Potencial elétrico e capacitores. Baseado no 8.02T MIT-opencourse

Potencial elétrico e capacitores

Baseado no 8.02T MIT-opencourse

!g =

_−G

M

r

2

ˆr

!

F

_g

= m!g

!

E = k

_e

q

r

2

ˆr

!

F

_e

= q !

E

Gravidade x eletricidade

Massa M

Carga(+/-q)

Campos

Forças

Energia potencial x potencial

Gravidade

Gravidade: força e trabalho

!

F

_g

= −G

M m

r

2

ˆr

Gravidade: força e trabalho

!

F

_g

= −G

M m

r

2

ˆr

Força exercida em m devido a M

W

_g

=

!

B

A

!

F

_g

_{· d!s}

Trabalho exercido pela gravidade ao

mover m de A a B

integral de trajetória

12 P04 !

Work Done by Earth’s Gravity

"#$%&'#()&*+&,$-./0+&1#./(,&1&2$#1&A 0#&B: g g W !

#

!! " d "!

$

%

2 ˆ ˆ ˆ B A GMm r dr rd& ' ( ) ! _{+ ,} " * - .

#

# # ! 1 1 B A GMm r r ( ) ! ₊ ' _, - . 2 B A r r GMm dr r ! '

#

B A r r GMm r !

/

0

1

2

3

4

"3-0&/4&03)&4/,(&1#./(,&2$#1&$₅ 0#&$₆7 W_g = ! B A ! F_{g · d!s} = = ! B A " −G M m r2 ˆr # · (drˆr + rdθˆθ) = ! B A −G M m r2 dr = $ G M m r %r_B r_A = GMm " 1 r_B − 1 r_A #

Trabalho realizado pela gravidade terrestre

Trabalho realizado pela gravidade ao

mover m de A a B

Trabalho depende apenas dos pontos A e B!

Forças conservativas

Mecânica:

W

_A→B

=

_∆E

_cin

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W !

'

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mg

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mg

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( ! " ! "

'

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Forças conservativas:

6

Forças conservativas

Mecânica:

W

_A→B

=

_∆E

_cin

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W !

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mg

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'

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( ! " ! "

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Forças conservativas:

Forças conservativas

Mecânica:

W

_A→B

=

_∆E

_cin

!"#$%&'(%&

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W

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W

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B

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W !

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mg

%

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( ! " ! "

'

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Forças conservativas:

6

∆U

_g

= U

_B

_{− U}

_A

_{= −}

!

B

A

!

F

_g

_{· d!s = −W}

_g

= W

_ext

∆U

_g

= U

_B

_{− U}

_A

_{= −}

!

B A

!

F

_g

_{· d!s = −W}

_g

= W

_ext

!

F

_g

= −G

M m

r

2

ˆr → U

g

= G

M m

r

+ U

0

Energia potencial x potencial

∆U

_g

= U

_B

_{− U}

_A

_{= −}

!

B A

!

F

_g

_{· d!s = −W}

_g

= W

_ext

!

F

_g

= −G

M m

r

2

ˆr → U

g

= G

M m

r

+ U

0

Energia potencial x potencial

U0: constante que depende do pto de referência Apenas tem significado físico ∆U_{g → ∆Vg}

∆U

_g

= U

_B

_{− U}

_A

_{= −}

!

B A

!

F

_g

_{· d!s = −W}

_g

= W

_ext

!

F

_g

= −G

M m

r

2

ˆr → U

g

= G

M m

r

+ U

0

∆V

_g

=

∆U

g

m

= −

!

B A

( !

F

_g

/m)

_{· d!s = −}

!

B A

!g

_{· d!s}

Energia potencial x potencial

U0: constante que depende do pto de referência Apenas tem significado físico ∆U_{g → ∆Vg}

Definição da diferença de potencial gravitacional

∆U

_g

= U

_B

_{− U}

_A

_{= −}

!

B A

!

F

_g

_{· d!s = −W}

_g

= W

_ext

!

F

_g

= −G

M m

r

2

ˆr → U

g

= G

M m

r

+ U

0

∆V

_g

=

∆U

g

m

= −

!

B A

( !

F

_g

/m)

_{· d!s = −}

!

