Apostila de Eletromagnetismo Verso 2011

8/16/2019 Apostila de Eletromagnetismo Verso 2011

1/167

UNIVERSIDADE FEDERAL DE UBERLNDIA UFUFACULDADE DE ENGENHARIA ELTRICA FEELTP I N S

A

E

U

2011


2/167

U F U

F E E

E

P. I N S

2

SUMRIO GERAL

C C P

1 A V 03

2 L C I C E 20

3 D F E, L G D 35

4 E P P E 51

5 C, D C 71

6 E P L 96

7 C M E 107

8 F M, M I 132

9 C V T E M 157


3/167

U F U

F E E

E

P. I N S

3

1 ANLISE VETORIAL

1.1 E V

O

( ). E : , , ,

, , (), .

U , . E

: , , , .

U . U

, ,

.

1.2 V

A , .

A , .

A , , A B B A+ = +

. A

, , ( ) ( ) A B C A B C + + = + +

.

A ,

A B−

( ) A B+ −

; , , ,

.

V . O ,

, ,

. A

,


4/167

U F U

F E E

E

P. I N S

4

( )( ) ( ) ( )r s A B r A B s A B

rA rB sA sB

+ + = + + +

= + + +

A

.

A .

1.3 S C C

P , , , ,

. H ,

. O

. N

, , .

N () ,

, ,

, , x , y z . N

0 x = , 0 y = 0 z = .

T ( )1,2,3P ( )2, 2,1Q − ,

() . P , ,

1 x = , 2 y = 3 z = , Q

2 x = , 2 y = − 1 z = .


5/167

U F U

F E E

E

P. I N S

5

P, () , ( ), ,P x y z

( )' , ,P x dx y dy z dz+ + + . O P

'P 6 , ,

dv dx d y dz= ; dS dxdy ,

dydz dzdx . E , dL P 'P

( ) ( ) ( )2 2 2

dx dy dz+ + .

1.4

C V V UP ,

r

P . S r

x

, y

z

, r x y z= + +

, () .

O : ,

, ,

.


6/167

U F U

F E E

E

P. I N S

6

C,

. A () ̂xa , ̂ya ̂za

, , . C Pr

( )1,2,3P ,

: ˆ ˆ ˆ2 3P x y zr a a a= + +

. C

Qr

( )2, 2,1Q − , ˆ ˆ ˆ2 2Q x y zr a a a= − +

. U PQ

R

( )1,2,3P ( )2, 2,1Q − :

( ) ( ) ( )ˆ ˆ ˆ2 1 2 2 1 3

ˆ ˆ ˆ4 2

PQ Q P x y z

x y z

R r r a a a

a a a

= − = − + − − + −

= − −

O () .

E, B

, ˆ ˆ ˆ x x y y z z

B B a B a B a= + +

. E B

,

B

, B ,


7/167

U F U

F E E

E

P. I N S

7

2 2 2

x y z B B B B= + +

C

. C, ,

.

E B

2 2 2 ̂B

x y z

B Ba

B B B B= =

+ +

A

, .

E 01:E , ,

( )2, 2, 1P − − .

E 02:

D ( )

1, 2,1 M − ,( )3, 3,0 N −

( )2, 3, 4P − − − , :

) MN

R

;

) MN MP

R R+

;

) M r

;

) ̂MPa ;

) 2 3P N r r +

.

1.5 I C

U ( )

. N

.

E ,

,

. C

, .


8/167

U F U

F E E

E

P. I N S

8

S, , r

, G

, , ( )G r

; T ( )T r

.

P

, ,

. P ,

,

.

E 03:

U S

( ) ( ) ( )( ) ( ) ( )2 2 2

125ˆ ˆ ˆ1 2 1

1 2 1 x y zS x a y a z a

x y z = − + − + +

− + − + +

.

) C S

( )2,4,3P ;

) D S

P ;

) E ( ), , f x y z 1S =

.

1.6

P E

D A

B

, , ,

A

, B

. A,

cos AB A B A B θ ⋅ =

O , A

B . O , ,

, .

A B B A⋅ = ⋅

A

. P ,

. U

ˆ ˆ ˆ x x y y z z

A A a A a A a= + +


B B a B a B a= + +

. O

, , A B⋅

, . E,


9/167

U F U

F E E

E

P. I N S

9

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

ˆ ̂ ˆ ̂ ˆ ̂

ˆ ̂ ˆ ̂ ˆ ̂

ˆ ̂ ˆ ̂ ˆ ̂

x x x x x y x y x z x z

y x y x y y y y y z y z

z x z x z y z y z z z z

A B A B a a A B a a A B a a

A B a a A B a a A B a a


⋅ = ⋅ + ⋅ + ⋅

+ ⋅ + ⋅ + ⋅

+ ⋅ + ⋅ + ⋅

90,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0 x y y x x z z x y z z y

a a a a a a a a a a a a⋅ = ⋅ = ⋅ = ⋅ = ⋅ = ⋅ =

O ,

,

x x y y z z A B A B A B A B⋅ = + +

.

O ,

22 A A A A⋅ = =

, , ˆ ˆ 1 A Aa a⋅ = .

U

. P () B

a

cos cos Ba Ba B a B a Bθ θ ⋅ = =

N . A, B a⋅

B

a

,

.

P B

a

,

() a , , ( ) B a a⋅

.


10/167

U F U

F E E

E

P. I N S

10

E 04:

C ˆ ˆ ˆ2,5 3 x y z

G ya xa a= − +

( )4,5,2Q . D :

) O G

Q ;) A G

Q ( )13ˆ ˆ ˆ ˆ2 2 N x y za a a a= + − ;

) A G

Q ̂N a ;

) O Gaθ ( )QG r

̂N a .

E 05:

O ( )6, 1, 2 A − , ( )2,3, 4 B − − ( )3,1,5C − .

D:

) AB R

;)

AC R

;

) O BAC θ A ;

) A AB R

AC R

;

) O AB R

AC R

.

1.7 P V

D A

B

, , , A

B

, , A B×

, A B .

O A B×

; A B×

A

, B

A

B

; A B×

A

B

A

B

, .


11/167

U F U

F E E

E

P. I N S

11

N ,

̂N AB A B a A B senθ × =

O ̂N a . O

, ( ) A B B A× = − ×

.

S ̂xa ̂ya ,

ˆ ˆ ˆ x y z

a a a× = , ,

̂xa ̂ya z

. D , ˆ ˆ ˆ y z x

a a a× = ˆ ˆ ˆ z x y

a a a× = .

O

,

A

B

, .

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

ˆ ̂ ˆ ̂ ˆ ̂

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ̂ ˆ ̂ ˆ ̂

x x x x x y x y x z x z

y x y x y y y y y z y z

z x z x z y z y z z z z

A B A B a a A B a a A B a a



× = × + × + ×

+ × + × + ×

+ × + × + ×

J ˆ ˆ ˆ x y z

a a a× = , ˆ ˆ ˆ y z x

a a a× = ˆ ˆ ˆ z x y

a a a× = . O

, ,

. E


12/167

U F U

F E E

E

P. I N S

12

( ) ( ) ( )ˆ ˆ ˆ y z z y x z x x z y x y y x z A B A B A B a A B A B a A B A B a× = − + − + −

, ,

ˆ ˆ ˆ x y z

x y z

x y z

a a a

A B A A A

B B B

× =

U

,

.

E 06:

O ( )6, 1,2 A − , ( )2,3, 4 B − − ( )3,1,5C − .

D:

) AB AC R R×

;

) A ;) O .

1.8

S C C C

O , , ,

.

O ( ,

, )

. N

, ρ

φ , 0φ = . U

z 0 z = , 0 ρ = .

