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Transcript of Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia...
![Page 1: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/1.jpg)
Potencial elétrico e capacitores
Baseado no 8.02T MIT-opencourse
1
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!g = !GM
r2r
!Fg = m!g
!E = keq
r2r
!Fe = q !E
Gravidade x eletricidade
Massa M Carga(+/-q)
Campos
Forças
2
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Energia potencial x potencial
Gravidade
3
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Gravidade: força e trabalho
!Fg = !GMm
r2r
Força exercida em m devido a M
4
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Gravidade: força e trabalho
!Fg = !GMm
r2r
Força exercida em m devido a M
Wg =! B
A
!Fg · d!s
Trabalho exercido pela gravidade ao mover m de A a B
integral de trajetória
4
![Page 6: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/6.jpg)
12P04 !
Work Done by Earth’s Gravity"#$%&'#()&*+&,$-./0+&1#./(,&1&2$#1&A 0#&B:
ggW d! "#! "!
!
$ %2ˆ ˆˆ
B
A
GMm
rdr rd&'( )! " *+ ,
- .# # # !
1 1
B A
GMmr r
( )! '+ ,
- .
2
B
A
r
r
GMmdr
r! '#
B
A
r
r
GMm
r
! / 01 23 4
"3-0&/4&03)&4/,(&1#./(,&2$#1&$5 0#&$67
Wg =! B
A
!Fg · d!s
= =! B
A
"!G
Mm
r2r
#· (drr + rd"")
=! B
A!G
Mm
r2dr =
$G
Mm
r
%rB
rA
= GMm
"1rB! 1
rA
#
Trabalho realizado pela gravidade terrestre
Trabalho realizado pela gravidade ao mover m de A a B
Trabalho depende apenas dos pontos A e B!
5
![Page 7: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/7.jpg)
Forças conservativas
Mecânica: !EcinWA!B =
!"#$%&'(%& )&*+#&,-./&%-(#&01&'(*#.('$&'3#(*&456+&'4&1-57&8+#1&4"9:$1&%"!!#.&01&'&
(#3'*";#&4"3(<& 7&&
#2*W
#2*gW W! "
&
=#'.& >'.*+?4& 45.!'6#)& *+#& 3.';"*'*"-('$& !"#$%& !!
&"4& '::.-2"9'*#$1& 6-(4*'(*)& ,"*+& '&
9'3("*5%#& )& ,+#.#& &"4& *+#& .'%"54& -!& >'.*+7& 8+#&,-./& %-(#& 01&
3.';"*1&"(&9-;"(3&'(&-0@#6*&!.-9&+#"3+*&
A AB C7D9BEg GM r! # 4 Er
Ay &*-& &EF"35.#&G7H7AI&"4&By
&
& 6-4 6-4 E IB
A
B B y
g g B AA A y
W d mg ds mg ds mg dy mg y y$ %! & ! ! " ! " ! " "' ' ' '" #!
!
& EG7H7JI&
&&
"$!%&'()*+*,&K-;"(3&'&9'44&m&!.-9&A&*-&B7&
&
&
8+#& .#45$*& '3'"(& "4& "(%#:#(%#(*& -!& *+#& :'*+)& '(%& "4& -($1& '& !5(6*"-(& -!& *+#& 6+'(3#& "(&
;#.*"6'$&+#"3+*& 7&B Ay y"
& & &L(&*+#'9:$#4&'0-;#)&"!&*+#&:'*+&!-.94&'&6$-4#%&$--:)&4-&*+'*&*+#&-0@#6*&9-;#4&'.-5(%&
'(%&*+#(&.#*5.(4&*-&,+#.#&"*&4*'.*4&-!!)&*+#&(#*&,-./&%-(#&01&*+#&3.';"*'*"-('$&!"#$%&,-5$%&
0#&M#.-)&'(%&,#&4'1&*+'*&*+#&3.';"*'*"-('$&!-.6#&"4&6-(4#.;'*";#7&K-.#&3#(#.'$$1)&'&!-.6#&"!
&
"4&4'"%&*-&0#&conservative&"!&"*4&$"(#&"(*#3.'$&'.-5(%&'&6$-4#%&$--:&;'("4+#4<&
&
& Nd& !' " #! !
" & EG7H7OI&
&
P+#(&%#'$"(3&,"*+&'&6-(4#.;'*";#&!-.6#)&"*&"4&-!*#(&6-(;#("#(*&*-&"(*.-%56#&*+#&6-(6#:*&-!&
:-*#(*"'$&#(#.31&U7&8+#&6+'(3#&"(&:-*#(*"'$&#(#.31&'44-6"'*#%&,"*+&'&6-(4#.;'*";#&!-.6#&
&'6*"(3&-(&'(&-0@#6*&'4&"*&9-;#4&!.-9&A&*-&B&"4&%#!"(#%&'4<&""!
&
&B
B AA
U U U d W( ! " ! " & ! "' " #!
!
& EG7H7QI&
&
,+#.#&W &"4&*+#&,-./&%-(#&01&*+#&!-.6#&-(&*+#&-0@#6*7&&L(&*+#&6'4#&-!&3.';"*1)&gW W! &'(%&
!.-9&>R7&EG7H7GI)&*+#&:-*#(*"'$&#(#.31&6'(&0#&,."**#(&'4&
&
&Ng
GMmU
rU! " ) & EG7H7SI&
& A
Forças conservativas:
6
![Page 8: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/8.jpg)
Forças conservativas
Mecânica: !EcinWA!B =
!"#$%&'(%& )&*+#&,-./&%-(#&01&'(*#.('$&'3#(*&456+&'4&1-57&8+#1&4"9:$1&%"!!#.&01&'&
(#3'*";#&4"3(<& 7&&
#2*W
#2*gW W! "
&
=#'.& >'.*+?4& 45.!'6#)& *+#& 3.';"*'*"-('$& !"#$%& !!
&"4& '::.-2"9'*#$1& 6-(4*'(*)& ,"*+& '&
9'3("*5%#& )& ,+#.#& &"4& *+#& .'%"54& -!& >'.*+7& 8+#&,-./& %-(#& 01&
3.';"*1&"(&9-;"(3&'(&-0@#6*&!.-9&+#"3+*&
A AB C7D9BEg GM r! # 4 Er
Ay &*-& &EF"35.#&G7H7AI&"4&By
&
& 6-4 6-4 E IB
A
B B y
g g B AA A y
W d mg ds mg ds mg dy mg y y$ %! & ! ! " ! " ! " "' ' ' '" #!
!
