libro electro ejerciciosresueltos garrido...
TRANSCRIPT
!"#$%&' (
!"#$"%&#!'$%
!"#$%&' ()
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*"$+,-./0
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+/ Q 1#' )# 8%+ 2,/&, +!"#!&',' ), $(;+'+!"(, $+ 1#&+!"(,) 1,', #2&+!+' )# 8%+ 2%/",7#/4 <#'
$+9!("(=!3 ), $(;+'+!"(, $+ 1#&+!"(,) +/
∆V = Vb − Va = −∫ b
a
~E · d~r
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IJ
! !"#$%&' () '*+,*-!+'.,-
"# $%&'$#()$* $#(+,-.,+* %$ /-0.* 1&+ 2$3. +*($ *-0+(,)$ +4 5&3. 6., 4$* ($6$* '+4 7-4-#',. +*
#&4.8 9., 4. ($#(.:
φ =
∫
S
~E · ndS =
∫
manto
~E · ndS = E2πrℓ
;$ 7$,<$ -#(+,-., qint +* Q8 =64-7$#'. 4$ 4+% '+ <$&** /+0.* 1&+ E2πrℓ = Qǫ0
% 6., 4. ($#(.
~E =Q
2πǫ0rℓr
9$,$ 4$ 7$*7$,$ 7-4)#',-7$ 7$,<$'$ ($02-># *+ (-+#+ 4$ 0-*0$ *-0+(,)$? 6+,. 7.0. 4$ 7$,<$ +*($
'+6.*-($'$ (.($40+#(+ +# *& *&6+,@7-+? 4$ 7$,$ -#(+,-., *+,A #&4$ % 6., 4. ($#(. *+ (+#',A 1&+
+4 7$06. +4>7(,-7. $6.,($'. 6., +44$ 6$,$ r < b ($02-># *+,A #&4.8
B$ 7$47&4$'. +4 7$06. +4>7(,-7. '+2+0.* +4+<-, &#$ (,$%+7(.,-$8 "4-<-,+ 4$ 0$* .2/-$? 1&+ +* -,
+# '-,+77-C# ,$'-$4 '+*'+ a D$*($ b8 E.# +*(.? /+0.* 1&+
~E · d~r = Er · d~r = Edr =Q
2πǫ0rℓdr
F+ +*($ 0$#+,$ -#(+<,$#'. +*($ +G6,+*-C# '+*'+ r = a D$*($ r = b:
Vb − Va = −∫ b
a
Q
2πǫ0rℓdr = − Q
2πǫ0ℓ
∫ b
a
1
rdr
Vb − Va = −Q
2πǫ0ℓln(
b
a)
=D.,$? 7.#*-'+,$0.* ∆V 7.0. 4$ 0$<#-(&' '+ 4$ '-H+,+#7-$ '+ 6.(+#7-$4 % '+ +*($ H.,0$ '+
4$ +G6,+*-C# 6$,$ 4$ 7$6$7-'$' /+0.* 1&+:
C =Q
∆V=
QQ ln( b
a)
2πǫ0ℓ
=2πℓǫ0ln(b/a)
!"#$%&' ((
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0"# d >> R8 9% $"# 0,(6,&"$ , )#, &%-'('#0%, &' ."*'#0%,1 V04 0"# 0,(6,$ Q 7 −Q4 '#0)'#*('
1, 0,.,0%&,& ."( )#%&,& &' 1"#6%*)& ,.(":%+,&, .,(, '1 $%$*'+, &' 0%1%#&("$8
)"$*+,-./
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0"+" $% '1 "*(" #" '$*)B%'$'C8 D&'+@$4 1, &%-'('#0%, &' ."*'#0%,1 '#*(' ,+5"$ 0%1%#&("$ '$
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'$*@# , 1" 1,(6" &' 1, 12#', ?)' )#' $)$ '3'$ A" 1, &%-'('#0%, &' ."*'#0%,1 '#*(' &"$ .)#*"$
0),1'$?)%'(, &' ,+5"$ 0%1%#&("$4 .)'$ ,1 '$*,( '# '?)%1%5(%" ,+5"$ 0"#&)0*"('$4 $)$ $).'(/0%'$
$"# '?)%."*'#0%,1'$C8
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'1 .("51'+, F &' 1, ,7)&,#*2, G4 '1 0),1 '$
~E(r) =
~0 r < Rσǫ0
(
Rr
)
· ρ R ≤ r
&"#&' ρ '$ '1 B'0*"( )#%*,(%" ?)' B, &'$&' '1 '3' &' $%+'*(2, &'1 0%1%#&("4 #"(+,1 , <$*'4 ,1
.)#*" '# 0)'$*%=#8
;"#6,+"$ '1 "(%6'# '# '1 .)#*" +'&%" &'1 $'6+'#*" ?)' )#' ,+5"$ '3'$4 7 '1 0%1%#&(" &' 0,(6,
."$%*%B, '# x < 08 >' '$*, -"(+,4 1, &%-'('#0%, &' ."*'#0%,1 '#*(' 1"$ .)#*"$ +'#0%"#,&"$ '$
V0 = V = −∫
Γ
~E(~r) · d~r
= −∫ −d+R
d−R
(
σRx
ǫ0(d+ x)− σR · −x
ǫ0(d− x)
)
· dxx
= −σR
ǫ0
∫ −d+R
d−R
(
1
d+ x+
1
d− x
)
dx
= −σR
ǫ0ln
(
x+ d
d− x
)∣
∣
∣
∣
−d+R
d−R
=2σR
ǫ0· ln
(
2d−R
R
)
.'(" σ = Q(2πR)L 4 ."( 1" ?)'
V =Q
ǫ0πL· ln
(
2d−R
R
)
=⇒ |V | = Q
ǫ0πL· ln
(
2d−R
R
)
;'("4 *'#'+"$ ?)' 1, 0,.,0%*,#0%, ."( )#%&,& &' 1"#6%*)& '$ C = Q|V |L 4
=⇒ C =ǫ0π
ln(
2d−RR
)
∼= ǫ0π
ln(
2dR
)
!! !"#$%&' () '*+,*-!+'.,-
"#$% d >> R& '()$*(% +$, -#$ ./ 0/"/012/301/ )$"$3)$ )$ ./ 4$(*$2,5/ )$. %1%2$*/ 6 )$.
*$)1( $3 $. -#$ $%27&
!
!"#$%&' ()
"#$%&$' $# (&')*# '+,)' $#- .$#%#- /' &+ %0+/'+-#/0) /' .$#%#- .#)#$'$#- /' 1)'# A 2 -'.#)#%34+
d5 6#)# '-,07 #$'8' &+ .09&3,0 :3+;+3,'-3<#$<'+,'= $#- .$#%#- /'$ %0+/'+-#/0) 2 /3># 9&? .#-#
%0+ $# (&')*# '+ $0- -3>&3'+,'- %#-0-@
#= A$ %0+/&%,0) '-,1 #3-$#/0 %0+ %#)># Q
B= A$ %0+/'+-#/0) '-,1 %0+'%,#/0 # &+# B#,')C# /' /3(')'+%3# /' .0,'+%3#$ V05
*"$+,-./0
#= D'+'<0- 9&' $# %#.#%3,#+%3# 2 $# '+')>C# #%&<&$#/# /'$ %0+/'+-#/0) -0+
C(x) = ǫ0A
xU =
1
2
Q2
C=⇒ U(x) =
1
2
Q2
Aǫ0x
60) ,#+,07 $# (&')*# '-
~F = −~∇U = −dU
dxx = −1
2
Q2
Aǫ0x
2 $#- .$#%#- -' #,)#'+ :+0,' 9&' $# (&')*# +0 /'.'+/' /' E=5
B= D'+'<0- 9&' $# %#.#%3,#+%3# 2 $# '+')>C# #%&<&$#/# /'$ %0+/'+-#/0) -0+
C(x) = ǫ0A
x
1
2C(x)(V0)
2 =⇒ U(x) =1
2
A
xǫ0(V0)
2
A+ '-,' %#-07
~F = +~∇U = +dU
dxx = −Aǫ0(V0)
2
2
1
x2x
2 $#- .$#%#- -' #,)#'+ :+0,' 9&' $# (&')*# /'.'+/' /' E=5
F3 G#%'<0- x = d7
~F = −Aǫ0(V0)2
2
1
d2x = −CV0
V02d
x = −12
Q2
Aǫ0x
!" !"#$%&' () '*+,*-!+'.,-
#$% &$ '() &* +()%,* )- &* ./-.* 01$.$ 2)3) -)%45
6) 2)7* *& &)18$% )91$98%*% )-8*- 2$- ):#%)-/$9)- 1$9-/2)%*92$ '() &* +()%,* #$% (9/2*2
2) ;%)* -$3%) &* -(#)%<1/) 2) (9 1$92(18$% )-
~F/A = − σ2
2ǫ0n
!"
!"#$%&' ()
#$%&'()*) +%, &+-)*./') /$%(+/0$*, )&12*'/, () *,('$ a /$%/2%0*'/, , $0*, () *,('$ b3 /$%a < b 4,56,& &+-)*./')& 1$*5,% +% /$%()%&,($*78 9% ): )&-,/'$ )%0*) ,56$& /$%(+/0$*)& &)
'%0*$(+/) +% ,*5,;<% 5)0=:'/$ )&12*'/$ () *,('$ '%0)*'$* c > )?0)*'$* d3 /$%/2%0*'/$ , ,56,&
&+-)*./')& > %)+0*$8 @' '%'/',:5)%0) :, &+-)*./') '%0)*'$* 0')%) /,*A, +Q > :, )?0)*'$* −Q3
('&/+0, B+2 $/+**) /$% :, /,-,/'0,%/', (): &'&0)5, :+)A$ B+) &) -$%) ): ,*5,;<% > )%/+)%0*)
:, /,-,/'0,%/', '%'/',: > .%,: (): &'&0)5,8
*"$+,-./0
C$()5$& )%/$%0*,* ): /,5-$ ):2/0*'/$ )%0*) :,& ($& &+-)*./')&3 ,%0)& B+) &) -$%A, ): ,*5,;<%3
+&,%($ :, :)> () D,+&&8 E)%)5$& B+) ): /,5-$ 0')%) &'5)0*F, )&12*'/, > G, )% ): &)%0'($ ()
r3 )& ()/'*3
~E(~r) = E(r)r8 E$5,%($ /$5$ &+-)*./') () '%0)A*,/'<% +%, &+-)*./') )&12*'/,
/$%/2%0*'/, , :,& &+-)*./')&3 () *,('$ a < r < b3 0)%)5$&
∮
Γ
~E(~r) · ndS = 4πr2E(r) =Q
ǫ0=⇒ ~E(~r) =
Q
4πǫ0
r
r2
H&F3 :, ('1)*)%/', () -$0)%/',: )%0*) :,& &+-)*./')& )&
V1 = −∫ a
b
~E(~r) · d~r = − Q
4πǫ0
∫ a
b
dr
r2=
Q
4πǫ0
1
r
∣
∣
∣
∣
a
b
=Q
4πǫ0
(
1
a− 1
b
)
($%() ): /,5'%$ 0$5,($ 1+) *,(',:8
C$* 0,%0$3 :, /,-,/'0,%/', '%'/',: )&
Ci =Q
V1=
4πǫ0(
1a − 1
b
)
H: '%0*$(+/'* ): ,*5,;<%3 ): -$0)%/',: )%0*) :,& &+-)*./')& ('&5'%+>)3 -$* :$ /+,: )&-)*,5$&
B+) :, /,-,/'0,%/', ,+5)%0)8
9% :, &+-)*./') '%0)*'$* (): ,*5,;<% &) ('&0*'6+>) +%'1$*5)5)%0) +%, /,*A, −Q8 9&0$ &)
()&-*)%() () :, :)> () D,+&&8 9% )1)/0$3 +&,%($ /$5$ &+-)*./') () '%0)A*,/'<% +%, &+-8
)&12*'/, /<%/)%0*'/, , :,& $0*,& > ()%0*$ (): ,*5,;<%3 /$5$ ): ,*5,;<% )& /$%(+/0$* > )&0= )%
)B+':'6*'$3 ): /,5-$ ()%0*$ () 2: ()6) &)* %+:$3 -$* 0,%0$
∮
Γ
~E(~r) · ndS = 0 =Q+Qc
ǫ0=⇒ Qc = −Q
> /$5$ ): ,*5,;<% )& %)+0*$3 , &+ G); &) ('&0*'6+>) +%'1$*5)5)%0) )% :, &+-)*./') )?0)*'$* ():
,*5,;<% +%, /,*A, Q8 9% ): '%0)*'$* () :, -*'5)*, &+-)*./')3 )& ()/'*3 -,*, a ≤ r3 ): /,5-$ )&
%+:$ 4-+)& %$ I,> /,*A, ()%0*$ () 2:3 -$* :)> () D,+&&7 ,: 'A+,: B+) -,*, r > b8 C,*, )%/$%0*,*
): /,5-$ )% d < r < b > a < r < c +&,5$& ): 5'&5$ -*$/)('5')%0$ B+) )% ): /,&$ ,%0)*'$*3
-$* 0,%0$3 $60)%)5$& B+) ): /,5-$ ):2/0*'/$ )&
~E(~r) =
~0 b < r, a > rQ4πǫ0
rr2 c < r < b, a < r < c
!" !"#$%&' () '*+,*-!+'.,-
#$%&'( %) *'+%,-./) %,+0% )/1 1$*%02-.%1 %3+%0,/ % .,+%0,/ %1
V2 =
= −∫ a
b
~E(~r) · d~r =∫ b
a
~E(~r) · d~r
=
∫ c
a
~E(~r) · d~r +∫ d
c
~E(~r) · d~r +∫ b
d
~E(~r) · d~r
=
∫ c
a
~E(~r) · d~r +∫ b
d
~E(~r) · d~r
=Q
4πǫ0
(∫ c
a
dr
r2+
∫ b
d
dr
r2
)
=Q
4πǫ0
(
1
a+1
d− 1
c− 1
b
)
4'0 +/,+'( )/ -/*/-.+/,-./ 2,/) %1
Cf =Q
V2=
4πǫ0(
1a +
1d − 1
c − 1b
)
4'5%6'1 7%0 8$% Ci < Cf 5% )' 1.&$.%,+%9 :%,%6'1 8$%
c < d =⇒ 1
d− 1
c< 0 =⇒ 1
a− 1
b+1
d− 1
c<1
a− 1
b=⇒ 4πǫ0
1a − 1
b
= Ci < Cf =4πǫ0
1a − 1
b +1d − 1
c
Cf *'50;/6'1 </=%0)' -/)-$)/5' 0>*.5/6%,+% ,'+/,5' 8$% /) .,+0'5$-.0 %) /06/?@, +%,%6'1
5'1 -',5%,1/5'0%1 %, 1%0.%9 4'0 +/,+'( -'6' -','-%6'1 )/ -/*/-.+/,-./ %,+0% 5'1 1$*%02-.%1
%1AB0.-/1 -',-B,+0.-/1( *'5%6'1 %,-',+0/0 )/ -/*/-.+/,-./ %8$.7/)%,+% 5%) 1.1+%6/9 C, %A%-+'(
1%/ C2 )/ -/*/-.+/,-./ 5% )/1 5'1 1$*%02-.%1 %3+%0.'0%1 D C1 )/ 5% )/1 .,+%0.'0%19 E% %1+/ A'06/
+%,%6'1
C2 =4πǫ01d − 1
b
, C1 =4πǫ01a − 1
c
4'0 +/,+'( )/ -/*/-.+/,-./ %8$.7/)%,+% %1+> 5/5/ *'0
1
Cf=
1
C1+
1
C2=
1
4πǫ0
(
1
a− 1
b+1
d− 1
c
)
=⇒ Cf =4πǫ0
1a − 1
b +1d − 1
c
8$% %1 %) 6.16' 0%1$)+/5' '=+%,.5' /,+%0.'06%,+%9
!"