N

,

. E : ( ρ = ),

(φ = ) ( z = ),

() .


13/167

U F U

F E E

E

P. I N S

13

O

,

: â ρ

, âφ

̂za . A () .

O ,

, ,

, ˆ ˆ ̂z

a a a ρ φ

× = ,

, , ρ , φ z .

U

ρ , φ z d ρ , d φ dz . O ρ d ρ ρ + , φ d φ φ +

z z dz+ , () . N

d ρ dz , d φ ; d ρ φ . O

d d dz ρ φ ρ , ,

, .

A

. T


14/167

U F U

F E E

E

P. I N S

14

cos x

y sen z z

ρ φ

ρ φ

=

==

D , x , y z

( )2 2 0

arctan

x y

y

x

z z

ρ ρ

φ

= + ≥

=

=

O φ x y , .

D


A A a A a A a= + +

, ,

ˆ ˆ ˆ z z

A A a A a A a ρ ρ φ φ = + +

P

. A,

ˆ

ˆ

ˆ z z

A A a

A A a

A A a

ρ ρ

φ φ

= ⋅

= ⋅

= ⋅

,

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

x x y y

x x y y

z z

A A a a A a a

A A a a A a a

A A

ρ ρ ρ

φ φ φ

= ⋅ + ⋅

= ⋅ + ⋅

=

A , ̂xa â ρ φ ,

, ˆ ˆ cos x

a a ρ

φ ⋅ = ; ̂y

a â ρ

90º φ − , ˆ ˆ y

a a sen ρ

φ ⋅ = . O

.


15/167

U F U

F E E

E

P. I N S

15

P

â ρ âφ ̂za

̂xa ⋅ cosφ senφ − 0

̂ya ⋅ senφ cosφ 0

̂za ⋅ 0 0 1

A

, ,

E 07:

T ( ) ˆ ˆ ˆ x y z

B ya xa za= − +

.

E 08:P:

) D ( 4, 4; 115º; 2)C z ρ φ = = − = ;

) D ( 3,1; 2,6; 3) D x y z= − = = − ;

) D C D .

E 09:P:

) E 2 2

ˆ ˆ x y xa ya

D x y

+=

+

;

) C D

( )2; 0,2 ; 5 z ρ φ π = = = .


16/167

U F U

F E E

E

P. I N S

16

E 10:

T :) ˆ ˆ ˆ10 8 6

x y zF a a a= − +

( )10, 8,6P − ;

) ( ) ( )ˆ ˆ2 4 x yG x y a y x a= + − −

( ), ,P z ρ φ ;

) D ˆ ˆ ˆ20 10 3 z

H a a a ρ φ = − +

( )5, 2, 1P x y z= = = − .

1.9 S C E

A () .

I, r . A r =

.

A θ z

. A θ = .


17/167

U F U

F E E

E

P. I N S

17

A φ , φ

. E x 0 z = . A φ = z .

P

,

() .

O ˆr a , âθ âφ . O

,

ˆ ˆ ˆr

a a aθ φ

× = . P , ,

, r , θ φ , () . N

φ , , 3

2.

U

r , θ φ dr , d θ d φ , () . A

r r dr + dr ; θ

d θ θ + r d θ φ d φ φ +

r sen d θ φ . O 2r sen dr d d θ θ φ .

A

cos

cos

x r sen

y r sen sen

z r

θ φ

θ φ

θ

=

=

=

A

( )

( )

2 2 2

2 2 2

0

arccos 0º 180º

arctan

r x y z r

z

x y z

y

x

θ θ

φ

= + + ≥

= ≤ ≤

+ +

=

A

. O

. O .


18/167

U F U

F E E

E

P. I N S

18

P

ˆr a âθ âφ

̂xa ⋅ sen cosθ φ cos cosθ φ senφ −

̂ya ⋅ sen senθ φ cos senθ φ cosφ

̂za ⋅ cosθ senθ − 0

P, ,

, ,

cos

r sen

z r

ρ θ

φ φ

θ

=

=

=

A

( )

( )

2 20

arccos 0º 180º

r z r

z

ρ

ρ θ θ

φ φ

= + ≥

= ≤ ≤

=

J

. E

.

P

ˆr a âθ âφ

â ρ ⋅ senθ cosθ 0

âφ ⋅ 0 0 1

̂za ⋅ cosθ senθ − 0

E 11:

C ̂x

xzG a

y=

( ) .


19/167

U F U

F E E

E

P. I N S

19

E 12:

D , ( )3,2,1 A − ( )5, 20º , 70º B − , :) A A ;

) A B ;

) A A . B

E 13:T :

) ˆ10 xa ( )3,2,4P − ;

) ˆ10 ya ( )5,30º,4Q ;

) ˆ10 za ( )4,110º,120º M .

E 14:T :

) ˆ15a ρ ( )1, 3,5P − ;

) ˆ15aφ ( )2, 10º, 3Q − − ;

) ˆ15 za ( )3,45º,60º M .


20/167

U F U

F E E

E

P. I N S

20

2 LEI DE COULOMB E INTENSIDADE DE CAMPO ELTRICO

2.1 L C

L C: ,

, ,

1 2

2

Q QF k

R=

[N]

O k C. E

, ,

.

A k

0

1

4k

πε =

0ε , ,

SI (S I),

12 2 2

0 8,85418781762 10 / C N mε −= × ⋅

9 2 28,99 10 / k N m C = × ⋅

A C

1 22

04Q QF

Rπε =

P C,

. P, ,

.


21/167

U F U

F E E

E

P. I N S

21

N , 1r

1Q 2r

2Q . E

12 2 1 R r r = −

1Q 2Q , . O 2F

2Q 1Q 2Q . A

C

1 22 122

0 12

ˆ4

Q QF a

Rπε =

12â = 12 R

. E

, ( ).

E 01:

S 41

3.10Q C −= ( )1,2,3 M 42 10Q C −

= −

( )2,0,5 N . E 1Q 2Q .

E 02:

U 20 AQ C µ = − ( )6,4,7 A − 50 BQ C µ =

( )5,8, 2 B − . S ,

AQ BQ .

2.2 I C E

S 1Q ,

, t Q , ,

; ,

. A


22/167

U F U

F E E

E

P. I N S

22

112

0 1

ˆ

4

t t t

t

Q QF a

Rπε

=

A

t F

t Q . U E

,

112

0 1

ˆ4

t t

t t

F Q E a

Q Rπε = =

A N C, , . D ,

2

0

ˆ4

R

Q E a

Rπε =

R R R

,

Q E

, ̂Ra

R

.

S Q , ̂Ra

ˆr a , R r . A,

2

0

ˆ4

r

Q E a

r πε =

J ,

ˆ ˆ ˆ x y z

R r xa ya za= = + +

2 2 2

ˆ ˆ ˆˆ x y z R xa ya zaa x y z

+ +=

+ +; ,

( )2 2 2 2 2 2 2 2 2 2 2 20ˆ ˆ ˆ

4 x y z

Q x y z E a a a

x y z x y z x y z x y zπε

= + + + + + + + + + +

E, ,

, .

S , , . P Q


23/167

U F U

F E E

E

P. I N S

23

ˆ ˆ ˆ' ' ' ' x y z

r x a y a z a= + +

, ,

ˆ ˆ ˆ x y zr xa ya z a= + + , R

'r r −

,

( ) 20

'

'4 '

Q r r E r

r r r r πε

−=

−−

( ) ( )

3

0

'

4 '

Q r r E r

r r πε

−=

−

( ) ( ) ( ) ( )

( ) ( ) ( )3/2

2 2 2

0

ˆ ˆ ˆ' ' '

4 ' ' '

x y zQ x x a y y a z z a

E r x x y y z zπε

− + − + −

= − + − + −

P

, ,

,

( ) 1 21 22 2 20 1 0 2 0

ˆ ˆ ˆ4 4 4

nn

n

QQ Q E r a a ar r r r r r πε πε πε

= + + +− − −

A

P 1Q 2Q .


