* *-%' ,'/*$'
~E(~0) =kQ
2R2k
!
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+$@5.)/50 +3.%)B +# 1/)+55/9# j
"$ 5%,40 +$@5.)/50 4)01(5/10 40) (#% 5%)'% dq 3+)B dE = kdqr2 6 C0#1+ r +3 $% 1/3.%#5/% 1+31+
$% 5%)'% dq (-/5%1% +# xi ?%3.% +$ 4(#.0 1+ %4$/5%5/9# djD r =√
x2 + d26 E$ /'(%$ 50,0 $0
?+,03 ?+5?0 %#.+3* $% 5%)'% dq $% 401+,03 +35)/-/) 50,0 dq = λ1dx 10#1+ λ1 +3 $% 1+#3/1%1
$/#+%$ 1+ 5%)'% 2 $% 1+&#/,03 50,0 λ1 = Q2L 6 E3/* +$ 5%,40 ;(+1% +F4)+3%10 50,0
dE =kλ1dx
x2 + d2
G% 1/</,03 ;(+ +$ 5%,40 +3.%)B +# 1/)+55/9# j 2% ;(+ $%3 50,40#+#.+3 +# x 3+ %#($%)%#6 >%3
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cos(θ) =d
√
(x2 + d2)
! !"#$%&' () *+$,-.!&,/ 0, '%&'12
"# #$%& '&(#)&
dEy =kλ1ddx
(x2 + d2)3
2
*(%#+)&(,- ,#$,# x = −L .&$%& x = L/
Ey = kλ1d
∫ L
−L
1
(x2 + d2)3
2
dx = kλ1d(x
(d2√
x2 + d2
∣
∣
∣
∣
L
−L
) =2kλ1L
d√
L2 + d2
"# #$%& '&(#)& 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 $#)<
dF = dqE = dq2kλ1L
d√
L2 + y2
=$%# &0&'7)# %8#(# ,8$%8(%& ,#($8,&, ,# 6&)+& >& 42# $8 78#( %8#(# 0& '8$'& 6&)+& %-%&0 &0'&?
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λ2 =Q
L
C$8@ dq = λ2dy > 0& 12#)3& 42#,& #D9)#$&,& 6-'-
dF = λ2dy2kλ1L
y√
L2 + y2
*(%#+)&(,- ,#$,# y = L .&$%& y = 2L
F = 2kλ1λ2L
∫ L
−L
1
y√
L2 + y2dy
E# %8#(# 42#
∫
1
y√
L2 + y2dy =
1
Lln(
√
L2 + y2 − L
y)
F-) 0- %&(%-
F = 2kλ1λ2L(1
Lln(
√
L2 + y2 − L
y)
∣
∣
∣
∣
∣
2L
L
)
"#$&))-00&(,-
F = 2kλ1λ2 ln(
√5− 1
2(√
2− 1))
G##'90&3&(,- 0-$ ;&0-)#$ ,# λ1 > λ2 #( 12(68H( ,# Q/
F =kQ2
L2ln(
√5− 1
2(√
2− 1))
!"#$%&' ()
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+57(& 1"# &#&2&-15 &+
~dF21 = dq2
∫ ∫
Ω
dq1 · −~r1
4πǫ0|~r1|3, con dq1 = σdxdy
= −dq2
∫ +∞
−∞
∫ b+d
d
σdxdy · xx
4πǫ0(x2 + y2)3
2
− dq2
∫ +∞
−∞
∫ b+d
d
σdxdy · yy
4πǫ0(x2 + y2)3
2
= −dq2σy
4πǫ0
∫ +∞
−∞
∫ b+d
d
dxdy · y(x2 + y2)
3
2
=dq2σy
4πǫ0
∫ +∞
−∞
(
1√
x2 + y2
)∣
∣
∣
∣
∣
b+d
d
· dx
=dq2σy
4πǫ0
∫ +∞
−∞
(
1√
x2 + (b + d)2− 1√
x2 + d2
)
· dx, pero dq2 = λdx2
=⇒~dF21
dx2=
λσy
4πǫ0
∫ +∞
−∞
(
1√
x2 + (b + d)2− 1√
x2 + d2
)
· dx
=λσy
2πǫ0· ln
(
x +√
x2 + (b + d)2
x +√
x2 + d2
)∣
∣
∣
∣
∣
+∞
0
=λσy
2πǫ0·
lımx→+∞
ln
1 +
√
1 + ( b+dx )
2
1 +
√
1 + ( dx)
2
− ln
(
b + d
d
)
=λσy
2πǫ0· ln
(
d
b + d
)
=⇒~dF21
dx2=
λσy
2πǫ0· ln
(
d
b + d
)
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R /#-# %( 3( ($ ,+ 24*)+5 6-.#% (%07$ *$&8#)-(-($0( /+)4+'#% /#$ '($%&'+' ,&$(+, '( /+)4+
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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 = λdx A ;#) ,# 0+$0#B
d ~E =kλdx(−xi + dj)
(x2 + d2)3
2
E$0(4)+$'# '(%'( −∞ + +∞B
~E =
∫ +∞
−∞
kλdx(−xi + dj)
(x2 + d2)3
2
= kλ((−∫ +∞
−∞
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
∫ +∞
0g(x)dx
80. 10 2#&209 1# *=-.*+()& -#.# *1 '#,-0 +* .*4%'* #
~E = 2kdλ
∫ +∞
0
1
(x2 + d2)3
2
j
@* 2(*&* A%*
∫
1
(x2+d2)3
2
= x
d2(x2+d2)1
2
80. 10 2#&20
~E = 2kdλ(x
d2(x2 + d2)3
2
∣
∣
∣
∣
∣
∞
0
)j =2kλ
dj
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# λds9 40&4* ds *+ %& -*A%*H0 2.0G0 4* #1#,;.* '(.'%1#. *1 '%#1 -04*,0+ *+'.(;(. *& $%&'()&
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dF = dqE = λRdθ · 2kλ
R + R sin(θ)=
2kλ2dθ
1 + sin(θ)
M&2*3.#&40 4*+4* θ = 0 # θ = π 0;2*&*,0+ 1# $%*.G# 202#1 +0;.* *1 #1#,;.* +*,('(.'%1#.7 8#.#
*+20 %2(1(G#,0+ *1 C*'C0 4* A%*?
! !"#$%&' () *+$,-.!&,/ 0, '%&'12
∫
1
1 + sin(θ)=
sin(θ)− 1
cos(θ)
"#$ %&'# '%$%(#& )*% +, -*%./, .%&*+',$'% %&
F = 4kλ2
!"#$%&' (
!" #! $%&''
!"#$%&' ()
!" #"$%" &'!('") q > 0 *+(" $,-*"-" &,$ '!" +'&*$.#/* #*$$"-" 0,$1"-" &,$ '! 1"!(, #2!/#,-* $"-/, R 3 ")('$" H4 3 '!" +'&*$.#/* +*1/*+05$/#" #,!#5!($/#" #,! )" #"$%"4 +*%6! +* ,7+*$8"*! )" .%'$"9 :")#')* *) ;'<, -* #"1&, *)5#($/#, " ($"85+ -*) 1"!(, #2!/#,9
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φ =
∫
S
~E · d~S =q
ǫ0(1)
A-*1B+4 *) ;'<, >'* "($"8/*+" )" +'&*$.#/* #*$$"-" C>'* ))"1"1,+ DE +*$B /%'") " )" +'1" -*
),+ ;'<,+ >'* &"+*! &,$ )" +*1/*+0*$" C>'* ))"1"1,+ SeE 3 ),+ ;'<,+ >'* &"+*! &,$ *) 1"!(,
#2!/#,C>'* ))"1"1,+ ScE9 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 =
∫
Se
~E · d~S
F) #"1&, *)5#($/#, &$,-'#/-, &,$ )" #"$%" q *+ E = 14πǫ0
qR2 r9 G,!-* r *+ *) 8*#(,$ '!/("$/,
*! -/$*##/2! $"-/")9 F) 8*#(,$ d~S ("17/5! *+(" *! -/$*##/2! $"-/")4 3" >'* d~S = ndS 3 n *+&"$")*), " r #,1, 1'*+($" )" .%'$"@
HI
! !"#$%&' () &*+ ,* -!%..
"# #$%&' $# %#()*+ ,-#
~E · d~S =1
4πǫ0
q
R2r · ndS =
1
4πǫ0
q
R2dS
.(%#/*0()& $# &1%2#(# #3 4-5& 60*0 30 $#72#$8#*09
φe =
∫
Se
~E · d~S =
∫
Se
1
4πǫ0
q
R2dS =
1
4πǫ0
q
R2
∫
Se
dS
:0 2(%#/*03
∫
SedS ;&**#$6&()# 03 +*#0 )# 30 $#72#$8#*0< =(%&(;#$
∫
SedS = 2πR2
< "# #$%0
70(#*0 #3 4-5& $#*+9
φe =1
4πǫ0
q
R2· 2πR2 =
q
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-$$@<
*"$+,-./0
"# 30 :#B )# I0-$$ $01#7&$ ,-# #3 4-5& )#3 ;076& #3H;%*2;& #$ 2()#6#()2#(%# )# 30 $-6#*C;2#
-$0)0 60*0 #(;#**0* ;2#*%0 ;0(%2)0) )# ;0*/0 >72#(%*0$ #$%0 ;0(%2)0) $#0 30 72$70@ B ,-#
#$ )2*#;%07#(%# 6*&6&*;2&(03 0 30 ;0*/0 #(;#**0)0' ;&( ;&($%0(%# )# 6*&6&*;2&(032)0)
1ǫ0< =$
)#;2*'
!
∮
Ω
~E · ndS =Qint
ǫ0
!" # Qint #$ %& '&()& #"'#((& & *!( %& $+*#(,'-# Ω. /$0& 1!(2& -"0#)(&% # %& 3#4 #
5&+$$ $# *+# # #6*(#$&( #" 1!(2& -1#(#"'-&% +$&" ! #% 0#!(#2& # %& -7#()#"'-& 8%! '+&% $#
#9& &% %#'0!(:; !<0#"-#" ! %& #6*(#$-=" #>+-7&%#"0#
~∇ · ~E =ρ
ǫ0
/$ *!$-<%# +$&( %& 3#4 # 5&+$$ *&(& '&%'+%&( #% '&2*! #%?'0(-'! <&9! &%0&$ '!" -'-!"#$ #
$-2#0(@&; %! '+&% $# '+2*%# #" #$0# '&$!. A!")&2!$ '!2! #9# B #% #9# # $-2#0(@& #% '-%-" (!.
A!( %& $-2#0(@& '-%-" (-'&; #% '&2*! #%?'0(-'! "! #*#" # # θ 4 *!( $#( -","-0! "! #*#" # #
z. C$@ & *(-!(-; $&<#2!$ >+#
~E(~r) = E(r)r; !" # r $# 2- # #$ # #% #9# &% *+"0! #" '+#$0-="
4 r #$ #% 7#'0!( +"-0&(-! >+# 7& #$ # #% #9# &% *+"0!.
D#<#2!$ #%#)-( +"& $+*#(,'-# # -"0#)(&'-=" !" # | ~E| "! #*#" & # %&$ 7&(-&<%#$ >+# #E
$'(-<#" %& $+*#(,'-# 8#$ #'-(; | ~E| #<# $#( '!"$0&"0# $!<(# %& $+*#(,'-#:; *&(& *! #( $&'&(
| ~E| 1+#(& # %& -"0#)(&%. A!( #%%!; '!"$- #(#2!$ +"& $+*#(,'-# '-%@" (-'& # (& -! ( '!" #9# #
$-2#0(@& #% #9# B 84 #% #9# #% '-%-" (! '&()& !:. A! #2!$ -$0-")+-( !$ '&$!$F
&: G- r < R./" #$0# '&$! 0#"#2!$ >+# "! H&4 '&()& #"'#((& & *!( #% '-%-" (!; *!( %! '+&%
∮
Ω
~E · ndS =Qint
ǫ0= 0
C #2I$;
∮
Ω
~E · ndS =
∫
mantoE(r)r · ndS +
∫
tapasE(r)r · ndS
= E(r)
∫
mantodS
= E(r)(2πr)h
=⇒ E(r)(2πr)h = 0 =⇒ E(r) = 0 =⇒ ~E = ~0
/$ -"0#(#$&"0# #% H#'H! >+# #% '&2*! #%?'0(-'! $#& "+%! #"0(! #% '-%-" (!.