B A

!g

_{· d!s}

Energia potencial x potencial

U0: constante que depende do pto de referência Apenas tem significado físico ∆U_{g → ∆Vg}

Definição da diferença de potencial gravitacional

!

F

_g

_{→ !g}

Campo Força

∆U

_g

_{→ ∆V}

_g Potencial Energia

Potencial gravitacional

Potencial de planeta +sol

Gravidade x eletricidade

!

E = k

_e

q

r

2

ˆr

!

F

_e

= q !

E

Carga(+/-q)

Massa M

!g =

_−G

M

r

2

ˆr

!

F

_g

= m!g

∆U

_g

_{= −}

!

B A

!

F

_g

_{· d!s}

∆V

_g

_{= −}

!

B

!g

_{· d!s}

Ambas as forças são conservativas, então:

∆U = −

!

B

A

!

F

_e

_{· d!s}

∆V

_{= −}

!

B A

!

E

_{· d!s}

Potencial e energia

Unidades: Joules/Coulomb =Volts 10

∆V

_{= −}

!

B A

!

E

_{· d!s}

Potencial e energia

Unidades: Joules/Coulomb =Volts

W

_ext

= ∆U = U

_B

_{− U}

_A

= q∆V

_Joules

Trabalho realizado pela gravidade ao

mover m de A a B:

Potencial

V (!r) = V

₀

+ ∆V = V

₀

₋

!

B A

!

E

_{· d!s}

Cargas geram potenciais

Potencial

V (!r) = V

₀

+ ∆V = V

₀

₋

!

B A

!

E

_{· d!s}

Cargas geram potenciais

28 P04 !

Potential Landscape

Positive Charge Negative Charge

q positiva

q negativa

Potencial

V (!r) = V

₀

+ ∆V = V

₀

₋

!

B A

!

E

_{· d!s}

Cargas geram potenciais

U (!r) = qV (!r)

Cargas sentem potenciais

28 P04 !

Potential Landscape

Positive Charge Negative Charge

q positiva

q negativa

11

26 P04 !

Potential Created by Pt Charge

! ! " #! # dr ˆ " r d

!

ˆ B B A _A

V V

V

d

$ #

%

# %

'

$ "

!

&

!

2

ˆ

r

kQ

!

$

!

#

2 2

ˆ

B B A A

dr

kQ

d

kQ

r

r

# %

'

!

&

"

!

# %

'

1

1

B A

kQ

r

r

(

)

#

_*

%

₊

,

-"#$%&V '&(&#)&r '&!*

r

kQ

r

V

_{Point Charge}

(

)

#

∆V

= V

_B

_{− V}

_A

_{= −}

!

B A

!

E

_{· d!s}

= −

!

B A

"

k

Q

r

2

ˆr

#

· d!s = −

!

B A

k

Q

r

2

dr

=

$

k

Q

r

%

rB rA

= kQ

" 1

r

_B

−

1

r

_A

#

26 P04 !

Potential Created by Pt Charge

! ! " #! # dr ˆ " r d

!

ˆ B B A _A

V V

V

d

$ #

%

# %

'

$ "

!

&

!

2

ˆ

r

kQ

!

$

!

#

2 2

ˆ

B B A A

dr

kQ

d

kQ

r

r

# %

'

!

&

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!

# %

'

1

1

B A

kQ

r

r

(

)

#

_*

%

₊

,

-"#$%&V '&(&#)&r '&!*

r

kQ

r

V

_{Point Charge}

(

)

#

∆V

= V

_B

_{− V}

_A

_{= −}

!

B A

!

E

_{· d!s}

= −

!

B A

"

k

Q

r

2

ˆr

#

· d!s = −

!

B A

k

Q

r

2

dr

=

$

k

Q

r

%

rB rA

= kQ

" 1

r

_B

−

1

r

_A

#

Vcarga pontual(r) = k

Q

_r

V (r =

_{∞) = 0}

Potencial criado por uma carga pontual

Potencial: princípio da superposição

Soma direta. Potencial é um

escalar!

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:

densidade

linear de carga superficial de cargadensidade volumétrica de cargadensidade

Calculando E a partir de V

30 P04 !

Deriving E from V

ˆ

x

! " !

!

!

"

"#$#%&'(')*'#+$%&,!&'(')* B A

V

d

! " #

%

#

!