24/167

U F U

F E E

E

P. I N S

24

E 03:

D ( )1,1,1P 3 nC

( )1 1,1,0P , ( )2 1,1,0P − , ( )3 1, 1,0P − − ( )4 1, 1,0P − , .

E 04:

U 3 C µ − ( )25, 30,15 A − ( cm )

0,5 C µ ( )10,8,12 B cm− . D E

:

) N ;

) E ( )15,20,50P cm .

E 05:

U 100 nC ( )1,1,3 A − . P:

) E ( ), ,P x y z 500 / x E V m= ;

) D 1 y ( )12, , 3 A y− .


25/167

U F U

F E E

E

P. I N S

25

2.3 C D D V C

C

C ,

v ρ , C

(C/3).

U Q∆ v∆

vQ v ρ ∆ = ∆

A ,

v

vol

Q dv ρ = ∫

N , dv

, , .

E 06:

D 2 cm .


26/167

U F U

F E E

E

P. I N S

26

E 07:

C :

) 1 , , 2 x y z≤ ≤ 3 3 3

1v

x y z ρ = ;

) 0,1;0 ;2 4o z ρ φ π ≤ ≤ ≤ ≤ ≤ ≤ ( )2 2sen 0,6v z ρ ρ φ = ;

) U 2

2

r

v

e

r ρ

−

= .

2.4 C L C

C ,

L ρ , C (C/).

C z

( ) −∞ ∞ , . D

E

L ρ .

E ( )0, ,0P y y . A

P ' LdQ dz ρ = ,

( )3

0

'

4 '

Q r r E

r r πε

−=

−

( )


27/167

U F U

F E E

E

P. I N S

27

( )3

0

' '

4 '

L dz r r dE

r r

ρ

πε

−=

−

( )

ˆ ˆ

ˆ' '

y

z

r ya a

r z a

ρ ρ = =

=

ˆ ˆ' ' z

r r a z a ρ

ρ − = −

P,

( )

( )3/2

2 2

0

ˆ ˆ' '

4 '

L zdz a z a

dE z

ρ ρ ρ

πε ρ

−=

+

P , , z E

,

E ρ

,

( )3/2

2 2

0

'

4 '

L dzdE z

ρ

ρ ρ

πε ρ =

+

( )3/2

2 2

0

'

4 '

L dz E z

ρ

ρ ρ

πε ρ

∞

−∞=

+∫

I ,

2 2 20

1 '

4 '

L z

E z

ρ

ρ ρ

πε ρ ρ

∞

−∞

= +

02

L E ρ ρ

πε ρ

=


28/167

U F U

F E E

E

P. I N S

28

, :

0

ˆ2

L E a ρ ρ

πε ρ =

E , .

D

z . C , z

6 x = , 8 y = , . D E

( ), ,P x y z .

N ,

ρ P , ( ) ( )2 2

6 8 R x y= − + −

â ρ

̂Ra . A,

( ) ( )2 2

0

ˆ2 6 8

L R E a

x y ρ

πε =

− + −

( ) ( )

( ) ( )2 2

ˆ ˆ6 8ˆ

6 8

x y

R

x a y aa

x y

− + −=

− + −

P,


29/167

U F U

F E E

E

P. I N S

29

( ) ( )

( ) ( )2 20

ˆ ˆ6 8

2 6 8

x y L x a y a

E x y

ρ

πε

− + −=

− + −

N, , z . D ,

:

0

ˆ2

L R E a

R

ρ

πε =

E 08:

U 16 / nC m 2 y = − ,

5 z = . D ( )1,2,3P .

E 09:

D 5 / nC m x y

. D :

) ( )0,0,4 AP ;

) ( )0,3,4 BP .

2.5 C L C

O S ρ

2 / C m . E .

C yz , .


30/167

U F U

F E E

E

P. I N S

30

U , ,

, ,

. A ' L S dy ρ ρ = . A

,

0

ˆ2

L R E a

R

ρ

πε =

( )

0

'ˆ

2

S R

dydE a

R

ρ

πε =

( )

ˆ

ˆ' '

x

y

r xa

r y a

=

=

ˆ ˆ' ' x y R r r xa y a= − = −

2 2' R R x y= = +

2 2

ˆ ˆ'ˆ

'

x y

R

xa y a Ra

R x y

−= =

+

P,

( )

( )2 20

ˆ ˆ' '

2 '

S x ydy xa y adE

x y

ρ

πε

−=

+

A , y E

, x E

,

( )2 20

'

2 '

S x

x dydE

x y

ρ

πε =

+

( )2 2

0

'

2 '

S x

x dy E

x y

ρ

πε

∞

−∞=

+

∫


31/167

U F U

F E E

E

P. I N S

31

I ,

0

'

2

S x

y E arctg

x

ρ

πε

∞

−∞

=

02

S x E

ρ

ε =

S P x ,

02

S x E

ρ

ε = −

, .

E ̂N a ,

. E,

0

ˆ

2

S N E a

ρ

ε

=

S , S ρ − ,

x a= ,

. N x a> ,

0 0

02 2

S S x x

E E E a a ρ ρ

ε ε + −

= + = − =

0 x < ,

( ) ( )0 0

02 2

S S x x E E E a a

ρ ρ

ε ε + −

= + = − − − =

0 x a< < ,

( )0 0 02 2

S S S x x x E E E a a a

ρ ρ ρ

ε ε ε + −

= + = − − =

E ,

, .


32/167

U F U

F E E

E

P. I N S

32

E 10:

T :23 / nC m 4 z = − ;

26 / nC m 1 z = 28 / nC m− 4 z = . D

:

) ( )2,5, 5 AP − ;

) ( )4,2, 3 BP − ;

) ( )1, 5,2C P − − ;

) ( )2,4,5 DP − .

2.6

L F E C

A

. E .

P , :

• A .

• A

.

• A

.

C, ,

, ,

, .


33/167

U F U

F E E

E

P. I N S

33

N , 0 z E = . A

z . N ,

, x E y E .

A

y

x

E dy

E dx=

C

02 L ρ πε = ,

1ˆ E a ρ

ρ =

E ,

2 2 2 2ˆ ˆ

x y x y E a a

x y x y= +

+ +


34/167

U F U

F E E

E

P. I N S

34

A,

y

x

E dy y

dx E x= =

dy dx

y x=

P,

ln ln y x C = + ln ln ln y x C = +

,

y C x=

S ,

C.

E 11:

D ( )1,4, 2P − :

) 2

2

8 4ˆ ˆ

x y

x x E a a

y y

−= +

;

) ( )5 ˆ ˆ2 5 1 x x y E e y x a xa = + +

.


35/167

U F U

F E E

E

P. I N S

35

3 DENSIDADE DE FLUO ELTRICO, LEI DE GAUSS E

DIVERGNCIA

3.1 D F E

A F

. A

Q+ , ,

. C F , ,

.

E

,

. O Ψ () , ,

QΨ =

O C. P , ,

, .

A

. T D

. A D

,

. A

D

, , C (

, ).


36/167

U F U

F E E

E

P. I N S

36

N, ,

2 ˆ

4 r

r a

Q D a

aπ ==

( )

2 ˆ

4 r

r b

Q D a

bπ ==

( )

r , a r b≤ ≤ ,

2

ˆ4 r

Q D a

r π =

S Q,

r

.

C

2

0

ˆ4

r

Q E a

r πε =

, ,

0 D E ε =

E ,

, ,

, .