<: G- R ≤ r./" #$0# '&$! 0#"#2!$ >+#
∮
Ω
~E · ndS = E(r)(2πr)h =(2πR)hσ
ǫ0=⇒ E(r) =
σ
ǫ0
(
R
r
)
A!( 0&"0!;
~E(r) =
~0 r < Rσǫ0
(
Rr
)
· ρ R ≤ r
J!0# >+# #% $&%0! # -$'!"0-"+- & #% 2= +%! #% '&2*! #%?'0(-'! 8>+# #$ "!(2&% & %& $+*#(E
,'-#: #$ + σǫ0.
! !"#$%&' () &*+ ,* -!%..
!"#$%&' ()
"# $%#&# '&( )%*$+%,'-%.& #*/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+
U =ǫ02
∫
R3
~E2d3x
)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
~E(~r) = E(r)rA )2&)# r #* 6( )%*$(&-%( )#6 -#&$+2 )# 6( #*/#+( (6 7'&$2 #& -'#*$%.& 4 r #* #6
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('** $#&#;2*
∮
Ω
~E · ndS =Qint
ǫ0∮
ΩE(r)r · ndS =
1
ǫ0
∫
Θρd3x
E(r)4πr2 =1
ǫ0
∫ 2π
0
∫ π
0
∫ r
0αr3sen(ϕ)drdϕdθ
E(r)4πr2 =4π
ǫ0
∫ r
0αr3dr
E(r)4πr2 =παr4
ǫ0
=⇒ E(r) =α
4ǫ0r2
,> R ≤ r 92&*%)#+#;2* &'#5(;#&$# -2;2 *'7#+C-%# )# %&$#1+(-%.& '&( -?*-(+( #*/0+%-( )#
+()%2 + -2& ;%*;2 -#&$+2 <'# 6( )%*$+%,'-%.& )# -(+1( Θ8 @2+ 6( 6#4 )# D('** $#&#;2*
∮
Ω
~E · ndS =Qint
ǫ0
E(r)4πr2 =4π
ǫ0
∫ R
0αr3dr
E(r)4πr2 =παR4
ǫ0
=⇒ E(r) =α
4ǫ0
(
R
r
)2
R2
!
"#$ %&'%#( %)')*#+ ,-)
~E(r) =
α4ǫ0
r2r r < Rα
4ǫ0
(
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&$6& Θ:
U =ǫ02
∫
R3
~E2d3x
=ǫ02
∫ 2π
0
∫ π
0
∫ +∞
0E(r)2r2sen(ϕ)drdϕdθ
=4πǫ0
2
∫ +∞
0E(r)2r2dr
=4πǫ0
2
(∫ R
0
α2
16ǫ20r6dr +
∫ +∞
R
α2
16ǫ20
R8
r2dr
)
=πα2
8ǫ0·(
r7
7
∣
∣
∣
∣
R
0
− R8 1
r
∣
∣
∣
∣
+∞
R
)
=πα2
8ǫ0·(
R7
7+ R7
)
=πα2
7ǫ0R7
! !"#$%&' () &*+ ,* -!%..
!"#$%&' ()
"#$%&'()(*#% '#% (%+(),% $# -#$-.$/)&-,% '( ),'&# R0 -#$ '($%&','(% '( -,)1, 2#34*./)&5
-,% ρ 6 −ρ 4$&+#)*(%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, '(
'($%&',' '( -,)1, 4$&+#)*( ρ (%
~E(r) = ρr3ǫ0· r7
@( 3, %&*(/)C, '( 3, '&%/)&94-&?$0 , =)&#)& %,9(*#% ;4( ,'(*:% =,), R ≤ r (3 -,*=#
-4*=3(
~E(~r) = E(r)r7 <& 9&($ $# (% $(-(%,)&# =,), 3, )(%#34-&?$ '(3 =)#93(*, -#$#-() (3
-,*=# (3.-/)&-# +4(), '( 3, (%+(),0 ($-#$/).*#%3# 4%,$'# 3, 3(6 '( D,4%%7
"#$%&'()(*#% -#*# %4=()E-&( '( &$/(1),-&?$ 4$, -:%-,), (%+.)&-, '( ),'&# r > R7 F($5
(*#% ;4(
∮
Ω
~E · ndS =Qint
ǫ0
E(r)4πr2 =4πR3ρ
3ǫ0
=⇒ ~E(~r) =R3ρ
3ǫ0r2r
>#) /,$/#0 /($(*#% ;4( (3 -,*=# (3.-/)&-# 1($(),'# =#) 3, (%+(), *,-&B, (%
~E(r) =
ρr3ǫ0· r r < R
R3ρ3ǫ0r2 r R ≤ r
>#$1,*#% (3 #)&1($ '( -##)'($,',% ($ (3 -($/)# '( 3, (%+(), '( '($%&',' '( -,)1, ρ7 G%C0/($(*#% ;4( 3#% -,*=#% (3.-/)&-#% ($ 3#% &$/()&#)(% '( -,', (%+(), (%/:$ ','#% =#)
~E1(r) =ρ
3ǫ0· ~r para |~r| < R
~E2(r) = − ρ
3ǫ0· (~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) =ρ
3ǫ0· ~r − ρ
3ǫ0· (~r − ~d) =
ρ
3ǫ0· ~d
(3 -4,3 (% -#$%/,$/(0 /,3 -#*# %( ;4()C, =)#9,)7
!
!"#$%&' ()
"#$ %&'()&*+,&-# %. ,$)/$ 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 =+'$#%1 2.> %. ?$+'';:
*; @. ,121,$ +#$ ,$)/$ 8+#(+$2 −Q < 0 .# .2 ,.#()1 %. 2$ .'7.)$A > '. 2$ %.6$ 2&*).: BC+.%$
1 #1 .# .D+&2&*)&1 2$ ,$)/$ .# .2 ,.#()1E @& .' $5)3$(&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) =
ρr3ǫ0· r r < R
R3ρ3ǫ0r2 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 ρr3ǫ0· 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 = −Qρr
3ǫ0· r
~a + Qρr
3mǫ0· r = ~0
=⇒ r + Qρ
3mǫ0· r = 0
r + w2 · r = 0
M1) ($#(1A .2 310&3&.#(1 %. 2$ ,$)/$ .' +# 310&3&.#(1 $)3-#&,1 '&382. > '+ 8.)I1%1 %.
1',&2$,&-# .' T = 2πw = 2π
√
3mǫ0Qρ
,; M1) .2 8)&#,&8&1 %. '+8.)81'&,&-#A .2 ,$381 .24,()&,1 ).'+2($#(. .'
~E(r) =
ρr3ǫ0· r + E0x r < R
R3ρ3ǫ0r2 r + E0x R ≤ r
!"#$%&' () &*+ ,* -!%..
!""#$%&'&"()* "+ $,-.* "+/$0%&$* , +* +,%1* )"+ "2" 34 %"#5+0,
~E(r) =
( ρx3ǫ0
+ E0) · x −R < x < R
( R3ρ3ǫ0x2 + E0) · x R ≤ x
(E0 − R3ρ3ǫ0x2 ) · x x ≤ −R
6($*(0%"-*# +,# (5"7,# .*#&$&*("# )" "85&+&'%&*9
&: −R < x < R ;5"%"-*# 85"
ρx
3ǫ0+ E0 = 0 =⇒ x1 =
−3ǫ0E0
ρ
<,%, 85" x1 "3�"4 #" )"'" $5-.+&% 85"
x1 =−3ǫ0E0
ρ> −R =⇒ E0 <
Rρ
3ǫ0
6#0, $*()&$&=( )"'" $5-.+&% E0 .,%, 85" >,?, 5( .5(0* )" "85&+&'%&* )"(0%* )" +,
"#@"%,9
A,'"-*# 85" #& "+ +,.+,$&,(* )" +, "("%1B, .*0"($&,+ "+/$0%&$, ∇2U "( "+ .5(0*
)" "85&+&'%&* "# -,?*% 85" $"%*4 "(0*($"# 0,+ .5(0* "# "#0,'+"C #& "# -"(*% 85"
$"%*4 "(0*($"# 0,+ .5(0* "# &("#0,'+"4 ? 85" U = −qV 4 )*()" D "# +, )&@"%"($&, )"
.*0"($&,+ "+/$0%&$*9 E#B4 0"("-*# 85"
V (x) = −∫
~E(x) · xdx + c
= −∫
(
ρx
3ǫ0+ E0
)
dx + c
= −E0x−ρx2
6ǫ0+ c
U(x) = −QV (x)
= QE0x + Qρx2
6ǫ0−Qc
=⇒ ∇2U =d2U
dx2=
Qρ
3ǫ0> 0
<*% +* 0,(0*4 "# .5(0* )" "85&+&'%&* "#0,'+"9
&&: R ≤ x ;5"%"-*# 85"
R3ρ
3ǫ0x2+ E0 = 0 =⇒ x2 imaginario
.*% +* 85" (* >,? .*#&$&=( )" "85&+&'%&* .,%, R ≤ x9
!
"""# x ≤ −R $%&'&()* +%&
E0 −R3ρ
3ǫ0x2= 0 =⇒ x3 = −R
√
ρR
3ǫ0E0
,-'- +%& x1 &."*/&0 *& 1&2& 3%(45"' +%&
x3 = −R
√
ρR
3ǫ0E0< −R =⇒
√
ρR
3ǫ0E0> 1 =⇒ E0 <
Rρ
3ǫ0
+%& &* 5- ("*(- 3)61"3"76 +%& &63)6/'-()* -6/&'")'(&6/& 4-'- +%& 8%2"&'- &+%"9
5"2'"): ;*/& 4%6/) 1& &+%"5"2'") &* "6&*/-25&0 *& 1&<- -5 5&3/)' =&'">3-'5):
?- 4'%&2- 1& 5) *"@%"&6/& *& 1&<- -5 5&3/)': A&()* ="*/) +%& 4-'- E0 < Rρ3ǫ0
&."*/&6 1)* 4%6/)*
1& &+%"5"2'")0 %6) &*/-25& B &5 )/') "6&*/-25&: ,-'- E0 = 0 &."*/& %6- *)5- 4)*"3"76 1& &+%"5"2'")0
&6 &5 3&6/') 1& 5- &*C&'- B &* &*/-25&: ,-'- E0 = Rρ3ǫ0
&."*/& %6- *)5- 4)*"3"76 1& &+%"5"2'")0 &*
&6 x = −R B &* "6&*/-25&: D 4-'- E0 > Rρ3ǫ0
6) 8-B 4)*"3")6&* 1& &+%"5"2'"): E)1) &*/) *& 4%&1&
=&' 35-'-(&6/& 8-3"&61) %6 @'F>3) 1& ;G.# =&'*%* . G*& 1&<- -5 5&3/)'#:
! !"#$%&' () &*+ ,* -!%..