& EG7H7JI&
&&
"$!%&'()*+*,&K-;"(3&'&9'44&m&!.-9&A&*-&B7&
&
&
8+#& .#45$*& '3'"(& "4& "(%#:#(%#(*& -!& *+#& :'*+)& '(%& "4& -($1& '& !5(6*"-(& -!& *+#& 6+'(3#& "(&
;#.*"6'$&+#"3+*& 7&B Ay y"
& & &L(&*+#'9:$#4&'0-;#)&"!&*+#&:'*+&!-.94&'&6$-4#%&$--:)&4-&*+'*&*+#&-0@#6*&9-;#4&'.-5(%&
'(%&*+#(&.#*5.(4&*-&,+#.#&"*&4*'.*4&-!!)&*+#&(#*&,-./&%-(#&01&*+#&3.';"*'*"-('$&!"#$%&,-5$%&
0#&M#.-)&'(%&,#&4'1&*+'*&*+#&3.';"*'*"-('$&!-.6#&"4&6-(4#.;'*";#7&K-.#&3#(#.'$$1)&'&!-.6#&"!
&
"4&4'"%&*-&0#&conservative&"!&"*4&$"(#&"(*#3.'$&'.-5(%&'&6$-4#%&$--:&;'("4+#4<&
&
& Nd& !' " #! !
" & EG7H7OI&
&
P+#(&%#'$"(3&,"*+&'&6-(4#.;'*";#&!-.6#)&"*&"4&-!*#(&6-(;#("#(*&*-&"(*.-%56#&*+#&6-(6#:*&-!&
:-*#(*"'$&#(#.31&U7&8+#&6+'(3#&"(&:-*#(*"'$&#(#.31&'44-6"'*#%&,"*+&'&6-(4#.;'*";#&!-.6#&
&'6*"(3&-(&'(&-0@#6*&'4&"*&9-;#4&!.-9&A&*-&B&"4&%#!"(#%&'4<&""!
&
&B
B AA
U U U d W( ! " ! " & ! "' " #!
!
& EG7H7QI&
&
,+#.#&W &"4&*+#&,-./&%-(#&01&*+#&!-.6#&-(&*+#&-0@#6*7&&L(&*+#&6'4#&-!&3.';"*1)&gW W! &'(%&
!.-9&>R7&EG7H7GI)&*+#&:-*#(*"'$&#(#.31&6'(&0#&,."**#(&'4&
&
&Ng
GMmU
rU! " ) & EG7H7SI&
& A
Forças conservativas:
6
![Page 9: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/9.jpg)
Forças conservativas
Mecânica: !EcinWA!B =
!"#$%&'(%& )&*+#&,-./&%-(#&01&'(*#.('$&'3#(*&456+&'4&1-57&8+#1&4"9:$1&%"!!#.&01&'&
(#3'*";#&4"3(<& 7&&
#2*W
#2*gW W! "
&
=#'.& >'.*+?4& 45.!'6#)& *+#& 3.';"*'*"-('$& !"#$%& !!
&"4& '::.-2"9'*#$1& 6-(4*'(*)& ,"*+& '&
9'3("*5%#& )& ,+#.#& &"4& *+#& .'%"54& -!& >'.*+7& 8+#&,-./& %-(#& 01&
3.';"*1&"(&9-;"(3&'(&-0@#6*&!.-9&+#"3+*&
A AB C7D9BEg GM r! # 4 Er
Ay &*-& &EF"35.#&G7H7AI&"4&By
&
& 6-4 6-4 E IB
A
B B y
g g B AA A y
W d mg ds mg ds mg dy mg y y$ %! & ! ! " ! " ! " "' ' ' '" #!
!
& EG7H7JI&
&&
"$!%&'()*+*,&K-;"(3&'&9'44&m&!.-9&A&*-&B7&
&
&
8+#& .#45$*& '3'"(& "4& "(%#:#(%#(*& -!& *+#& :'*+)& '(%& "4& -($1& '& !5(6*"-(& -!& *+#& 6+'(3#& "(&
;#.*"6'$&+#"3+*& 7&B Ay y"
& & &L(&*+#'9:$#4&'0-;#)&"!&*+#&:'*+&!-.94&'&6$-4#%&$--:)&4-&*+'*&*+#&-0@#6*&9-;#4&'.-5(%&
'(%&*+#(&.#*5.(4&*-&,+#.#&"*&4*'.*4&-!!)&*+#&(#*&,-./&%-(#&01&*+#&3.';"*'*"-('$&!"#$%&,-5$%&
0#&M#.-)&'(%&,#&4'1&*+'*&*+#&3.';"*'*"-('$&!-.6#&"4&6-(4#.;'*";#7&K-.#&3#(#.'$$1)&'&!-.6#&"!
&
"4&4'"%&*-&0#&conservative&"!&"*4&$"(#&"(*#3.'$&'.-5(%&'&6$-4#%&$--:&;'("4+#4<&
&
& Nd& !' " #! !
" & EG7H7OI&
&
P+#(&%#'$"(3&,"*+&'&6-(4#.;'*";#&!-.6#)&"*&"4&-!*#(&6-(;#("#(*&*-&"(*.-%56#&*+#&6-(6#:*&-!&
:-*#(*"'$&#(#.31&U7&8+#&6+'(3#&"(&:-*#(*"'$&#(#.31&'44-6"'*#%&,"*+&'&6-(4#.;'*";#&!-.6#&
&'6*"(3&-(&'(&-0@#6*&'4&"*&9-;#4&!.-9&A&*-&B&"4&%#!"(#%&'4<&""!
&
&B
B AA
U U U d W( ! " ! " & ! "' " #!
!
& EG7H7QI&
&
,+#.#&W &"4&*+#&,-./&%-(#&01&*+#&!-.6#&-(&*+#&-0@#6*7&&L(&*+#&6'4#&-!&3.';"*1)&gW W! &'(%&
!.-9&>R7&EG7H7GI)&*+#&:-*#(*"'$&#(#.31&6'(&0#&,."**#(&'4&
&
&Ng
GMmU
rU! " ) & EG7H7SI&
& A
!"#$%&'(%& )&*+#&,-./&%-(#&01&'(*#.('$&'3#(*&456+&'4&1-57&8+#1&4"9:$1&%"!!#.&01&'&
(#3'*";#&4"3(<& 7&&
#2*W
#2*gW W! "
&
=#'.& >'.*+?4& 45.!'6#)& *+#& 3.';"*'*"-('$& !"#$%& !!
&"4& '::.-2"9'*#$1& 6-(4*'(*)& ,"*+& '&
9'3("*5%#& )& ,+#.#& &"4& *+#& .'%"54& -!& >'.*+7& 8+#&,-./& %-(#& 01&
3.';"*1&"(&9-;"(3&'(&-0@#6*&!.-9&+#"3+*&
A AB C7D9BEg GM r! # 4 Er
Ay &*-& &EF"35.#&G7H7AI&"4&By
&
& 6-4 6-4 E IB
A
B B y
g g B AA A y
W d mg ds mg ds mg dy mg y y$ %! & ! ! " ! " ! " "' ' ' '" #!