!"#$%&' ()
#$ %&'&%()*+ )(,$, '-&%&. %/&0+&0&.1 %&0& /$& 0, -&0* a1 2*+3&$0* /$ 4$5/-* θ %*3* 3/,.)+&
-& 65/+&7 8,3/,.)+, 9/,1 '&+& θ ',9/,:*1 -& %&'&%()&$%(& ,.)& 0&0& '*+
C =ǫ0a
2
d(1− aθ
2d)
*"$+,-./0
;* 9/, <&+,3*. ,. 0(=(0(+ -&. '-&%&. %/&0+&0&. ,$ -43($&. ',9/,:&. 0, &$%<* dx > -&+5* a )&-
%*3* 3/,.)+& -& 65/+&?
@*0,3*. %*$.(0,+&+ ,.)&. '-&%&. %*3* '&+&-,-&.7 @&+& /$ %*$0,$.&0*+ 0, '-&%&. '&+&-,-&. .,
)(,$, 9/, -& %&'&%()&$%(& ,.
C =ǫ0A
H
7 8*$0, H ,. -& 0(.)&$%(& ,$)+, -&. '-&%&.7 ;/,5*1 '&+& -&. ',9/,:&. '-&%&. ($0(%&0&. ),$0+,3*.
9/, ,- 4+,& ,. A = adx > -& 0(.)&$%(& H =&+(&+& &- =&+(&+ x? H = H(x)7 A*$.(0,+,3*. -&
.(5/(,$), %*$65/+&%(B$?
A*3* =,3*. ., %/3'-, -& +,-&%(B$
h(x)
x=
a sin(θ)
a
!" !"#$%&' () '*+,*-!+'.,-
#$% &$ '()'$ h(x) = x sin(θ)* +,%$ -$.$ θ ,/ +,01,2$* +$3,.$/ -$)/43,%(% sin(θ) = θ -$) &$
-1(& 01,3( h(x) = xθ5 6, &( 781%( ()',%4$% 9,.$/ 01, H(x) = d+ h(x) : %,,.+&(;()3$ -$)
&( %,&(-4<) ()',%4$% %,/1&'(
h(x) = d+ θx
=$) ,/'$* ',)3%,.$/ 01, &( -(+(-43(3 3, ,/', +,01,2$ -$)3,)/(3$% ,/
dC =ǫ0adx
d+ θx
>)',8%()3$ 3,/3, x = 0 ( x = a?
C =
∫ a
0
ǫ0adx
d+ θx=
ǫ0a
θln(
d+ θa
d)
=$.$ +$3,.$/ 9,%* (& /,% θ +,01,2$* /, -1.+&, 01,
d+θad ≈ 15 #$3,.$/ (@$%( (+%$A4.(% +$%
'(:&$%B/,81)3$ $%3,)C &( ,A+%,/4<) ln(d+θad )5 D& 3,/(%%$&&$ 3, '(:&$% +(%( ln(x) (&%,3,3$% 3,
,/ ?
ln(x) = (x− 1)− 1
2(x− 1)2
D9(&1()3$ ,) x = d+θad ?
ln(d+ θa
d) =
aθ
d− 1
2
(aθ)2
d2
E,,.+&(;()3$ ,/', 9(&$% ,) &( ,A+%,/4<) ,)-$)'%(3( +(%( &( -(+(-4'()-4( C %,/1&'(?
C =ǫ0a
θ(aθ
d− 1
2
(aθ)2
d2) =
ǫ0a
θ(aθ
d(1− aθ
2d))
C =ǫ0a
2
d(1− aθ
2d)
F1, ,/ &$ 01, 01,%G(.$/ 3,.$/'%(%5
!"
!"#$%&' ()
#$% &'(&)% *& )%*+, a '& -%).% % /,0&$-+%1 V0 2 '& %3'1%4 5,'0&)+,)6&$0& '& -,$&-0% % 0+&))%
% 0)%78' *& 9$ -,$*&$'%*,) -92% -%/%-+*%* &' C: -,6, 69&'0)% 1% ;.9)%4<5,) *&;$+-+=$ 1%
0+&))% &'0% % /,0&$-+%1 -&), +$*&/&$*+&$0& *& 1% -%).% >9& %*>9+&)%?
'* @%1-91& &1 /,0&$-+%1 ;$%1 *& 1% &'(&)% 2 1% -%).% ;$%1 &$ 1% &'(&)% 2 &$ &1 -,$*&$'%*,)4
#* A@9B$0% &$&).3% '& *+'+/, %1 C%-&) 1% -,$&D+=$ % 0+&))%E
+"$,-./01
'*F1 /,0&$-+%1 *& 9$% &'(&)% *& )%*+, a &$ '9 '9/&);-+& &' V = kQa *,$*& Q &' 1% -%).%
%16%-&$%*% &$ 1% &'(&)%4 5%)% $9&'0), /),G1&6% '& 0+&$& >9& +$+-+%16&$0& 1% &'(&)% &'0% % 9$
/,0&$-+%1 V04 5,) 1, *+-C, %$0&)+,)6&$0& '& 0&$*)B >9&
V0 =kQ0
a(1)
H,$*& Q0 &' 1% -%).% >9& 0+&$& 1% &'(&)% +$+-+%16&$0&4
H&'/98' *& -,$&-0%) 1% &'(&)% %1 -,$*&$'%*,): 1% &'(&)% >9&*% -,$ 9$% -%).% Qf 2 '9 /,0&$-+%1
'&)B &$0,$-&'
Vf =kQf
a(2)
I &1 -,$*&$'%*,) >9&*% -,$ -%).% Qc 2 /,0&$-+%1 Vf : 2% >9& &'0% -,$&-0%*%, -,$ 1% &'(&)% %
0)%7&J *& 9$ -,$*9-0,)4 5,) *&;$+-+=$ C = Qc
V /,) 1, 0%$0,
Vf =Qc
C(3)
K*&6B': /,) -,$'&)7%-+=$ *& -%).%
Q0 = Qf +Qc (4)
H& <L? 2 <M?N Qc =kCQf
a
F$ <O?N Q0 = Qf +kCQf
a =⇒ Qf =aQ0
a+kC
P&&6/1%J%$*, &'0% &D/)&'+=$ &$ <L?N Vf =kQ0
a+kCH& 1, %$0&)+,) /,*&6,' '%G&) VC 2 QcN
VC = Vf =kQ0
a+ kC
QC = kCQ0
a+ kC
!" !"#$%&' () '*+,*-!+'.,-
! #$%&%'()*$+* (' *$*,-.' /* 0*1* ' (' 2,*/*$&%' 0*( &')23 *(4&+,%&3 *$ (' */5*,'6 7* +%*$* 89*
U =1
2ǫ0
∫
VE2dV
:/ &(',3 89* *( &')23 *(4&+,%&3 2,309&%03 23, (' */5*,' &',-'0' &3$ Q0 ' 9$' 0%/+'$&%' r > a*/
E =kQ0
r2
:/&,%1%)3/ *( *(*)*$+3 0* ;3(9)*$ &3)3< dV = 4πr2dr=* */+' )'$*,' +*$*)3/ 89* (' *$*,-.' '/3&%'0' /*,>
dUi =1
2ǫ0E
2dV =1
2ǫ0
kQ0
r24πr2dr = ǫ0
2πk2Q02
r2dr
#$+*-,'$03 ,*/9(+' 89*
Ui =
∫ ∞
aǫ02πk2Q0
2
r2dr =
2πk2Q02ǫ0
a=
kQ02
2a
?' *$*,-.' @$'( /* 0*1* '( &')23 *(4&+,%&3 0* (' */5*,' A '( &3$0*$/'03,6 ?' *$*,-%' '/3&%'0'
'( $9*;3 &')23 *(4&+,%&3 /* &'(&9(' 0* %-9'( 53,)' A /*,>
kQf2
2a 6 B3, 3+,3 ('03C (' &',-' '/3&%'0'
' 9$ &3$0*$/'03, */ UC =Q2
C
2C 6 =* */+' )'$*,' (' *$*,-.' @$'( /*,><
Uf =kQf
2
2a+
QC2
2C
D**)2('E'$03 (3/ ;'(3,*/ 0* */+' &',-' *$ 59$&%3$ 0* Q0<
Uf =kQ0
2
2(a+ kC)
=* */+' )'$*,'C (' *$*,-.' 0%/%2'0' /*,>
∆U = Uf − Ui =kQ0
2
2(
1
a+ kC− 1
a)
!"
!"#$%&' ()
#$%&'()*) )+ ,'*,-'.$ /-) &) 0-)&.*1 )% +1 23-*14 5*'0)*$ &) ,1*31 )+ ,161,'.$* C17 ,)**1%($ )+
'%.)**-6.$* S14 8-)3$ &) ,1*31 )+ ,$%()%&1($* C2 ,)**1%($ S24 #1+,-+) +1 ,1*31 2%1+ () ,1(1
,$%()%&1($* 9()&6-:& () ,)**1* S2; < )+ ,10='$ () )%)*3>1 )% +1& &'3-')%.)& &'.-1,'$%)&?
'* @+ '%.)**-6.$* S1 6)*01%),'A ,)**1($4
#* @+ '%.)**-6.$* S1 &) 1=*'A 1%.)& () ,)**1* S24
+"$,-./0
'* B+ ,)**1* S1 &) .)%(*C /-) +1 ('D)*)%,'1 () 6$.)%,'1+ 1 +1 ,-1+ )&.1 &$0).'($ )+ ,$%()%&1($*
C1 &)*C ∆V 4 B&' +1 ,1*31 /-) 1(/-')*) )&) &'06+)0)%.)
Q1 = C1∆V
5$* $.*1 61*.) )+ ,$%()%&1($* C2 %$ &) ,1*317 6$* +$ .1%.$ Q2 = 0 E) )&.1 01%)*17 +1 )%)*3>1
'%','1+ ()+ &'&.)01 &)*C
Ei =1
2C1∆V 2
@&,*'.1 )% D-%,'A% () Q1?
Ei =1
2
Q21
C1
BF$*17 ,)**10$& )+ '%.)**-6.$* S2 01%.)%')%($ S1 ,)**1($4 81 ('D)*)%,'1 () 6$.)%,'1+ 61*1
10=$& ,$%()%&1($*)& &)*C +1 0'&01? ∆V 4 G 1&>7 &'06+)0)%.) +1& ,1*31& &)*C%?
Q′1 = ∆V C1
Q′2 = ∆V C2
81 )%)*3>1 ()+ &'&.)01 &)*C +1 16$*.1(1 6$* C1 < C2? Ef =12C1∆V 2 + 1
2C2∆V 2
B&>7 )+ ,10='$ () )%)*3>1 &)*C?
Ef − Ei =1
2C2∆V 2
#*
H' 1=*'0$& S1 1%.)& () ,)**1* S2 .)%(*)0$& /-) +1 ('D)*)%,'1 () 6$.)%,'1+ ∆V <1 %$ '%I-<)4
#$0$ C2 )&.1 ()&,1*31($ &) .)%(*C /-) +1 ,1*31 Q1 &) *)('&.*'=-'*C () .1+ 01%)*1 /-) )+
&'&.)01 /-)(1 )% )/-'+'=*'$4 8+101*)0$& Q′1 < Q′2 +1& ,1*31 /-) 1(/-')*)% +$& ,$%()%&1($*)&
C1 < C27 *)&6),.'J10)%.)4 @% )&.) ,1&$ .)%)0$& -%1 ,$0='%1,'A% )% 61*1+)+$ () ,$%()%&1($*)&
< &1=)0$& /-) )% )&.1 &'.-1,'A% +1 ('D)*)%,'1 () 6$.)%,'1+ )% 10=$& ,$%()%&1($*)& )& +1 0'&017
6$* +$ .1%.$ &) .)%(*C +1 &'3-')%.) ),-1,'A%?
V1 = V2
! !"#$%&' () '*+,*-!+'.,-
Q′1C1
=Q′2C2
"# $%&# '()#(*+, -)' ,'('.*%#/$. .' $0%*',' 1$& '2 1&*,(*1*$ 3' ($,.'&4#(*+, 3' (#&5#6 '2
($,3',.#3$& C1 -)' '.%# *,*(*#2/',%' (#&5#3$ ($, Q1 1$& 2$ %#,%$ 2# .)/# 3' 2#. ,)'4#.
(#&5#. %',3&7, -)' .'& *5)#2 # '.%#8
Q′1 +Q′2 = Q1
9' '.%# /#,'&# ($/0*,#,3$ #/0#. '()#(*$,'. %','/$. -)'
Q′1 =C1
C1 + C2Q1
Q′2 =C2
C1 + C2Q1
9' '.%# /#,'&# 1$3'/$. '.(&*0*& 2# ','&5:# 3'2 .*.%'/# -)' .'&7 2# #1$&%#3# 1$& (#3# ($,;
3',.#3$&8
Ef =1
2
Q′21C1
+1
2
Q′22C2
=1
2(
C1
C1 + C2)2
Q21
C1+1
2(
C2
C1 + C2)2
Q21
C2
= Q21(1
2
C1
(C1 + C2)2+1
2
C2
(C1 + C2)2)
<2 (#/0*$ 3' ','&5:# .'&7 ',%$,('.8
Ef − Ei =Q21
2(
C1
(C1 + C2)2+
C2
(C1 + C2)2)− 1
2
Q21
C1
=Q21
2(
C1
(C1 + C2)2+
C2
(C1 + C2)2− 1
C1)
!"#$%&' ()
!" #$"%"% #&"' ()%*"%'+*)&"' *" (+,+($*+*"' C1 = C0-C2 = 2C0 . C3 = 4C0- . '" ()%"(#+% +
/%+ 0+#"&1+ 2/" "%#&"3+ /%+ *$4"&"%($+ *" ,)#"%($+5 V0- ()6) 6/"'#&+ 5+ 73/&+8
'* 9+5(/5" 5+ (+&3+ 2/" +*2/$"&" (+*+ ()%*"%'+*)& 6$"%#&+' "5 $%#"&&/,#)& S "'#+ +0$"&#)8
#* 9+5(/5" 5+ (+&3+ 2/" +*2/$"&" (+*+ ()%*"%'+*)& (/+%*) "5 $%#"&&/,#)& S "'#+ ("&&+*)8
+* :;)&+ '" *"'()%"(#+ 5+ 0+#"&1+ *"5)' 0)&%"' . < *" 5+ 73/&+ ()% "5 $%#"&&/,#)& S ("&&+*)8
=9/>%#) ?+&1+% 5+' (+&3+' "% (+*+ ()%*"%'+*)&@
,*A% "5 ($&(/$#) '$% 5+ 0+#"&1+- '" #)6+ /% ()%*"%'+*)& C1 . '$% 2/" ,$"&*+ '/ (+&3+ '"
$%?$"&#"- *" #+5 4)&6+ 2/" 5+ ,5+(+ ,)'$#$?+ 2/"*+ "% "5 5/3+& *)%*" "'#+0+ 5+ %"3+#$?+- .