E 01:

E D

8 / nC m z .

E 02:

C D

( )2, 3,6P −

:

) 55Q mC = ( )2,3, 6 M − − ;

) 20 / L mC m ρ = x ;

) 2120 / S C m ρ µ = 5 z = − .


37/167

U F U

F E E

E

P. I N S

37

3.2 L G

I , ,

, . S

Q+ C, Q C ,

D

S

D

, S D

, S

D

.

E , ,

S ∆

,

( ∆Ψ ) :

S normal S D S D S ∆Ψ = ∆ = ⋅ ∆

O

S ∆

,

S S D dS Ψ = ⋅∫

( )

E , , . T . T, ,

G,

carga envolvidaS

S D dS QΨ = ⋅ = =∫

A , ,

, , . C

, L

G


38/167

U F U

F E E

E

P. I N S

38

S vS vol D dS dv ρ ⋅ =∫ ∫

.

P G,

F Q

r . T

( )0 0 20

2

2 2 2

ˆ4

ˆ 44 4 4

S r S S S

r S S

Q D dS E dS a dS

r

Q Q Qa dS dS r Q

r r r

ε ε πε

π π π π

⋅ = ⋅ = ⋅

= ⋅ = = =

∫ ∫ ∫

∫ ∫

Q C ,

, Q C. A

S

D

dS

, ,

J

22 2

0 04esferaS r sen d d r

φ π θ π

φ θ θ θ φ π

= =

= == =∫ ∫

, ,

.

V

S S D dS Ψ = ⋅∫

( )


39/167

U F U

F E E

E

P. I N S

39

E 03:

D 60 C µ , :

) 26r cm= 0 /2θ π < < 0 /2φ π < < ;

) 26 cm ρ = 26 z cm= ± ;

) 26 z cm= .

E 04:

D 2 2ˆ0,3 / r D r a nC m=

, :

) E

( )2,25º,90ºP .

) 3r = .) 4r = .

E 05:

C , , 5 x y z = ±

:

) , 0,1 C µ ( )1, 2,3− 17 C µ ( )1,2, 2− − ;

) / C mπ µ 2 x = − 3 y = ;

) 20,1 / C m µ 3 y x= .

3.3 A L G: A D S

C

A G

:

1. S

D

,

S D dS ⋅

S D dS ⋅ , .

2. N S

D dS ⋅

,S D .

I S D dS

S D . A , ,

S

D

, .

V Q

. A

, r


40/167

U F U

F E E

E

P. I N S

40

.S

D

;S D

.

T, ,

22

0 0

24

S S S esfera

S S esfera

S

Q D dS D dS

D dS D r sen d d

r D

φ π θ π

φ θ θ θ φ

π

= =

= =

= ⋅ =

= =

=

∫ ∫

∫ ∫ ∫

24S Q

D r π =

C r S D

,

2 ˆ

4 r

Q D a

r π =

2

0

ˆ4

r

Q E a

r πε =

C.

C ,

z −∞ +∞ .

N , D ρ

z . A

ρ 0 z = z L= .


41/167

U F U

F E E

E

P. I N S

41

A G

2

0 00 0 2

S S S S S S cilindro lateral topo base

z L

S S S lateral z

Q D dS D dS D dS D dS D dS

D dS D d dz D Lφ π

φ ρ φ πρ

= =

= =

= ⋅ = ⋅ = ⋅ + ⋅ + ⋅

= + + = =

∫ ∫ ∫ ∫ ∫

∫ ∫ ∫

2

S

Q D

Lπρ =

E :2 2

L LS

L D

L

ρ ρ

πρ πρ = = . R ,

ˆ2

L D a ρ ρ

πρ =

0

ˆ2

L E a ρ ρ

πε ρ =

E C.

U . S

, a b ,

, . C S ρ

. A .

U L ρ , a b ρ < < ,

,

2S Q D Lπρ =

2

S

Q D

Lπρ =


42/167

U F U

F E E

E

P. I N S

42

E :2

2

S S S

aL a D

L

ρ π ρ

πρ ρ

= = . P ,

, LQ L ρ = , ,2 2

L LS

L D

L

ρ ρ

πρ πρ = = . E

,

ˆ2

L D a ρ ρ

πρ =

0

ˆ2

L E a ρ ρ

πε ρ =

.

C b ρ > , , . U

a ρ < ,

, .

E 06:

C 50 cm 1 mm

4 mm , . A

30 nC . C , G,

, .

E 07:

U 0,25 C µ ,

: 22 / mC m 1r cm= 20,6 / mC m−

1,8r cm= . C :

) 0,5r cm= ;

) 1,5r cm= ;

) 2,5r cm= ;

) Q 3r cm=

3,5r cm= ?

3.4 A L G: E D V

A G

. P

, G,

D

,


43/167

U F U

F E E

E

P. I N S

43

D

D

T.

C P , ,

. O D

,0 0 0 0


D D a D a D a= + +

.

E , P ,

x∆ , y∆ z∆ , G,

S S frente atrás esquerda direita topo base D dS Q⋅ = + + + + + =∫ ∫ ∫ ∫ ∫ ∫ ∫

, , .

C ,

,

ˆ

frente frente frente

frente x

x frente

D S

D y z a D y z

= ⋅ ∆

= ⋅ ∆ ∆

= ∆ ∆

∫

x D . A

/ 2 x∆ P ,

, 0

0

taxa de variação de com2

2

x frente x x

x x

x D D D x

D x D

x

∆= + ×

∂∆= +

∂


44/167

U F U

F E E

E

P. I N S

44

T,

02

x x

frente

D x D y z

x

∂∆ = + ∆ ∆

∂ ∫

C ,

( )

,

ˆ

atrás atrásatrás

atrás x

x atrás

D S

D y z a

D y z

= ⋅ ∆

= ⋅ −∆ ∆

= − ∆ ∆

∫

, ,

, 02

x x atrás x

D x D D

x

∂∆= −

∂

0

2

x x

atrás

D x D y z

x

∂∆ = − + ∆ ∆

∂ ∫

S ,

x frente atrás

D x y z x

∂+ = ∆ ∆ ∆

∂∫ ∫

U ,

y

direita esquerda

D x y z

y

∂+ = ∆ ∆ ∆

∂∫ ∫

z

topo base

D x y z

z

∂+ = ∆ ∆ ∆

∂∫ ∫

S x y z v∆ ∆ ∆ = ∆ , :

y x zS

S

D D D D dS Q v

x y z

∂ ∂ ∂⋅ = = + + ∆

∂ ∂ ∂ ∫

A v∆ .

E 08:

D 9 310 m−

, ( ) ( ) 2ˆ ˆ ˆsen cos 2 / x x x y z D e y a e y a za C m− −= − +

.


45/167

U F U

F E E

E

P. I N S

45

E 09:

N , 4 2 4 2 3 2ˆ ˆ ˆ8 4 16 / x y z D xyz a x z a x yz a pC m= + +

. D:

) 2 z = , 0 2 x< < , 1 3 y< <

̂za ;

) ( )2, 1,3P − ;

)

( )2, 1,3P − 12 310 m− .

3.5 D

N ,

y x zS

S

D D D D dS Q v

x y z

∂ ∂ ∂⋅ = = + + ∆

∂ ∂ ∂ ∫

O , v∆

. E

S y x S z D dS D D D Q

x y z v v

⋅∂ ∂ ∂+ + = =

∂ ∂ ∂ ∆ ∆

∫

P, ,

0 0lim lim

S y x S z

v v

D dS D D D Q

x y z v v∆ → ∆ →

⋅∂ ∂ ∂+ + = =

∂ ∂ ∂ ∆ ∆

∫

v ρ ,

0lim

S y x S zv

v

D dS D D D

x y z v ρ

∆ →

⋅∂ ∂ ∂+ + = =

∂ ∂ ∂ ∆

∫

P , , ,

0lim

S y x S z

v

D dS D D D

x y z v∆ →

⋅∂ ∂ ∂+ + =

∂ ∂ ∂ ∆

∫

.