!
& EG7H7JI&
&&
"$!%&'()*+*,&K-;"(3&'&9'44&m&!.-9&A&*-&B7&
&
&
8+#& .#45$*& '3'"(& "4& "(%#:#(%#(*& -!& *+#& :'*+)& '(%& "4& -($1& '& !5(6*"-(& -!& *+#& 6+'(3#& "(&
;#.*"6'$&+#"3+*& 7&B Ay y"
& & &L(&*+#'9:$#4&'0-;#)&"!&*+#&:'*+&!-.94&'&6$-4#%&$--:)&4-&*+'*&*+#&-0@#6*&9-;#4&'.-5(%&
'(%&*+#(&.#*5.(4&*-&,+#.#&"*&4*'.*4&-!!)&*+#&(#*&,-./&%-(#&01&*+#&3.';"*'*"-('$&!"#$%&,-5$%&
0#&M#.-)&'(%&,#&4'1&*+'*&*+#&3.';"*'*"-('$&!-.6#&"4&6-(4#.;'*";#7&K-.#&3#(#.'$$1)&'&!-.6#&"!
&
"4&4'"%&*-&0#&conservative&"!&"*4&$"(#&"(*#3.'$&'.-5(%&'&6$-4#%&$--:&;'("4+#4<&
&
& Nd& !' " #! !
" & EG7H7OI&
&
P+#(&%#'$"(3&,"*+&'&6-(4#.;'*";#&!-.6#)&"*&"4&-!*#(&6-(;#("#(*&*-&"(*.-%56#&*+#&6-(6#:*&-!&
:-*#(*"'$&#(#.31&U7&8+#&6+'(3#&"(&:-*#(*"'$&#(#.31&'44-6"'*#%&,"*+&'&6-(4#.;'*";#&!-.6#&
&'6*"(3&-(&'(&-0@#6*&'4&"*&9-;#4&!.-9&A&*-&B&"4&%#!"(#%&'4<&""!
&
&B
B AA
U U U d W( ! " ! " & ! "' " #!
!
& EG7H7QI&
&
,+#.#&W &"4&*+#&,-./&%-(#&01&*+#&!-.6#&-(&*+#&-0@#6*7&&L(&*+#&6'4#&-!&3.';"*1)&gW W! &'(%&
!.-9&>R7&EG7H7GI)&*+#&:-*#(*"'$&#(#.31&6'(&0#&,."**#(&'4&
&
&Ng
GMmU
rU! " ) & EG7H7SI&
& A
Forças conservativas:
6
![Page 10: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/10.jpg)
!Ug = UB ! UA = !! B
A
!Fg · d!s = !Wg = Wext
Energia potencial x potencial
7
![Page 11: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/11.jpg)
!Ug = UB ! UA = !! B
A
!Fg · d!s = !Wg = Wext
!Fg = !GMm
r2r " Ug = G
Mm
r+ U0
Energia potencial x potencial
7
![Page 12: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/12.jpg)
!Ug = UB ! UA = !! B
A
!Fg · d!s = !Wg = Wext
!Fg = !GMm
r2r " Ug = G
Mm
r+ U0
Energia potencial x potencial
U0: constante que depende do pto de referência
Apenas tem significado físico
!Ug ! !Vg
7
![Page 13: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/13.jpg)
!Ug = UB ! UA = !! B
A
!Fg · d!s = !Wg = Wext
!Fg = !GMm
r2r " Ug = G
Mm
r+ U0
!Vg =!Ug
m= !
! B
A(!Fg/m) · d!s = !
! B
A!g · d!s
Energia potencial x potencial
U0: constante que depende do pto de referência
Apenas tem significado físico
!Ug ! !Vg
Definição da diferença de potencial gravitacional
7
![Page 14: Potencial elétrico e capacitores - if.ufrj.brtgrappoport/aulas/aula-magna3.pdf · Energia potencial x potencial!g á d !s U0: constante que depende do pto de referência Apenas tem](https://reader035.fdocument.pub/reader035/viewer/2022062402/5c4cec4c93f3c34c550ac41d/html5/thumbnails/14.jpg)
!Ug = UB ! UA = !! B
A
!Fg · d!s = !Wg = Wext
!Fg = !GMm
r2r " Ug = G
Mm
r+ U0
!Vg =!Ug
m= !
! B
A(!Fg/m) · d!s = !
! B
A!g · d!s
Energia potencial x potencial
U0: constante que depende do pto de referência
Apenas tem significado físico
!Ug ! !Vg
Definição da diferença de potencial gravitacional
!Fg ! !gCampoForça
!Ug ! !VgPotencialEnergia
7
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Potencial gravitacional
Potencial de planeta +sol
8
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Gravidade x eletricidade
!E = keq
r2r
!Fe = q !E
Carga(+/-q)Massa M
!g = !GM
r2r
!Fg = m!g
!Ug = !! B
A
!Fg · d!s
!Vg = !! B
A!g · d!s
Ambas as forças são conservativas, então:
!U = !! B
A
!Fe · d!s
!V = !! B
A
!E · d!s
9
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!V = !! B
A
!E · d!s
Potencial e energia
Unidades: Joules/Coulomb
=Volts
10
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!V = !! B
A
!E · d!s
Potencial e energia
Unidades: Joules/Coulomb
=Volts
Wext = !U = UB ! UA
= q!VJoules
Trabalho realizado pela gravidade ao mover m de A a B:
10
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Potencial
V (!r) = V0 + !V = V0 !! B
A
!E · d!s
Cargas geram potenciais
11
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Potencial
V (!r) = V0 + !V = V0 !! B
A
!E · d!s
Cargas geram potenciais
28P04 !
Potential Landscape
Positive Charge
Negative Charge
q positiva
q negativa
11
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Potencial
V (!r) = V0 + !V = V0 !! B
A
!E · d!s
Cargas geram potenciais
U(!r) = qV (!r)Cargas sentem potenciais
28P04 !
Potential Landscape
Positive Charge
Negative Charge
q positiva
q negativa
11
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26P04 !
Potential Created by Pt Charge
!!"# ˆˆ !drdr "#!
B
B AA
V V V d$ # % # % &' $ "!
!
2
ˆ
rkQ
!$ #!
2 2
ˆB B
A A
drkQ d kQr r
# % & # %' '!
"!
1 1
B A
kQr r
( )# %* +
, -
"#$%&V '&(&#)&r '&!*
r
kQrV #)(ChargePoint
!V = VB ! VA = !! B
A
!E · d!s
= !! B
A
"k
Q
r2r
#· d!s = !