?$("?"&'+8 9+5(/5" 5+ (+&3+ "% 5)' ()%*"%'+*)&"' 5/"3) *"5 %/"?) "2/$5$0&$) "5B(#&$()
-"$.+/012
'* !$ "5 $%#"&&/,#)& S "'#+ +0$"&#)- ')5) '" (+&3+&+ "5 ()%*"%'+*)& C38 C/"3)- ()6) 5+ *$4"&"%($+
*" ,)#"%($+5 "' V0 . '+0"6)' '/ (+,+($*+*- 5+ (+&3+ +56+("%+*+ '"&>D
Q3 = V0C3
9)6) C3 = 4C0- )0#"%"6)'
Q3 = 4C0V0
#* :5 ("&&+& "5 $%#"&&/,#)& S '" (+&3+&+% #)*)' 5)' ()%*"%'+*)&"' ()6) 6/"'#&+ 5+ 73/&+
A5 ()%*"%'+*)& C3 '" 6+%#"%*&> ()% $3/+5 (+&3+ .+ 2/" 5+ *$4"&"%($+ *" ,)#"%($+5 "' 5+ 6$'6+D
Q3 = 4C0V08E+&+ (+5(/5+& 5+' (+&3+' Q1 . Q2 /#$5$F+&"6)' "5 ()6,)&#+6$"%#) *" ()%*"%'+*)&"' "% '"&$"
5) (/+5 %)' "%#&"3+&+ *)' "(/+($)%"' ,+&+ "'#+' *)' ?+&$+05"'8 C+ *$4"&"%($+ *" ,)#"%($+5 2/"
! !"#$%&' () '*+,*-!+'.,-
"#$%&" "'&(" ) ! "% V0* +,"-. /.0. C1 ) C2 "%&1' "' %"($"2 34 %,04 5" 3.% 6.3&47"% 5" /454
/.'5"'%45.( %"(1 V08
V1 + V2 = V0
9"(.2 :.( 5";'$/$<' &"'"0.% =," V1 =Q1
C1) V2 =
Q2
C2* >" "%&4 04'"(4
Q1
C1+
Q2
C2= V0
?5"01%2 "3 /.'5,/&.( =," &.04 34 :34/4 5"("/@4 5" C1 ) 34 :34/4 $A=,$"(54 5" C2 "%&4 '",&(.2 43
/4(-4(%" %" &"'5(1 =," "%&" /.'5,/&.( %" :.34($A4 /.' /4(-4% −Q1 ) Q22 :"(. :.( /.'%"(64/$<'
5" 34 /4(-4 "%&4% 5"B"' %,04( /"(. :4(4 =," "3 /.'5,/&.( %$-4 '",&(.8
Q2 −Q1 = 0
C.' "%&4% ! "/,4/$.'"% &"'5("0.% =,"
Q1 = Q2 =C1C2
C1 + C2V0
9"(.2 '.% 5$/"' =," C1 = C0 ) C2 = 2C0* ?%$2 34% /4(-4% %.'
Q1 = Q2 =2
3C0V0
! ?3 %4/4( 34 B4&"(D4 34% /4(-4% %"-,$(1' %$"'5. 34% 0$%04% =," /43/,340.% 4'&"($.(0"'&" )4
=," %" /,0:3" 34 /.'%"(64/$<' 5" 34 /4(-4 ) "' "3 /$(/,$&. /"((45. 34 %,04 5" 5$E"("'/$4% 5"
:.&"'/$43 "% ',34*
"! +4 %$&,4/$<' %" :,"5" 4:("/$4( "' 34 %$-,$"'&" ;-,(48
+4 /4(-4 %" /.'%"(64(42 5" "%&4 04'"(4 &"'"0.% =," %" /,0:3" 34 ("34/$<'
Q′1 +Q′2 = Q1 +Q2 (1)
Q3 −Q1 = Q′3 −Q′1 (2)
+4 %,04 5" &.5.% 3.% :.&"'/$43"% 5"3 /$(/,$&.F"% /"((45.G "% ',348
V ′1 + V ′2 + V ′3 = 0
Q′1C1
+Q′3C3
+Q′2C2
= 0 (3)
!
"#$# %&$#' (&)&$#' ! &*+,*-#)&' . ! %,/-,01&' 2#/ 1# (,)(# 2#3&$#' &)*#)(/,/ 1#' %,1#/&'
3& 1,' *,/4,' 3& *,3, *#)3&)',3#/5
6& 1, &*+,*-7) 8 9 '& (-&)&
Q′2 = Q1 +Q2 −Q′1
6& 1, &*+,*-7) 8:9 '& (-&)&
Q′3 = Q3 −Q1 +Q′1
;(-1-<,)3# &'(#' %,1#/&' &) 8!9
Q′1C1
+Q3 −Q1 +Q′1
C3− Q1 +Q2 −Q′1
C2= 0
Q′1(1
C2+
1
C2+
1
C3) +
Q3 −Q1
C3− Q1 +Q2
C2= 0
=) 1,' 2/&4+)(,' ,)(&/-#/&' ., #0(+%-$#' 1#' %,1#/&' -)-*-,1&' 3& 1,' *,/4,'>
Q1 = Q2 =2
3C0V0
Q3 = C3V0 = 4C0V0
;(-1-<,)3# &'(#' %,1#/&'? (&)&$#'
Q′17
4C0− 5
6V0 −
2
3V0 = 0
Q′1 =2
21C0V0
"#) &'(# #0(&)&$#' 1,' 3&$@' *,/4,'>
Q′2 =10
7C0V0
Q′3 =24
7C0V0
! !"#$%&' () '*+,*-!+'.,-
!"#$%&' (
!"#"$%&!$'(
!"#$%&' ()
! "#!$! %$ &'$(!$)*('+ (! ,-*&*) ,*+*-!-*) (! *+!* A . )!,*+*&#'$ d/ . )! &*+0* (! 1'(' "*-
2%! )% ,-*&* )%,!+#'+ *(2%#!+! &*+0* Q/ . -* #$3!+#'+ −Q4 5*$"!$#!$(' !- &'$(!$)*('+ *#)-*('/
)! #$"+'(%&!$ 6-'2%!) (! 1*"!+#*- (#!-7&"+#&'/ (! ,!+1#"#8#(*(!) ǫ1 = 2ǫ0 . ǫ2 = 4ǫ0 9*)"* -*
1#"*( (!- &'$(!$)*('+/ &'1' 1%!)"+* -* :0%+*4
'* ;!"!+1#$! -* $%!8* &*,*&#(*( (!- &'$(!$)*('+/ &'$ -') 6-'2%!) (#!-7&"+#&') !$ )% #$"!+#'+4
#* <*-&%-! &'1' )! (#)"+#6%.! -* &*+0* !$ -*) ,-*&*) &'$(%&"'+*) ,'+ !- 9!&9' (! #$"+'(%&#+
-') (#!-7&"+#&')4
+"$,-."/0
'* =- )#)"!1* 1')"+*(' )! ,%!(! 8!+ &'1' %$ )#)"!1* (! > &'$(!$)*('+!) &'1' 1%!)"+* -*
:0%+*?
@@A
! !"#$%&' () *+,&, $-+ '.
"#$%&#' ()*(+*), *)' ()-)(.$)$%' $% ()$) (#/$%/')$#, $% 0#,&) $.,%(1)2
C1 =A/2
2/3dǫ1 =
3A
2dǫ0
C2 =A/2
1/3dǫ2 =
6A
2dǫ0
C3 =A/2
dǫ0 =
A
2dǫ0
3+%4#5 -),) ()*(+*), *) ()-)(.$)$ $%* '.'1%&) 6)'1) +1.*.7), *)' ,%4*)' -),) (#&6./)(.#/%'
$% (#/$%/')$#,%'8 3#' (#/$%/')$#,%' C1 9 C2 %'1)/ %/ '%,.%5 $% %'1) &)/%,) *) ()-)(.$)$
%:+.;)*%/1% -),) %'1#' < (#/$%')$#,%' (+&-*.,)2
1
C ′eq=
1
C1+
1
C1=
13A2d ǫ0
+1
6Ad ǫ0
=5
6
d
Aǫ0
=% *# :+% '% #61.%/% :+% C ′eq =65
Ad ǫ08
>?#,)5 C ′eq !"# $ %#&#' '( )($ C3 * %(& '( "#$"( '# )#%#)+,#, -.+/#' $" ! &# '# !.0# ,
#01#! )#%#)+,#, !2
Ceq = C ′eq + C3 =6
5
A
dǫ0 +
A
2dǫ0 =
17
10
A
dǫ0
! 3# %& ! $)+# , ' ,+ ' )"&+)( 4#&5 -. '# )#&6# Q * −Q , %(!+"#,# $ '#! %'#)#! ! & ,7
+"&+1.*#$ , .$# , " &0+$#,# 0#$ &# # , 8$+&9 : 8$+0(! q3 '# )#&6# -. #,-.+ & ' )($, $7
!#,(& C3 * q '# )#&6# $ '(! )($, !#,(& ! C1 * C2; ! '# 0+!0# $ #01(! )($, !#,(& ! %(&
!"#& $ ! &+ < )(0( 0. !"&# '# 86.
=(0( '# )#&6# ! )($! &/# ! " $,&5 '# ).#)+($2
q1 + q2 = Q (1)
>(& ("&( '#,(? ' )+&).+"( ! ) &&#,( * %(& '( "#$"(2
∆V3 −∆V1 −∆V2 = 0q3C1− q
C2− q
C3= 0 (2)
: ;@<2 q3 = Q− q A 0%'#B#$,( !" /#'(& $ '# ).#)+($ ;C<2
1
C3(Q− q)− q(
1
C1+
1
C2) = 0
!
−q(1
C3+
1
C1+
1
C2) +
Q
C3= 0
"##$%&'(')*+ &+, -'&+.#, *# &', /'%'/01'/0', #)/+)1.'*', #) &' %'.1# '23 .#,4&1' 54#6
q =12
17Q
q3 =5
17Q
! !"#$%&' () *+,&, $-+ '.
!"#$%&' ()
"# $%&$%'(# )&*+',$- .%$,/- 0,)1+$2',$-3 0) '%0,-& ,#2)'#- a 4 )52)'#- b3 )& &-.)2,0- 6%7- 8#
$%.9- )52)'#- :8) 9'-08$) )# +1 8# ;)$2-' 9-1%',/%$,(#
~P (~r) =k
rr
0-#0) k )& 8#% $-#&2%#2) %'6,2'%',% 4 r 1% 0,&2%#$,% %1 $)#2'- 0)1 $%&$%'(#< =-2) :8) #- >%4
$%'?% 1,6') )# )&2) 9'-61).%< @#$8)#2') )1 $%.9- )1+$2',$- 0) 9-1%',/%$,(# )# 2-0- )1 )&9%$,-<
*"$+,-./0 A%0- 1% &,.)2'B% )&*+',$% 0)1 0,)1+$2',$- 4 )1 ;)$2-' 9-1%',/%$,(#3 )& $1%'- :8) )1
;)$2-' $%.9- )1+$2',$- 0) 9-1%',/%$,(# )& 0) 1% *-'.%
~E(~r) = E(r)r< C)9%').-& )1 9'-61).%
)# 2')& ')?,-#)&<
%D r < a@& $1%'- :8) )# )&2% ')?,(#3 9-' 1)4 0) E%8&&3 )1 ;)$2-' $%.9- )1+$2',$- )&
~E(~r) = ~0
6D a < r < bC%6).-& :8)
~D(~r) = ǫ0 ~E(~r) + ~P (~r)
F-.- #- >%4 $%'?% 1,6')3 2)#).-& :8)
∮
Ω
~D(~r) · ndS = 4πr2D(r) = Qlibre = 0 =⇒ ~D(~r) = ~0
)# 2-0- )1 )&9%$,-3 9-' 2%#2-
~E(~r) = − 1
ǫ0~P (~r)
@# )&2% ')?,(#3 )1 ;)$2-' 9-1%',/%$,(# )&
~P (~r) =k
rr
9-' 1- $8%1
~E(r) = − k
ǫ0rr
@&2- 980- $%1$81%'&) 2%.6,+# 0) -2'% *-'.% G%8#:8) .H& 1%'?%D< I)#).-& 9-' 1)4 0)
E%8&&
∮
Ω
~E(~r) · ndS =Qencerrada
ǫ0
0-#0) 1% $%'?% )#$)''%0% )# )&2) $%&- )&2H 0%0% 9-' 1% 0)#&,0%0 0) $%'?% &89)'J$,%1 σp
4 ;-18.+2',$% ρp 0) 9-1%',/%$,(#3 4 >).-& 2-.%0- 8#% &89)'J$,) 0) ,#2)?'%$,(# )&*+',$%
!
Ω "# $%"&' a < r < b( )*+'*,$#-'./'.( 0%1#-'. 23# σp = ~P (~r) · n 4 ρp = −~∇ · ~P (~r)56'$ /' 23#
σp(a) = ~P (~a) · n
=k
ar · −r
= −k
a
σp(b) = ~P (~b) · n
=k
br · r
=k
b
ρp(r) = −~∇ · ~P (~r)
= − 1
r2∂(r2Pr)
∂r− 1
rsen(φ)
∂(sen(φ)Pφ)
∂φ− 1
rsen(φ)
∂(Pθ)
∂θ
= − 1
r2∂(r2Pr)
∂r
= − 1
r2∂rk
∂r
= − k
r2
7# #.,% 8'$-%5 ,#*#-'. 23#
Qencerrada = σp(a)4πa2 +
∫
Ωρp(r)d
3x
= −4πka+ 4π
∫ r
aρpr
2dr
= −4πka− 4πk
∫ r
adr
= −4πka− 4πk(r − a)
= −4πkr
9'$ ,%*,'5
∮
Ω
~E(~r) · ndS = 4πr2E(r) =Qencerrada
ǫ0=−4πkr
ǫ0=⇒ ~E(r) = − k
ǫ0rr
!" !"#$%&' () *+,&, $-+ '.
#$ r > b%& '()* +',-.& '/ 0'#)1+ 21/*+-3*#-.& '( &4/15 '( 6'#-+5
~P (~r) = ~05 21+ /1 74'
~E(~r) = − 1
ǫ0~P (~r) = ~0
8*9:-;& 4(*&61 /* /'< 6' =*4((5 )'&'91( 74'
∮
Ω
~E(~r) · ndS =Qencerrada
ǫ0
< '& '()' #*(1 /* #*+,* '&#'++*6* '(
Qencerrada = σp(b)4πb2 + σp(a)4πa2 +
∫
Ωρp(r)d
3x
= 4πkb− 4πka+ 4π
∫ b
aρpr
2dr
= 4πk(b− a)− 4πk
∫ b
adr
= 4πk(b− a)− 4πk(b− a)
= 0
21+ /1 74' >&*/9'&)'
∮
Ω
~E(~r) · ndS = 4πr2E(r) = 0 =⇒ ~E(~r) = ~0
?1+ )*&)15 +'(49-'&61 '/ #*921 '/;#)+-#1 6' 21/*+-3*#-.& '(
~E(r) =
~0 r < a
− kǫ0r r a < r < b~0 r > b
!
!"#$%&' ()
"# $%#&'#()&%* &' +,)$)( +)*),',)( $%# +,)$)( &' )*') A - ('+)*)$.%# &' +,)$)( d /.'#' ,)
*'0.%# '#/*' '(/)( ,,'#) $%# &%( 1)/'*.),'( &.','$/*.$%(2$%1% (' 3' '# ,) 405*)6 75+%#0) 85'
d ≫ L - d ≫ W'* 9'/'*1.#' ,) $)+)$./)#$.)
#* 9'15'(/*' 85' $5)#&% κ1 = κ2 = κ2 (5 *'(5,/)&% (' 35',3' ', 1.(1% 85' ', $%**'(+%#&.'#/'
) 5# $)+)$./%* 85' $%#/.'#' 5# (%,% &.','$/*.$%: C = κǫ0Ad 6
+"$,-."/0
;%&'1%( $%#(.&'*)* '(/' $%#&'#()&%* &' 1)#'*) '85.3),'#/' ) 5# 3)*.%( $%#&'#()&%*'( '#
+)*),',% '# ', 85' ,) +*%+%*$.%# &' &.','$/*.$% 3) 3)*.)#&%6 ;*.1'*% 85' /%&% $),$5,)*'1%( ,)
$)+)$.&)& &' 5# $%#&'#()&%* &' +,)$)( +)*),',%( $%# ! &.','$/*.$%( '# (5 .#/'*.%* $%1% 15'(/*)
,) 405*):
!! !"#$%&' () *+,&, $-+ '.