!"#$%&' ((
"#$%&'()( *$ +&,&$')# -*. ,/)0# '( )/'&# R 1*( %( +/)0/ ($ %* &$2()&#) +#$ *$/ '($%&'/'
ρ = ρ0(1 − rR)3 '#$'( ρ0 (% *$/ +#$%2/$2( 4#%&2&5/3 %&($'# r ,/ '&%2/$+&/ -('&'/ '(%'( (, (6(
'(, +&,&$')#7 8$+*($2)( / 1*( '&%2/$+&/ '(, (6( (, +/-4# (,9+2)&+# (% -:;&-# . +/,+*,( (%2/
-/0$&2*' -:;&-/7
)"$*+,-./
</'# 1*( (, +&,&$')# (% -*. ,/)0#3 (, +/-4# (,9+2)&+# (%2/): ($ '&)(++&=$ )/'&/, ($ +##)'($/'/%
+&,>$')&+/%7 "#$ (%2# 4#'(-#% +#$%&'()/) +#-# %*4()?+&( '( 0/*%% *$ +&,&$')# '( )/'&# r < R3
+#$ &0*/, (6( 1*( (, +&,&$')# '( )/'&# R7 8%2# %( 4*('( /4)(+&/) ($ ,/ %&0*&($2( ?0*)/@
A#'(-#% ($2#$+(% (%+)&B&) (, +/-4# (,9+2)&+# +#-#
~E = Er3 '#$'( r (% (, 5(+2#) *$&2/)&#
)/'&/, ($ +##)'($/'/% +&,>$')&+/%7 A#) ,/ ,(. '( 0/*%% %( 2($'): 1*(@
∫
S
~E · ndS =
∫
tapas
~E · ndS +
∫
manto
~E · ndS =qinterior
ǫ0(1)
8, 5(+2#) $#)-/, n '( ,/% 2/4/% (% 4()4($'&+*,/) / r7 "#$ (%2# %( 2($'): 1*( r · n = 0 . 4#)
,# 2/$2#
∫
tapas
~E · ndS =
∫
tapasEr · ndS = 0 (2)
8, 5(+2#) n3 $#)-/, /, -/$2#3 %(): 4/)/,(,# / r . 4#) ,# 2/$2# r · n = 1 ,# 1*( &-4,&+/ 1*(@
∫
manto
~E · ndS =
∫
mantoEr · ndS = E ·
∫
mantodS
C/ &$2(0)/,
∫
manto dS +#))(%4#$'( /, :)(/ '(, -/$2#@
∫
manto dS = 2πrL7 A#) ,# 2/$2#@
∫
manto
~E · ndS = E · 2πrL (3)
D((-4,/E/$'# FGH . FIH ($ FJH )(%*,2/ 1*(@
E · 2πrL =qinterior
ǫ0(4)
!
"# $%& '()*( +(,( ,&-#).&, &-*& +,#/)&0( &- 1()1%)(, )( 1(,2( $%& 3(4 &5 &) 65*&,6#, 7& )( -%+&,8
916& 7& 2(%--: ;&5&0#- %5( 7&5-67(7 .#)%0<*,61( 7& 1(,2( &)<1*,61( 7&5*,# 7& &-*( -%+&,916&:
=&0#- $%& -& 1%0+)6,> )( -62%6&5*& ,&)(16?5@
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%,(@
"%&2#C dV -&,> -60+)&0&5*& &) .#)%0&5 &5*,& )#- 7#- 16)657,#-C )# $%& ,&-%)*(@
dV = 2πr′dr′L (6)
D&&0+)(E(57# FGH &5 FIH@
dq = 2πr′ · dr′ · L · p(r′)
dq = 2πr′ · dr′ · L · ρ0(1−r′
R)
=⇒ dq = 2πL · ρ0(r′ − r′2
R)dr′
J5*&2,(57#@
q = 2πLρ0
∫ r
0(r′ − r′2
R)dr′ = 2πLρ0 · (
∫ r
0r′dr′ −
∫ r
0
r′2
Rdr′)
=⇒ q = 2πLρ0 · (r2
2− r3
3R) = 2πLρ0 · (
3Rr2 − 2r3
6R)
=⇒ q =πLρ0
3R(3Rr2 − 2r3) (7)
K& F H 4 F!H@
E · 2πrL =πLρ0
3Rǫ0(3Rr2 − 2r3)
K&-+&B(57# E@
E =ρ0
6Rǫ0(3Rr − 2r2) (8)
! !"#$%&' () &*+ ,* -!%..
"#$%$&'( ()*$% +)%) ,#$ -).'% /$ r $. 0)&+' $.102%30' $( &)43&'5 6)%) $(2' /$%3-)&'( E%$(+$02' ) %7
dE
dr=
ρ0
6Rǫ0(3R− 4r)
89#).):/' ) ; ($ '*23$:$ ,#$ $. -).'% /$ r *#(0)/' $(7
r =3R
4
<$ $(2) &):$%)
|E|max =3ρ0R
16ǫ0
!
!"#$%&' ()
"#$%& '() *+,#() -#.#-$() *,%,+&+() /0& )& 
&#$%,# , 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-&#$&A
B,%, (#$%,% &+ 1,5*( &+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.%&$#' #$
%#)*+ 4 0( (1-+
~E · n = 0 $+ ;&( .)*$.%# ;&(
∫
~E · ndS = 0
@./ ()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) · n = E $+ ;&( .)*$.%# ;&(
∫
S
~E · ndS = E
∫
SdS
9( (1-# )#/('#= #*$.%#/0+ $# $(4 0( 2#&11= 1( -(/0'8 ;&( ($ >&7+ -+-#$ 1+A'( $# %#7# 1('8 ($
>&7+ # -'#?,1 0( $#1 %#'#1 $#-('#$(15
φ = E
∫
S1
dS + E
∫
S2
dS = EA + EA = 2EA =qint
ǫ0(1)
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
2EA =ρ2xA
ǫ0
E =ρx
ǫ0
J 0#0# $# 1.)(-'K# ?()+1 ;&( ($ %#)*+ (1-# (/ 0.'(%%.:/ i= *+' $+ -#/-+ 1( -.(/( ;&(5
~Eρ =ρx
ǫ0i, si 0 < x < a
!
• 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
φ = 2EA =ρ2aA
ǫ0
E =ρa
ǫ0
B- ',/$& #53, #2 ()%#'')12 i $&% -& 3,23&9
Eρ =ρa
ǫ0i, si x > a
C&3,% 0+# #- ',/$& #2 #53, %#*)12 #5 '&253,23#?
!"#$% &'()*+,)% $+%-.),-% $%+ '" )"/)"+" &/0(+,)" 1Eσ2 2
"#$,%,%#/&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
! !"#$%&' () &*+ ,* -!%..
"# $%&' ( )*(+,- ./ /-)( -%0/*123/ -/*4 φ =∫
S~E · ndS5 0/*' /# +/2)'* n /- 0(*(#/#' (# 2(67
0'8(69'- *(.3(#/-: ; 0'* #' <%/
~E · n = E ; 0'* #' )(=)' φ = E∫
S dS = E4πr2>
?'* ')*' #(.'5 #( 2(*@( 3=)/*3'* /- =%#( ;( <%/ -'#' A(; 2(*@( /= #( 2(-2(*(B qint = 0> ?'* #( #/;
./ @(%-- -/ )/=.*4 <%/
φ = E4πr2 =qint
ǫ0= 0
E = 0, si r < R
• r > R
C'=-3./*(6'-5 (# 3@%(# <%/ /= /# 2(-' (=)/*3'*5 %=( 2(-2(*( /-D,*32( ./ *(.3' r > R 2'6'
-%0/*123/ @(%--3(=(> E(&' #'- 63-6'- (*@%6/=)'- ./ -36/)*F( (=)/*3'* +/6'- <%/ /# 2(60'
/#,2)*32' -/*4 *(.3(# ; 0'* #' )(=)'5 0(*(#/#' (# +/2)'* ='*6(# ; ./ /-)( 6(=/*(B
φ =
∫
S
~E · ndS = E
∫
SdS = E4πr2
G( 2(*@( 3=)/*=( qint -/*4 #( 2(*@( 2'=)/=3.( /= )'.( #( -%0/*123/5 #( 2%(# -/ '9)3/=/ 6%#)30#37
2(=.' /# 4*/( )')(# 0'* #( ./=-3.(. -%0/*123(# ./ 2(*@(B qint = σ4πR2>
?'* #( #/; ./ @(%--5 -/ )/=.*4 <%/
φ = E4πr2 =qint
ǫ0=
σ4πR2
ǫ0
H/-0/&(=.' /# 2(60' /#,2)*32' -/ )3/=/ <%/
E =σR2
ǫ0r2
"-)/ 2(60' /-)(*4 /= .3*/223I= *(.3(#5 0'* #' )(=)'B
~Eσ =σR2
ǫ0r2r, si r > R
H'=./ r /- /# +/2)'* %=3)(*3' /= .3*/223I= *(.3(#>
JA'*( <%/ /=2'=)*(6'- #'- .'- 2(60'- /#,2)*32'- 0'./6'- /=2'=)*(* /# 2(60' */-%#)(=)/
-%0/*0'=3,=.'#'-> ?(*( /-)'5 2'=-3./*(*/ K */@3'=/-B
L: 0 < x < aLL: a < x < a + 2RLLL: x > a + 2R
L: 0 < x < aB "= /-)( */@3I= #'- 2(60'- /#,2)*32'- -'=B
~Eρ = ρxǫ0
i ; ~Eσ = σR2
ǫ0r2 r> JA'*( )/=/6'-
<%/ /M0*/-(* r ; r ./ D'*6( 2'=+/=3/=)/ 0(*( 0'./* -%6(* (69'- 2(60'-> C'=-3./*/6'- #(
-3@%3/=)/ 1@%*(B
N/6'- <%/ -/ 2%60#3*4 <%/ a + R = x + r 2'= #' <%/ */-%#)( r = a + R − x> J./64-5 +/6'-
<%/ r = −i ; 0'* #' )(=)' 0'./6'- /-2*393* /# 2(60' /#,2)*32' 2'6'B
~Eσ =−σR2
ǫ0(a + R− x)2i
!
"# #$%& '&(#)&* #+ ,&'-. #+/,%)0,. )#$1+%&(%# #( #$%& )#203( #$
~E = ~Eρ + ~Eσ* #$ 4#,0)
~E = (ρx
ǫ0− σR2
ǫ0(a + R− x)2)i
556 a < x < a + 2R7 8( #$%& )#203( #+ ,&'-. +. &-.)%& $.+. Eρ 9& :1# #+ ,&'-. -).41,04. -.)
+& ,&$,&& #$;#)0,& #$ (1+. #( #+ 0(%#)0.) #++&< =.) +. %&(%.7
~E = ~Eρ =ρa
ǫ0i, si a < x < a + 2R
5556 x > a + 2R 8( #$%& )#20.* &'>.$ ,&'-.$ #$%&)?( #( 40)#,,03( 9 $#(%04. i< =&)& .>%#(#)
+& #@-)#$03( 4#
~Eσ #( ;1(,03( 4# @* ,.($04#)&'.$ +& $0210#(%# A21)&7
B#'.$ :1# $# ,1'-+# +& )#+&,03( x = a+R+ r* ,.( +. :1# )#$1+%& :1# r = x−a−R< C4#'?$*
r = i< "# #$%& ;.)'&* %#(4)#'.$ :1#7
~Eσ =σR2
ǫ0(x− a−R)2i
D1#2.* -.) -)0(,0-0. 4# $1-#)-.$0,03(* #+ ,&'-. #+/,%)0,. #( #$%& )#203( $#)?7
~E = (ρa
ǫ0+
σR2
ǫ0(x− a−R)2)i, si x > a + 2R
E0(&+'#(%# #+ ,&'-. #+/,%)0,. )#$1+%&(%# -.) &'>&$ 40$%)0>1,0.(#$ 4# ,&)2& $#)?7
! !"#$%&' () &*+ ,* -!%..