! B
Ak
Q
r2dr
=$k
Q
r
%rB
rA
= kQ
"1rB! 1
rA
#
Potencial criado por uma carga pontual
12
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26P04 !
Potential Created by Pt Charge
!!"# ˆˆ !drdr "#!
B
B AA
V V V d$ # % # % &' $ "!
!
2
ˆ
rkQ
!$ #!
2 2
ˆB B
A A
drkQ d kQr r
# % & # %' '!
"!
1 1
B A
kQr r
( )# %* +
, -
"#$%&V '&(&#)&r '&!*
r
kQrV #)(ChargePoint
!V = VB ! VA = !! B
A
!E · d!s
= !! B
A
"k
Q
r2r
#· d!s = !
! B
Ak
Q
r2dr
=$k
Q
r
%rB
rA
= kQ
"1rB! 1
rA
#
Vcarga pontual(r) = kQ
r
V (r =!) = 0
Potencial criado por uma carga pontual
12
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Potencial: princípio da superposição
Soma direta.Potencial é um
escalar!
Potencial devido a um conjunto de cargas:
13
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Potencial: princípio da superposição
Soma direta.Potencial é um
escalar!
Potencial devido a um conjunto de cargas:
Potencial devido a uma distribuição contínua de cargas:
densidadelinear de carga
densidadesuperficial de carga
densidadevolumétrica de carga
13
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Calculando E a partir de V
30P04 !
Deriving E from V
ˆx! " !! "!
"#$#%&'(')*'#+$%&,!&'(')*
B
A
V d! " # $%# !!
!
( , , )
( , , )
x x y z
x y z
V d
&!
! " # $% # !!
!
' # $!# !!
! ˆ( ) xx E x" # $ ! " # !# "!
x
V VE
x x
! (' # ) #
! (
Ex = Rate of change in V
with y and z held constant
!V = !! B
A
!E · d!s
A = (x, y, z), B = (x + !x, y, z)!!s = !xı
14
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!V = !! (x+!x,y,z)
(x,y,z)
!E · d!s " !E · !!s = ! !E · (!xı) = !Ex!x
Calculando E a partir de V
30P04 !
Deriving E from V
ˆx! " !! "!
"#$#%&'(')*'#+$%&,!&'(')*
B
A
V d! " # $%# !!
!
( , , )
( , , )
x x y z
x y z
V d
&!
! " # $% # !!
!
' # $!# !!
! ˆ( ) xx E x" # $ ! " # !# "!
x
V VE
x x
! (' # ) #
! (
Ex = Rate of change in V
with y and z held constant
!V = !! B
A
!E · d!s
A = (x, y, z), B = (x + !x, y, z)!!s = !xı
14
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!V = !! (x+!x,y,z)
(x,y,z)
!E · d!s " !E · !!s = ! !E · (!xı) = !Ex!x
Ex ! "!V
!x# "!V
!x
Calculando E a partir de V
30P04 !
Deriving E from V
ˆx! " !! "!
"#$#%&'(')*'#+$%&,!&'(')*
B
A
V d! " # $%# !!
!
( , , )
( , , )
x x y z
x y z
V d
&!
! " # $% # !!
!
' # $!# !!
! ˆ( ) xx E x" # $ ! " # !# "!
x
V VE
x x
! (' # ) #
! (
Ex = Rate of change in V
with y and z held constant
!V = !! B
A
!E · d!s
A = (x, y, z), B = (x + !x, y, z)!!s = !xı
14
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Calculando E a partir de V
!E = !!
"V
"xi +
"V
"yj +
"V
"zk
"
= !!
"
"xi +
"
"yj +
"
"zk
"V
15
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Calculando E a partir de V
!E = !!
"V
"xi +
"V
"yj +
"V
"zk
"
= !!
"
"xi +
"
"yj +
"
"zk
"V
!! =!
"
"xi +
"
"yj +
"
"zk
"
Operadorgradiente
15
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Calculando E a partir de V
!E = !!
"V
"xi +
"V
"yj +
"V
"zk
"
= !!
"
"xi +
"
"yj +
"
"zk
"V
!! =!
"
"xi +
"
"yj +
"
"zk
"
!E = !!"VOperadorgradiente
15
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V !"#$%&'%'( &'( &( )*'!+!,%( #-&$.%(/*,%'( &0*".( '*/%( 1!$%#+!*"2( '&3( x2( 4!+-( 2(
+-%"(+-%$%(!'(&("*"5,&"!'-!".(#*/)*"%"+(*6(!
7 8V x! ! "!"
(!"(+-%(*))*'!+%(1!$%#+!*"( 9 :(;"(+-%(
#&'%(*6(.$&,!+32(!6(+-%(.$&,!+&+!*"&0()*+%"+!&0(!"#$%&'%'(4-%"(&(/&''(!'(0!6+%1(&(1!'+&"#%(h2(
+-%(.$&,!+&+!*"&0(6*$#%(/<'+(=%(1*4"4&$1:(
8>xE# $
(
;6(+-%(#-&$.%(1!'+$!=<+!*"()*''%''%'(')-%$!#&0('3//%+$32(+-%"(+-%($%'<0+!".(%0%#+$!#(6!%01(!'(
&( 6<"#+!*"( *6( +-%( $&1!&0( 1!'+&"#%( r2( !:%:2( ?rE%! "
"
:( ;"( +-!'( #&'%2( :rdV E dr% # (;6( !'(
@"*4"2(+-%"(! (/&3(=%(*=+&!"%1(&'(
9 >V r"
(
( ?r
dVE
dr
& '% % #( )
* +! " ?"!"
( 9A:B:C>(
(
D*$(%E&/)0%2(+-%(%0%#+$!#()*+%"+!&0(1<%(+*(&()*!"+(#-&$.%(q(!'( 89 > 7 FV r q r,-% :(G'!".(+-%(
&=*,%(6*$/<0&2(+-%(%0%#+$!#(6!%01(!'('!/)03(H
8?9 F >q r,-%! # "
!"
:((
(
(
$%&%'()"*+,-./(*.+(!01,23/-./,*45(
((
I<))*'%( &( '3'+%/( !"( +4*( 1!/%"'!*"'( -&'( &"( %0%#+$!#( )*+%"+!&0( 9 2 >V x y :( J-%( #<$,%'(
#-&$&#+%$!K%1( =3( #*"'+&"+ 9 2 >V x y &$%( #&00%1( %L<!)*+%"+!&0( #<$,%':( ME&/)0%'( *6(
%L<!)*+%"+!&0(#<$,%'(&$%(1%)!#+%1(!"(D!.<$%(A:B:N(=%0*4:(
(
((
6,71"-($%&%'(ML<!)*+%"+!&0(#<$,%'(
(
;"( +-$%%( 1!/%"'!*"'( 4%( -&,%( %L<!)*+%"+!&0( '<$6&#%'( &"1( +-%3( &$%( 1%'#$!=%1( =3(
9 2 2 >V x y z O#*"'+&"+:( I!"#%( (4%( #&"( '-*4( +-&+( +-%( 1!$%#+!*"( *6( !!"