!"#$%&' ()
"#$%&#'(& )* $*+*$,-*- -&) $.#-&#/*-.( -& )* 01%(*2
*"$+,-"./
!"
!"#$%&' ((
#$%&'%( %$ &$)$&*+$+ +( ', &-,+(,.$+-/ +( )%$&$. )$/$%(%$. +( $/($ 0 1 .()$/$&*-, d &'$,+-
2.3( .( %%(,$ &-4)%(3$4(,3( &-, ', 4$3(/*$% +*(%(&3/*&- +( )(/4*.*5*+$+ 5$/*$6%( ǫ(z) = ǫ0(1+dz )7
)"$*+,"-.
!" !"#$%&' () *+,&, $-+ '.
!"#$%&' ()
#$ %&$'$ (' )*'+$',-+*. )&/0'+.&)* +$ .-+&*, &'%$.'* a 1 $2%$.'* b3 +$ /-.4* L >> b, a3 1)-.4- 56- Q 7/- 8/-)- &'%$.'- $, 8*,&%&9- 1 /- $2%$.'- '$4-%&9-:; </ )*'+$',-+*. $,%= //$'* )*'
(' >/*?($ +&$/@)%.&)* +$ 8$.A&%&9&+-+ ǫ 1 A-,- m3 ?($ +$,/&B- ,&' .*)$ $' /-, 8-.$+$, +$ /*,
)&/&'+.*,; <')($'%.$ /- )*'+&)&C' ,*>.$ $/ /-.4* L +$/ )*'+$',-+*. 8-.- ?($ $2&,%- 8*,&)&C' +$
$?(&/&>.&* 1 +$%$.A&'$ +&)D- 8*,&)&C'3 )(-'+* $/ )*'+$',-+*. ,$ 56- 9$.%&)-/A$'%$ 7+$ E*.A-
?($ $/ 8$,* +$,/&B- $/ >/*?($:;
*"$+,-./0
F-+* ?($ L >> a, b3 8*+$A*, +$,8.$)&-. /*, $E$)%*, +$ >*.+$ +$/ )*'+$',-+*.; <' $,%-,
)*'+&)&*'$,3 %$'$A*, ?($ $/ ,&,%$A- %&$'$ ,&A$%.0- )&/0'+.&)-3 8*. /* ?($ $/ )-A8* 8.$,$'%- /-
E*.A-
~E(~r) = E(r)r3 )*' . /- +&,%-')&- '*.A-/ +$,+$ $/ $6$ +$ ,&A$%.0- +$/ )*'+$',-+*. -/
8('%* $' )($,%&C'3 1 r $/ 9$)%*. ('&%-.&* )*' +&)D* ,$'%&+*;
G,-'+* /- /$1 +$ H-(,,3 //$4-A*, - ?($ $/ )-A8* $, 7$,%* ,$ +$6- -/ /$)%*.:
~E(r) =
~0 r < aQ
2πǫ0Lrr a < r < b
~0 r > b
8*. /* ?($ /- +&E$.$')&- +$ 8*%$')&-/ $'%.$ 8/-)-, $,
V =Q
2πǫ0L
∫ b
a
dr
r=
Q
2πǫ0Lln(b/a) =⇒ C =
2πǫ0L
ln(a/b)
$, /- )-8-)&%-')&- +$/ )*'+$',-+*. )&/0'+.&)*;
I/ 56-. 9$.%&)-/A$'%$ $/ )*'+$',-+*.3 $/ >/*?($ +&$/@)%.&)* +$,)&$'+$ 8*. $/ 8$,* ?($+-'+* ('-
8-.%$ +$/ )*'+$',-+*. ,&' +&$/@)%.&)*3 8*. /* ?($ /- )-8-)&%-')&- +$/ ,&,%$A- )-A>&-; #$- x /-
+&,%-')&- ?($ $/ >/*?($ +&$/@)%.&)* +$,)&$'+$3 A$+&+- +$,+$ /- 8-.%$ ,(8$.&*. +$/ )*'+$',-+*.
D-,%- /- 8-.%$ ,(8$.&*. +$/ +&$/@)%.&)*3 1 x $/ 9$)%*. ('&%-.&* $' $/ ,$'%&+* A$')&*'-+*; I/
+$,)$'+$. x $/ +&$/@)%.&)*3 ,$ E*.A-' +*, )*'+$',-+*.$, $' 8-.-/$/* +$ )-8-)&%-')&-,
C1 =2πǫ0x
ln(b/a)C2 =
2πǫ(L− x)
ln(b/a)
8*. /* ?($ /- )-8-)&%-')&- +$/ )*'+$',-+*. )*A8/$%* $,
C(x) = C1 + C2 =2πǫ0x
ln(b/a)+2πǫ(L− x)
ln(b/a)=
2π
ln(b/a)(ǫL− x(ǫ− ǫ0))
8-.- 0 < x < LJ- $'$.40- 8*%$')&-/ -/A-)$'-+- $' $/ )*'+$',-+*. $,
U(x) =1
2
Q2
C(x)
#->$A*, ?($ /- E($.B- ,*>.$ $/ +&$/@%.&)* $6$.)&+- 8*. $/ )*'+$',-+*.3 )(-'+* /- )-.4- $,
)*',%-'%$3 $,
~F = −~∇U 3 8*. /* ?($ /- E($.B- %*%-/ ,*>.$ $/ >/*?($ +&$/@)%.&)* $,
~F = −dU
dxx+mgx = ~0
!"
#$%&'( )*+,&)&-+ ./0 10 ,020 )/'3$&4 0+ 0$ 0./&$&24&*5 6*4 %(+%*7
mg =dU
dx
= −12
(
Q
C
)2 dC
dx
=Q2ln(b/a)(ǫ− ǫ0)
4π (ǫL− x(ǫ− ǫ0))2
=⇒ (ǫL− x(ǫ− ǫ0))2 =
Q2ln(b/a)(ǫ− ǫ0)
4πmg
|ǫL− x(ǫ− ǫ0)| = Q
√
ln(b/a)(ǫ− ǫ0)
4πmg
(ǫL− x(ǫ− ǫ0)) = ±Q
√
ln(b/a)(ǫ− ǫ0)
4πmg
=⇒ x =ǫL
(ǫ− ǫ0)∓ Q
(ǫ− ǫ0)
√
ln(b/a)(ǫ− ǫ0)
4πmg
8020'*1 (3$&)(4 $( 401%4&))&-+ 0 < x < L ( ('2(1 1*$/)&*+015 90+0'*1 34&'04('0+%0 ./0
x+ =ǫL
(ǫ− ǫ0)+
Q
(ǫ− ǫ0)
√
ln(b/a)(ǫ− ǫ0)
4πmg< L
=⇒ Q
(ǫ− ǫ0)
√
ln(b/a)(ǫ− ǫ0)
4πmg< L
(
1− ǫ
(ǫ− ǫ0)
)
= − Lǫ0(ǫ− ǫ0)
< 0
$* )/($ 01 /+( )*+%4(,&))&-+ 3/01
Q(ǫ−ǫ0)
√
ln(b/a)(ǫ−ǫ0)4πmg > 0
6*4 %(+%*7 $( #+&)( 3*1&)&-+ ,0 0./&$&24&* 3*4 $* 34*+%* 3*1&2$0 01
x0 =ǫL
(ǫ− ǫ0)− Q
(ǫ− ǫ0)
√
ln(b/a)(ǫ− ǫ0)
4πmg
:3$&)(+,* $(1 401%4&))&*+017 %0+0'*1 ./0
x0 =ǫL
(ǫ− ǫ0)− Q
(ǫ− ǫ0)
√
ln(b/a)(ǫ− ǫ0)
4πmg> 0
=⇒ L >Q
ǫ
√
ln(b/a)(ǫ− ǫ0)
4πmg
01 /+( )*+,&)&-+ ./0 ,020 )/'3$&4 0$ $(4;* ,0$ )*+,0+1(,*4 3(4( ./0 0<&1%( 3*1&)&-+ ,0 0./&=
$&24&*5
> $( #$%&'( 401%4&))&-+ 01
!" !"#$%&' () *+,&, $-+ '.
x0 =ǫL
(ǫ− ǫ0)− Q
(ǫ− ǫ0)
√
ln(b/a)(ǫ− ǫ0)
4πmg< L
=⇒ L <Q
ǫ0
√
ln(b/a)(ǫ− ǫ0)
4πmg
#$%&'( () $(*% +,*)&% -'./)-)0. $( -'&1%*)2,$ -'. ,% %.*$3)'34 5%3% 67$ $8)(*% 1'()-)0. /$
$67),)23)' /$2$ -7&1,)3($ ()&7,*9.$%&$.*$ 67$
Q
ǫ
√
ln(b/a)(ǫ− ǫ0)
4πmg< L <
Q
ǫ0
√
ln(b/a)(ǫ− ǫ0)
4πmg
=⇒ Q
ǫ<
Q
ǫ0
,' -7%, $( :$3/%/$3'4
5'3 *%.*'; ,% 1'()-)0. /$ $67),)23)' $(
x0 =ǫL
(ǫ− ǫ0)− Q
(ǫ− ǫ0)
√
ln(b/a)(ǫ− ǫ0)
4πmg
!"
!"#$%&' ()
#$ %&$'$ (') $*+$,) -.'/(-%.,) /$ ,)/&. a 0 -),1) 2.*&%&3) q4 ,./$)/) 2., (') -5*-),) /&$67-8
%,&-)4 /$ ,)/&. &'%$,&., 2a 0 ,)/&. $9%$,&., 2, 5a 0 /$ 2$,:&%&3&/)/ 3),&);6$ ǫ(r) = (1 + ra)ǫ0<
=)6-(6$>
'* ?6 -):2. $67-%,&-. 0 $6 3$-%., /$*26)@):&$'%. $67-%,&-. $' %./. $6 $*2)-&.<
#* ?6 3$-%., /$ 2.6),&@)-&A' /$6 /&$67-%,&-.<
+* B)* /$'*&/)/$* /$ -),1) 3.6(:7%,&-) 0 *(2$,C-&)6 /$6 /&$67-%,&-.<
,"$-+./0
'* D&3&/&:.* $6 $*2)-&. $' E ,$1&.'$* 0 -)6-(6):.* $6 -):2. $67-%,&-. 0 $6 3$-%., /$*26)@)8
:&$'%. $' -)/) -)*.>
• r < a1 B) $*+$,) /$ ,)/&. ) $* -.'/(-%.,) 0 2., 6. %)'%. '. F)0 -):2. $6$-%&,-. $' *( &'%$,&.,>
~E = 0, si r < a
D$ $*%) :)'$,) $6 3$-%., /$*26)@):&$'%. *$,5>
~D = ǫ0 ~E = 0, r < a
• a < r < 2a1 ?' $*%$ -)*. (%&6&@):.* 6$0 /$ 1)(**>
∫
S
~E · ndS =qenc
ǫ0
G*H4 /)/) 6) *&:$%,H) $*+7,&-) 0 I($ 6) -),1) $' 6) $*+$,) -.'/(-%.,) $* q .;%$'$:.* $6 ,$*(6%)/.
/$ *&$:2,$>
~E =q
4πǫ0r2r, si a < r < 2a
J),) $6 3$-%., /$*26)@):&$'%.>
~D = ǫ0 ~E =q
4πr2r, si a < r < 2a
• 2a < r < 2, 5a1 G26&-):.* 6$0 /$ 1)(** 2),) /&$67-%,&-.*>
∫
S
~D · ndS = qenc
!" !"#$%&' () *+,&, $-+ '.
#$ %&'()* +&,-$./.01&2() &, -.*.$&$) .$ '.0-) &$3'(*1') 456& &, *.+1.$78 9& &,(. 0.2&*.
~D =q
4πr2r
:;)*.< ,.=&0), 56&
~D = ǫ ~E< +)2+& ǫ &, $. -&*01(1%1+.+ +&$ +1&$3'(*1') 56& %.*1. &2 >62'1?2
+& r8 9& &,(. 0.2&*.< &$ '.0-) &$3'(*1') ,&*@A
~E =q
4πǫ0r2(1 +ra)
r, si 2a < r < 2, 5a
• r > 2, 5a B(1$1/.2+) $&C +& D.6,, )=(&2&0), +1*&'(.0&2(& 56&A
~E =q
4πǫ0r2r
~D =q
4πr2r
!" #$ %&'()* -)$.*1/.'1?2 &E1,(1*@ &%1+&2(&0&2(& ,)$) &2 $. *&D1?2 +)2+& ,& &2'6&2(*. &$ +1&$3'F
(*1')8 G10), &2 &$ *&,60&2 56& D = ǫ0 ~E + ~P C D = ǫ ~E8 9& &,(. 0.2&*.
~P = (ǫ− ǫ0) ~E
H)2 ǫ = ǫ0(1 +ra) C &$ %.$)* +&$ '.0-) &$3'(*1') &2 &$ +1&$3'(*1') &2')2(*.+) &2 .7A
~P = (ǫ0(1 +r
a)− ǫ0)
q
4πǫ0(1 +ra)r
2r
I10-$1J'.2+)A
~P =q
4πar(1 + ra)
r
#" H)2)'1&2+) &$ %&'()* -)$.*1/.'1?2 -)+&0), '.$'6$.* +1*&'(.0&2(& $., +&2,1+.+&, +& '.*D.
,6-&*J'1.$ C %)$603(*1'. +&$ +1&$3'(*1')A
$%&'()*) '+,%-.#(*/ σp G10), 56& $. +&2,1+.+ +& '.*D. ,6-&*J'1.$ ,& '.$'6$. -)*A
σp = ~P · (n)
K&2+*&0), 62. +&2,1+.+ ,6-&*J'1.$ &2 &$ 12(&*1)* +&$ +1&$3'(*1') C &2 &$ &E(&*1)* +&$ +1&$3'(*1')8
L.*. &$ 12(&*1)* +&$ +1&$3'(*1') 4r = 2a7
σpint =~P · −r =
q
24πa2
#2 &$ &E(&*1)* +&$ +1&$3'(*1') 4r = 2, 5a7A
σpext =~P · r = q
35πa2
$%&'()*) 01/+234-(#* ρp M. +&2,1+.+ +& '.*D. %)$603(*1'. ,& '.$'6$. ')0)
ρp = −~∇ · ~P
!"
#$%$ &'%$( )* +, -,./, ,01 )+ &)2/$. -$+,.'3,2'4* )( .,5',+ 6-,.,+)+$ ,+ 2,%-$ )+72/.'2$0 8 -$.
+$ /,*/$ +, 5'&).9)*2', 5) )(/) &)2/$. () 2,+2:+, 2$%$;
ρp = −~∇ · ~P = − 1
r2∂
∂r(r2P )
<)(,..$++,*5$ )(/, )=-.)('4*1 $>/)*)%$(;
ρp = −qa
4πr2(a+ r)2
!" !"#$%&' () *+,&, $-+ '.
!"#$%&' ()
#$ %&$'($)*'&+ '( ,-*%*) ,*+*-(-*). )(,*+*'*) /$* '0)1*$%0* d. 10($( /$ '0(-2%1+0%& ($ )/
0$1(+0&+. */)($1( '( %*+3*) -04+(). %/5* ,(+601070'*' (-2%1+0%* () ǫ 5 '(,($'( '( -* '0)1*$%0* *
/$* '( -*) ,-*%*)8 9*-%/-*+ -* %*,*%0'*' '(- %&$'($)*'&+ )0 ǫ(x) 70($( '*'& ,&+:
ǫ(x) = ǫ0(1
1− y2
3d2
)
*"$+,-".