$

!

!

( , , ) ( , , ) x x y z x y z V d &! ! " #

%

#! $ !! ' # $!#! !! " # $ !#! ( x"ˆ) " # !E x_x x

V

V

E

x

x

!

(

' #

) #

!

(

E_x = 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 = ∆xˆı

∆V _{= −} ! (x+∆x,y,z) (x,y,z) ! E _{· d!s " !}E _{· ∆!s = − !}E _{· (∆xˆı) = −Ex}∆x

Calculando E a partir de V

30 P04 !

Deriving E from V

ˆ

x

! " !

!

!

"

"#$#%&'(')*'#+$%&,!&'(')* B A

V

d

! " #

%

#

!

$

!

!

( , , ) ( , , ) x x y z x y z V d &! ! " #

%

#! $ !! ' # $!#! !! " # $ !#! ( x"ˆ) " # !E x_x x

V

V

E

x

x

!

(

' #

) #

!

(

E_x = 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 = ∆xˆı 14

∆V _{= −} ! (x+∆x,y,z) (x,y,z) ! E _{· d!s " !}E _{· ∆!s = − !}E _{· (∆xˆı) = −Ex}∆x

E

_x

_{! −}

∆V

∆x → −

∂V

∂x

Calculando E a partir de V

30 P04 !

Deriving E from V

ˆ

x

! " !

!

!

"

"#$#%&'(')*'#+$%&,!&'(')* B A

V

d

! " #

%

#

!

$

!

!

( , , ) ( , , ) x x y z x y z V d &! ! " #

%

#! $ !! ' # $!#! !! " # $ !#! ( x"ˆ) " # !E x_x x

V

V

E

x

x

!

(

' #

) #

!

(

E_x = 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 = ∆xˆı

Calculando E a partir de V

!

E

= −

!

∂V

∂x

ˆi+ ∂V

∂y

ˆj + ∂V

∂z

ˆk

"

= −

!

∂

∂x

ˆi+ ∂

∂y

ˆj + ∂

∂z

ˆk

"

V

15

Calculando E a partir de V

!

E

= −

!

∂V

∂x

ˆi+ ∂V

∂y

ˆj + ∂V

∂z

ˆk

"

= −

!

∂

∂x

ˆi+ ∂

∂y

ˆj + ∂

∂z

ˆk

"

V

!

∇ =

!

∂

∂x

ˆi+ ∂

∂y

ˆj + ∂

∂z

ˆk

"

Operador gradiente

Calculando E a partir de V

!

E

= −

!

∂V

∂x

ˆi+ ∂V

∂y

ˆj + ∂V

∂z

ˆk

"

= −

!

∂

∂x

ˆi+ ∂

∂y

ˆj + ∂

∂z

ˆk

"

V

!

∇ =

!

∂

∂x

ˆi+ ∂

∂y

ˆj + ∂

∂z

ˆk

"

!

E =

_−!

_∇V

Operador gradiente 15

V !"#$%&'%'( &'( &( )*'!+!,%( #-&$.%( /*,%'( &0*".( '*/%( 1!$%#+!*"2( '&3( x2( 4!+-( 2( +-%"(+-%$%(!'(&("*"5,&"!'-!".(#*/)*"%"+(*6( ! 7 8 V x ! ! " !" (!"(+-%(*))*'!+%(1!$%#+!*"( 9 :(;"(+-%( #&'%(*6(.$&,!+32(!6(+-%(.$&,!+&+!*"&0()*+%"+!&0(!"#$%&'%'(4-%"(&(/&''(!'(0!6+%1(&(1!'+&"#%(h2( +-%(.$&,!+&+!*"&0(6*$#%(/<'+(=%(1*4"4&$1:( 8> x E # $ ( ;6(+-%(#-&$.%(1!'+$!=<+!*"()*''%''%'(')-%$!#&0('3//%+$32(+-%"(+-%($%'<0+!".(%0%#+$!#(6!%01(!'( &( 6<"#+!*"( *6( +-%( $&1!&0( 1!'+&"#%( r2( !:%:2(!" % E_r"? :( ;"( +-!'( #&'%2( dV % #E dr_r : (;6( !'( @"*4"2(+-%"( ! (/&3(=%(*=+&!"%1(&'( 9 > V r " ( ( E_r? dV dr & ' % _{% #( )} * + !!" " ?" ( 9A:B:C>( ( D*$(%E&/)0%2(+-%(%0%#+$!#()*+%"+!&0(1<%(+*(&()*!"+(#-&$.%(q(!'(V r9 > % q 7 F,-₈r :(G'!".(+-%( &=*,%(6*$/<0&2(+-%(%0%#+$!#(6!%01(!'('!/)03(!!" % 9 Fq# ,-₈rH>"? :(( ( ( $%&%'()"*+,-./(*.+(!01,23/-./,*45( ((