~E =
(ρxǫ0− σR2
ǫ0(a+R−x)2)i 0 < x < a
ρaǫ0
i a < x < a + 2R
(ρaǫ0
+ σR2
ǫ0(x−a−R)2)i 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#+ '$*'(+ )+,%A+ &' &'#$%&+&
&' ,+(0+ 3#%*"()' ρ '$
~E(r) = ρr3ǫ0· r5
>' 2+ $%)'.(B+ &' 2+ &%$.(%73,%=#/ + ;(%"(% $+7')"$ 93' +&')8$ ;+(+ R ≤ r '2 ,+);" ,3);2'
~E(~r) = E(r)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'
∮
Ω
~E · ndS =Qint
ǫ0
E(r)4πr2 =4πR3ρ
3ǫ0
=⇒ ~E(~r) =R3ρ
3ǫ0r2r
<"( .+#."/ .'#')"$ 93' '2 ,+);" '2-,.(%," 0'#'(+&" ;"( 2+ '$*'(+ )+,%A+ '$
~E(r) =
ρr3ǫ0· r r < R
R3ρ3ǫ0r2 r R ≤ r
<"#0+)"$ '2 "(%0'# &' ,""(&'#+&+$ '# '2 ,'#.(" &' 2+ '$*'(+ &' &'#$%&+& &' ,+(0+ ρ5 F$B/
.'#')"$ 93' 2"$ ,+);"$ '2-,.(%,"$ '# 2"$ %#.'(%"('$ &' ,+&+ '$*'(+ '$.8# &+&"$ ;"(
~E1(r) =ρ
3ǫ0· ~r para |~r| < R
~E2(r) = − ρ
3ǫ0· (~r − ~d) para |~r − ~d| < R
6"$ 1',."('$ '# 2+ %#.'($',,%=# &' +)7+$ '$*'(+$ $+.%$*+,'# $%)32.8#'+)'#.'
|~r−~d| < R 4 |~r| < R5 F$B/ ;"( '2 ;(%#,%;%" &' $3;'(;"$%,%=#/ '2 ,+);" '2-,.(%," '# 2+ %#.'($',,%=#
&' +)7+$ '$*'(+$ '$
~E(~r) = ~E1(~r) + ~E2(~r) =ρ
3ǫ0· ~r − ρ
3ǫ0· (~r − ~d) =
ρ
3ǫ0· ~d
'2 ,3+2 '$ ,"#$.+#.'/ .+2 ,")" $' 93'(B+ ;("7+(5
! !"#$%&' () &*+ ,* -!%..
!"#$%&' (
!"#$%&'( )(#%"*!#+"'"&%!
!"#$%&' ()
!"#$%&'& (" )*+"! $","$-! %& %&"#$%+% %& .+'/+ #()&',.$+* ("$0!'1& σ > 0 "!'1+* +* &2& 3%& &.(+.$4" x = 05 6" ax #& &".(&"-'+ ("+ .+'/+ )("-(+* −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$-+.$!"+*K
*"$+,-./0
+7 !"!.&1!# &* .+1)! &*8.-'$.! /&"&'+%! )!' &* )*+"! : *+ .+'/+ −q #!9'& &* &2& 3 ?(& *!#("&5 C+*&# .+1)!# #!"
~E1 =σ
2ǫ0x
~E2 =1
4πǫ0
−q
(a− x)2· −x =
1
4πǫ0
q
(a− x)2· x
D#=E &* .+1)! &*8.-'$.! &"-'& ; : −q A?(& &#-FE %$/+1!#E &" &* )("-! D7 &#
~E = ~E1 + ~E2 =
(
σ
2ǫ0+
1
4πǫ0
q
(a− x)2
)
· x
G&'! :+ ?(&
~E = −~∇V E &"-!".&# &" Ω A*+ '&/$4" )&%$%+7
∂V
∂z= 0 ∂V
∂y = 0 −∂V
∂x=
(
σ
2ǫ0+
1
4πǫ0
q
(a− x)2
)
=⇒ V (x) = −(
σx
2ǫ0+
1
4πǫ0
q
(a− x)
)
+ C
97 6* .+1)! &*&.-'!#-F-$.! &# .!"#&'B+-$B!E +#= ?(& )!%&1!# (#+' .!"#&'B+.$4" %& &"&'/=+
&"-'& &"&'/=+ )!-&".$+* : .$"8-$.+5 H+ .+'/+ −e )+'-& %&* '&)!#! %&#%& a/2x I+.$+ &*
)*+"!5 G!' .!"#&'B+.$4" %& &"&'/=+E -&"&1!# ?(&
JK
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
−eV (a/2) =mv2
2− eV (0)
=⇒ K =mv2
2= e (V (0)− V (a/2))
= e
(
− 1
4πǫ0
q
a+ C +
1
4πǫ0
2q
a+
σa
4ǫ0− C
)
= e
(
1
4πǫ0
q
a+
σa
4ǫ0
)
!
!"#$%&' ()
"# $%#&#& '() #)*+,-) .(&'/.$(,-) '# ,-'%() r1, r2 0 .-,1-) q1, q2 )#2-,-'-) 2(, /&- 1,-& '%)3
$-&.%- d >> r1, r24 "% -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 = kq1
r1, V2 = k
q2
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,%(: $#() ;/#
q1 + q2 = (q1)f + (q2)f
C8 -8.-&?-, #8 #;/%8%6,%(: #) '#.%,: ./-&'( '#F- '# D-6#, $,-&)*#,#&.%- '# .-,1-) #&$,# 8-) #)*#,-):
$#() ;/# 8- '%*#,#&.%- '# 2($#&.%-8 #8+.$,%.( #&$,# -56-) #) &/8(: #) '#.%,:
V1 = V2
k(q1)f
r1= k
(q2)f
r2
4π(r1)2(σ1)f
r1=
4π(r2)2(σ2)f
r2
=⇒ (σ2)f
(σ1)f=
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$#()
(q1)f + (q2)f = q1 + q2
4π(r1)2(σ1)f + 4π(r2)
2(σ2)f = q1 + q2
4πr1(σ1)f (r1 + r2) = q1 + q2
=⇒ (σ1)f =1
4πr1
q1 + q2
r1 + r2
(σ2)f =1
4πr2
q1 + q2
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%+,-:
*"$+,-./0
6-.$+*!&!"-$ #. $!,%-& "#; "#; /!'#!<- *! () $#/!&4,+! ,!.%&)*- !. ~r5 %). /!'#!<-'#! $!) ,)$+ /().-5 ; $+%#!"-$ ) %&)27$ *! !(() #. ,+(+.*&- Ω *! )(%#&) 0 < 2h << 1 ; 9&!)0 < A << 15 %)( '#! () $#/!&4,+! (- ,&#=! /-& () "+%)*: >-& () (!; *! 0)#$$5 %!.!"-$
∮
Ω
~E(~r) · ndS =Qint
ǫ0
>!&-
Qint∼= σ(~r)A =⇒ Qint
ǫ0∼= σ(~r)A
ǫ0
?!.!"-$ )*!"9$
∮
Ω
~E(~r) · ndS =
∫
manto
~E(~r) · ndS +
∫
carasup
~E1(~r) · n1dS +
∫
carainf
~E2(~r) · n2dS
>-*!"-$ !@/&!$)& !( ,)"/- !(7,%&+,- ,-"- () $#") *! $# ,-"/-.!.%! %).0!.,+)( ; .-&")(
) () $#/!&4,+!5 !$ *!,+&5
~E = ~E⊥ + ~E//: A*!"9$5 ,-"- !( 9&!) !$ "#; /!'#!<)5 !( "1*#(-
*!( ,)"/- !(7,%&+,- $-8&! 7( $! ").%+!.! ,)$+ ,-.$%).%!5 /-& (- '#! (- /-*!"-$ )/&-@+")& /-&
| ~E(~r)| !. %-*) !( 9&!): ?)"8+7.5 n1 = −n2 6-. !$%- !. "!.%!5 %!.!"-$ '#!
∫
carasup
~E1(~r) · n1dS +
∫
carainf
~E2(~r) · n2dS = ~(E1)⊥A− ~(E2)⊥A
= ( ~(E1)⊥ − ~(E2)⊥)A
!@/&!$+1. '#! .- *!/!.*! *! h B)( +0#)( '#! σ(~r)Aǫ0C: ?)"8+7. %!.!"-$ '#!
∫
manto
~E(~r) · ndS → 0 cuando h → 0
!
"#$% &'(')*# +(,-)'(&'
( ~(E1)⊥ − ~(E2)⊥)A =σ(~r)A
ǫ0
E⊥A =σ(~r)A
ǫ0
=⇒ E⊥(~r) = +σ(~r)
ǫ0
.,/')*# 01' 1( 2*(312&*4 '( '015-5/45* '-62&4*#&7&52* (* 84'#'(&, 2,49, '( #1 5(&'45*4% '-
2,)8* '-62&452* '( #1 5(&'45*4 '# (1-* : &*3, #1 2,49, '#&7 35#&45/153, '( -, #18'4+25' 3' &,-
;*4), 01' '- 2,)8* '-62&452* '( -, #18'4+25' '# (*4),- , -, )5#), <81'# #5 &1=5'4, 2*)8*('(&'
&,(9'(25,- , '--,% -,# 2,49,# #' )*='4$,( : (* '#&,4$,( '( '015-5/45* '-'2&4*#&7&52*>? "#$% 2*(@
#53'4,(3* '#&* '( '- ,(7-5#5# ,(&'45*4% &'(')*# 01' '- 2,)8* '-62&452* '( ~r #*/4' -, #18'4+25'
'#
~E(~r) = +σ(~r)
ǫ0n
2*( n '- ='2&*4 (*4),- 'A&'45*4 , -, #18'4+25' '( ~r?
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!"#$%&' () "# $%#&#& '() *+,*'-+*) #)./+%0*)1 ,#$23%0*)1 4-#0*) 5 0(&0/&$+%0*) '# +*'%() *
5 6 768*91 5 '# #):#)(+ '#):+#0%*63# *-&;-# <&%$(= >* *+,*'-+* %&$#+&* )# 0*+?* 0(& -&* 0*+?*
q0 > 0= "# )-:(&# ;-# 3* $%#++* 7;-# )# $(,* 0(,( (+%?#& '# :($#&0%*3#)9 #)$2 %&<&%$*,#&$#
*3#@*'*=
A= >* *+,*'-+* #B$#+&* )# 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%?#&$#,#&$#=
AAA= "# '#)0(�$* 3* *+,*'-+* #B$#+&* '# 3* $%#++*1 5 3-#?( )# *0#+0* -&* 0*+?* q > 0 4*)$*
-&* '%)$*&0%* c > b '#3 0#&$+( '#3 0(&'#&)*'(+= H#0%'* )% 3* *00%F& '# 3* 0*+?* ; ,('%<0*
( &(I
'9 3* 0*+?* $($*3 '# 0*'* -&* '# 3*) *+,*'-+*)
#9 3* '#&)%'*' '# 0*+?* #& #33*)
.9 3* .-&0%F& :($#&0%*3 5 3* '%.#+#&0%* '# :($#&0%*3 #&$+# #33*)=
JB:3%;-# 6+#C#,#&$# )-) +#):-#)$*)=
*"$+,-./0
A= J3 :+(63#,* $%#&# )%,#$+D* #)./+%0* 0(& 0#&$+( '# 0((+'#&*'*) #3 0#&$+( '# 3*) 02)0*+*)
#)./+%0*)1 '#6%'( * ;-# 3* $%#++* #)$2 %&<&%$*,#&$# 3#@*&* 7+#0(+'*+ ;-# 3* $%#++* )# :%#&)*
0(,( -& )-,%'#+( 5 .-#&$# %&<&%$* '# 0*+?*9= K('*) 3*) '#&)%'*'#) '# 0*+?* )#+2&
-&%.(+,#)= H*'* 3* )%,#$+D* #)./+%0*1 #3 0*,:( 5 :($#&0%*3 #3/0$+%0() '#:#&'#+2& )F3( '#
3* '%)$*&0%* *3 0#&$+( '# 3*) 02)0*+*) 5 #3 0*,:( *:-&$*+2 #& #3 )#&$%'( '# r1 #) '#0%+1
V = V (r) 5
~E(~r) = E(r)r=
*9 "#*& q1, q2, q3, q4 3*) 0*+?*) '# 3*) )-:#+<0%#) %&$#+%(+ 5 #B$#+%(+ '# 3*) *+,*'-+*)
%&$#+&* 5 #B$#+&*1 +#):#0$%C*,#&$#= K#&#,() ;-# q0 = q1 + q2=
L:3%0*&'( #3 $#(+#,* '# M*-)) 0(& -&* )-:#+<0%# #)./+%0* 0(&0/&$+%0* 5 #&$+# 3*)
)-:#+<0%#) '# 3* :+%,#+* *+,*'-+*1 $#&#,()
∮
~E · ndS = 0 =q1
ǫ0=⇒ q1 = 0 =⇒ q2 = q0
'(&'#
~E = ~0 :-#) #3 0*,:( #3/0$+%0( #& #3 %&$#+%(+ '# -& 0(&'-0$(+ #& #;-%3%6+%(
#3#0$+()$2$%0( #) &-3(=
!