!'( &04&3'(
)%$)%"1!#<0&$( +*( +-%( %L<!)*+%"+!&0( +-$*<.-( +-%( )*!"+:( P%0*4( 4%( .!,%( &( )$**6( !"( +4*(
1!/%"'!*"':(Q%"%$&0!K&+!*"(+*(+-$%%(1!/%"'!*"'(!'('+$&!.-+6*$4&$1:(
2V% #.!"
(
8"339:(
(
R%6%$$!".( +*( D!.<$%( A:B:H2( 0%+( +-%( )*+%"+!&0( &+( &( )*!"+( 9 2 >P x y =%( 9 2 >V x y :(S*4(/<#-( !'(
#-&".%1(&+(&("%!.-=*$!".()*!"+(V 9 2P x dx >y dy/ / T(U%+(+-%(1!66%$%"#%(=%(4$!++%"(&'(
(
( N8
Superfícies equipotenciais
Superfícies de mesma energiaV=constante
•E perpendicular às equipotenciais:
• Nenhum trabalho é necessário para mover uma carga ao longo de uma superfície equipotencial
• Componente tangencial de E é zero ao longo das equipotenciais
!E = !!"V
16
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V !"#$%&'%'( &'( &( )*'!+!,%( #-&$.%(/*,%'( &0*".( '*/%( 1!$%#+!*"2( '&3( x2( 4!+-( 2(
+-%"(+-%$%(!'(&("*"5,&"!'-!".(#*/)*"%"+(*6(!
7 8V x! ! "!"
(!"(+-%(*))*'!+%(1!$%#+!*"( 9 :(;"(+-%(
#&'%(*6(.$&,!+32(!6(+-%(.$&,!+&+!*"&0()*+%"+!&0(!"#$%&'%'(4-%"(&(/&''(!'(0!6+%1(&(1!'+&"#%(h2(
+-%(.$&,!+&+!*"&0(6*$#%(/<'+(=%(1*4"4&$1:(
8>xE# $
(
;6(+-%(#-&$.%(1!'+$!=<+!*"()*''%''%'(')-%$!#&0('3//%+$32(+-%"(+-%($%'<0+!".(%0%#+$!#(6!%01(!'(
&( 6<"#+!*"( *6( +-%( $&1!&0( 1!'+&"#%( r2( !:%:2( ?rE%! "
"
:( ;"( +-!'( #&'%2( :rdV E dr% # (;6( !'(
@"*4"2(+-%"(! (/&3(=%(*=+&!"%1(&'(
9 >V r"
(
( ?r
dVE
dr
& '% % #( )
* +! " ?"!"
( 9A:B:C>(
(
D*$(%E&/)0%2(+-%(%0%#+$!#()*+%"+!&0(1<%(+*(&()*!"+(#-&$.%(q(!'( 89 > 7 FV r q r,-% :(G'!".(+-%(
&=*,%(6*$/<0&2(+-%(%0%#+$!#(6!%01(!'('!/)03(H
8?9 F >q r,-%! # "
!"
:((
(
(
$%&%'()"*+,-./(*.+(!01,23/-./,*45(
((
I<))*'%( &( '3'+%/( !"( +4*( 1!/%"'!*"'( -&'( &"( %0%#+$!#( )*+%"+!&0( 9 2 >V x y :( J-%( #<$,%'(
#-&$&#+%$!K%1( =3( #*"'+&"+ 9 2 >V x y &$%( #&00%1( %L<!)*+%"+!&0( #<$,%':( ME&/)0%'( *6(
%L<!)*+%"+!&0(#<$,%'(&$%(1%)!#+%1(!"(D!.<$%(A:B:N(=%0*4:(
(
((
6,71"-($%&%'(ML<!)*+%"+!&0(#<$,%'(
(
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9 2 2 >V x y z O#*"'+&"+:( I!"#%( (4%( #&"( '-*4( +-&+( +-%( 1!$%#+!*"( *6( !!"
!'( &04&3'(
)%$)%"1!#<0&$( +*( +-%( %L<!)*+%"+!&0( +-$*<.-( +-%( )*!"+:( P%0*4( 4%( .!,%( &( )$**6( !"( +4*(
1!/%"'!*"':(Q%"%$&0!K&+!*"(+*(+-$%%(1!/%"'!*"'(!'('+$&!.-+6*$4&$1:(
2V% #.!"
(
8"339:(
(
R%6%$$!".( +*( D!.<$%( A:B:H2( 0%+( +-%( )*+%"+!&0( &+( &( )*!"+( 9 2 >P x y =%( 9 2 >V x y :(S*4(/<#-( !'(
#-&".%1(&+(&("%!.-=*$!".()*!"+(V 9 2P x dx >y dy/ / T(U%+(+-%(1!66%$%"#%(=%(4$!++%"(&'(
(
( N8
Superfícies equipotenciais
Superfícies de mesma energiaV=constante
•E perpendicular às equipotenciais:
• Nenhum trabalho é necessário para mover uma carga ao longo de uma superfície equipotencial
• Componente tangencial de E é zero ao longo das equipotenciais
!E = !!"V
Gravidade: mapa topográfico mostra superfícies equipotenciais :Vg=gz
!"#$%&'%#&()#*$'+$#,-)%'(#.()/0$*-&+/1#*$1/.$2#$*-33/&)4#5$/*$+'00'6*7$
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,-./01&'()23'45"67%/18'9:.%#&;'<7;'
'
L'.*)5#&$/$.'.?1'.5-1().:$&'5$'+$0#.:("$ ! $"/@).:$/$-.)+'&3$1"/&:#$5#.*)(=" ;$D).5$("#$
#0#1(&)1$%'(#.()/0$/( >$/$%#&%#.5)1-0/&$5)*(/.1#$P y $/2'@#$("#$3)5%').($'+$("#$&'5;$
$
$$
!"#$%&'()*)*$C$.'.?1'.5-1().:$&'5$'+$0#.:("$ $/.5$-.)+'&3$1"/&:#$5#.*)(=! " ;$$
$ MN
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Equipotenciais
Carga pontual Dipolo elétrico Placas paralelas
17
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Equipotenciais e linhas de campo
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!"#$% -
Conductors in Equilibrium
Conductors are equipotential objects:
1) E = 0 inside
2) Net charge inside is 0
3) E perpendicular to surface
4) Excess charge on surface
$!