I<))*'%( &( '3'+%/( !"( +4*( 1!/%"'!*"'( -&'( &"( %0%#+$!#( )*+%"+!&0( 9 2 >V x y :( J-%( #<$,%'(

#-&$&#+%$!K%1( =3( #*"'+&"+ V x y &$%( #&00%1( %L<!)*+%"+!&0( #<$,%':( ME&/)0%'( *6(9 2 > %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:( 2 V % #. !" ( 8"339:( (

R%6%$$!".( +*( D!.<$%( A:B:H2( 0%+( +-%( )*+%"+!&0( &+( &( )*!"+( 9 2 >P x y =%( 9 2 >V x y :( S*4( /<#-( !'(

#-&".%1(&+(&("%!.-=*$!".()*!"+(

V P x dx9 / 2 y dy/ >T(U%+(+-%(1!66%$%"#%(=%(4$!++%"(&'( (

Superfícies equipotenciais

Superfícies de mesma energia

V=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

V !"#$%&'%'( &'( &( )*'!+!,%( #-&$.%( /*,%'( &0*".( '*/%( 1!$%#+!*"2( '&3( x2( 4!+-( 2( +-%"(+-%$%(!'(&("*"5,&"!'-!".(#*/)*"%"+(*6( ! 7 8 V x ! ! " !" (!"(+-%(*))*'!+%(1!$%#+!*"( 9 :(;"(+-%( #&'%(*6(.$&,!+32(!6(+-%(.$&,!+&+!*"&0()*+%"+!&0(!"#$%&'%'(4-%"(&(/&''(!'(0!6+%1(&(1!'+&"#%(h2( +-%(.$&,!+&+!*"&0(6*$#%(/<'+(=%(1*4"4&$1:( 8> x E # $ ( ;6(+-%(#-&$.%(1!'+$!=<+!*"()*''%''%'(')-%$!#&0('3//%+$32(+-%"(+-%($%'<0+!".(%0%#+$!#(6!%01(!'( &( 6<"#+!*"( *6( +-%( $&1!&0( 1!'+&"#%( r2( !:%:2(!" % E_r"? :( ;"( +-!'( #&'%2( dV % #E dr_r : (;6( !'( @"*4"2(+-%"( ! (/&3(=%(*=+&!"%1(&'( 9 > V r " ( ( E_r? dV dr & ' % _{% #( )} * + !!" " ?" ( 9A:B:C>( ( D*$(%E&/)0%2(+-%(%0%#+$!#()*+%"+!&0(1<%(+*(&()*!"+(#-&$.%(q(!'(V r9 > % q 7 F,-₈r :(G'!".(+-%( &=*,%(6*$/<0&2(+-%(%0%#+$!#(6!%01(!'('!/)03(!!" % 9 Fq# ,-₈rH>"? :(( ( ( $%&%'()"*+,-./(*.+(!01,23/-./,*45( ((

I<))*'%( &( '3'+%/( !"( +4*( 1!/%"'!*"'( -&'( &"( %0%#+$!#( )*+%"+!&0( 9 2 >V x y :( J-%( #<$,%'(

#-&$&#+%$!K%1( =3( #*"'+&"+ V x y &$%( #&00%1( %L<!)*+%"+!&0( #<$,%':( ME&/)0%'( *6(9 2 > %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:( 2 V % #. !" ( 8"339:( (

R%6%$$!".( +*( D!.<$%( A:B:H2( 0%+( +-%( )*+%"+!&0( &+( &( )*!"+( 9 2 >P x y =%( 9 2 >V x y :( S*4( /<#-( !'(