"#$%&'()* '+*,' -$ .-*,-/' )- 0'122 &*( 1(' 21#-,3&%- -245,%&' &*(&5(.,%&' 6 -(.,-
$'2 21#-,3&%-2 )- $' 2-71()' ',/')1,'8 .-(-/*2
∮
~E · ndS = 0 =q1 + q2 + q3
ǫ0=⇒ q1 + q2 + q3 = 0 =⇒ q3 = −q2 = −q0
"#$%&'()* )- (1-9* -$ .-*,-/' )- 0'122 &*( 1(' 21#-,3&%- -245,%&' &*(&5(.,%&' )-
,')%* r > b 6 :1- &*(.-(7' $' 2-71()' ',/')1,'8 .-(-/*2
∮
~E · ndS = E(r)4πr2 =q1 + q2 + q3 + q4
ǫ0=⇒ ~E(r) =
q4
4πǫ0
r
r2
;*, *.,' #',.-8 2'<-/*2 :1- $' )%4-,-(&%' -(.,- $' ',/')1,' -=.-,(' 6 $' .%-,,' >-(
-$ %(3(%.*? -2 V08 -2.* -2
V (b)− V (+∞) = V (b) = V0 = −∫ b
+∞~E · d~r
V0 = −∫ b
+∞
q4
4πǫ0
dr
r2
V0 =q4
4πǫ0
1
b=⇒ q4 = 4πǫ0bV0
)*()- .*/'/*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∮
~E · ndS = E(r)4πr2 =q1
ǫ0= 0 =⇒ ~E = ~0
%%? a < r < b >,-7%D( -(.,- ',/')1,'2?
G2 -9%)-(.- :1- -=%2.- &'/#* -$5&.,%&* -( -2.' ,-7%D(8 #1-2 +'6 1(' )%4-,-(&%'
)- #*.-(&%'$ -(.,- ',/')1,'2A C-(-/*2 -( -2.- &'2* :1-
∮
~E · ndS = E(r)4πr2 =q0
ǫ0=⇒ ~E(r) =
q0
4πǫ0
r
r2
%%%? r = b >/-.'$ )- $' ',/')1,' -=.-,('?
G( -2.- #1(.*8
~E(b) = ~08 #1-2 -$ &'/#* -$5&.,%&* -( -$ %(.-,%*, )- 1( /-.'$
-2 2%-/#,- (1$*A
%9? r > b >,-7%D( -=.-,%*, ' $'2 ',/')1,'2?
C'/<%5( ':1@ -2 -9%)-(.- :1- -=%2.- 1( &'/#* -$5&.,%&*8 #1-2 +'6 1(' )%4-,H
-(&%' )- #*.-(&%'$ -(.,- $' ',/')1,' -=.-,(' 6 -$ %(3(%.* >#1(.* )- ,-4-,-(&%'
)-$ #*.-(&%'$?A "2@8 .-(-/*2 :1-
∮
~E · ndS = E(r)4πr2 =q4
ǫ0=
4πǫ0bV0
ǫ0=⇒ ~E(r) =
bV0
r2· r
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
"#$%&'#()*+ #, -.&/* #,0-12'-* #$
~E(r) =
~0 r ≤ aq0
4πǫ0· r
r2 a < r < b~0 r = b
bV0
r2 · r b ≤ r
-3 4.&5'0( )'6').&*$ #, #$/.-'* #( 6.2'.$ 2#7'*(#$8
'3 a ≤ r ≤ b
4#(#&*$
V (r) = −∫
~E · dr + C
= −∫
q0
4πǫ0
dr
r2+ C
=q0
4πǫ0· 1r
+ C
)*()# #, -.&'(* )# '(1#72.-'9( %$.)* :%# 2.)'.,8
;.2. )#1#2&'(.2 < '&/*(#&*$ ,. -*()'-'9( )# 5*2)#8 =.5#&*$ >%# V (b) = V0+
/*2 1.(1* C = V0 − q0
4πǫ0· 1
b + #(1*(-#$
V (r) =q0
4πǫ0
(
1
r− 1
b
)
+ V0
;*2 1.(1*+ 1#(#&*$ >%# ,. )':#2#(-'. )# /*1#(-'., #(12# ,.$ .2&.)%2.$ #$
V (a)− V (b) =q0
4πǫ0
(
1
a− 1
b
)
''3 r ≤ a?, /*1#(-'., #( r = a @.2&.)%2. '(1#2(.3 #$
V (a) =q0
4πǫ0
(
1
a− 1
b
)
+ V0
<*&* #, -.&/* #,0-12'-* #( #$1. 2#7'9( #$ (%,*+ #(1*(-#$ 1*). ,. 2#7'9( )#5#
1#(#2 #, &'$&* /*1#(-'., @)#5# $#2 #>%'/*1#(-'.,3+ /*2 ,* >%#
V (r ≤ a) =q0
4πǫ0
(
1
a− 1
b
)
+ V0
'''3 b ≤ r
V (r) = −∫
~E · dr + C‘
= −∫
q4
4πǫ0
dr
r2+ C‘
=q4
4πǫ0· 1r
+ C‘
=bV0
r+ C‘
!
"#$% V (b) = V0 =⇒ C‘ = 0
=⇒ V (r) =bV0
r
&#'()*#+,%- #. /%0#+1*2. #.310$*1% #'
V (r) =
q0
4πǫ0
(
1a − 1
b
)
+ V0 r ≤ aq0
4πǫ0
(
1r − 1
b
)
+ V0 a < r < bbV0
r b ≤ r
445 6. 1%$0%1*$1(*02$ .2 720#$82- #' ,#1*$- 921#$ V0 = 0- .2 2$)2,($2 #:0#$+2 #'02$; 2. )*')%
/%0#+1*2. <(# .2 0*#$$25 =+ 1%+'#1(#+1*2- +% 927$; 12)/% >(#$2 ,# .2 '#?(+,2 2$)2,($2-
/(#' '@.% #:*'0# 12)/% 1(2+,% 92A (+2 ,*>#$#+1*2 ,# /%0#+1*2.5 (#$,# <(#
~E = −~∇V 5
B%' 1;.1(.%' '%+ *,3+0*1%' 2 .%' 2+0#$*%$#'- 2'8 72'02 *+0$%,(1*$ V0 = 0 A 1%+'*,#$2$ .%
,*19% 2+0#$*%$)#+0#5 C# #'02 >%$)2- 0#+#)%'
12$?2D
q1 = 0
q2 = q0
q3 = −q0
q4 = 0
12)/%D
r ≤ a : ~E = ~0
a < r < b : ~E =q0
4πǫ0· r
r2
b ≤ r : ~E = ~0
/%0#+1*2.D
r ≤ a : V (r) =q0
4πǫ0
(
1
a− 1
b
)
a < r < b : V (r) =q0
4πǫ0
(
1
r− 1
b
)
b ≤ r : V (r) = 0
E .2 ,*>#$#+1*2 ,# /%0#+1*2. #+0$# 2$)2,($2' #' V (a) − V (b) = q0
4πǫ0
(
1a − 1
b
)
- .2
)*')2 <(# #+ #. 12'% 2+0#$*%$5
4445 6. ,#'1%+#102$ .2 2$)2,($2 ,# .2 0*#$$2- '*+ 21#$12$ 2F+ .2 12$?2 q > 0- .2' 12$?2' #+ .2'
2$)2,($2' +% '(>$#+ )%,*G121*@+5
!"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!" #$ %&'(&%( $% &%()% *+ $% &%()% ,-,%$ !' &%!% %(.%!/(% 01)/' 01'2!- $% .10.%+
3-( &-20'(4%&152 !' &%()% '$6&,(1&%7 8'%.-0 */6 3%0% &-2 q1, q2, q3, q47 9-( $': !'
;%/00+ ,'2'.-0 */'
∮
~E · ndS = 0 =q1
ǫ0=⇒ q1 = 0 =⇒ q2 = q0
!-2!' '$ </=- '0 &'(- 3/'0 '$ &%.3- !'2,(- !' /2 &-2!/&,-( '2 '*/1$1>(1- 01'.3('
'0 2/$- ?$% 0/3'(@&1' !' 12,')(%&152 A/' '0A6(1&%+ &-2&62,(1&% % $%0 %(.%!/(%0 :
!'2,(- !' $% 3(1.'(% %(.%!/(%"7
∮
~E · ndS = 0 =q1 + q2 + q3
ǫ0=
q0 + q3
ǫ0=⇒ q3 = −q0 =⇒ q4 = 0
9-( ,%2,-+ $%0 &%()%0 '2 $%0 0/3'(@&1'0 !' $%0 &%0&%(-2'0 2- &%.>1%2 ?$- */' '(% !'
'03'(%(0'"7
'" B% !'201!%! !' &%()% !' $% 0/3'(@&1' !' $% %(.%!/(% 'C,'(2% 0' .-!1@&%+ 3/'0 $%
&%()% q 12,(-!/&' /2% %01.',(D% '2 '$ '03%&1-+ : $%0 &%()%0 '2 $-0 .',%$'0 ?&-2E
!/&,-('0" '0,F2 $1>('0 !' .-4'(0' ?$% !'201!%! !' &%()% !' $% 0/3'(@&1' 'C,'(2% 0'
('!10,(1>/:' !' ,%$ A-(.% !' %2/$%( '$ &%.3- 3(-!/&1!- 3-( q '2 '$ 12,'(1-( !' $%
%(.%!/(% 'C,'(1-("7 9'(- $%0 !'201!%!'0 !' $%0 0/3'(@&1'0 12,'(2%0 2- 0/A('2 &%.E
>1-7 G1 05$- &-201!'(%.-0 %(.%!/(%0+ $% &%()% !' $% %(.%!/(% 'C,'(2% 0'(F %,(%D!%
3-( $% &%()% q > 0+ : $% !'201!%! 0/3'(@&1%$ ,'2!(F /2 .FC1.- ?!' &%()% 2')%,14%"
'2 $% 3%(,' !' $% '0A'(% .F0 &'(&%2% % q7
A" B% %(.%!/(% 'C,'(2% 0' 3-2!(F % /2 3-,'2&1%$ !10,12,- !' &'(- ?01 01)/' &-201!E
'(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
0')/1(F 01'2!- $% .10.% ?'0,- 0' !'03('2!' !'$ I'&I- */' $% )'-.',(D% !' $%0 %(E
.%!/(%0 2- I% &%.>1%!-+ 3-( $- */' 0/ &%3%&1,%2&1% 2- I% &%.>1%!-+ : ,%.3-&-
I% &%.>1%!- $% &%()% '2 '$$%0"7
!