46/167

U F U

F E E

E

P. I N S

46

A D

,

A

(, , , .). P

0lim

y x S z

v

A dS A A A

x y z v∆ →

⋅∂ ∂ ∂+ + =

∂ ∂ ∂ ∆

∫

E

, . A A

0Divergência de div lim S

v

A dS A Av∆ →⋅= =

∆∫

A

. E A

. A :

A A

.

P ,

. O , ,

, .

P .

E,

, ,

, . A

, , . J ,

.

U

. D ,

(). C ,

.

A

div y x z

D D D D

x y z

∂ ∂ ∂= + +

∂ ∂ ∂


47/167

U F U

F E E

E

P. I N S

47

E . C

, .

( )1 1

div z D D

D D z

φ

ρ ρ

ρ ρ ρ φ

∂ ∂∂= + +

∂ ∂ ∂

( ) ( )221 1 1

div r D

D r D sen Dr r r sen r sen

φ

θ θ θ θ θ φ

∂ ∂ ∂= + +

∂ ∂ ∂

A , ,

,

.

E 10:

D div D

( ) ( ) 2ˆ ˆ ˆsen cos 2 / x x x y z D e y a e y a za C m− −

= − +

.

E 11:

D div D

:) ( ) ( )2 2 2 2ˆ ˆ ˆ2 2 / x y z D xyz y a x z xy a x ya C m= − − − +

( )2,3, 1P − ;

) ( ) ( ) ( )2 2 2 2 2 2ˆ ˆ ˆ2 sen sen 2 2 sen / z D z a z a z a C m ρ φ ρ φ ρ φ ρ φ = − +

( )2,110º, 1P − ;

) ( ) ( ) ( ) ( ) ( ) 2ˆ ˆ ˆ2 sen cos cos cos sen / r D r a r a r a C mθ φ θ φ θ φ φ = + −

( )1,5;30º;50ºP .

3.6 P E M (E)

A

0div lim

S S

v

D dS D

v∆ →

⋅=

∆

∫

div y x z

D D D D

x y z

∂∂ ∂= + +

∂ ∂ ∂

div v D ρ =

A ;

,


48/167

U F U

F E E

E

P. I N S

48

. E

( ) Lei de GaussS S D dS Q⋅ =∫

S S D dS Q

v v

⋅=

∆ ∆

∫

0 0lim lim

S S

v v

D dS Q

v v∆ → ∆ →

⋅==

∆ ∆

∫

divv D ρ =

E M. E

. A M

G. D , G

M.

A ;

.

E 12:D :

) 2 2

2

2

4 2 2ˆ ˆ ˆ / x y z

xy x x y D a a a C m

z z z= + +

;

) ( ) ( ) ( ) 2ˆ ˆ ˆsen cos sen / z D z a z a a C m ρ φ φ φ ρ φ = + +

;

) ( ) ( ) ( ) ( ) ( ) 2ˆ ˆ ˆsen sen cos sen cos / r D a a a C mθ φ θ φ θ φ φ = + +

.

3.7 O O V ∇

(N) T D

D ∇

,

:

ˆ ˆ ˆ x y za a a

x y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂

C ∇

D

,


49/167

U F U

F E E

E

P. I N S

49

( )ˆ ˆ ˆ ˆ ˆ ˆ x y z x x y y z z D a a a D a D a D a x y z

∂ ∂ ∂∇ ⋅ = + + ⋅ + +

∂ ∂ ∂

y x z D D D

D x y z

∂∂ ∂∇ ⋅ = + +

∂ ∂ ∂

I D

,

div y x z

D D D D D

x y z

∂∂ ∂= ∇ ⋅ = + +

∂ ∂ ∂

O D∇ ⋅ div D . A ,

D∇ ⋅

.

O ∇

. S

D

, D∇ ⋅

D

, , ∇

.

I . E

. P G,

S D dS Q⋅ =∫

vvol

Q dv ρ = ∫

v ρ ,

v D ρ ∇ ⋅ =

vS vol vol D dS Q dv D dv ρ ⋅ = = = ∇ ⋅∫ ∫ ∫

A ,

S vol D dS D dv⋅ = ∇ ⋅∫ ∫

:


50/167

U F U

F E E

E

P. I N S

50

A

.

N,

. S

.

O ,

,

. A ,

, ,

. E , , ,

.

E 13:

C 2 2ˆ ˆ2 / x y D xya x a C m= +

0 1 x e= , 0 2 y e= 0 3 z e= .

E 14:

D ( ) ( ) 21 12 2ˆ ˆ6 sen 1,5 cos / D a a C m ρ φ ρ φ ρ φ = +

2 ρ = , 0 eφ π = 0 5 z e= .


51/167

U F U

F E E

E

P. I N S

51

4 ENERGIA POTENCIAL E POTENCIAL ELTRICO

4.1 T E M C I

C E

S ,

,

. J

, ; , .

A Q E

E F QE =

A dL

ˆ ˆ EL E L L

F F a QE a= ⋅ = ⋅

̂La dL

.

A

, ,

ˆaplicada L

F QE a= − ⋅

J

. P

Q ̂La

( )ˆ ˆ L LdW QE a dL QE a dL QE dL= − ⋅ = − ⋅ = − ⋅

̂La dL dL

.

O

final

inicialW Q E dL= − ⋅

∫


52/167

U F U

F E E

E

P. I N S

52

. C, ,

.

E 01:

D ( ) ( )2 221

ˆ ˆ ˆ8 4 4 / x y z E xyza za x ya V m z

= + −

,

6 nC 2 m µ ,

( )2, 2,3P − :

) 6 3 2

ˆ ˆ ˆ ˆ7 7 7

L x y za a a a= − + + ;

) 6 3 2

ˆ ˆ ˆ ˆ7 7 7

L x y za a a a= − − .

4.2 I L

A , ,

. S ,

final

LinicialW Q E dL= −

∫

L E = E

dL

.

O ,

B A

. O .


53/167

U F U

F E E

E

P. I N S

53

O Q B A , ,

( )1 1 2 2 6 6 L L LW Q E L E L E L= − ∆ + ∆ + + ∆…

, ,

( )1 1 2 2 6 6W Q E L E L E L= − ⋅ ∆ + ⋅ ∆ + + ⋅ ∆

…

,

( )1 2 6W QE L L L= − ⋅ ∆ + ∆ + + ∆

…

A ,

, BA

L

. P,

( )uniforme BAW QE L E = − ⋅

P ,

Q, E

BA

L

. E

, , B A

.

N dL

,

.

ˆ ˆ ˆ (coordenadas cartesianas)

ˆ ˆ ˆ (coordenadas cilíndricas)

ˆ ˆ ˆ (coordenadas esféricas)

x y z

z

r

dL dxa dya dza

dL d a d a dza

dL dra rd a r sen d a

ρ φ

θ φ

ρ ρ φ

θ θ φ

= + +

= + +

= + +

P , , .