#-&".%1(&+(&("%!.-=*$!".()*!"+(

V P x dx9 / 2 y dy/ >T(U%+(+-%(1!66%$%"#%(=%(4$!++%"(&'( (

( N8

Superfícies equipotenciais

Superfícies de mesma energia

V=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$ $

8)9 !"#$ #0#1(&)1$ +)#05$ 0).#*$ /&#$ %#&%#.5)1-0/&$ ('$ ("#$ #,-)%'(#.()/0*$ /.5$ %').($ +&'3$ "):"#&$('$0'6#&$%'(#.()/0*;$

$

8))9 <=$*=33#(&=>$("#$#,-)%'(#.()/0$*-&+/1#*$%&'5-1#5$2=$/$%').($1"/&:#$+'&3$/$+/3)0=$ '+$ 1'.1#.(&)1$ *%"#&#*>$ /.5$ +'&$ 1'.*(/.($ #0#1(&)1$ +)#05>$ /$ +/3)0=$ '+$ %0/.#*$ %#&%#.5)1-0/&$('$("#$+)#05$0).#*;$

$

8)))9 !"#$ (/.:#.()/0$ 1'3%'.#.($ '+$ ("#$ #0#1(&)1$ +)#05$ /0'.:$ ("#$ #,-)%'(#.()/0$ *-&+/1#$ )*$ 4#&'>$ '("#&6)*#$ .'.?@/.)*").:$ 6'&A$ 6'-05$ 2#$ 5'.#$ ('$ 3'@#$ /$ 1"/&:#$ +&'3$ '.#$ %').($'.$("#$*-&+/1#$('$("#$'("#&;$

$

8)@9 B'$6'&A$)*$&#,-)&#5$('$3'@#$/$%/&()10#$/0'.:$/.$#,-)%'(#.()/0$*-&+/1#;$ $

C$ -*#+-0$ /./0':=$ +'&$ #,-)%'(#.()/0$ 1-&@#*$ )*$ /$ ('%':&/%")1$ 3/%$ 8D):-&#$ E;F;G9;$ H/1"$ 1'.('-&$0).#$'.$("#$3/%$&#%&#*#.(*$/$+)I#5$#0#@/()'.$/2'@#$*#/$0#@#0;$J/("#3/()1/00=$)($)*$ #I%&#**#5$/*$ ;$K).1#$("#$:&/@)(/()'./0$%'(#.()/0$.#/&$("#$*-&+/1#$'+$ H/&("$)*$ >$("#*#$1-&@#*$1'&&#*%'.5$('$:&/@)(/()'./0$#,-)%'(#.()/0*;$ 8 > 9 1'.*(/.( z ! f x y ! g V ! zg $ $ _$ !"#$%&'()*)+$C$('%':&/%")1$3/%$ $ $ ,-./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 16

Equipotenciais

Equipotenciais e linhas de campo

!" #$%

-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 = σ/ε

₀

Condutores

• E perpendicular à superfície do condutor

• E=0 dentro do condutor

Potencial em um condutor

No condutor E=0: variação do potencial = 0

Campo elétrico = variação do potencial _{V constante}

no condutor

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

Capacitores

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 #

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

Capacitor de placas paralelas

!" #$%

-Parallel Plate Capacitor

top bottom V d ! " #

%

E S! $ ! $ Q d A& " Ed " d A V Q C " &$ ! "

C depends only on geometric factors A and d

Integral de trajetória para encontrar V

∆V

_{= −}

!

d 0

!

E

_{· d!s = Ed =}

σ

ε

₀

d =

Q

Aε

₀

d

Capacitor de placas paralelas

!" #$%

-Parallel Plate Capacitor

top bottom V d ! " #

%

E S! $ ! $ Q d A& " Ed " d A V Q C " &$ ! "

C depends only on geometric factors A and d

Integral de trajetória para encontrar V

∆V

_{= −}

!

d 0

!

E

_{· d!s = Ed =}

σ

ε

₀

d =

Q

Aε

₀

d

C =

Q

|∆V |

=

Aε

₀

d

Capacitor de placas paralelas

!" #$%

-Parallel Plate Capacitor

top bottom V d ! " #

%

E S! $ ! $ Q d A& " Ed " d A V Q C " &$ ! "

C depends only on geometric factors A and d

Integral de trajetória para encontrar V

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

dW = dq∆V = dq q V = 1 C qdq W = ! dW = ! Q 0 1 C qdq W = 1 C 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

U =

1

C

Q

2

2

=

1

2

C

|∆V |

2

Energia armazenada no capacitor

C =

Q

U =

1

C

Q

2

2

=

1

2

C

|∆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

C

"

!