!"#$%&' ()
"#$%&'()( *$+ ,+)&--+ '(-.+'+ '( '($%&'+' -&$(+- *$&/#)0( λ 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) =
∫
Ω
dq
4πǫ0|~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) =
∫ L
0
λdx1
4πǫ0√
(x1 − x)2 + y2 + z2
3$4#$5)(0#%
∫
dx√x2+1
2 9& x = tan(θ)< ($5#$4(%
∫
dx√x2 + 1
=
∫
sec2(θ)dθ√
tan2(θ) + 1=
∫
sec(θ)dθ = ln(sec(θ)+tan(θ)) = ln(x+√
x2 + 1) = arccosh(x)
3=5($'&($'# (%5# +
∫
dx1√(x1−x)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) =
∫ L
0
λdx1
4πǫ0√
(x1 − x)2 + y2 + z2
=λ
4πǫ0
∫ L
0
dx1√
(x1 − x)2 + y2 + z2
=λ
4πǫ0· ln(x1 − x +
√
(x1 − x)2 + y2 + z2)∣
∣
∣
L
0
=λ
4πǫ0· ln
(
L− x +√
(L− x)2 + y2 + z2
−x +√
x2 + y2 + z2
)
!"#$%&' 12
"#$%&'()( (- '&6#-# (-745)&4# '( -+ @.*)+2
3$4*($5)( 6+)+ (- '&6#-# (-745)&4# 0&4)#%4C6&4# >(% '(4&)< 6+)+ '&%5+$4&+ 0*4D# 0;% .)+$'(%
8*( 2d< -+ %(6+)+4&C$ ($5)( 4+).+%AE
+A 3- 6#5($4&+- (-745)&4# ($ 5#'# (- (%6+4&# > !"# E F%( 4##)'($+'+% 6#-+)(%< 1 *%( -+ -(1 '(
-#% 4#%($#% 6+)+ (=6)(%+) -+% '&%5+$4&+% )(-(,+$5(% ($ 5+-(% 4##)'($+'+%A2
:A 3- 4+06# (-745)&4# ($ 5#'# (- (%6+4
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
"# $% &'&()*% +,-&'".%/ 0&/ 0.+,/, 1&2-,3 +%(% 4' 0.+,/, &' )&'&(%/#5
!"#$%&'(
%# 6,( &/ +(.'".+., 0& 24+&(+,2.".7'3 &/ +,-&'".%/ &/8"-(.", &' &/ +4'-, 6 &2-9 0%0, +,(
V =q
4πǫ0
(
1
r+− 1
r−
)
6,( /% /&: 0&/ ",2&',3 -&'&;,2 <4&
r2+ = r2 + d2 − 2rd · cos
(π
2− θ
)
= r2 + d2 − 2rd · sen(θ)
=⇒ 1
r+=
1√
r2 + d2 − 2rd · sen (θ)
=1
r· 1√
1 + (dr )2 − 2d
r · sen (θ)
6&(, &2-%;,2 "%/"4/%'0, &/ +,-&'".%/ +%(% r >> d3 +,( /, <4&
dr << 1 :
1
r+=
1
r· 1√
1 + (dr )2 − 2d
r · sen (θ)
∼= 1
r·(
1− 1
2
(
(
d
r
)2
− 2d
r· sen(θ)
))
=1
r·(
1− 1
2
(
d
r
)2
+d
r· sen(θ)
)
= 24 >&?3 -&'&;,2 +%(% r−
!
r2− = r2 + d2 − 2rd · cos
(π
2+ θ
)
= r2 + d2 + 2rd · sen (θ)
" #$ %&'() )*+,&-) ) ,& )*.$'/&'
1
r−∼= 1
r·(
1− 1
2
(
d
r
)2
− d
r· sen(θ)
)
0&' ,& 12$ $, 3&.$*4/), 3)') r >> d '$52,.)
V (r, θ) =q
4πǫ0
(
1
r+− 1
r−
)
∼= q
4πǫ0r·((
1− 1
2
(
d
r
)2
+d
r· sen(θ)
)
−(
1− 1
2
(
d
r
)2
− d
r· sen(θ)
))
=q
4πǫ0r·(
2d
r· sen(θ)
)
=2qd
4πǫ0· sen(θ)
r2
=p
4πǫ0· sen(θ)
r2
V (r, θ) =1
4πǫ0· ~p · r
r2
#&*#$ ~p $5 $, (&($*.& #/3&,)' $,64.'/4& #$, #/3&,&7
89 :)8$(&5 12$
~E = −∇V = −∂V
∂rr − 1
r
∂V
∂θθ
;5<= .$*$(&5 12$
∂V
∂r= − 1
2πǫ0
psen(θ)
r3
1
r
∂V
∂θ=
1
4πǫ0
pcos(θ)
r3
=⇒ ~E(r, θ) =p
4πǫ0r3
(
2sen(θ)r − cos(θ)θ)
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))
= qdV
= q
(
∂V
∂xdx +
∂V
∂ydy +
∂V
∂zdz
)
=∂V
∂xpx +
∂V
∂ypy +
∂V
∂zpz
= ~p · ~∇V
U = −~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	
*"$+,-./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&>+$'#?
Vp = −∫ p
∞~E · d~ℓ
"#$%&'()+$'# (7 6#3($0&+7 &,*+7 + 0()# ($ (7 &$@$&3#9
A7 6)#47(5+ 3&($( %&5(3)B+ (%./)&0+ 8 6#) 7# 3+$3# %( 3($'); 2*(
~E = Er1 '#$'( r (% (7 C(03#)
*$&3+)&# ($ '&)(00&-$ )+'&+79
"#$%&'()+)(5#% D )(,&#$(%?
EF r < aEEF a < r < bEEEF 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# + r9 :( (%3+ 5+$()+ %( 3($'); 2*( (7 0+56#
~E (% 6+)+7(7# +7 C(03#) $#)5+7 n '( 7+ %*6()@0&( H 6#) 7# 2*(
~E · n = E 8 6#) 7# 3+$3#?
φ =
∫
S
~E · ndS = E
∫
SdS = E4πr2
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+ @,*)+?
J7 &,*+7 2*( ($ (7 0+%# +$3()&#) %( 3($'); 2*(
~E · n = E 8 6#) 7# 3+$3#?
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
φ =
∫
S
~E · ndS = E4πr2
"# $#%&# '()*%'+% qint ,* -.*/* $#0$.0#% 12$'03*()* 4# 5.* 0# /*(,'/#/ /* $#%&# *, $+(,)#()*6
7* )*(/%2 5.* qint = ρ0V 6 V *, *0 8+0.3*( *()%* 0# *,1*%# /* %#/'+ r 4 0# *,1*%# /* %#/'+ a $+(
0+ 5.* %*,.0)#9
qint = ρ04
3π(r3 − a3)
:-0'$#(/+ 0# 0*4 /* &#.,,9
E4πr2 = ρ04
3ǫ0π(r3 − a3)
;+( 0+ 5.* ,* +<)'*(*9
E =ρ0(r
3 − a3)
3r2ǫ0
;+3+ $+3*()#3+, #()*%'+%3*()* *,)* $#3-+ *,)# *( /'%*$$'=( %#/'#0 r> -+% 0+ )#()+9
~E =ρ0(r
3 − a3)
3ǫ0r2r
• ! r > b"
?%+$*/'*(/+ /* '&.#0 3#(*%# 5.* *( 0# %*&'=( @@A 8*%*3+, 5.* 0# $#%&# #03#$*(#/# *( 0#
/',)%'<.$'=( 5.* %+/*# #0 $+(/.$)+% ,*%29
qint =4
3πρ0(b
3 − a3)
B* *,)# 3#(*%# #-0'$#(/+ 0# 0*4 /* &#.,, )*(/%*3+, 5.*9
E4πr2 =ρ0(b
3 − a3)
3rǫ0
E =ρ0(b
3 − a3)
3ǫ0r2
C*$)+%'#03*()* 5.*/# *D-%*,#/+ $+3+9
~E(r) =ρ0(b
3 − a3)
3ǫ0r2r
!
"#$%&'#()*+ #, -.&/* #,0-12'-* #( 1*)* #, #$/.-'* 2#$%,1.
~E =
0 r < aρ0(r3−a3)
3ǫ0r2 r a < r < bρ0(b3−a3)
3ǫ0r2 r 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. %(. )'$1.(-'. r > bA*&* /*)#&*$ =#2
~E · d~ℓ = Er · d~ℓ = Edr6 7# #$1. &.(#2. #, /*1#(-'., #,0-12'-* $#2BC
V (r) = −∫ r
∞~E · d~ℓ = −
∫ r
∞Edr = −
∫ r
∞
ρ0(b3 − a3)
3ǫ0r2)dr
D(1#92.()* $# *81'#(# 5%#C
V (r) =ρ0(b
3 − a3)
3ǫ0r, si r > b
34*2. 5%#2#&*$ -.,-%,.2 #, /*1#(-'., . %(. )'$1.(-'. r+ )*()# a < r < b6 E*2 )#@('-'<(
1#(#&*$ 5%#
V (r) = −∫ r
∞~E · d~ℓ
7#$-*&/*(#&*$ #$1. '(1#92., )# ,?(#. $#/.2.()* ,. 12.:#-1*2'. )#$)# #, '(@('1* 4.$1. r+ #( ,.$
12.:#-1*2'.$ )#$)# '(@('1* 4.$1. b &.$ ,. 12.:#-1*2'. )#$)# b 4.$1. r6
V (r) = −∫ b
∞~E · d~ℓ +−
∫ r
b
~E · d~ℓ
F. /2'. '(1#92., 2#$%,1. )# #=.,%.2 #, /*1#(-'., .(1#2'*2 #( r = bC
V (b) = −∫ b
∞~E · d~ℓ =
ρ0(b3 − a3)
3ǫ0b
E*2 *12* ,.)*+ /.2. ,. *12. '(1#92., 1#()2#&*$
−∫ r
b
~E · d~ℓ = −∫ r
b
ρ0(r3 − a3)
3ǫ0r2dr
= − ρ0
3ǫ0· (∫ b
∞rdr − a3
∫ r
b
1
r2dr)
= − ρ0
3ǫ0· (( r2
2
∣
∣
∣
∣
r
b
)− a3(−1
r
∣
∣
∣
∣
r
b
))
=ρ0
3ǫ0· (b
2 − r2
2+ a3 (r − b)
rb)
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
"# #$%& '&(#)&* #+ ,-%#(./&+ #( r )#$0+%& $#)
V (r) =ρ0
3ǫ0(b3 − a3
b+
b2 − r2
2+ a3 (r − b)
rb), si a < x < b
1&)& +& )#2/3( r < a 4#$.-',-(#'-$ +& %)&5#.%-)/& 4#$4# #+ /(6(/%- 7&$%& ) #( 8 %)&5#.%-)/&$9
:#(4)#'-$ ;0# #+ ,-%#(./&+ #$
V (r) = −∫ r
∞~E · d~ℓ = −
∫ b
∞~E · d~ℓ +−
∫ a
b
~E · d~r +−∫ r
a
~E · d~ℓ
<+ $#) (0+- #+ .&',- #+=.%)/.- ,&)& r < a %#(4)#'-$ ;0#
−∫ r
a
~E · d~ℓ = 0
> 4# #$%& '&(#)& #+ ,-%#(./&+ )#$0+%& $#)?
V (r) =ρ0
3ǫ0(b3 − a3
b+
b2 − a2
2+ a3 (a− b)
ab), si r < a
!