;( ,0'( %*-%/-*+ -* %*,*%0'*' '(- )0)1(6* 5 )*4(6&) </(
C =Q
V
9*-%/-*6&) ($&$%() (- ,&1($%0*- * ,*+10+ '(- %*6,& (-(%1+0%& 5 ,*+* ()1&) )/,&$(6&) </( -*)
%*+3*) )( '01+04/5($ /$0=&+6(6($1( ($ %*'* ,-*%*8 ;( 1($'+* ($1&$%() </( -* '($)0'*' '( %*+3*
-04+( ()
σL =Q
S
>?&+* ,&'(6&) *,-0%*+ -* -(5 '( 3*/)) ,*+* '0(-(%1+0%&)8 @*+* ()1& %&$)0'(+*6&) %&6& )/,(+A%0(
3*/))0*$* /$ %0-0$'+& %&$ 1*,*) '( *+(* S (- %/*- 10($( -* 1*,* )/,(+0&+ ($ (- 0$1(+0&+ '(- /$*
'( -*) ,-*%*) %&$'/%1&+*) 5 -* &1+* 1*,* ($ (- '0(-(%1+0%&8 B(%&+'*6&) </( '(),+(%0*$'& -*)
%&$'0%0&$() '( 4&+'(. (- %*6,& (-(%1+0%& () ,(+,($'0%/-*+ * -*) ,-*%*) 5 ,&+ -& 1*$1& (- 7(%1&+
'(),-*C*60($1& (-(%1+0%& 1*640($ -& ()8 >'(6*). -* %*+3* -04+( ($%(++*'* )(+D qenc = σLS
∫
S
~D · d~S = ~D1 · ~S1 + ~D2 · ~S2 = qenc = σLS
E( -* A3/+* 7(6&) </( -&) ,+&'/%1&) ($1+( -&) 7(%1&+() '(),-*C*60($1& 5 -&) 7(%1&+() '( *+(*
%/6,-( %&$:
~D1 · ~S1 + ~D2 · ~S2 = −D1s+D2S = σLS
−D1 +D2 = σL
F$ (- 0$1(+0&+ '(- %&$'/%1&+ (- %*6,& (-2%1+0%& () $/-& 5 ,&+ -&. ,*+* (- 7(%1&+ '(),-*C*60$+1&
(-(%1+0%& ($ (- %&$'/%1&+ )( 10($
D1 = 0
@&+ -& 1*$1& 1($'+(6&) -* )03/0($1( (%/*%0&$:
D2 = ǫE2 = σL
F- %*6,& (-(%10+%& )(+D ($1&$%() E2 =σL
ǫ 8
>?&+*. (7*-/*$'& -&) 7*-&+() '( σL 5 ǫG'*'& ($ (- ($/$%0*'&H8
E2 =Q
Sǫ0(1
1− y2
3d2
)
>?&+* ()1* (I,+()0&$ -* ,&'(6&) +(()%+040+ %&6&:
E2 =Q
3Sǫ0d2(3d2 − y2)
!
"#$% &' ($)*+*,-$' (* ./0*,-$'& #* -'&-1&'+' -/2/
V =
∫ d
0Es(y)dy =
∫ d
0
Q
3Sǫ0d2(3d2 − y2)
3* *#0/ #* /40$*,* 51*
V =8
9
Qd
Sǫ0
6' -'.'-$('( *# *,0/,-*#
C =Q
V=9
8
Sǫ0d
!" !"#$%&' () *+,&, $-+ '.
!"#$%&' ()
#$%&'( )$ *% +(%)$%,-)(' )$ ./-+-, .-'-/$/-,0 )$ ,$++12% A 3 $,.$,(' d0 1%&'()*+14(, *%
)1$/5+&'1+( )$ .$'41&161)-) 6-'1-7/$0 ,1$%)( y /- )1'$++12% .$'.$%)1+*/-' - /-, ./-+-,8 #$,.'$9
+1-%)( /(, $:$+&(, )$ 7(')$ 3 $% +-,( )$ %( $;1,&1' +-'<-, /17'$, -/ 1%&$'1(' )$/ )1$/5+&'1+(0
+-/+*/-'=
'* >/ +-4.( $/5+&'1+(0 $/ )$,./-?-41$%&( $/5+&'1+( 3 $/ 6$+&(' )$ .(/-'1?-+12%0 +*-%)( -./19
+-4(, *%- )1:$'$%+1- )$ .(&$%+1-/ V0 $%&'$ /-, ./-+-,8
#* @-, )$%,1)-)$, )$ +-'<- )$ .(/-'1?-+12%8
+* @- +-.-+1)-) )$/ +(%)$%,-)('8
ǫ = ǫ0(1 +y
d)
,"$-+."/
'* >% $/ 1%&$'1(' )$/ )1$/+&'1+( %( &$%$4(, +-'<- /17'$ 3 .(' /( &-%&( /- )$%,1)-) )$ +-'<- /17'$
$, %*/-= ρL = 08 A(4( B$4(, 61,&( )*'-%&$ $/ +*',(0 )$,.'$+1-%)( /-, +(%)1+1(%$, )$ 7(')$0
$/ +-4.( $/$+&'1+( ,$'- .$'.$%)1+*/-' - /-, ./-+-, 3 .(' /( &-%&(0 )$ /- $+*-+1(% C"D0 $/ 6$+&('
)$,./-?-41$%&( $/5+&'1+( &-471$% /( ,$'-8 #$ $,&- 4-%$'-0 )$ -/ :('4- )1:$'$%+1-/ )$ /- /$3 )$
E-*,, &$%)'$4(,=
~∇ · ~D =∂
∂yD = ρP = 0
#$ $,&(0 '$,*/&- F*$ $/ 6$+&(' )$,./-?-41$%&( $/$+&'1+( $, +(%,&-%&$8
G$%$4(, )-)( *% .(&$%+1-/ V0 F*$ +*4./1'- /- ,1<*1$%&$ $+*-+1(%=
V0 =
∫
~E · d~l
H$'( )$ /- $+*-+1(% C"D &$%$4(, F*$
~E =~Dǫ 8#(%)$ ǫ = ǫ(y)8 H(' /( &-%&(=
V0 =
∫ ~D
ǫ· d~l = D
∫
1
ǫdy
#(%)$ D .*$)$ ,-/1' )$ /- 1%&$<'-/ -/ ,$' +(%,&-%&$ 3 +(%,1)$'-4(, dy = dl8 I$$4./-?-%,( $/
6-/(' )$ ǫ )-)( $% $/ $%*%+1-)( $ 1%&$<'-%)( &$%)'$4(, F*$=
V0 =Dd ln(2)
ǫ0
#$,.$J-%)( D=
D =V0ǫ0
d ln(2)
KB('- F*$ &$%$4(, $/ 6$+&(' )$,./-?-41$%&(0 .()$4(, +-/+*/-' $/ 6$+&(' +-4.,( $/$+&'1+( )$
/- $+*-+1(% C"D=
E =D
ǫ=
V0d ln(2)(1 + y
d)
#$ /- $+*-+1(% C D .()$4(, ,-7$' +*-/ $, $/ 6$+&(' )$ .(/-'1?-+1(%=
!!
~P = ~D − ǫ0 ~E =V0ǫ0
d ln(2)
y
y + d
! "#$# %&'&$#( &) *&+%#, -#).,/0.+/#'1 #2%&'&$#( ).( +.,3.( 4& -#).,/0.+/#' 4/,&+%.$&'%&
4& &+5.+/#' 6789
:. +.-.+/4.4 ). +.)+5).$#( +#$# (/&$-,&; C = QV09 < 4/=&,&'+/. 4&) &>&,+/+/# .'%&,/#,1 ?.
%&'&$#( ). 4/=&,&'+/. 4& -#%&'+/.)9 @#, )# %.%'# %&'&$#( A5& *&, .)35'. $.'&,. 4& +.)+5)., ).
+.,3. Q &' =5'+/#' 4& V09 @.,. &(%#1 .-)/+.,&$#( ). :&? 4& B.5(( .) /35.) A5& &' &) -,#2)&$.
;
∫
S
~D · d~S = ~D1 · ~S1 + ~D2 · ~S2 = qenc = σLS
C& ). D35,. *&$#( A5& )#( -,#45+%#( &'%,& )#( *&+%#,&( 4&(-).0.$/&'%# ? )#( *&+%#,&( 4& .,&.
+5$-)& +#';
~D1 · ~S1 + ~D2 · ~S2 = −D1s+D2S = σLS
−D1 +D2 = σL
E' &) /'%&,/#, 4&) +#'45+%#, &) +.$-# &)F+%,/+# &( '5)# ? -#, )#1 -.,. &) *&+%#, 4&(-).0.$/',%#
&)&+%,/+# &' &) +#'45+%#, (& %/&'
D1 = 0
@#, )# %.'%# %&'4,&$#( ). (/35/&'%& &+5.+/#';
D2 = σL
:. +.,3. 4&) +#'4&'(.4#, (&,G ). 4&'(/4.4 4& +.,3. -#, ). (5-&,D+/& 4& ).( -).+.(6H<89 @,#
+#$# ). 4&'(/4.4 4& +.,3. )/2,& &( /35.) .) *&+%#, 4&(-).0.$/&'%# &)F+%,/+#;
Q = SσL = SD2 = SV0ǫ0
d ln(2)
C& &(%. $.'&,. ). +.-.+/4.4 (&,G;
C =Q
V0= S
ǫ0d ln(2)
!" !"#$%&' () *+,&, $-+ '.
!"#$%&' ()
#$ %&$'$' ! $()$*+( ,-'./,%-*+( ,-',0'%*&,+( .$ *+.&-( a1 b 2 c3 45 $(6+,&- $'%*$ 5+( .-( 6*&7$*+(
$(%+ 55$'+ .$ 7+%$*&+5 .&$50,%*&,- .$ ,-'(%+'%$ κ3 8'&,&+57$'%$ 5+ $()$*+ .$ *+.&- a $(%+ .$(,+*9
:+.+ 2 5+( $()$*+ .$ *+.&-( b 2 c %&$'$' ,+*:+( &'&,&+5$( q1 2 q21 *$(6$,%&;+7$'%$3 <+ $()$*+
&'%$*&-* .$ *+.&- + ($ ,-'$,%+ ,-' 5+ $()$*+ .$ *+.&- c ,-' /' ,+=5$ +&(5+.- .$5:+.-3
'* >+5,/5+* 5+ ,+*:+ 5&=*$ $' ,+.+ $()$*+3
#* >+5,/5+* 5+ ,+*:+ .$ 6-5+*&?+,&@' $' 5+ (/6$*A,&$ $B%$*'+ 2 $' 5+ (/6$*A,&$ &'%$*'+ .$5
.&$50,%*&,-3
!"
!"#$%&' ()
#$ %&$'$' ()* +)'($'*,()-$* ($ ./,+,* .,-,/$/,* +)' &01,/$* 2-$,* S3 $ &01,/$* (&*%,'+&,* $'%-$
/,* ./,+,* d4 5/ .-&6$- +)'($'*,()- %&$'$ ,&-$ $'%-$ *1* ./,+,*7κ = 18 9 +,.,+&(,( C04 5/
*$01'() +)'($'*,()- %&$'$ 1', +,.,+&(,( X4
'* #& $/ .-&6$- +)'($'*,()- %&$'$ &'&+&,/6$'%$ 1', +,-0, Q3 +,/+1/$ $/ +,6:&) ($ $'$-0;,
$/$+%-)*%2%&+,3 $' <1'+&=' ($ X3 *& ,6:)* +)'($'*,()-$* *$ +)'$+%,' $' .,-,/$/)4
#* #& $/ (&$/>+%-&+) ($/ *$01'() +)'($'*,()- %&$'$ 1', +)'*%,'%$ (&$/>+%-&+, ?,-&,:/$ $' $/
$*.,+&) ($ ?,/)- κ = κ0(1 +xd ) +,/+1/$ /, +,.,+&(,( ($ $*%$ +)'($'*,()-4
!" !"#$%&' () *+,&, $-+ '.
!"#$%&' ()
#$ %&'&$()* %*$(+%,*) -+.%* /&$ ./0./*) (. )1(&* r1 2 '1)3* ℓ4 ''.51 +$1 %1)31 +q +$&6*)7.8
7.$,. (&/,)&9+&(1: ;').(.(*) (. .'4 *,)* %&'&$()* %*$(+%,*) -+.%* /&$ ./0./*)4 %*1<&1' %*$ .'
1$,.)&*)4 (. )1(&* r2 2 7&/7* '1)3* ℓ 4 ,1' %*7* &$(&%1 '1 =3+)14 ''.51 +$1 %1)31 −q4 ,179&>$
+$&6*)7.7.$,. (&/,)&9+&(1: ?. ''.$1 '1 ).3&@$ .$,). '*/ (*/ %&'&$()*/ %*$ +$ 71,.)&1' %+21 %*$8
/,1$,. (&.'>%,)&%1 ./ 6+$%&@$ (. '1 (&/,1$%&1 1' .A. (. '*/ %&'&$()*/4 κ = κ(r): B*$/&(.). ℓ 7+%-*
712*) C+. r1 2 r2:
'* D$%+.$,). +$1 .<0)./&@$ 01)1 κ(r) C+. ''.51 1 +$ %170* .'.%,)*/,E,&%* )1(&1' &$(.0.$(&.$,.
(. '1 (&/,1$%&1 1' .A.:
#* B1'%+'. '1 %101%&(1( (.' %*$(.$/1(*) %&'F$()&%* %*))./0*$(&.$,.:
+* B1'%+'. '1 (.$/&(1( (. %1)31 (. 0*'1)&G1%&@$ /+0.)=%&1' (.' 71,.)&1' (&.'>%,)&%*4 1 r = r1 2
r = r24 01)1 .' 51'*) (. κ(r) %1'%+'1(* .$ .' &,.7 a):,* B*$/&(.). C+. .$ %**)(.$1(1/ %&'F$()&%1/
~∇ · ~A = 1r
∂∂r (rAr) !"#!$#"% #"& '()&*'"'(&
+,#$-./%*!"& '( !"%0"& '( 1,#"%*2"!*3) () (# '*(#.!/%*!,4
! 5"#!$#"% #" !")/*'"' /,/"# '( !"%0"& '( 1,#"%*2"!*3) *)!#$6.)',&( !"%0"& () #" &$1(%7!*( 6
() (# +,#$-() '(# '*(#.!/%*!,4
"! 5"#!$#( #" ()(%08" 1,/()!*"# (#(!/%,&/9/*!" "#-"!()"'" () (# !,)'()&"',%4
!"#$%&' (
!""#$%&$ '($)&"#)*
!"#$%&' ()
!"#$ %&' (&!%)("&#$' $'*+#,(&' (&!(+!"#,(&' %$ #-%,&' a . b/ '$ 00$!- (&! )! 1-"$#,-0 (&!%)("&#%$ (&!%)(",2,%-% 2-#,-30$ g(r) = g0b
r 4 5-0()0$ 0- #$','"$!(,- $!"#$ 0-' %&' 60-(-' (&!%)("&#-'4
*"$+,-./
70 8-3$# )!- %,*$#$!(,- %$ 6&"$!(,-0 $!"#$ -13&' (&!%)("&#$' $'*+#,(&'/ '$ 9$!$#- )!- (&##,$!"$
~J :)$ $' #-%,-0;
~J = J(r)r
<&%$1&' $'(#,3,# 0- (&##,$!"$ $! *)!(,=! %$
~J 1$%,-!"$ 0- #$0-(,=!;
I =
∫
S
~J · d~S = J
∫
SdS = J4πr2
>?@
!" !"#$%&' () '**+,-$, ,&, $*+ !