"#E
E = !/"0
Condutores
•E perpendicular à superfície do condutor
• E=0 dentro do condutor
• Condutores são objetos equipotenciais
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Potencial em um condutor
No condutor E=0: variação do potencial = 0
Campo elétrico = variação do potencial V constante no condutor
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Potencial em um condutor
No condutor E=0: variação do potencial = 0
Campo elétrico = variação do potencial V constante no condutor
Mas qual o valor de V ?
Valor que ele tem na superfície
V é uma função contínua
20
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Capacitores
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Capacitance and Dielectrics
5.1 Introduction
A capacitor is a device which stores electric charge. Capacitors vary in shape and size,
but the basic configuration is two conductors carrying equal but opposite charges (Figure
5.1.1). Capacitors have many important applications in electronics. Some examples
include storing electric potential energy, delaying voltage changes when coupled with
resistors, filtering out unwanted frequency signals, forming resonant circuits and making
frequency-dependent and independent voltage dividers when combined with resistors.
Some of these applications will be discussed in latter chapters.
Figure 5.1.1 Basic configuration of a capacitor.
In the uncharged state, the charge on either one of the conductors in the capacitor is zero.
During the charging process, a charge Q is moved from one conductor to the other one,
giving one conductor a charge Q! , and the other one a charge . A potential
difference is created, with the positively charged conductor at a higher potential than
the negatively charged conductor. Note that whether charged or uncharged, the net charge
on the capacitor as a whole is zero.
Q"
V#
The simplest example of a capacitor consists of two conducting plates of area A , which
are parallel to each other, and separated by a distance d, as shown in Figure 5.1.2.
Figure 5.1.2 A parallel-plate capacitor
Experiments show that the amount of charge Q stored in a capacitor is linearly
proportional to , the electric potential difference between the plates. Thus, we may
write
V#
|Q C V |$ # (5.1.1)
2
C =Q
|!V |
Capacitores
Dois condutores com cargas iguais e opostas separados por uma distância d e com uma diferença de potencial ∆V entre eles.
Armazenamento de Energia!
Unidade: Coulomb/Volt
Farad
=
22
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Capacitor de placas paralelas
!"#$% -
Parallel Plate Capacitor
top
bottom
V d! " # $% E S!!
$
Qd
A&"Ed"
d
A
V
QC $
&"
!"
C depends only on geometric factors A and d
Integral de trajetória para encontrar V
23
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!V = !! d
0
!E · d!s = Ed ="
#0d =
Q
A#0d
Capacitor de placas paralelas
!"#$% -
Parallel Plate Capacitor
top
bottom
V d! " # $% E S!!
$
Qd
A&"Ed"
d
A
V
QC $
&"
!"
C depends only on geometric factors A and d
Integral de trajetória para encontrar V
23
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!V = !! d
0
!E · d!s = Ed ="
#0d =
Q
A#0d
C =Q
|!V | =A!0
d
Capacitor de placas paralelas
!"#$% -
Parallel Plate Capacitor
top
bottom
V d! " # $% E S!!
$
Qd
A&"Ed"
d
A
V
QC $
&"
!"
C depends only on geometric factors A and d
Integral de trajetória para encontrar V
23
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Energia necessária para carregar capacitor
!!"#$ -
Energy To Charge Capacitor
1. Capacitor starts uncharged.
2. Carry +dq from bottom to top.
Now top has charge q = +dq, bottom -dq
3. Repeat
4. Finish when top has charge q = +Q, bottom -Q
+q
-q
• Capacitor inicialmente descarregado
• +dq sai da placa inferior e vai para a superior
• Uma placa fica com +dq e a outra com -dq
• Processo ocorre até uma placa ter +Q e a outra -Q
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dW = dq!V = dqq
V=
1C
qdq
W =!
dW =! Q
0
1C
qdq
W =1C
Q2
2
Trabalho realizado para carregar capacitor
!!"#$ -
Energy To Charge Capacitor
1. Capacitor starts uncharged.
2. Carry +dq from bottom to top.
Now top has charge q = +dq, bottom -dq
3. Repeat
4. Finish when top has charge q = +Q, bottom -Q
+q
-q
25
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U =1C
Q2
2=
12C|!V |2
Energia armazenada no capacitor
C =Q
|!V |
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U =1C
Q2
2=
12C|!V |2
5.4.1 Energy Density of the Electric Field
One can think of the energy stored in the capacitor as being stored in the electric field
itself. In the case of a parallel-plate capacitor, with 0 /C A d!" and | |V Ed# " , we have
$ % $22 00
1 1 1| |
2 2 2E
AU C V Ed E Ad
d%2!
!" # " " (5.4.3)
Since the quantity Ad represents the volume between the plates, we can define the electric
energy density as
2
0
1
Volume 2
EE
Uu !" " E (5.4.4)
Note that is proportional to the square of the electric field. Alternatively, one may
obtain the energy stored in the capacitor from the point of view of external work. Since
the plates are oppositely charged, force must be applied to maintain a constant separation
between them. From Eq. (4.4.7), we see that a small patch of charge
Eu
( )q A&# " # experiences an attractive force 2
0( ) / 2F A& !# " # . If the total area of the
plate is A, then an external agent must exert a force 2
ext 0/ 2F A& !" to pull the two plates
apart. Since the electric field strength in the region between the plates is given by
0/E & !" , the external force can be rewritten as
20ext
2F E A
!" (5.4.5)
Note that is independent of d . The total amount of work done externally to separate
the plates by a distance d is then
extF
2
0ext ext ext
2
E AW d F d
!' (" ) " " *
+ ,- F s d.!
!
(5.4.6)
consistent with Eq. (5.4.3). Since the potential energy of the system is equal to the work
done by the external agent, we have . In addition, we note that the
expression for is identical to Eq. (4.4.8) in Chapter 4. Therefore, the electric energy
density can also be interpreted as electrostatic pressure P.
2
ext 0/Eu W Ad E!" " / 2
Eu
Eu
Interactive Simulation 5.2: Charge Placed between Capacitor Plates
This applet shown in Figure 5.4.2 is a simulation of an experiment in which an aluminum
sphere sitting on the bottom plate of a capacitor is lifted to the top plate by the
electrostatic force generated as the capacitor is charged. We have placed a non-
13
Energia armazenada no capacitor
C =Q
|!V |
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U =1C
Q2
2=
12C|!V |2
5.4.1 Energy Density of the Electric Field
One can think of the energy stored in the capacitor as being stored in the electric field
itself. In the case of a parallel-plate capacitor, with 0 /C A d!" and | |V Ed# " , we have
$ % $22 00
1 1 1| |
2 2 2E
AU C V Ed E Ad
d%2!