54/167

U F U

F E E

E

P. I N S

54

O ,

0

ˆ ˆ2

L E E a a ρ ρ ρ ρ

πε ρ = =

D 1 ρ ,

(), 1 ˆdL d aφ ρ φ =

, :

2

1

0 0 1

2

00

ˆ ˆ

2

ˆ ˆ 02

final

inicial

L

L

W Q E dL

Q a d a

Q d a a

π

ρ φ

π

ρ φ

ρ ρ φ

πε ρ ρ

φ πε

= − ⋅

= − ⋅

= − ⋅ =

∫

∫

∫

C a ρ = b ρ =

, () . A, ˆdL d a ρ

ρ =

0

0

0

ˆ ˆ2

2

ln2

b

a

b

a

L

L

b L

a

W Q a d a

d Q

Q

ρ

ρ ρ ρ

ρ

ρ

ρ ρ

πε ρ

ρ ρ

πε ρ

ρ ρ

πε ρ

= − ⋅

= −

= −

∫

∫

Cb

ρ a ρ , ,

, , . J b ρ

a ρ , ,

0

ln2

b L

a

QW

ρ ρ

πε ρ =

N , ( )

.

E 02:

D ˆ ˆ ˆ2 x y z E ya xa a= + +

,

2C ( )1,0,1 B ( )0,8;0,6;1 A 2 2

1 x y+ = 1 z = .


55/167

U F U

F E E

E

P. I N S

55

E 03:

C 4C ( )1,0,0 B ( )0,2,0 A 2 2 , 0 y x z= − = :

) ˆ5 / x E a V m=

;

) ˆ5 / x E xa V m=

;

) ˆ ˆ5 5 / x y

E xa ya V m= +

.

E 04:

( ) C ( )ˆ5 / x E xa V m=

,

3C ( )1,3,5 B ( )2,0,3 A :

) ( )1,3,5 B ( )2,3,5 ( )2,0,5 ( )2,0,3 A ;

) ( )1,3,5 B ( )1,3,3 ( )1,0,3 ( )2,0,3 A .

4.3 D D P P E

D ( )

( )1Q = ,

Diferença de Potencial final

inicialV E dL= = − ⋅∫

A ABV A B ,

B A . A,

( ) V A

AB B

V E dL= − ⋅∫

, ,

, .

N ,

Q b ρ a ρ

0

ln2

b L

a

QW

ρ ρ

πε ρ =

A, b

ρ a

ρ


56/167

U F U

F E E

E

P. I N S

56

0

ln

2

b Lab

a

W V

Q

ρ ρ

πε ρ

= =

J , A B

Ar

Br

, Q ,

2

0

ˆ ˆ4

r r r

Q E E a a

r πε = =

ˆr

dL dr a=

2

0 0

1 1

4 4

A

B

A r

AB B r

A B

Q QV E dL dr

r r r πε πε

= − ⋅ = − = −

∫ ∫

S B Ar r > , ABV ,

Br

Ar . I

.

M , ,

,

,

.

O

, T. O

. E

. M ,

. N, ,

.

S A AV B BV ,

AB A BV V V = −

AV BV

. O ABV

AB B Ar r r = −

.


57/167

U F U

F E E

E

P. I N S

57

E 05:

U ( )2 ˆ ˆ ˆ6 6 4 / x y z E x a ya a V m= + +

. D:

) MN V ( )2,6, 1 M − ( )3, 3,2 N − − ;

) M V 0V = ( )4, 2, 35Q − − ;

) N V 2V = ( )1,2, 4P − .

4.4 C P C P

N ,

Ar r = Br r = , Q

.

0

1 1

4 AB A B

A B

QV V V

r r πε

= − = −

C . A,

A B θ φ , .

O dL

r , θ φ ,

. T, , ,

2

0 0

1 1

4 4

A A

B B

A r r

AB r B r r

A B

Q QV E dL E dr dr

r r r πε πε

= − ⋅ = − = − = −

∫ ∫ ∫

O , ,

.

A, 0V = , Ar


58/167

U F U

F E E

E

P. I N S

58

04

A

A

QV

r πε

=

, A ,

04

QV

r πε =

P

. N

, ,

, .

O: , ,

B A

abW QV =

E 06:

U 15nC . C 1V

( )1 2,3, 1P − − :

) 0V = ( )6,5,4 ;) 0V = ;

) 5V V = ( )2,0,4 .

4.5 C P S C P

C C P

O , 1Q 1r

,

r . P

,

( ) 1

0 14

QV r

r r πε =

−

O , 1Q 1r

2Q 2r

,

, ,

( ) 1 20 1 0 24 4Q QV r

r r r r πε πε = +

− −


59/167

U F U

F E E

E

P. I N S

59

C , n

( ) 1 2

0 1 0 2 04 4 4

n

n

QQ QV r

r r r r r r πε πε πε = + + +

− − −

…

S

v v ρ ∆ ,

( ) ( ) ( ) ( )1 1 2 2

0 1 0 2 04 4 4

v v v n n

n

r v r v r vV r

r r r r r r

ρ ρ ρ

πε πε πε

∆ ∆ ∆= + + +

− − −

…

F , :

( ) ( )

0

' '

4 '

v

vol

r dvV r

r r

ρ

πε =

−∫

E . P

, , ,

( )

( )

0

' '

4 '

L r dL

V r r r

ρ

πε =

−∫

( ) ( )

0

' '

4 '

S

S

r dS V r

r r

ρ

πε =

−∫

C

. P V z

L ρ a ρ = 0 z = ,

.


60/167

U F U

F E E

E

P. I N S

60

T, :

' 'dL d a d ρ φ φ = = ; ̂zr z a=

; ˆ'r a a ρ

=

E,

2 2'r r a z− = +

2

2 2 2 20

0 0

'

4 2

L La d aV a z a z

π ρ φ ρ

πε ε = =

+ +∫

P , :

1. O

, .

2. O

.

3. O

.

R, ,

,

0 E dL⋅ =∫


61/167

U F U

F E E

E

P. I N S

61

L : A

AB B

V E dL= − ⋅∫

.

A

. E G , ,

.

A , , E

.

Q ( ,

), . O

( ) . U ,

()

() .

E 07:

T , ( )0,2,2

:

) 12 / nC m 2,5m ρ = , 0 z = ;

) 18nC ( )1,2, 1− ;) 12 nC 2,5 y m= , 0 z = .

E 08:

C ( ) ˆsen E aφ πρ =

1r ρ = . V

.

4.6 G C P E

T ,

. E,

, ,

.

J ,

.

J


62/167

U F U

F E E

E

P. I N S

62

V E dL= − ⋅∫

E L∆

E

, V ∆ ,

V E L∆ = − ⋅ ∆

C , , E

V

.

S L∆

E

θ , ,

cosV E L θ ∆ = − ∆

M ,

cosdV E dL θ = −

D / dV dL

cosdV

E dL

θ = −

D : L∆

V ∆ ? L E

L∆

. A L∆ . E, ,

máxV ∆ cos θ 1− , , L∆

E

. P ,

máx

dV E

dL

=


63/167

U F U

F E E

E

P. I N S

63

E E

V

:

1. A

.

2. O

E

, , E

.

A .

C , E

,

, ( ). T,

L∆

, V ∆

, ,

0V E L∆ = − ⋅ ∆ =

E

L∆

, E

L∆

, ,

, .

C ˆ N

a

,

̂N

máx

dV E a

dL= −

E E

V

(

).


64/167

U F U

F E E

E

P. I N S

64

C / máx

dV dL L∆

̂N a ( ),

máx

dV dV

dL dN =

̂N dV

E adN

= −

E

. A V E

. O

T

ˆGradiente de grad N dT

T T adN

= =

̂N a

T . U ,

grad E V = −

C V x , y , z , ,

( ), , x y z ,

V V V dV dx dy dz

x y z

∂ ∂ ∂= + +

∂ ∂ ∂

,

.