#$%&' ($%)*&'+,-$) ($-$ S .%, )./'+0(*' ')12+*(,3 #' ')4, -,%'+, '5 6'(4$+ &'%)*&,& &'
($++*'%4' )' /.'&' ')(+*7*+8
~J =I
4πr2r
9, 5': &' ;<- %$) /'+-*4' +'5,(*$%,+ '5 (,-/$ '52(4+*($ ($% '5 6'(4$+ &'%)*&,& &' ($++*'%4'3
=% ')4' (,)$ ($-$ 5, ($%&.(4*6*&,& ') g(r) = g0br )' 4'%&+>8
~J = g(r) ~E
#')/'?,%&$ '5 (,-/$ '52(4+*($ : .4*5*@,%&$ '5 6,5$+ '%($%4+,&$ /,+,
~J
~E =I
4πbg0rr
A<$+,B )' /.'&' (,5(.5,+ 5, &*1'+'%(*, &' /$4'%(*,5 '%4+' ,-7$) ($%&.(4$+') :, C.' 4'%'-$)
.%, 'D/+')*E% /,+, '5 (,-/$ '52(4+*($38
V0 =
∫ b
a
~E · d~r = I
4πbg0
∫ b
a
1
rdr =
I
4πbg0ln(
b
a)
F,7*'%&$ 5, &*1'+'%(*, &' /$4'%(*,5 : 5, *%4'%)*&,& &' ($++*'%4' I /$&'-$) (,5(.5,+ 5, +')*)4'%(*,
-'&*,%4' 5, +'5,(*E% R = V0
I 3 G$+ 5$ 4,%4$B /,+, '5 )*)4'-, -$)4+,&$ 5, +')*)4'%(*, ')8
R =V0I=
ln( ba)
4πbg0
!"
!"#$%&' ()
#$ %$&'()*+ ,-+ .+/0+ Q0 $- ,-+ &1+.+ .'-%,.*'/+ .,+%/+%+ %$ 1+%' a2 3,$ $(*4 ($&+/+%+ ,-+
%)(*+-.)+ d %$ '*/+ &1+.+ )0,+12 %$(.+/0+%+ 5 .'-$.*+%+ + *)$//+ 6.'-()%$/$ 3,$ a >> d &+/+
%$(&/$.)+/ $7$.*'( %$ 8'/%$9: ;-*/$ $11+( $<)(*$ ,- =+*$/)+1 %$ &$/=)*)>)%+% ǫ 5 .'-%,.*)>)%+%
µ2 +=8+( .'-(*+-*$(:
+9 ?$*$/=)-$ 1+ .'//)$-*$ 3,$ .)/.,1+ $- 7,-.)@- %$1 *)$=&':
89 ;-.,$-*/$ 1+ /$()(*$-.)+ %$1 ()(*$=+ 5 >$/)A3,$ 3,$ RC = ǫµ :
.9 B+1.,1$ 1+ $-$/0C+ %)()&+%+ $- $1 &/'.$(' .'=&1$*'2 $( %$.)/2 +1 &+(+/ *'%+ 1+ .+/0+ %$
,-+ &1+.+ + '*/+:
*"$+,-./0
+9 #+8$='( %$ 1+( +5,%+-*C+( +-*$/)'/$( 3,$ $1 .+=&' $1D.*/).' 0$-$/+%' &'/ ,- &1+-' %$
%$-()%+% %$ .+/0+ (,&$/A.)+1 .'-(*+-*$ σ $(
~E =σ
2ǫ0x
.'- x -'/=+1 +1 &1+-':
;- $(*$ .+('2 %$(&/$.)+-%' $7$.*'( %$ 8'/%$ %$8)%' + 3,$ a >> d2 5 &'/ 1+ &/$($-.)+ %$1
=+*$/)+1 %$ &$/=)*)>)%+% ǫ $-*/$ &1+.+(2 $1 .+=&' $1D.*/).' $-*/$ &1+.+( $(
~E =σ
2ǫx =
Q
2ǫa2x
#+8$='( 3,$
~J = µ~E2 &'/ 1' .,+1
~J =µQ
2ǫa2x = Jx
!" !"#$%&' () '**+,-$, ,&, $*+ !
#$ %&$'$ '()*'+$,$ $ +-*..$ /* 0$)+-*)* $ %(+*)'-$& *&1'+.-'( '()/+$)+* -23$& $ '*.(4 %(.
&( 53* &3*2( ,* ,*%(/-+$,$ &$ '$.2$ Q0 *) &$ (+.$ %&$'$4 %(. &$ ,-6*.*)'-$ ,* %(+*)'-$&4 /*
%.(,3'* 3) '$0%( *&1'+.-'( 753* 8$ 9*0(/ *)'()+.$,(: 8 3)$ '(..-*)+* ,* 3)$ %&$'$ $
(+.$ 7%3*/ *& '$0%( *&1'+.-'( 03*;* &$ '$.2$ -)-'-$& ,* 3)$ %&$'$ $ (+.$4 2*)*.$),( 3)$
,*)/-,$, ,* '(..-*)+* *) *& 0$+*.-$&: 9$/+$ 53* &$ %&$'$ -)-'-$&0*)+* '$.2$,$ /* ,*/'$.2$
8 +(,( *& /-/+*0$ /* ;3*&;* *53-%(+*)'-$&<
#$ '(..-*)+* *&1'+.-'$ */+= ,$,$ %(.
i =
∮
Ω
~J · ndS
= Ja2
=µQ
2ǫ
= −dQ
dt
,(),* *& /-2)( 0*)(/ */ ,*>-,( $ 53* &$ %&$'$ /* */+= ,*/'$.2$),( 8 &$ /3%*.?'-* ,*
-)+*2.$'-@) 63* 3) '3$,.$,( ,* &$,( a %$.$&*&( $ &$/ %&$'$/<
A* &( $)+*.-(.4 +*)*0(/ &$ *'3$'-@) ,-6*.*)'-$&
−µQ
2ǫ=
dQ
dt
ssi 0 =dQ
dt+
µQ
2ǫ
ssi 0 =dQ
dtexp
(
µt
2ǫ
)
+µQ
2ǫexp
(
µt
2ǫ
)
ssi 0 =d
dt
(
Q · exp
(
µt
2ǫ
))
=⇒ Q(t) = Q0 · exp(
− µ
2ǫt)
%3*/ Q(0) = Q0<
B(. +$)+(4 &$ '(..-*)+* *&1'+.-'$ *) 63)'-@) ,*& +-*0%( */
i(t) = −dQ
dt=
µQ0
2ǫ· exp
(
− µ
2ǫt)
>: A*& -+10 $)+*.-(.4 /$>*0(/ 53*
~E =Q(t)
2ǫa2x = E(t)x
#$ .*/-/+*)'-$ ,* 3) (>C*+( ,*%*),* ,* &$ 2*(0*+.D$ ,*& 0-/0(4 ,* &$ 6(.0$ *) 53* /*
$%&-53* &$ ,-6*.*)'-$ ,* %(+*)'-$& 8 ,*& 0$+*.-$& ,*& 53* */+= 9*'9( 7&( 53* /* +.$,3'* *)
&$ .*/-/+-;-,$, ( '(),3'+-;-,$, ,*& (>C*+(:< E) */+* '$/( +*)*0(/ 53* &$ '(),3'+-;-,$, µ
!
"# $%&#'(&'") *%+ ,% -." #" $./*," ,( ,"0 1" 23/) *%+ ,% $.(, ,( +"#4#'"&$4( "# $%&#'(&'"
0 &% 1"*"&1" 1", 5%,'(6" &4 1" ,( $%++4"&'" -." *(#( *%+ ", %76"'%8 9( ,"0 1" 23/ &%#
14$" -."
V = i ·R*%+ ,% -." "&$%&'+(&1% V (t)) i(t) "& .& 4&#'(&'" 1(1% *%1"/%# 3(,,(+ R8
:&$%&'+"/%# ,( 14;"+"&$4( 1" *%'"&$4(,
V (t) =
∫ d
0E(t)x · xdx = E(t)d =
Q(t)d
2ǫa2
<"&"/%# (1"/=# -."
i(t) =µ
2ǫQ(t) =⇒ Q(t) =
2ǫ
µi(t)
>%+ '(&'%)
V (t) =Q(t)d
2ǫa2=
2ǫd
2µǫa2i(t) =
1
µ
d
a2i(t) = R · i(t) =⇒ R =
1
µ
d
a2
?(7"/%# 1" (0.1(&'@(# (&'"+4%+"# -." ,( $(*($4'(&$4( 1" .& $%&1"&#(1%+ 1" *,($(# *(+A
(,",(# 1" =+"(# A) #"*(+($4B& d 0 #"*(+(1(# *%+ .& /"14% 14",C$'+4$% 1" *"+/4'4541(1 ǫ"#
C = ǫa2
d
>%+ '(&'%)
RC =1
µ
d
a2· ǫa2
d=
ǫ
µ
$D 9( "&"+E@( 14#4*(1( "& ", *+%$"#% "#
U =
∫ +∞
0i(t)V (t)dt
=µd
4a2ǫ2
∫ +∞
0Q2(t)dt
=µQ0
2d
4a2ǫ2
∫ +∞
0exp
(
−µ
ǫt)
dt
=Q0
2d
4a2ǫ· exp
(
−µ
ǫt)∣
∣
∣
0
+∞
=Q0
2d
4a2ǫ
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$ %&$'$' ()* *+,$-./&$* /)'(+/%)-0* $*12-&/0* 3 /)'/2'%-&/0*4 ($ -0(&)* 0 3 5 6/)' a < b7480'%$'&()* 0 +'0 (&1$-$'/&0 ($ ,)%$'/&09 V0 6/)' $9 803)- ,)%$'/&09 $' 90 *+,$-./&$ &'%$-&)-7:
;9 $*,0/&) $'%-$ 90* *+,$-./&$* *$ 99$'0 /)' +' 80%$-&09 /)'(+/%)- <)8)=2'$) ($ /)'(+/%&>&(0(
/)'*%0'%$ µ: ?$%$-8&'$ 90 /)--&$'%$ $92/%-&/04 $9 /08,) $92/%-&/) $'%-$ *+,$-./&$* 3 90 -$*&*%$'@
/&0 ($9 *&*%$804 3 >$-&.A+$ '+$>08$'%$ A+$ RC = ǫµ :
*"$+,-./0
?$ 90 *&8$%-B0 ($9 ,-)59$804 %$'$8)* A+$
~E(~r) = E(r)r4 ()'($ r $* 90 /))-($'0(0 /)--$*,)'@
(&$'%$ ($ /))-($'0(0* $*12-&/0*: C)8)
~J = µ~E4 $>&($'%$8$'%$
~J = J(r)r: ?$ $*%) >$8)* A+$
90 /0-=0 *$ %-0'*,)-%0 -0(&098$'%$4 $* ($/&-4 90 /)--&$'%$ $* -0(&09: D($8E*4 (0() A+$ µ $*
/)'*%0'%$4 *$ /+8,9$ 90 9$3 ($ )<8 3 V0 = iR4 /)' R /)'*%0'%$4 ,)- 9) A+$ 90 /)--&$'%$ i $*
/)'*%0'%$:
;'/)'%-$8)* 90 /)--&$'%$:
i =
∫
Ω
~J · ndS = J(r)
∫
ΩdS = J(r)4πr2 =⇒ ~J(r) =
i
4π
r
r2
()'($ +*08)* ($ *+,$-./&$ ($ &'%$=-0/&F' +'0 *+,$-./&$ $*12-&/0 ($ -0(&) - /)'/2'%-&/0 0
90* *+,$-./&$* /)'(+/%)-0*:
?$ $*%) %$'$8)* A+$
~E(r) =1
µ~J(r) =
i
4πµ
r
r2
D<)-04 ,0-0 $'/)'%-0- 90 -$*&*%$'/&0 <099$8)* 90 (&1$-$'/&0 ($ ,)%$'/&09 $'%-$ *+,$-./&$*:
V0 = −∫
Γ
~E · dr
=i
4πµ
∫ b
a
dr
r2
=i
4πµ
1
r
∣
∣
∣
∣
a
b
=i
4πµ
(
1
a− 1
b
)
= iR
=⇒ R =
(
1a − 1
b
)
4πµ
D($8E*4 %$'$8)* A+$ 90 /)--&$'%$ $*
i =4πµV0(
1a − 1
b
)
3 $9 /08,) $92/%-&/)
~E(r) =i
4πµ
r
r2=
V0(
1a − 1
b
)
r
r2
!"
#$ %&'(%)*+%, %)*$-./-$, ,%0$1/, 2'$ 3% 4%5%4.*%)4.% ($ (/, ,'5$-64.$, $,78-.4%, 4/)48)9
*-.4%, ($ -%(./, a & b 4/) a < b: 33$)%, $)*-$ ,+ ($ ') 1%*$-.%3 (.$384*-.4/ ($ 5$-1.*.;.(%( ǫ:$,
C =4πǫ
(
1a − 1
b
)
</- *%)*/:
RC =
(
1a − 1
b
)
4πµ· 4πǫ(
1a − 1
b
) =ǫ
µ
!! !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ((
"#$%&$' $# (')*)+',%*# -' &, )'.*+/(/ -' )'%%*0, %&#-(#-# -' $#-/ a 1 (#-*/ *,+'(*/( b2 )* ')+34'%4/ -' &, .#+'(*#$ %/, %/,-&%+*5*-#- µ %/,)+#,+'6
)"$*+,-./
"/,'%+'./) &,# 7#+'(*# -' -*8'(',%*# -' 9/+',%*#$ V0 # $#) %#(#) -'$ )'.*+/(/6 :/( $#
)*.'+(;# -'$ +/(/2 '$ %#.9/ '$<%+(*%/ -'7' ')+#( -*(*=*-/ ', '$ )',+*-/ -' θ6 > 9(*.'(# 5*)+#2
+','./) ?&'
~E = E(r, θ)θ2 -/,-' r2 θ )/, $#) %/,/%*-#) %//(-',#-#) %*$;,-(*%#)6 @/)+('./)
?&' )* $# %/((*',+' ')+3 ', (<=*.', ')+#%*/,#(*/2 ',+/,%') E = E(r)6 A, '8'%+/2 ', (<=*.',
')+#%*/,#(*/ B/ 9'(.#,',+'C +','./) ?&'
~∇ · ~J =1
r
∂
∂r(rJr) +
1
r
∂Jθ
∂θ+
∂Jz
∂z= 0
:'(/
~J = µ~E = µE(r, θ)θ = Jθ =⇒ Jθ = J
:/( +#,+/
~∇ · ~J =1
r
∂J
∂θ= 0 =⇒ J = f(r) + C = J(r)
9/( $/ ?&' 5'./) ?&'
~J = J(r)θ 1
~E = E(r)θ6
D# %/((*',+' ')
i =
∫
Ω
~J · ndS
=
∫ a
0
∫ b+a
bJ(r)drdz
= aµ
∫ b+a
bE(r)dr
-/,-' 4'./) +/.#-/ %/./ )&9'(E%*' -' *,+'=(#%*0, &,# %#(# -'$ +/(/6
D# -*8'(',%*# -' 9/+',%*#$ ',+(' $#) %#(#) 'F+('.#) -'$ +/(/ ') V06 >);2 +','./) ?&'
V0 =
∫
Γ
~E · dr
=
∫ π
0E(r)θ · rdθθ
= E(r)rπ
=⇒ E(r) =V0πr
!"