!" # " " (5.4.3)
Since the quantity Ad represents the volume between the plates, we can define the electric
energy density as
2
0
1
Volume 2
EE
Uu !" " E (5.4.4)
Note that is proportional to the square of the electric field. Alternatively, one may
obtain the energy stored in the capacitor from the point of view of external work. Since
the plates are oppositely charged, force must be applied to maintain a constant separation
between them. From Eq. (4.4.7), we see that a small patch of charge
Eu
( )q A&# " # experiences an attractive force 2
0( ) / 2F A& !# " # . If the total area of the
plate is A, then an external agent must exert a force 2
ext 0/ 2F A& !" to pull the two plates
apart. Since the electric field strength in the region between the plates is given by
0/E & !" , the external force can be rewritten as
20ext
2F E A
!" (5.4.5)
Note that is independent of d . The total amount of work done externally to separate
the plates by a distance d is then
extF
2
0ext ext ext
2
E AW d F d
!' (" ) " " *
+ ,- F s d.!
!
(5.4.6)
consistent with Eq. (5.4.3). Since the potential energy of the system is equal to the work
done by the external agent, we have . In addition, we note that the
expression for is identical to Eq. (4.4.8) in Chapter 4. Therefore, the electric energy
density can also be interpreted as electrostatic pressure P.
2
ext 0/Eu W Ad E!" " / 2
Eu
Eu
Interactive Simulation 5.2: Charge Placed between Capacitor Plates
This applet shown in Figure 5.4.2 is a simulation of an experiment in which an aluminum
sphere sitting on the bottom plate of a capacitor is lifted to the top plate by the
electrostatic force generated as the capacitor is charged. We have placed a non-
13
Energia armazenada no capacitor
C =Q
|!V |
5.4.1 Energy Density of the Electric Field
One can think of the energy stored in the capacitor as being stored in the electric field
itself. In the case of a parallel-plate capacitor, with 0 /C A d!" and | |V Ed# " , we have
$ % $22 00
1 1 1| |
2 2 2E
AU C V Ed E Ad
d%2!
!" # " " (5.4.3)
Since the quantity Ad represents the volume between the plates, we can define the electric
energy density as
2
0
1
Volume 2
EE
Uu !" " E (5.4.4)
Note that is proportional to the square of the electric field. Alternatively, one may
obtain the energy stored in the capacitor from the point of view of external work. Since
the plates are oppositely charged, force must be applied to maintain a constant separation
between them. From Eq. (4.4.7), we see that a small patch of charge
Eu
( )q A&# " # experiences an attractive force 2
0( ) / 2F A& !# " # . If the total area of the
plate is A, then an external agent must exert a force 2
ext 0/ 2F A& !" to pull the two plates
apart. Since the electric field strength in the region between the plates is given by
0/E & !" , the external force can be rewritten as
20ext
2F E A
!" (5.4.5)
Note that is independent of d . The total amount of work done externally to separate
the plates by a distance d is then
extF
2
0ext ext ext
2
E AW d F d
!' (" ) " " *
+ ,- F s d.!
!
(5.4.6)
consistent with Eq. (5.4.3). Since the potential energy of the system is equal to the work
done by the external agent, we have . In addition, we note that the
expression for is identical to Eq. (4.4.8) in Chapter 4. Therefore, the electric energy
density can also be interpreted as electrostatic pressure P.
2
ext 0/Eu W Ad E!" " / 2
Eu
Eu
Interactive Simulation 5.2: Charge Placed between Capacitor Plates
This applet shown in Figure 5.4.2 is a simulation of an experiment in which an aluminum
sphere sitting on the bottom plate of a capacitor is lifted to the top plate by the
electrostatic force generated as the capacitor is charged. We have placed a non-
13
Densidade de energia
Energia armazenada no campo!
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Aumentando a capacitância
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Dielétricos (visão microscópica)
Dielétricos polares
Dielétricos com momento de dipolo permanente
Ex: água
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Dielétricos (visão microscópica)
Dielétricos polares
Dielétricos com momento de dipolo permanente
Ex: água
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Dielétricos não polares (visão microscópica)
Dielétricos com momento de dipolo induzido pelo campo elétrico
Ex: CH4
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Dielétricos não polares (visão microscópica)
Dielétricos com momento de dipolo induzido pelo campo elétrico
Ex: CH4
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Dielétricos (visão macroscópica)
1
1
Volume
N
i
i!
! "P p!
!
(5.5.2)
In the case of our cylinder, where all the dipoles are perfectly aligned, the magnitude of
is equal to P!
Np
PAh
! (5.5.3)
and the direction of is parallel to the aligned dipoles. P!
Now, what is the average electric field these dipoles produce? The key to figuring this
out is realizing that the situation shown in Figure 5.5.4(a) is equivalent that shown in
Figure 5.5.4(b), where all the little ± charges associated with the electric dipoles in the
interior of the cylinder are replaced with two equivalent charges, PQ# , on the top and
bottom of the cylinder, respectively.
Figure 5.5.4 (a) A cylinder with uniform dipole distribution. (b) Equivalent charge
distribution.
The equivalence can be seen by noting that in the interior of the cylinder, positive charge
at the top of any one of the electric dipoles is canceled on average by the negative charge
of the dipole just above it. The only place where cancellation does not take place is for
electric dipoles at the top of the cylinder, since there are no adjacent dipoles further up.
Thus the interior of the cylinder appears uncharged in an average sense (averaging over
many dipoles), whereas the top surface of the cylinder appears to carry a net positive
charge. Similarly, the bottom surface of the cylinder will appear to carry a net negative
charge.
How do we find an expression for the equivalent charge PQ in terms of quantities we
know? The simplest way is to require that the electric dipole moment PQ produces,
PQ h , is equal to the total electric dipole moment of all the little electric dipoles. This
gives , or PQ h Np!
P
NpQ
h! (5.5.4)
18
QP = Carga induzida
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Dielétricos em capacitores
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Dielétricos em capacitores
C =Q
|!V |Aumento da capacitância com diminuição de ∆V
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Dielétricos em capacitores
C =Q
|!V |Aumento da capacitância com diminuição de ∆V
∆V diminui porque a polarização do dielétrico diminui o campo elétrico
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Constante dielétrica κ
dielétricos diminuem o campo elétrico original por um fator κ
Constante dielétrica
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Constante dielétrica κ
Constantes dielétricas Vácuo 1.0 Papel 3.7 Vidro Pyrex 5.6 Água 80
dielétricos diminuem o campo elétrico original por um fator κ
Constante dielétrica
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Lei de Gauss num dielétrico
The capacitance becomes
00
0 0| | | |
ee
QQC
V VC
!!" " "
# # (5.5.17)
which is the same as the first case where the charge Q0 is kept constant, but now the
charge has increased.