E

1ˆ ˆgrad N N

dV V V V E V a dx dy dz a

dN dN x y z

∂ ∂ ∂= − = − = − + +

∂ ∂ ∂

1ˆ ˆ ˆ

x x x x

V V V V E a dx dy dz a a

dx x y z x

∂ ∂ ∂ ∂= − + + = −

∂ ∂ ∂ ∂

1ˆ ˆ ˆ

y y y y


dy x y z y

∂ ∂ ∂ ∂= − + + = −

∂ ∂ ∂ ∂


65/167

U F U

F E E

E

P. I N S

65

1ˆ ˆ ˆ

z z z z


dz x y z z

∂ ∂ ∂ ∂= − + + = −

∂ ∂ ∂ ∂

P,

x

V E

x

∂= −

∂ y

V E

y

∂= −

∂ z

V E

z

∂= −

∂

E

ˆ ˆ ˆ grad x y zV V V

E a a a V

x y z

∂ ∂ ∂= − + + = −

∂ ∂ ∂

E

ˆ ˆ ˆgrad x y z

V V V V a a a

x y z

∂ ∂ ∂= + +

∂ ∂ ∂

O ∇

()

ˆ ˆ ˆ x y za a a

x y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂

P,

ˆ ˆ ˆgrad x y zV V V

V V a a a x y z

∂ ∂ ∂= ∇ = + +

∂ ∂ ∂

C , ( )

. R ,

E V = −∇

O

, ,

ˆ ˆ ˆ x y z

V V V V a a a

x y z

∂ ∂ ∂∇ = + +

∂ ∂ ∂

1ˆ ˆ

̂z

V V V V a a a

z ρ φ

ρ ρ φ

∂ ∂ ∂∇ = + +

∂ ∂ ∂


66/167

U F U

F E E

E

P. I N S

66

1 1ˆ ˆ ˆ

r

V V V V a a a

r r r senθ φ

θ θ φ

∂ ∂ ∂∇ = + +

∂ ∂ ∂

E 09:

D 22 5V x y z= − ( )4,3,6P − , :

) V P ;

) E

P ;

) D

P ;

) v ρ P .

E 10:

D ( )2

100cos

1V V

z ρ φ =

+ ( )3,60º,2P , :

) V P ;

) E

P ;

) v ρ .

4.7 D E

U , ,

,

P ,

.

P V , (

, ) , , E

.


67/167

U F U

F E E

E

P. I N S

67

C, , Q Q− P 1 R 2 R ,

, P

2 1

0 1 2 0 1 2

1 1

4 4

R RQ QV

R R R Rπε πε

−= − =

P 1 2 R R≅ , 1 2 R R

2r , 1 R 2 R , ,

1 2 cos R R d θ − =

O , ,

2

0

cos

4

Qd V

r

θ

πε =

N 0 z = , 90ºθ = .

U ,

1 1ˆ ˆ ˆ

r

V V V E V a a a

r r r senθ φ

θ θ φ

∂ ∂ ∂= −∇ = − + +

∂ ∂ ∂

3 3

0 0

2 cos senˆ ˆ

4 4r

Qd Qd E a a

r r θ

θ θ

πε πε

= − − −


68/167

U F U

F E E

E

P. I N S

68

( )30

ˆ ˆ2cos sen

4 r

Qd E a a

r θ θ θ

πε

= +

E . O

. P

Q− Q+ d

, , Qd

,

p

. A,

p Qd =

A p

.C m . C ˆ

cosr d a d θ ⋅ =

,

2

0

ˆ

4r

p aV

r πε

⋅=

E ,

,

2

0

1 '

'4 '

r r V p

r r r r πε

−= ⋅

−−

E 11:U

( )ˆ ˆ ˆ3 2 x y z p a a a nC m= − + ⋅

. D:

) V ( )2,3,4 AP ;

) V ( )2,5;30º;40º BP .

E 12:

U ( )ˆ6 z p a nC m= ⋅

. D:

) V ( )4,20º,0ºP ;

) E

P .

4.8 D E C E

P ,

.


69/167

U F U

F E E

E

P. I N S

69

P . T 1Q

, . O 2Q

1Q 2Q

1Q . R 2,1V ,

, . E,

2 2 2,1Trabalho para posicionar Q Q V =

D ,

:

3 3 3,1 3 3,2Trabalho para posicionar Q Q V Q V = +

4 4 4,1 4 4,2 4 4,3Trabalho para posicionar Q Q V Q V Q V = + +

. O E W :

2 2,1 3 3,1 3 3,2 4 4,1 4 4,2 4 4,3 E W Q V Q V Q V Q V Q V Q V = + + + + + + …

O ,

1 22 2,1 2 1 1 1,2

0 21 0 124 4

Q QQ V Q Q Q V

R Rπε πε = = =

S ,

1 1,2 1 1,3 2 2,3 1 1,4 2 2,4 3 3,4 E W Q V Q V Q V Q V Q V Q V = + + + + + + …

M 2

,

( )( )

( )

( )

1 1,2 1,3 1,4

2 2,1 2,3 2,4

3 3,1 3,2 3,4

4 4,1 4,2 4,3

2 E W Q V V V

Q V V V

Q V V V

Q V V V

= + + +

+ + + +

+ + + +

+ + + + +

…

…

…

… …

C

. E ,

1,2 1,3 1,4 1

V V V V + + + =…


70/167

U F U

F E E

E

P. I N S

70

O 1Q 2Q , 3Q , 4Q , …. T, ,

( )1 1 2 2 3 3 4 41 1

2 2 E n n

n

W Q V Q V Q V Q V Q V = + + + + = ∑…

P

, vdv ρ , ,

1

2 E v

volW V dv ρ = ∫

U M ,

( ) ( )1 1

2 2 E

vol volW D V dv D V dv= ∇ ⋅ = − ⋅ ∇∫ ∫

S E V = −∇

,

2

0

1 1

2 2 E

vol volW D E dv E dvε = ⋅ =∫ ∫

S ,

1

2 E dW D E dv= ⋅

,

1

2

E dW D E dv

= ⋅

, , .

E 13:D 2 3mm r mm< < , 0 90ºθ < < ,

0 90ºφ < < , :

) 200

V V r

= ;

) 2

300cosV V

r

θ = .


71/167

U F U

F E E

E

P. I N S

71

5 CONDUTORES, DIELTRICOS E CAPACITNCIA

5.1 C D C

C . A

(A) I , ,

dQ I

dt =

A , , ,

, .

O ,

. A

, J

S I 2 / A m . O

dI J

dS

=

A

S I J dS = ⋅∫

O I ∆ S ∆ ,

,

N I J S ∆ = ∆

C Q∆ , ,

x . E vQ v ρ ∆ = ∆ . C

x∆ t ∆ ,

v vv S xQ I t t t

ρ ρ ∆ ∆ ∆∆∆ = = =

∆ ∆ ∆


72/167

U F U

F E E

E

P. I N S

72

S ,

v x I S v ρ ∆ = ∆

xv x v

. E ,

,

x v x J v ρ =

, ,

v J v ρ =

E

. D , J

. N

. A

( )

(,

, ) .

E 01:

D ( )2 2 2ˆ ˆ10 4 cos / J za a A m ρ φ ρ ρ φ = −

, :

) ( )3,30º,2P ;

) 3 ρ = , 0 2φ π < < ,

2 2,8 z< < .

5.2 C C

O

, ( ) ( ).


73/167

U F U

F E E

E

P. I N S

73

A

. A

S I J dS = ⋅∫

(

) . S

iQ , / idQ dt −

i

S

dQ I J dS

dt = ⋅ = −

∫

A ,

, ,

:

( )S vol J dS J dv⋅ = ∇ ⋅∫ ∫

L J

D

, .