#$%#& '&($) *$(+#$ ,% -+(.%$ )&(.-./-,0+/ #& /+#.$ /1 -$% dr = rdθθ23& &)*+ 4$/(+1 *&%&($) 5,&
i = aµ
∫ b+a
bE(r)dr
=aµV0
π
∫ b+a
b
dr
r
=aµV0
πln
(
1 +a
b
)
=⇒ R =π
aµln(
1 + ab
)
6,&) V0 = iR
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$%&' ()* +,-.-* +-&-,',-* .)$(/.%)&-* (' 0&'-* A 1 *'+-&-.23$ d4 '52*%'$ ()* 6-%'&2-,'* ('
.)$(/.%272(-('* µ14 µ2 1 +'&62%272(-('* ǫ14 ǫ24 .)6) 6/'*%&- ,- 89/&- :+-&- ('*+&'.2-& ';'.%)*
(' <)&('4 .)$*2('&' =/' ,-* (26'$*2)$'* ,2$'-,'* (' ,-* +,-.-* *)$ 6/.>) 6/.>) 6-1)&'* =/'
(4 '* ('.2&4
√A >> d?@ A2 ,-* +,-.-* .)$(/.%)&-* *' .)$'.%-$ - /$- (2;'&'$.2- (' +)%'$.2-, V04
'*%-$() ,- +,-.- 2$;'&2)& - 6-1)& +)%'$.2-,4 '$./'$%&'B
-? C- &'*2*%'$.2- (', *2*%'6-@
<? C- ('$*2(-( (' .-&9- ,2<&' =/' *' -./6/,- '$ ,- 2$%'&;-D (' ,)* 6-%'&2-,'* '$ &E926'$
'*%-.2)$-&2)@
*"$+,-./0
-? F-(- ,- *26'%&G- (', +&)<,'6- :1 =/' +)('6)* ('*+&'.2-& ';'.%)* (' <)&(' +/'*
√A >>
d?4 %'$'6)* =/'
~E = E(z)z4 +)& ,) =/'
~J = J(z)z@#$ &E926'$ '*%-.2)$-&2) %'$'6)*
~∇ · ~J =∂Jx
∂x+
∂Jy
∂y+
∂Jz
∂z=
∂Jz
∂z= 0 =⇒ J = Jz = C
()$(' H '* /$- .)$*%-$%'@ I)& %-$%)4
~J = Jz '* /$ 7'.%)& .)$*%-$%'@
F-() =/'
~J = µ~E4 ,)* .-6+)* '$ .-(- &'923$ *)$
~E1 =J
µ1z ~E2 =
J
µ2z
J>)&-4 ', +)%'$.2-, '*
!"
V0 = −∫
Γ
~E · dr
=
∫ d2
0E2dz +
∫ d2+d1
d2
E1dz
=Jd2µ2
+Jd1µ1
= J
(
d2µ2
+d1µ1
)
=⇒ J =V0
(
d2
µ2+ d1
µ1
)
#$%& '()(*+$ ,-(
i =
∫
Ω
~J · ndS = JA =V0A
(
d1
µ1+ d2
µ2
) =V0R
.+/ 0+ 1-20
R =1
A
(
d1µ1
+d2µ2
)
34 5+)$67(/(*+$ -) 1606)7/+ /(1'+ 1+) (8( 7( $6*9'/62 () (0 $()'67+ 7( z& 1+) 12/2$ 16/1-02/($
7( :/(2 S& (0 1-20 ($ 1+/'27+ .+/ (0 6)'(/;2< ()'/( *2'(/620($= >02*(*+$ $- $-.(/?16( Ω=5+*+
~D = ǫ ~E& ()'+)1($
~D = D(z)z= @+/ 02 0(A 7( B2-$$ C()(/206<272& '()(*+$
∮
Ω
~D · ndS = Qlibre = σS n = z
(D1 −D2)S = σS
=⇒ σ = D1 −D2
= ǫ1E1 − ǫ2E2
=
(
ǫ1µ1− ǫ2
µ2
)
J
=
(
ǫ1µ1− ǫ2
µ2
)
V0(
d2
µ2+ d1
µ1
)
σ =
(
ǫ1µ2 − ǫ2µ1d1µ2 + d2µ1
)
V0
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$%&'()*) +% ,-,./*) ()-0,($ 1 *20'($ () -,*0$ 2L3 4$* )- 5+,- 6+1) +%, 5$**')%7) i8 9' 4,*7'.$&
)- ,-,./*) 4$* -, .'7,( 4$* +% 4-,%$ %$*.,- , :-3 )%5+)%7*) )- 5,.4$ .,0%:7'5$ &$/*) +% 4+%7$
)% )- 4-,%$ , +%, ('&7,%5', r ()- 4+%7$ () '%7)*&)55';% )%7*) ,-,./*) 1 4-,%$8 <+)0$ )%5+)%7*)
)- 5,.4$ 4*$(+5'($ 4$* +% ,-,./*) '%=%'7$ )% 7$($ )- )&4,5'$3 +&,%($ 5$%>)%')%7).)%7) )-
*)&+-7,($ ,%7)*'$*8
*"$+,-./0
?&).$& 5$$*()%,(,& 5'-2%(*'5,&3 5$% )- ,-,./*) &$/*) )- )@) z 1 ,- $*'0)% @+&7$ )% &+ .'7,(8
9),% ~r = rr3 ~r′ = zz )- 4+%7$ () $/&)*>,5';% &)0A% )- )%+%5',($ 1 )- >)57$* B+) ()&5*'/) )-
,-,./*)8 C)%).$& B+) )- 5,.4$ .,0%D7'5$ 0)%)*,($ 4$* +%, 5$**')%7) ('E)*)%5',-
~di = i~dl )&
d ~B(~r) =µ04π
i~dl × (~r − ~r′)
|~r − ~r′|3
=µ04π
idzz × (rr − zz)
(r2 + z2)3/2
=⇒ ~B(~r) =µ04π· i
∫ L
−L
rdzθ
(r2 + z2)3/2
=µ04π· irθ
∫ L
−L
dz
(r2 + z2)3/2z = rtan(α)
=µ02π· irθ
∫ arctan(L/r)
0
rsec2(α)dα
r3sec3(α)
=µ02π· iθ
r
∫ arctan(L/r)
0cos(α)dα
=µ02π· iθ
rsen(α)|arctan(L/r)
0
=µ02π· iθ
r
(
L/r√
1 + (L/r)2
)
=µ02π· iθ
r
(
L√r2 + L2
)
FG$*,3 4,*, )%5$%7*,* )- 5,.4$ .,0%:7'5$ () +% ,-,./*) '%=%'7$ %$& /,&7, 7$.,* L→∞
!"
#$% &$'(%)*#+ *,)$&-+&. /+, %+ /(*% +0)$,$1+'
~B(~r) =µ02π· i
rθ
/*12+ 3($ )-$,$ '-1$)&4* /-%4,#&-/*. /+1+ $&* #$ $'2$&*&'$5
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$%&'()*) ($& +,+-.*)& '%/%'0$& 1+*+,),$&2 &)1+*+($& 1$* 3%+ ('&0+%4'+ d2 4$% 4$**')%0)& i12i25)% ), -'&-$ &)%0'($67
+6 8%43)%0*) ,+ 93)*:+ 1$* 3%'(+( () ,+*;$ <3) &) )=)*4)% ,$& +,+-.*)&2 ().'($ + &3 '%0)*>
+44'?% -+;%@0'4+7
.6 #$%&'()*) +A$*+ ($& +,+-.*)& '%/%'0+-)%0) ,+*;$& 4$% ()%&'(+( ,'%)+, λ2 &)1+*+($& 3%+('&0+%4'+ d B -$C'@%($&) 1+*+,),$& 4$% C),$4'(+( ~v7 8%43)%0*) v 0+, <3) ,+ +0*+44'?%
-+;%@0'4+ &) 4$-1)%&) 4$% ,+ *)13,&'?% ),@40*'4+7
*"$+,-./0
+6 D$%;+-$& ,$& )=)& 4$$*()%+($& () 0+, 9$*-+ <3) z +13%0) )% ), &)%0'($ () ,+& 4$**')%0)&2
x )%0*) )% ,+ 1E;'%+ ) y C+B+ ()&() ), +,+-.*) 2 +, 17F), 1*$.,)-+ +%0)*'$*2 &+.)-$& <3) ), 4+-1$ -+;%@0'4$ ;)%)*+($ 1$* 3% +,+-.*) '%/%'0$
1$* ), 43+, 1+&+ 3%+ 4$**')%0) i )&
~B(~r) =µ02π· i
rθ
G&H2 ,+ 93)*:+ <3) ), 4+-1$ ;)%)*+($ 1$* ), +,+-.*) 1 )=)*4) &$.*) 3% ),)-)%0$ ('9)*)%4'+,
() 4$**')%0) (), +,+-.*) 2 )&0E (+($ 1$*
d ~F21 = i2 ~dl2 × ~B1
= i2dz2z ×µ02π· i1
ax
=µ02πa
· i1i2ydz2
=⇒ d ~F21dz2
=µ02πa
· i1i2y
F) ,$ 43+, C)-$& <3) +-.$& +,+-.*)& &) +0*+)% 43+%($ ,+& 4$**')%0)& 0')%)% -'&-$
&)%0'($ B &) *)1),)% 43+%($ 0')%)% &)%0'($& ('9)*)%0)&7
.6 8% )&0) 4+&$ 0)%)-$& <3)
i =dq
dt=
λdz
dt= λ
dz
dt= λv
1$* ,$ <3) i = i1 = i2 = λvD$* 0+%0$2 (), '0@- +%0)*'$*2 ,+ 93)*:+ -+;%@0'4+ 1$* 3%'(+( () ,+*;$ &$.*) ), +,+-.*)
12 ().'($ + ,+ '%0)*+44'?% -+;%@0'4+2 )&
d ~Fm
dz=
µ02πa
· i2y = µ02πa
· (λv)2y
!
"# $%&'(# &)*+,'-+# )# ./0&1/2 3#))#' %2#40/ )# )&5 0& 6#%227 82#40/ ,#) )&5 92& 0&:# #)
)&+,/'; 2- 4/ '&+%&'0# +<1/ 3#+&')/; '&=-2& 1- ,&'+&'# #5%0#4,>#?; &) +#1./ &)*+,'-+/ 0&
%4 #)#1@'& 0& 0&42-0#0 )-4&#) λ 0& +#'A# %4-$/'1& &2
~E(r) =λ
2πǫ0
r
r
B4 &2,& +#2/; )# $%&'(# &)*+,'-+# 2/@'& %4 &)&1&4,/ 0-$&'&4+-#) 0&) #)#1@'& &2
d ~Fe = dq ~E(r)
= λdz ~E(r)
= − λ2
2πǫ0a· ydz
=⇒ d ~Fe
dz= − λ2
2πǫ0a· y
C/' ,#4,/; )# +/40-+-<4 D%& 0&2/2 &2 D%&
d ~Fe
dz+
d ~Fm
dz= ~0
)/ D%& &D%-=#)& # D%&
λ2
2πǫ0a=
µ02πa
· (λv)2 =⇒ v =1√ǫ0µ0
= c
0/40& c &2 )# =&)/+-0#0 0& )# )%(7
E)#'#1&4,& =&1/2 D%& &2,& 4/ &2 %4 '&2%),#0/ +/3&'&4,& 4- .'F+,-+/; .%&2 )# =&)/+-0#0 D%&
4&+&2-,#1/2 .#'# D%& )# -4,&'#++-<4 1#A4*,-+# 2& -A%#)& +/4 )# &)*+,'-+# &2 )# =&)/+-0#0
0& )# )%(; =&)/+-0#0 D%& 4-4AG4 +%&'./ 1#,&'-#) .%&0& #)+#4(#' 9.%&2 4&+&2-,#'-#; 2&AG4
)# '&)#,-=-0#0 &2.&+-#); -4H4-,# &4&'A>#?7 I #0&1F2; &) ,'#,#1-&4,/ D%& 3&1/2 3&+3/ &2
+)F2-+/; 4/ '&)#,-=-2,#7
"/ D%& 2> ./0&1/2 0&2.'&40&' 0& &2,/; &2 D%& )#2 -4,&'#++-/4&2 &)*+,'-+#2 2/4 1%+3/
1F2 $%&',&2 D%& )#2 -4,&'#++-/4&2 1#A4*,-+#2 90& #))> D%& &4 <.,-+#; .#'# ,'#,#' /40#2
&)&+,'/1#A4*,-+#2; 2& 1#4&:& &) +#1./ &)*+,'-+/ #) ,'#,#' +/4 )# /40#?7
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$%&'()*) +%, )&-'*, .'*.+/,* -$* /, .+,/ 0+1) +%, .$**')%2) i 3.$4$ '%('., /, 56+*,78 9%.+)%2*)
)/ .,4-$ 4,6%:2'.$ &$;*) )/ )<) () &'4)2*=, () /, )&-'*,8
*"$+,-./0
>)%)4$& ?+) )/ .,4-$ 4,6%:2'.$@ $*'6'%,($ -$* /, )&-'*, () .$**')%2) Γ@ )&2A (,($ -$*
~B(~r) =µ04π
∮
Γ
i~dl × (~r − ~r′)
|~r − ~r′|3
9% )&2) .,&$ ~r = zz@ ~r′ = Rρ@ ~dl = dsθ = Rdθθ8 B$* 2,%2$@
~B(~r) =µ04π
∮
Γ
i~dl × (~r − ~r′)
|~r − ~r′|3
=µ04π
∫ 2π
0
iRdθθ × (zz −Rρ)
(R2 + z2)3/2
=µ04π
iR
∫ 2π
0
dθ(zρ+Rz)
(R2 + z2)3/2
=µ04π
iR
∫ 2π
0
dθ(zρ)
(R2 + z2)3/2+
µ04π
iR
∫ 2π
0
dθ(Rz)
(R2 + z2)3/2
=µ04π
iR2 2πz
(R2 + z2)3/2
=µ02
iR2z
(R2 + z2)3/2
($%() /, -*'4)*, '%2)6*,/ )& .)*$ -+)& ρ = cos(θ)x+sen(θ)y 1 ,4;,& C+%.'$%)& 2*'6$%$4:2*'D
.,& &$% .)*$ ,/ '%2)6*,*/,& &$;*) +% -)*=$($8
!"