!"#$%&' ()
"#$%&'()( (* %&%+(,- '-'# ($ *- ./0)-1 2( +&($($ 3 4&*&$')#% ,05 *-)/#%6 70(4#%6 4-'- 0$# '(
)-'&# r 5 '($%&'-'(% '( 4-)/-% %08().4&-*(% 4#$%+-$+(% σ 5 −σ9
'* :$40($+)( (* 4-,8# (*;4+)&4# %#<)( *- *=$(- AB6 >0( (>0&'&%+- '( *#% 4&*&$')#% ($ 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&*&$')# '( *-)/# L 5 )-'&# r
φ =
∫
S
~E · ndS =qint
ǫ0(1)
A(%-))#**-,#% *- &$+(/)-* 4#,#
∫
S
~E · ndS = E
∫
SdS = E2πrL (2)
"#,# *- '($%&'-' '( 4-)/- %08().4&-* (% σ 4#$%+-$+(D%( 7-4( '( &/0-* ,-$()- 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 %+< '$3+5 >=&=)*<+5 5+)
=@7' #) #& %7)4+ %#*=*+2 &'5 >+$%+)#)4#5 #) y 5# ')7&'<') / %+< &+ 4')4+ #& >'$%+ #&?>4<=>+
#54'<E #) *=<#>>=B) i / 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=#$%<# #54'<E #) *=<#>>=B) iF D+$+ ;#$+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/-',.&
!"#" $% &' ()*"+ ', -%#." ',/-01)-" '2 ', )20'1)"1 (', -),)2(1" &'13 24,"+ ."1 ," 0%20"
~E1 = 0, si 0 < x < a
54'1% (', -),)2(1" ', -%#." ',/-01)-" &' -%,-4,%1% (' )64%, #%2'1% -"#" ," 7)-)#"& '2 ,% .%10'
%8+ (' '&0% #%2'1% 0'2'#"& 94':
~E1 =σa
xǫ0i, si x > a
!"#$ %&'()*+($ #*$,-(+,$ #$* %& (+&+.,*$ ,%*%(/$
;2 ', )20'1)"1 (', -),)2(1" ', -%#." .1"(4-)(" ."1 /, &'13 24,":
~E2 = 0, si d− a < x < d
<"1 "01" ,%("+ -"2&)('1%2(" ,% =641% #"&01%(% %20'1)"1#'20'+ >'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 =σa
(d− x)ǫ0i
?' '&0% #%2'1%+ %, &4.'1."2'1 ,"& -%#."& ',/-01)-"& "@0'2)("&+ >'#"& 94' ', -%#."& 1'&A,0%0'
'&:
~E =
σa(d−x)ǫ0
i 0 < x < a
( σaxǫ0
+ σa(d−x)ǫ0
)i a < x < d− aσaxǫ0
i 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%$'-0"1)% #%& "@>)%: ', -%#)2" &"@1' ', '*' x:
∆V = −∫ d
0
~E · d~ℓ
= −∫ a
0
~E · d~ℓ +−∫ d−a
a
~E · d~ℓ +−∫ d
d−a
~E · d~ℓ
=
∫ a
0
σa
(d− x)ǫ0dx +
∫ d−a
a(σa
xǫ0+
σa
(d− x)ǫ0)dx +
∫ d
d−a
σa
xǫ0i
= (−σa
ǫ0( ln(d− x)|a0)) + (
σa
ǫ0( ln(x)− ln(d− x)|d−a
a )) + (σa
ǫ0(( ln(x)|dd−a)))
= (−σa
ǫ0ln(
d− a
d)) + (
2σa
ǫ0ln(
d− a
a)) + (
−σa
ǫ0ln(
d− a
d))
=2σa
ǫ0(ln(
d− a
a)− ln(
d− a
d))
?' ("2(' &' "@0)'2'
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
∆V =2σa
ǫ0ln(
d
a)
!
!"#$%&' ((
"# $%&'( '%)'*#+) $, )+$%( R $, #+ -.*)+ ,&/0 '+).+$( '(1 *1+ $,1&%$+$ σ = σ0(1 − Rr )2 '(1
σ0 > 02 &%,1$( r #+ $%&/+1'%+ $, *1 3*1/( '*+#4*%,)+ $,# $%&'( + &* ',1/)( O5
') "1'*,1/), ,# 3(/,1'%+# ,1 ,# 3*1/( P &(6), &* ,7,8OP = R95
#) :;+'%+ $(1$, &, <(=,)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$(
V =
∫
kdq
r
"# =+#() $, r #( (6/,1,<(& 3() 3%/+.()+&D
r =√
a2 + d2
"&/+ $%&/+1'%+ &, <+1/%,1, '(1&/+1/, A 3() #( /+1/( 3*,$, &+#%) $, #+ %1/,.)+#
V =k√
a2 + d2
∫
dq =kq√
a2 + d2
E@()+2 '(1&%$,)+), +1%##(& $,# $%&'( '(1 '+).+ dq5 F,# ),&*#/+$( +1/,)%() A '(1&%$,)+1$( d = RA a = r &, /,1$)0 4*,
dV =kdq√
r2 + R2
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
! "!#$! dq %& '! ()*&+)% &%"#,-,# ")+)
dq = σ(r) · dA
.)+) '! *&/%,*!* *& "!#$! *&(&/*& *& '! *,%0!/",! r 0&/*#&+)% 12& "!*! !/,'') *&' *,%") 12&")/%,*&#&+)% 0&/*#3 ,$2!' *&/%,*!* 4! 12& "!*! &'&+&/0) *& 5' &%0! ! ,$2!' *,%0!/",! r *&'
"&/0#) *&' *,%")6 7& &%0! +!/&#! ")/%,*&#!#& ")+) dA &' 3#&! *& 2/ !'!+-#& *& !/"8) dr9
dA = 2πrdr
7& &%0! +!/&#! )-0&/&+)%
dq = 2πσ0(r −R)dr
:' ()0&/",!' *& (#)*2",*) ()# &%0& !/,'') % &/0)/"&%9
dV =2kπσ0(r −R)dr√
r2 + R2
;/0&$#!/*) *&%*& r = 0 8!%0! r = R9
V = 2kπσ0
∫ R
0
(r −R)dr√r2 + R2
= 2kπσ0(
∫ R
0
r√r2 + R2
dr −R
∫ R
0
1√r2 + R2
dr)
= 2kπσ0((√
r2 + R2∣
∣
∣
R
0)− ( ln(
√
r2 + R2 + r)∣
∣
∣
R
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))
= 2kπσ0R(√
2− 1− ln(√
2 + 1))
<, ")/%,*&#!+)%
√2− 1− ln(
√2 + 1) ≈ 1
2 #&%2'0!9
V ≈ −kπσ0R
=)*&+)% &%"#,-,# &%0& #&%2'0!*) &/ >2/",?/ *& '! "!#$! 0)0!' Q *&' *,%")6 :%0! "!#$! '! ()*&+)%
"!'"2'!# ,/0&$#!/*) %)-#& &' *,%") *& '! &@(#&%,?/ dq = 2πσ0(r −R)dr
Q =
∫ R
02πσ0(r −R)dr
= 2πσ0((r2
2−Rr)
∣
∣
∣
∣
R
0
)
= −πσ0R2
7& &%0! +!/&#!
V ≈ Q
−πσ0R2− kπσ0R
!
V ≈ kQ
R
"#$# %&$#'( &) *#+&,-./) &,-#,+0/1# &' .23/) /) *#+&,-./) 43& *0#13-& 3,/ -/02/ *3,+3/) /
3,/ 1.'+/,-./ R 1& &))/5
! "/1/ &)&$&,+# 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$ I56() 7,.)+89 :( %.(0/+$ 2 *+($ -#) 0+ +**&4$ '( 0+
,)+6('+' + 0+ -#%&*&4$ II3 +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 +kq2
2ℓ
G+ ($(),D+ -#/($*&+0 ($ 0+ -#%&*&4$ HH
UII = 2mgℓ(1− cos(θ)) +kq2
2ℓ sin(θ)
"#1# 0+ ($(),D+ %( *#$%()6+3 -#'(1#% I+*() UI = UII 2 -#) 0# /+$/#
2mgℓ +kq2
2ℓ= 2mgℓ(1− cos(θ)) +
kq2
2ℓ sin(θ)
J(%-(F+$'# ?K
q = ±2ℓ
√
mg sin(θ)
k(sec θ − tan(θ))
!
!"#$%&' ()
"#$%& '() *)+) ,&-.&,/0%.*) 1& %)1.# R ,& 2.&(& '() 1.,2%.$'*.3( 1& *)%4) '(./#%-& σ5 6)7*'7&8'* 97 +#2&(*.)7 &7&*2%#,2:2.*# ) 7# 7)%4# 1&7 &;& Z< +)%) z > 05#* 97 *)-+# &70*2%.*# ) 7# 7)%4# 1&7 &;& Z< +)%) z > 05+* 97 *)-+# &70*2%.*# &( &7 +'(2# O5
,"$-+./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) ,'+&%?*.&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 = R2sin(γ)dγdθ
F) 1.,2)(*.) r &( &,2& *),# 7) #$2&(&-#, '2.7.G)(1# &7 2&#%&-) 1&7 *#,&(#8
r2 = R2 + z2 − 2zR cos(γ)
H,I< &7 +#2&(*.)7 +%#1'*.1# +#% dq &,
dV =kσR2 sin(γ)dγdθ
√
R2 + z2 − 2zR cos(γ)
F'&4#< .(2&4%)-#, &,2) &E+%&,.#( &7.4.&(1# 7#, 7.-.2&, 1& .(2&4%)*.#( 1& 2)7 /#%-) 1& %&*#%%&%
7) ,'+&%?*.&8
V =
∫ 2π
0
∫ π
π2
kσR2 sin(γ)dγdθ√
R2 + z2 − 2zR cos(γ)
9,2) .(2&4%)7 ,& +'&1& %&1'*.% )
V = 2kπσR2
∫ π
π2
sin(γ)√
R2 + z2 − 2zR cos(γ)
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
"#$# $%&'()%$ %&*'+ ,%-%.'& &#-%$ /#(/0(#$ (# 12*%3$#( ,% (# 4'$.#5
∫
sin γ√
A + B cos(γ)dγ
6#/%.'& %( /#.-1' ,% )#$1#-(% u2 = A + B cos(γ)7 8% %&*# .#2%$# dγ = 2udu−B sin(γ) 7 9 #&1
∫
sin γ√
A + B cos(γ)dγ = − 2
B
∫
1du = − 2
Bu
:'.' u =√
A + B cos(γ)5
∫
sin γ√
A + B cos(γ)dγ = − 2
B
√
A + B cos(γ)
"#$# %( /#(/0(' ,% ;'*%2/1#( *%2%.'& <0% A = R2 + z2= B = −2zR7 >%%.;(?#2,' %2 (#
12*%3$#(5
∫
sin γ√
R2 + z2 − 2zR cos(γ)dγ =
1
zR
√
R2 + z2 − 2Rz cos(γ)
@A'$#+ ;',%.'& /#(/0(#$ %( 12*%3$#( &12 ,1B/0(*#,%&5
V (z) =2kπσR2
zR(√
R2 + z2 − 2Rz cos(γ)∣
∣
∣
π
π2
) =2kπσR
z(z + R−
√
R2 + z2)
! @A'$# <0% *%2%.'& %( ;'*%2/1#( %(C/*$1/' %2 zk+ %( /#.;' %(C/*$1/' &% ;0%,% /#(/0(#$ /'.'5
~E = −∇V
D%.'& <0% %( ;'*%2/1#( %&*# &'(' %2 402/1'2 ,% (# )#$1#-(% z = ;'$ (' *#2*' *%2,$%.'& <0% %(
/#.;' %(%/*$1/' %&5
~E = −dV
dzk
8%$1)#.'& V (z)5
dV
dz= 2kπσR
d
dz(1
z(z + R−
√
R2 + z2))
= 2kπσ(−1
z2(z + R−
√
R2 + z2) +1
z(1 +
1
2
1√R2 + z2
2z))
= 2kπσ(−1
z− R
z−√
R2 + z2
z2+
1
z+
1√R2 + z2
)
= −2kπσR2(
√R2 + z2 −R
z2√
R2 + z2)
8% %&*# .#2%$# %( /#.;' %(%/*$1/' %2 %( ;02*' zk &%$E5
!
~E(z) = 2kπσR2(
√R2 + z2 −R
z2√
R2 + z2)k
! "#$%& '(# )* #+,-#&.%/ */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%$%=
2kπσR2(1− R√
z2+R2
z2)
>(#6%7 0%$*-#$%& #) ).$.0# 8(*/1% z → 0 1# )* ?(/8.%/ E(z)k@=
lımz→0
E(z) = 2kπσR2 lımz→0
(1− R√
z2+R2
z2)
A%$% :#$%&7 #&0# ).$.0# #& 1# )* ?%-$*
00 ,%- )% '(# (0.).B*-#$%& >CDE,.0*) ,*-* 8*)8()*-)%=
2kπσR2 lımz→0
(1− R√
z2+R2
z2) = 2kπσR2 lım
z→0((1− R√
z2+R2)′
(z2)′)
= 2kπσR2 lımz→0
(R(z2 + R2)−
3
2
2z)
= 2kπσR2 · 1
2R2
= kπσ
F# #&0* $*/#-*
~E(0) = kπσk
! !"#$%&' () "'$*+ ,!& *&* $-'*.$!$, '
!"#$%&' (
!"#$%&!'()
!"#$%&' ()
! "#!$!$ %&' !'()*+' ,&$%-,"&*+' %! *+%#&' r1, r2 . ,+*/+' q1, q2 '!0+*+%+' 0&* -$+ /*+$ %#'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
>+%& 9-! !6 '#'"!3+ !'"? !$ -$+ *!/#:$ +,&"+%+ %!6 !'0+,#&8 0&%!3&' "&3+* ,&3& 0-$"& %!