₀

A d

/

and

|

# "

V

|

Ed

, we have

$

%

2

$

2 ₀ 0

1

1

1

|

|

2

2

2

E

A

U

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

E E

U

u

"

"

!

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

E

u

(

)

q

&

A

# "

#

experiences an attractive force

# "

F

&

2

(

#

A

) / 2

!

₀

. If the total area of the

plate is A, then an external agent must exert a force

F

_ext

"

&

2

A

/ 2

!

₀

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

2 0

ext

2

F

"

!

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

ext

F

2 0

ext ext ext

2

E A

W

"

)

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 E

u

"

W

Ad

"

!

E

/ 2

E

u

E

u

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 |

U =

1

C

Q

2

2

=

1

2

C

|∆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

C

"

!

₀

A d

/

and

|

# "

V

|

Ed

, we have

$

%

2

$

2 ₀ 0

1

1

1

|

|

2

2

2

E

A

U

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

E E

U

u

"

"

!

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

E

u

(

)

q

&

A

# "

#

experiences an attractive force

# "

F

&

2

(

#

A

) / 2

!

₀

. If the total area of the

plate is A, then an external agent must exert a force

F

_ext

"

&

2

A

/ 2

!

₀

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

2 0

ext

2

F

"

!

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

ext

F

2 0

ext ext ext

2

E A

W

"

)

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 E

u

"

W

Ad

"

!

E

/ 2

E

u

E

u

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-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

C

"

!

₀

A d

/

and

| # "V | Ed

, we have

$

%

2

$

2 ₀ 0

1

1

1

|

|

2

2

2

E

A

U

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

E E

U

u

"

"

!

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

E

u

( )

q

&

A

# " #

experiences an attractive force

# "F

&

2 (#A) / 2

!

₀

. If the total area of the

plate is A, then an external agent must exert a force

F_ext "

&

2 A / 2

!

₀

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

2 0

ext

2

F

"

!

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

ext

F

2 0

ext ext ext

2

E A

W

"

)

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 E u " W Ad "

!

E / 2 E

u

E

u

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!

26

Aumentando a capacitância

Dielétricos (visão microscópica)

Dielétricos polares

Dielétricos com momento

de dipolo permanente

Dielétricos (visão microscópica)

Dielétricos polares

Dielétricos com momento

de dipolo permanente

Ex: água

Dielétricos não polares (visão microscópica)

Dielétricos com momento de dipolo

induzido pelo campo elétrico

Dielétricos não polares (visão microscópica)

Dielétricos com momento de dipolo

induzido pelo campo elétrico

Ex: CH4

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 P Ah ! (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, #Q_P , 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 Q in terms of quantities we _P

know? The simplest way is to require that the electric dipole moment Q produces, _P

P

Q h , is equal to the total electric dipole moment of all the little electric dipoles. This

gives Q h Np_P ! , or P Np Q h ! (5.5.4)

Q

P

= Carga induzida

30

Dielétricos em capacitores

Dielétricos em capacitores

C =

Q

|∆V |

Aumento da capacitância com diminuição de ∆V

Dielétricos em capacitores

C =

Q

|∆V |

Aumento da capacitância com diminuição de ∆V

∆V diminui porque a polarização do dielétrico diminui o campo elétrico

Constante dielétrica

_κ

dielétricos diminuem o campo

elétrico original por um fator

_κ

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

Lei de Gauss num dielétrico

The capacitance becomes 0 0 0 0 | | | | e e Q Q C V V C ! _! " " " # # (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 E! ₀ in the region between the plates can be found by using Gauss’s law:

0 0 0 0 , S Q d E A E $ % % & " " ' "

((

E A!" "

#

We have see that when a dielectric is inserted (Figure 5.5.8), there is an induced chargeQ of opposite sign on the surface, and the net charge enclosed by the Gaussian _P

surface is Q Q) _P .

Figure 5.5.8 Gaussian surface in the presence of a dielectric.