!"#$%&' ()
#$%&'()*) +% ('&,$ () ()%&'(-( () ,-*.- &+/)*0,'-1 σ +%'2$*3)3)%4) ('&4*'5+'(-6 71 ('&,$ &)
/$%) - .'*-* - 8)1$,'(-( -%.+1-* w )% 4$*%$ - &+ )9) () &'3)4*:-6 7%,+)%4*) )1 ,-3/$ 3-.%;4',$
<+) .)%)*- &$5*) &+ )9)6
*"$+,-./0
=1 /$%)*&) - .'*-* )1 ('&,$ ,$% ()%&'(-( () ,-*.- &+/)*0,'-1 σ> &) /*$(+,) +%- ,$**')%4)
&$5*) )1 3'&3$> () ?-,)& () ,$**')%4)& ,'*,+1-*)& ,$%,;%4*',$& @ () $*'.)% )1 $*'.)% ()1 ('&,$6
#$%&'()*-*)3$& 1- ,$**')%4) )% )1 ('&,$ ('8'('(- )% ?-,)& () ,$**')%4)& () )&/'*-& ,$%,;%4*',-&
)% )1 ('&,$> ,-(- +%- () 1-& ,+-1)& .)%)*- +% ,-3/$ 3-.%;4',$6 A) )&4- 2$*3-> +&-%($ )1
/*'%,'/'$ () &+/)*/$&','B% ()1 ,-3/$ 3-.%;4',$ C<+) &) ()&/*)%() () 1- 1'%)-1'(-( () 1-&
),+-,'$%)& () D-EF)11G> )%,$%4*-*)3$& )1 ,-3/$ 4$4-1 /*$(+,'($ /$* )1 ('&,$ &+/)*/$%')%($
1$& ,-3/$& .)%)*-($& /$* 1-& )&/'*-& () ,$**')%4)& &+,)&'8-&6
H)%)3$& <+) 1- ()%&'(-( () ,-*.- 1'%)-1 () 1-& )&/'*-& )&4I (-($ /$* λ = σdr6 J&-%($ )&4$>
4)%)3$& <+)
i =dq
dt=
λds
dt= λ
rdθ
dt= λr
dθ
dt= λrw
($%() i )& 1- ,$**')%4) <+) /-&- /$* ,-(- )&/'*- '%0%'4)&'3-1 @ r &+ *-('$6
A)1 /*$51)3- -%4)*'$*> 4)%)3$& <+) )1 ,-3/$ .)%)*-($ /$* ,-(- )&/'*- )&
d ~B(~r) =µ02
ir2z
(r2 + z2)3/2=
µ02
λwr3z
(r2 + z2)3/2=1
2µ0σw
r3drz
(r2 + z2)3/2
K$* 4-%4$> )1 ,-3/$ 3-.%;4',$ .)%)*-($ /$* )1 ('&,$ -1 .'*-* ,$% *-/'()L -%.+1-* w )&
~B(~r) =1
2µ0σwz
∫ R
0
r3dr
(r2 + z2)3/2u2 = r2 + z2 → udu = rdr
=1
2µ0σwz
∫
√R2+z2
|z|
(u2 − z2)udu
u3
=1
2µ0σwz
∫
√R2+z2
|z|1− z2
u2
=1
2µ0σwz
(
√
R2 + z2 − |z|+ z2
u
∣
∣
∣
∣
√R2+z2
|z|
)
=1
2µ0σwz
(
√
R2 + z2 +z2√
R2 + z2− 2|z|
)
=1
2µ0σwz
(
2z2 +R2
√R2 + z2
− 2|z|)
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$%&'()*) +% ,-.%$ ,$* )- /+.- 0+1) /$**')%2)3 /$% ()%&'(.( () /$**')%2) 4 &+,)*5/'.-6
~J = J0y78%/+)%2*) )- /.9,$ 9.:%;2'/$ )% 2$($ )- )&,./'$7
*"$+,-./0
<$* -. &'9)2*=. ()- ,*$>-)9.3 2)%)9$& ?+)
~B(x, y, z) = ~B(z)7 @$9)9$& /$9$ ,+%2$ () $>A
&)*B./'C% ~r = zz 1
~r′ = xx+ yy )- B)/2$* ?+) ()&/*'>. )- ,-.%$7 @)%)9$& ?+) -. ()%&'(.( ()
/$**')%2) &+,)*5/'.- )&2D (.(. ,$*
~J = J0y3 ,$* -$ ?+) -. /$**')%2) &) 2*.%&9'2) )% )- &)%2'($
() y7 E&=3 2)%)9$& ?+) di = ~J · ydS = J0dS7 <$* 2.%2$3 )- /.9,$ 9.:%;2'/$ :)%)*.($ ,$* +%
)-)9)%2*$ ('F)*)%/'.- () /$**')%2) )&
d ~B(~r) =µ04π
diy × (zz − xx− yy)
(x2 + y2 + z2)3/2=
µ04π
J0dS(xz + zx)
(x2 + y2 + z2)3/2
/$% dS = dxdy7 E&=3 )- /.9,$ 9.:%;2'/$ :)%)*.($ ,$* )- ,-.%$ )&
~B(~r) =µ04π
J0
∫ ∞
−∞
∫ ∞
−∞
(xz + zx)dxdy
(x2 + y2 + z2)3/2
=µ04π
J0
∫ ∞
−∞
∫ ∞
−∞
xzdxdy
(x2 + y2 + z2)3/2+
µ04π
J0
∫ ∞
−∞
∫ ∞
−∞
zxdxdy
(x2 + y2 + z2)3/2
=µ04π
J0zx
∫ ∞
−∞
∫ ∞
−∞
dxdy
(x2 + y2 + z2)3/2
=µ04π
J0zx
∫ ∞
0
∫ 2π
0
rdrdθ
(r2 + z2)3/2
=µ02
J0zx
∫ ∞
0
rdr
(r2 + z2)3/2
=1
2µ0J0
z
|z| x
($%() -. ,*'9)*. '%2):*.- )& /)*$ ,+)& )- '%2):*.%($ )& '9,.* 4.- '%2):*.* )% G6 1 -$&
-=9'2)& () '%2):*./'C% &$% &'9;2*'/$& *)&,)/2$ 07 H$2) -$ &'9'-.* ?+) )& )- 9C(+-$ /$% )- 9CA
(+-$ ()- /.9,$ )-;/2*'/$ :)%)*.($ ,$* +% ,-.%$ () ()%&'(.( &+,)*5/'.- /$%&2.%2) σ3 )- /+.- )&~E(~r) = σ
2ǫ0z|z| z7
8- I)/I$ ?+) )- /.9,$ 9.:%;2'/$ )&2+B')*. .,+%2.%($ I./'. x )*. () )&,)*.*&)7 <.*. B)* )&2$
/$%&'()*) )- ,-.%$ /$9$ +%. &+/)&'C% '%5%'2. () .-.9>*)& () /$**')%2) )% )- &)%2'($ y7 J'&+.-A
'K.%($ )- /.9,$ 9.:%;2'/$ ?+) :)%)*. /.(. +%$ () )&2$& .-.9>*)& '%5%'2)&'9.-)& &+/)&'B$&3 1
.,-'/.%($ &+,)*,$&'/'C%3 )& /-.*$ ?+) )- /.9,$ 9.:%;2'/$ ()>) )&2.* )% )- &)%2'($ x7 LM$:*.&B)*-$N7
!!
!"#$%&' ()
!"#$ %&' ()*)!%#&' (&!%+("&#$' (&!(,!"#)(&'- .+/ *0#1&'- %$ #0%)&' a / c- '$ **$!0 %$ .0"$#)0*%$ (&!%+(")2)%0%$' g1 / g2- / %$ 3$#.)")2)0%$' ǫ1 / ǫ2- (&.& .+$'"#0 *0 41+#05'* !(&!"#0# *0 #$')'"$!()0 3&# +!)%0% %$ *0#1& $!"#$ $* !6(*$& / $* (&!%+("&# $7"$#!&5
#* 8) '$ (&!$("0 +!0 %)9$#$!()0 %$ 3&"$!()0* V0 $!"#$ $* !6(*$& / $* (&!%+("&# $7"$#!&- (0*(+*$*0 %$!')%0% %$ (&##)$!"$
~J / *0 (&##)$!"$ I :+$ ()#(+*05+* ;0*(+*$ *0 %$!')%0% %$ (0#10 *)<#$ :+$ '$ 0(+.+*0 $! *0 )!"$#90= %$ 0.<&' .0"$#)0*$' >)5$ $!
r = b? $! (&!%)()&!$' 3$#.0!$!"$'- 0* $'"0# *0' 3*0(0' (&!$("0%0' 0 +!0 %)9$#$!()0 %$ 3&"$!()0*V05
,"$-+./0 '* @0#0 $!(&!"#0# *0 #$')'"$!()0 '$ %$<$ 3#&($%$# %$ )1+0* 9&#.0 :+$ $! $* 3#&<*$.0
A5 B$!$.&' :+$ *0 (&##)$!"$ '$ (0*(+*0 (&.&C
I =
∫
S
~J · ndS
* 2$("&# %$!')%0% %$ (&##)$!"$ $' #0%)0*C
~J(r) = J(r)r5 D')C
I = J2πrL
;&! $'"&- 3&%$.&' $'(#)<)# $* 2$("&# %$!')%0% %$ (&##)$!"$ $! 9+!()E! %$ *0 (&##)$!"$C
~J =I
2πrLr
;&! $'"$ #$'+*"0%& 3&%$.&' (0*(+*0# *&' (0.3& $*,("#)(&' $! 0.<&' .0"$#)0*$' +")*)=0!%& *0
(&!%+(")2)%0% %$ (0%0 +!&C
~E1 =I
2πrLg1
~E2 =I
2πrLg2
DF�- (0*(+*0.&' $* 3&"$!()0* $!"#$ r = a / r = cC
∆V =
∫ c
a
~E · d~r
! $'"0 #$1)E! $* (0.3& $*,("#)(& 20#)0 $! 0.<&' (&!%+("&#$' / 3&# *& "0!"& '$30#0.&' $'"0
)!"$1#0* %$ *G!$0 %$ *0 ')1+)$!"$ .0!$#0C
!" !"#$%&' () '**+,-$, ,&, $*+ !
∆V =
∫ b
a
~E1 · d~r +∫ c
b
~E2 · d~r =I
2πLg1ln(
b
a) +
I
2πLg2ln(
c
b)
#$%&%'()*+ r = ∆VI +,$-)-.+/0
R =1
2πLg1ln(
b
a) +
1
2πLg2ln(
c
b)
1+.+ 2+*-.+/ (23-4%(3 -) &( -523-/%6) *- 3-/%/$-)4%( /+&+ %)$-37%-)-) &+/ *($+/ 3-8-3-)$-/ (
&(/ 23+2%-*(*-/ *-& .($-3%(& 9 &( :-+.-$3;( *-& /%/$-.(<
! =% /- 4+)-4$( >)( *%8-3-)4%( *- 2+$-)4%(& δV = V0? /- $%-)- @>- &( 4+33%-)$- 4>.2&- 4+)
I = RV0< A--.2&('()*+ -) &( -523-/%6) *- 3-/%/$-)4%( -)4+)$3(*( ()$-3%+3.-)$-0
I =V02πL
ln( ba)
g1+
ln( cb)
g2
1+) -/$( 4+33%-)$- -/ *%3-4$+ -)4+)$3(3 -& 7(&+3 *- &( *-)/%*(* *- 4+33%-)$-<
"! B(3( 4(&4>&(3 &( *-)/%*(* *- 4(3:( &%,3- @>- /- (4>.>&( -) &( %)$-38(' *- (.,+/ .($-3%(&-/
>$%&%'(.+/ &( &-9 *- :(>// 2(3( *%-&C4$3%4+/< B(3( -/$+ 4+)/%*-3(.+/ 4+.+ />2-3D4%- :(>//%()(
>) .()$+ 4%&;)*3%4+ 4+) />/ 3-/2-4$%7(/ $(2(/ *- E3-( .>9 2-@>-F(
∫
S
~D · ndS = qenc
B+3 &( /%.-$3;( 2+*-.+/ *-/4+.2+)-3 -/$( %)$-:3(& -)0
~D2 · (An) + ~D1 · (−An) = σA
( ~D2 − ~D1) · n = σ
1+.+ 7%.+/ -) -& 3-/>.-)? -& 7-4$+3 *-/2&('(.%-)$+ 2(3( >) *%-&C4$3%4+ /- 4(&4>&( /%.2&-.-)$-
4+.+ &( 2-3.%$%7%*(* 2+3 -& 4(.2+ -&C4$3%4+0
~D1 = ǫ1 ~E1
~D2 = ǫ2 ~E2
1+) &+/ 7(&+3-/ *- 4(.2+ -&C4$3%4+ -)4+)$3(*+/ -) &( 2(3$- (G $-)-.+/ @>- &+/ 7-4$+3-/ *-H
/2&('(.%-)$+ -&C4$3%4+ /+)0
!"
~D1 =ǫ1I
2πrLg1r
~D2 =ǫ2I
2πrLg2r
#$ %&'()* *+,-+$ &. -/0+$ +$ %&'()* 1)*2+$ + $+ .03&*4'-&5 r = n6 7& &.(+ 2+1&*+ 3),&2).
'+$'0$+* $+ ,&1.-,+, ,& '+*/+ $-8*&5
σ =I
2πbL(ǫ2g2− ǫ1
g1)
9)1 &$ %+$)* ,& ')**-&1(& ,& $+ 3+*(& 8: *&.0$(+ 41+$2&1(& ;0& $+ ,&1.-,+, ,& '+*/+ $-8*& &.5
σ =V0
b(ln( b
a)
g1+
ln( cb)
g2)(ǫ2g2− ǫ1
g1)
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$%&'($) *+, '&-(-%(.(/$/ 0,(1+'2& ρ -& 1+'2$ *+2+ 0,$ *03$ /& )$ 2$,&'$ (,/(*$/$ &, )$
450'$6 #0&-%'& 70& )$ '&-(-%&,*($ &,%'& )$- *$'$- A 8 B /& &-%$ *03$ &-9
R = ρL
w(y2 − y1)ln(
y2y1)
*"$+,-./
:$ '&-(-%&,*($ -& /&4,& *+2+ R = ρℓA 6 :+ 70& ;$'&2+- &- /(.(/(' )$ *03$ &, <&70&3+- =)+70&-
/& $,*;+ dx 8 $)%0'$ H(x) *+2+ 20&-%'$ )$ 450'$>.(-%$ )$%&'$)?9
@+/&2+- *+,-(/&'$' &-%+- =)+70&- *+2+ <&70&3+- <$'$)&)&<A<&/+- /& B'&$ A(x) = wH(x)C)$'5+ dx 8 '&-(-%(.(/$/ ρ6 D&=&2+- &,*+,%'$' 0,$ &E<'&-(F, <$'$ H(x)6 @$'$ &-%$=)&*&' )$-
'&)$*(+,&- 5&+2G%'(*$- *+,-(/&'$'& )$ -(50(&,%& 450'$9
!"
#$%$&'( )*$ ($ +*&,-./01
y2 − y1L
=h(x)
x
h(x) =(y2 − y1)x
L
2*$3'4 H(x) = h(x)+y1 =(y2−y1)x
L +y15 6'% $(7$ /$(*-789' :$&'( )*$ A(x) = w( (y2−y1)xL +y1)
;$ $(78 &8%$/8 -8 ,$)*$<8 /$(.(7$%+.8 )*$ 8,'/78 ($/01
dR =ρdx
A(x)=
ρdx
w( (y2−y1)xL + y1)
=ρL
w
dx
(y2 − y1)x+ y1L
#$%$&'( )*$ $(78( /$(.(7$%+.8( $(70% $% ($/.$4 9$ $(78 &8%$/8 7$%$&'( )*$ (*&8/ -.%$8-&$%7$
7'98( -8( /$(.(7$%+.8( 8,'/7898( ,'/ +898 =-')*$ > '=7$%$&'( -8 /$(.(7$%+.8 7'78- 9$ -8 +*<85
?8/8 $(7'4 .%7$3/8&'( 9$(9$ x = 0 8 x = L1
R =ρL
w
∫ L
0
dx
(y2 − y1)x+ y1L
@*-7.,-.+8&'( ,'/ *% *%' +'%:$%.$%7$ ,8/8 ,'9$/ .%7$3/8/ 9./$+78&$%7$1
R =(y2 − y1)
(y2 − y1)
ρL
w
∫ L
0
dx
(y2 − y1)x+ y1L=
ρL
w(y2 − y1)
∫ L
0
(y2 − y1)dx
(y2 − y1)x+ y1L=
ρL
w(y2 − y1)ln(
(y2 − y1)L+ y1L
(y2 − y1) · 0 + y1L)
A.&,-.B+8%9' $(78 $C,/$(.D% ($ '=7.$%$ B%8-&$%7$
R = ρL
w(y2 − y1)ln(
y2y1)
E*$ $( -' )*$ )*$/F8&'( 9$&'(7/8/5
!" !"#$%&' () '**+,-$, ,&, $*+ !
!"#$%&' ()
#$%&'() *$+ +, -&'.+'/).&0 .+ ,) 1($0) +/2) 3'3-3),4+'2+ .+/-)0().&5'& 6)7 -&003+'2+ -$)'.&
+, 3'2+00$%2&0 S +/2) )83+02&9: ;'-$+'20+ ,)/ +-$)-3&'+/ *$+ 4&.+,)' ,) -)0() .+, -&'.+'/).&0
+' <$'-3&' .+, 23+4%& -$)'.& /+ -3+00) +, 3'2+00$%2&0: =>&4& /+ -&4%&02) ,) -&003+'2+ +'
<$'-3&' .+, 23+4%&?:
!
!"#$%&' ()
"#$%&#'(& )* +&(,-,* ,& &#&(.-* +/( $*)/( ,& 0/%)& ,&) $-($%-'/ ,& )* 1.%(*2 3&4%&5'(& 6%&
&5'& #%4&(/ #/ ,&+&#,& ,& )* (&5-5'&#$-* R2 7#-$-*)4&#'& &) $/#,*,/( ,& $*+*$-,*, C1 '-&#&
%#* $*(.* Q0 8 &) 5&.%#,/ $/#,*,/( &5'* ,&5$*(.*,/2