*!(!*!$,#+ %!6 0&"!$,#+6 !6 #$<$#"& ! #/-+6+* !6 0&"!$,#+6 + ,!*& +66@8 !' %!,#*8 V (+∞) = 02 A'@8!6 0&"!$,#+6 '&4*! 6+' '-0!*<,#!' %! 6+' !'(!*+' ,&$%-,"&*+' !'
V1 = kq1
r1, V2 = k
q2
r2
%&$%! B!3&' '-0-!'"& 9-! !6 0&"!$,#+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+' !' $-6&8 !' %!,#*8
V1 = V2
k(q1)f
r1= k
(q2)f
r2
4π(r1)2(σ1)f
r1=
4π(r2)2(σ2)f
r2
=⇒ (σ2)f
(σ1)f=
r1
r2
>! !'"& 5!3&' 9-! !$ /!$!*+68 6+' *!/#&$!' !$ 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"; %<)&/&#%-
(q1)f + (q2)f = q1 + q2
4π(r1)2(σ1)f + 4π(r2)
2(σ2)f = q1 + q2
4πr1(σ1)f (r1 + r2) = q1 + q2
=⇒ (σ1)f =1
4πr1
q1 + q2
r1 + r2
(σ2)f =1
4πr2
q1 + q2
r1 + r2
!
!"#$%&' ()
"#$ %&$%&'&$ ($)*'+%&$ %#,-.%/#'&$ %#,%*,/'+%&$ -( '&-+#$ a < b /+(,(, 0#/(,%+&1($ V1 2 V23
'($0(%/+4&5(,/(6
'* 7&1%.1( 1& %&'8& -( %&-& ($)('&6
#* 9:#'& /'&(5#$ -($-( (1 +,;,+/# .,& %&'8& Q -+$/'+<.+-& .,+)#'5(5(,/( $#<'( 1& $.0(';%+(
-( .,& ($)('& -( '&-+# c > b3 %#,%*,/'+%& %#, 1&$ &,/('+#'($6 =7.>,/# /'&<&?# ($ ,(%($&'+#
'(&1+@&' 0&'& %#1#%&' -+%:& %&'8&A
+"$,-./01
"(;,+5#$ %#5# Q1 2 Q2 1&$ %&'8&$ -( 1& ($)('& %#, 0#/(,%+&1 V2 2 V23 '($0(%/+4&5(,/(6
7#5# 2& :(5#$ 4+$/# (, &2.-&,/+&$ &,/('+#'($3 (1 %&50# (1(%/'+%# 0'#-.%+-# 0#' %&-& %&$%&'&
$( 0.(-( %&1%.1&' ./+1+@&,-# 1(2 -( 8&.$$ ./+1+@&,-# %#5# $.0(';(%+( 8&.$$+&,& .,& %&$%&'&
($)('+%& %#, '&-+# a < r < b6 B& %&'8& (,%(''&-& 0#' ($/& $.0(';%+( ($ 1& &0#'/&-& 0#' 1&
%&$%&'& -( '&-+# a3 ($ -(%+'3 Q1 0#' 1# /&,/#C
∫
S
~E · ndS =Q1
ǫ0
D('#
∫
S~E · ndS = E4πr2
6 "( ($/& 5&,('& (1 %&50# (1(%/'+%#EF.( 2& :(5#$ 4+$/# F.( /+(,(
-+'(%%+#, '&-+&1G ($C
~E =Q1
4πǫ0r2r
9:#'&3 1& -+)('(,%+& -( 0#/(,%+&1 (,/'( &5<&$ $.0('%+($ ($C
V2 − V1 = −∫ a
b
Q1
4πǫ0r2dr =
Q1
4πǫ0(1
b− 1
a)
7#, ($/#3 0#-(5#$ #</(,(' 1& %&'8& Q1C
Q1 = 4πǫ0(V2 − V1)ab
b− a
9:#'&3 %&1%.1&'(5#$ (1 %&50# (1(%/'+%# & .,& -+$/&,%+& r > b ./+1+@&,-# 1(2 -( H&.$$6 I1(8+5#$
.,& %#5# $.0(';%+( 8&.$$+&,& .,& %&$%&'& -( '&-+# r > b 2 0'#%(-(5#$ -( +8.&1 )#'5&6 B#
F.( -(<(5#$ %#,$+-('&' (, ($/( %&$#3 ($ F.( 1& %&8& (,%(''&-& 0#' ($/& $.0(';%+( ($ Q1 + Q26
"( ($/& 5&,('& 4(5#$ F.(C
E4πr2 =Q1 + Q2
ǫ0
! !"#$%&' () '*+% $',-.
~E =Q1 + Q2
4πǫ0r2
"#$ %&'#( )#*%+#& ,-,./-0 %/ )#'%$,1-/ %$ b2
V2 = −∫ b
∞
Q1 + Q2
4πǫ0r2dr =
Q1 + Q2
4πǫ0b
3% %&'- +-$%0-( /- ,-04- *% ,-&,-0- %5'%01#0 &%062
Q2 = 4πǫ0b(V1 + (V2 − V1)ab
a− b)
! 7-0- ,-/,./-0 %/ '0-8-9# '#'-/( ,#$&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%,'-+%$'%
V (c) =Q1 + Q2
4πǫ0c
A&1(
W =Q(Q1 + Q2)
4πǫ0
!
!"#$%&' ()
"#$% &'%&'#'% $%($#)&'% &*+,-&.*#'% &*+&$.#)&'% /-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'/<#$ =-$ $%.' ')%1',*
'1 3'%'# 3*# 1' &'%&'#' &$+.#'17 >?-'1 $% '@*#' $1 3*.$+&)'1 $1$&.#)&* ,$ &',' -+' ,$ 1'% &'%5
&'#'%A>?-'1 $% 1' &'%&'#' $+ &',' -+' ,$ 1'% &'%&'#'%A
,"$-./"0
! !"#$%&' () '*+% $',-.
!"#$%&' ()
"#$ %$&%$'$ (&)*'+%$ %,#-.%/,'$ -( '$-+, a 0 (&1(&,' δ ≫ a %,#/+(#( %$'2$ #(/$ Q3 4( -+&/'+5.0(
.#$ %$'2$ q (# (6 7,6.8(# +#/('+,' -(6 %$&%$',# -( '$-+, a9.# $+&6$#/( (# 6$ 1$'/( +#/('+,' -(6
%$&%$',# +81+-( :.( (&/$ -(#&+-$- -( %$'2$ &( 1$&( $6 %,#-.%/,';3 <,& -+%(# :.( (6 %$81,
(6*%/'+%, (# (6 +#/('+,' -(6 %$&%$',# (&/$ -$-, 1,'
~E = K(r
a)4r
=,#-( K (& .# %,#&/$#/( 1,' -(/('8+#$' 0 r (& (6 7(%/,' .#+/$'+, '$-+$63 4( 1+-( (#%,#/'$'>
'* ?$ -(#&+-$- -( %$'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$& %,8.# %,8,>
φ =
∫
S
~E · ndS =qint
ǫ0
4+# (85$'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
r2
∂
∂r(r2E(r))
=1
r2
∂
∂r(r2K(
r
a)4)
=1
r2
∂
∂r(Kr6
a4)
=6Kr3
a4
=( (&/$ 8$#('$D 6$ -(#&+-$- -( %$'2$ (# (6 +#/('+,' -(6 %$&%$',# &('C>
ρ(r) =6ǫ0Kr3
a4
!
"#$%&' ()*)+$, ()-)%+./&% 0& 1$/,-&/-) K2 3&*)+$, 45) 0& 1&%6& -$-&0 &0+&1)/&(& )/ )0
./-)%.$% ()0 1&,1&%$/ ), q' 7$% 0$ -&/-$ ,) 8& & 15+70.% 0& %)0&1.$/9
q =
∫
ρ(r)dV
= 4π
∫ a
0
6ǫ0Kr3
a44πr2dr
=24ǫ0K
a4
r6
6
∣
∣
∣
∣
a
0
)
= 4πa2ǫ0K
:) ),-& +&/)%&
K =q
4πǫ0a2
",;' 7$()+$, ),1%.*.% 0& ()/,.(&( () 1&%6& )/ <5/1.$/ () 0& 1&%6& -$-&0 &0+&1)/&(&9
ρ(r) =3
2
qr3
πa6
! 3&*)+$, 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
σint = − q
4πa2
>$/ ),-$ ? 5-.0.@&/($ 0& 1$/,)%8&1.$/ () 0& 1&%6& 8)+$, 45) )/ 0& ,57)%=1.) )A-)%.$% ,) -)/(%&
5/& 1&%6& Q+q2 >$/ ),-) %),50-&($ 7$()+$, 1&0150&% 0& ()/,.(&( () 1&%6& ,57)%=1.&0 )A-)%.$%9
σext =Q + q
4πa2
:$/() #)+$, (),7%)1.&($ δ2"! B&%& ,&*)% 7$-)/1.&0 )/ -$($ )0 ),7&1.$ ()*)+$, ,&*)% )0 1&+7$ )0C1-%.1$2 B&%& 0& %)6.$/
./-)%.$% ?& -)/)+$, )0 1&+7$ )0)1-%.1$9
~Eint =q
4πǫ0a2(r
a)4r, para r < a
D/ )0 )A-)%.$% ()0 1$/(51-$% -)/)+$, -)/)+$, 45) )0 1&+7$ )0)1-%.1$ ), ,.+70)+)/-)9
~Eext =Q + q
4πǫ0r2r, si r > a
",. )0 7$-)/1.&0 7&%& 5/& %)6.$/ )/ 45) r > a ),9
V (r) = −∫
~Eextd~r = −Q + q
4πǫ0
∫ r
∞
1
r2dr =
Q + q
4πǫ0r
"#$%&' 7&%& 0& %)6.$ ./-)%.$% Er < aF )0 7$-)/1.&0 0$ 1&0150&+$, 1$+$9
! !"#$%&' () '*+% $',-.
V = −∫
~E · d~r = −∫ b
∞~Eext · d~r +−
∫ r
b
~Eint · d~r, si r > a
V (r) = −∫
~E · d~r
= −∫ b
∞~Eext · d~r +−
∫ r
b
~Eint · d~r
=Q + q
4πǫ0a− q
20πa6ǫ0(r5 − a5)
"#$% #& '$&(& )*&
V (r) =Q + q
4πǫ0a− q
20πa6ǫ0(r5 − a5), si r < a
!
!"#$%&' ()
"#$%&$' '$ %#()* '$+%,-.%* )-*/&%./* )*- &0# /.1,-.2&%.30 /' %#-4# ρ(r) ,#$ 5&' '$ )*,'0%.#$
5&' '1,# /.1,-.2&%.30 )-*/&%' '1,# /#/* )*-
V (r) = qe−λr
4πr2ǫ0
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&'
~E = −∇V
A*/'(*1 '0%*0,-#- =B%.$('0,' '$ %#()* '$+%,-.%*C
~E = −∂V
∂rr =
q
4πǫ0r2e−λr(1 + λr)r
D1,' -'1&$,#/* 0*1 )'-(.,' '0%*0,-#- 1.0 /.E%&$,#/ $# /'01./#/ /' %#-4#8 D1,* 1' @#%' &,.$.<#0/*
$# =*-(# /.='-'0%.#$ /' $# $': /' F#&11C
~∇ · ~E =ρ
ǫ0
9'0'(*1 5&' '$ %#()* '1 -#/.#$ : )*- $* ,#0,*
~E = Er7 /' '1,# (#0'-#
~∇ · ~E =1
r2
∂
∂r(r2E(r))
=1
4πǫ0
q
r2(−λe−λr(1 + λr) + λe−λr)
= − 1
4πǫ0
q
rλ2e−λr
A*- $* ,#0,*
ρ(r) = − 1
4π
q
rλ2e−λr
! !"#$%&' () '*+% $',-.