5.5.4 Gauss’s Law for Dielectrics
Consider again a parallel-plate capacitor shown in Figure 5.5.7:
Figure 5.5.7 Gaussian surface in the absence of a dielectric.
When no dielectric is present, the electric field 0E!
in the region between the plates can be
found by using Gauss’s law:
0 0
0 0
,S
Qd E A E
$
% %& " " ' "(( E A!" "
#
We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced
charge PQ of opposite sign on the surface, and the net charge enclosed by the Gaussian
surface is PQ Q) .
Figure 5.5.8 Gaussian surface in the presence of a dielectric.
22
Lei de Gauss sem dielétricos
33
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The capacitance becomes
00
0 0| | | |
ee
QQC
V VC
!!" " "
# # (5.5.17)
which is the same as the first case where the charge Q0 is kept constant, but now the
charge has increased.
5.5.4 Gauss’s Law for Dielectrics
Consider again a parallel-plate capacitor shown in Figure 5.5.7:
Figure 5.5.7 Gaussian surface in the absence of a dielectric.
When no dielectric is present, the electric field 0E!
in the region between the plates can be
found by using Gauss’s law:
0 0
0 0
,S
Qd E A E
$
% %& " " ' "(( E A!" "
#
We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced
charge PQ of opposite sign on the surface, and the net charge enclosed by the Gaussian
surface is PQ Q) .
Figure 5.5.8 Gaussian surface in the presence of a dielectric.
22
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
Lei de Gauss num dielétrico
The capacitance becomes
00
0 0| | | |
ee
QQC
V VC
!!" " "
# # (5.5.17)
which is the same as the first case where the charge Q0 is kept constant, but now the
charge has increased.
5.5.4 Gauss’s Law for Dielectrics
Consider again a parallel-plate capacitor shown in Figure 5.5.7:
Figure 5.5.7 Gaussian surface in the absence of a dielectric.
When no dielectric is present, the electric field 0E!
in the region between the plates can be
found by using Gauss’s law:
0 0
0 0
,S
Qd E A E
$
% %& " " ' "(( E A!" "
#
We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced
charge PQ of opposite sign on the surface, and the net charge enclosed by the Gaussian
surface is PQ Q) .
Figure 5.5.8 Gaussian surface in the presence of a dielectric.
22
Lei de Gauss sem dielétricos
33
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The capacitance becomes
00
0 0| | | |
ee
QQC
V VC
!!" " "
# # (5.5.17)
which is the same as the first case where the charge Q0 is kept constant, but now the
charge has increased.
5.5.4 Gauss’s Law for Dielectrics
Consider again a parallel-plate capacitor shown in Figure 5.5.7:
Figure 5.5.7 Gaussian surface in the absence of a dielectric.
When no dielectric is present, the electric field 0E!
in the region between the plates can be
found by using Gauss’s law:
0 0
0 0
,S
Qd E A E
$
% %& " " ' "(( E A!" "
#
We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced
charge PQ of opposite sign on the surface, and the net charge enclosed by the Gaussian
surface is PQ Q) .
Figure 5.5.8 Gaussian surface in the presence of a dielectric.
22
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
Lei de Gauss num dielétrico
The capacitance becomes
00
0 0| | | |
ee
QQC
V VC
!!" " "
# # (5.5.17)
which is the same as the first case where the charge Q0 is kept constant, but now the
charge has increased.
5.5.4 Gauss’s Law for Dielectrics
Consider again a parallel-plate capacitor shown in Figure 5.5.7:
Figure 5.5.7 Gaussian surface in the absence of a dielectric.
When no dielectric is present, the electric field 0E!
in the region between the plates can be
found by using Gauss’s law:
0 0
0 0
,S
Qd E A E
$
% %& " " ' "(( E A!" "
#
We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced
charge PQ of opposite sign on the surface, and the net charge enclosed by the Gaussian
surface is PQ Q) .
Figure 5.5.8 Gaussian surface in the presence of a dielectric.
22
Lei de Gauss sem dielétricos
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
QP = Carga induzida
33
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The capacitance becomes
00
0 0| | | |
ee
QQC
V VC
!!" " "
# # (5.5.17)
which is the same as the first case where the charge Q0 is kept constant, but now the
charge has increased.
5.5.4 Gauss’s Law for Dielectrics
Consider again a parallel-plate capacitor shown in Figure 5.5.7:
Figure 5.5.7 Gaussian surface in the absence of a dielectric.
When no dielectric is present, the electric field 0E!
in the region between the plates can be
found by using Gauss’s law:
0 0
0 0
,S
Qd E A E
$
% %& " " ' "(( E A!" "
#
We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced
charge PQ of opposite sign on the surface, and the net charge enclosed by the Gaussian
surface is PQ Q) .
Figure 5.5.8 Gaussian surface in the presence of a dielectric.
22
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
Lei de Gauss num dielétrico
The capacitance becomes
00
0 0| | | |
ee
QQC
V VC
!!" " "
# # (5.5.17)
which is the same as the first case where the charge Q0 is kept constant, but now the
charge has increased.
5.5.4 Gauss’s Law for Dielectrics
Consider again a parallel-plate capacitor shown in Figure 5.5.7:
Figure 5.5.7 Gaussian surface in the absence of a dielectric.
When no dielectric is present, the electric field 0E!
in the region between the plates can be
found by using Gauss’s law:
0 0
0 0
,S
Qd E A E
$
% %& " " ' "(( E A!" "
#
We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced
charge PQ of opposite sign on the surface, and the net charge enclosed by the Gaussian
surface is PQ Q) .
Figure 5.5.8 Gaussian surface in the presence of a dielectric.
22
Lei de Gauss sem dielétricos
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
permissividade elétrica do meio
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
Gauss’s law becomes
0
P
S
Q Qd EA
!
"# $ $%% E A
!" "
# (5.5.18)
or
0
PQ QE
A!
"$ (5.5.19)
However, we have just seen that the effect of the dielectric is to weaken the original field
by a factor . Therefore, 0E e&
0
0 0
P
e e
E Q QQE
A A& & ! !
"$ $ $ (5.5.20)
from which the induced charge PQ can be obtained as
1
1P
e
Q Q&
' ($ ")
* +, (5.5.21)
In terms of the surface charge density, we have
1
1P
e
- -&
' ($ ")
* +, (5.5.22)
Note that in the limit , 1e& $ 0PQ $ which corresponds to the case of no dielectric
material.
Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss’s law with dielectric can be
rewritten as
0eS
Q Qd
& ! !# $ $%% E A
""
# (5.5.23)
where 0e! & !$ is called the dielectric permittivity. Alternatively, we may also write
S
d Q# $%% D A
""
# (5.5.24)
where 0! &$D
!""
E is called the electric displacement vector.
23
QP = Carga induzida
33