E , iQ

,

( ) i vvol vol

dQ d J dv I dv

dt dt ρ ∇ ⋅ = = − = −∫ ∫

S ,

,

( ) vvol vol

J dv dvt

ρ ∂∇ ⋅ = −∂∫ ∫

C , ,

( ) v J v vt

ρ ∂∇ ⋅ ∆ = − ∆

∂

,


74/167

U F U

F E E

E

P. I N S

74

v J

t

ρ ∂∇ ⋅ = −

∂

E , ,

.

E 02:

U ( )6 1,5 2ˆ10 / z J z a A m= −

0 20 m ρ µ ≤ ≤ ; 20 m ρ µ > , 0 J =

. P:

) 0,1 z m= ̂za ;

) 6

2 10 / m s⋅ 0,1 z m= , v ρ ;) 0,15 z m= 32000 / C m− ,

.

5.3 C M

E , ,

, , . S

,

, .

O . O () .

S, , ()

,

. E (). N ,

,

. A

.

O , (). P ,

. E

.


75/167

U F U

F E E

E

P. I N S

75

O . C E

, Q e= −

F eE = −

N , ( );

,

.

E d v

.

D µ ,

d ev E µ = −

e µ , . N

E

.

S

(v

J v ρ =

),

e e J E ρ µ = −

e ρ , . O e ρ

J

E

.

C, J

E

σ (),

J E σ =


76/167

U F U

F E E

E

P. I N S

76

σ ( ) / S m . S ( )S

S I, ( )Ω . C

;

O.

D ,

,

e eσ ρ µ = −

C ,

O. C O ,

. A O

, .

E, . A ,

( ) / mΩ ,

. P ,

K; . O

.

P ,

,

, , ,

.

A O .

I, J

E

,

.

P

S I J dS J S = ⋅ =∫


77/167

U F U

F E E

E

P. I N S

77

a a

ab ba abb b

V E dL E dL E L E L= − ⋅ = − ⋅ = − ⋅ = ⋅∫ ∫

V EL=

A,

I V J E

S Lσ σ = = =

LV I

S σ =

A

, ,

V RI =

.

D

L L R

S S

ρ

σ = =

ρ , , . E .

E 03:D 1,8mm 2 km .

E 04:D

76,17 107 / S mσ = ⋅ 20,0056 / e

m V s µ = ⋅ :

) 1,5 / m s µ ;

) 1 / mV m ;

) 2,5mm 0,4mV

;) 2,5mm 0,5 A .


78/167

U F U

F E E

E

P. I N S

78

E 05:

U 1,2 mm 360m . S dc 50 A .

) ;) ?

) dc ?) Q ?

5.4 P C C F

S . E , . I

,

. C

. A,

. E

.

O

, O: ,

, .

R:

. E,

.

I . S

, ,

, ,

,

.

E , ,

:

0 E dL⋅ =∫

abcda , .


79/167

U F U

F E E

E

P. I N S

79

A

0

b c d a

a b c d + + + =∫ ∫ ∫ ∫

L 0 E =

, a b c

d w∆ b c d a h∆ ( ),

N, em b N, em a

1 10

2 2t E w E h E h∆ − ∆ + ∆ =

0t E =

J N D

N E ,

. U , G

S D dS Q⋅ =∫

C h∆

S ∆ . F h∆ . E,

topo base lateral Q+ + =∫ ∫ ∫

,

, ,

h∆ . A, , ,

N S D S Q S ρ ∆ = = ∆

N S D ρ =

E ,


80/167

U F U

F E E

E

P. I N S

80

0t t D E = =

0 N N S D E ε ρ = =

U

.

P ,

1. A .

2. A ,

, .

3. A .

E 06:D

( ) ( )2 2100V x y V = −

( )2, 1,3P − ,

, V , E S ρ P

.

E 07:

C, , P .

E 08:

D , ( ) ( ) ( )100senh 5 sen 5V y y V =

( )0,1;0,2;0,3P , P :

) V ;

) E ;

) S ρ , P .


81/167

U F U

F E E

E

P. I N S

81

5.5 M I

U

. T

. O

0V = , , ,

.

A, ()

(),

. A , ,

.

C , ,

. E

. I

( 0V = ), .

C ,

( )2,5,0P 0 z = 30 / nC m 0 x = ,

3 z = , () . R


82/167

U F U

F E E

E

P. I N S

82

30 / nC m− 0 x = , 3 z = − , (). O P

.

O P ˆ ˆ2 3 x z R a a+ = −

, ˆ ˆ2 3 x z R a a− = +

.

A,

9

0 0

ˆ ˆ2 330 10ˆ

2 2 13 13

x z L R

a a E a

R

ρ

πε πε

−

+ +

+

−×= =

9

0 0

ˆ ˆ2 330 10ˆ

2 2 13 13

x z L R

a a E a

R

ρ

πε πε

−

− −

−

+− ×= =

S ,

9

0

ˆ180 10

ˆ249 / 2 13

z

z

a

E a V mπε

−− ×

= = −

E 09:U 4 x = ,

40 / nC m 6 x = , 3 y = . S 0V = .

E ( )7, 1,5P − , .

5.6 S

S

, . O

. O

. E

,

e , h µ

. A ; ,


83/167

U F U

F E E

E

P. I N S

83

. A , ,

,

e e h hσ ρ µ ρ µ = − +

P , , 0,12 0,025,

, , , , 0,36 0,17.

A .

, ,

. C ,

, . N

;

.

O

. M

,

. O

710

5

10 .

S O; ,

.

E 10:D

30,0029 / C m− 30,0029 / C m ,

, :

) ;) ;) .

5.7 N M D

U

. E . A

,


84/167

U F U

F E E

E

P. I N S

84

. E ,

.

A , , ,

, . E

.

O .

A

. P ,

. A , ,

p

, ,

p Qd =

Q d

.

S n v∆ , n v∆

, ,

1

n v

total i

i

p p∆

=

= ∑

S ,total p

. C, total p

.

D P

,

01

1limn v

iv

i

P pv

∆

∆ →=

=∆ ∑

C .

V . N

0P =

. E ,

S ∆ , ,

E

. O p Q d =

,

p

d

θ S ∆

, .


85/167

U F U

F E E

E

P. I N S

85

P S ∆ . C

( )1/2 cosd θ

S ∆ . A,

S ∆ ( )1/2 cosd θ S ∆ . A,

S ∆

( )1/2 cosd θ S ∆ . P, 3

moléculas / n m , (

)

( ) ( ) ( ) ( )2 /2 cos cosnQ v n Q v n Q d S n Q d S θ θ ∆ = ∆ = ∆ = ∆ ,

bQ n Q d S ∆ = ⋅ ∆

bQ∆ S ∆ , b

. E ,

0 01

1 1lim lim

n v

iv v

i

P p n v p n p nQ d v v

∆

∆ → ∆ →=

= = ∆ = =

∆ ∆ ∑

bQ P S ∆ = ⋅ ∆

S S ∆

, S ∆

bS

Q P dS = − ⋅∫


86/167

U F U

F E E

E

P. I N S

86

P

T bQ Q Q= +

Q S . A G

0T S

Q E dS ε = ⋅∫

C ,

,

( ) ( )0 0T bS S S

Q Q Q E dS P dS E P dS ε ε = − = ⋅ − − ⋅ = + ⋅∫ ∫ ∫

P D

,

0 D E Pε = +

H, , D

. A,

S Q D dS = ⋅∫

Q .

O

, ,

v D ρ ∇ ⋅ =

P ,

E

P

. E ,

, ,

E

P

. E

, , . E

, . A

.

P , E

P


87/167

U F U

F E E

E

P. I N S

87

0eP E χ ε =

e χ () .

U ,

( )0 0 01e e D E E E ε χ ε χ ε = + = +

A

1 R eε χ = +

E ,

. A,

0 R D E ε ε =

D E ε =

0 Rε ε ε =

ε .

M

.

E 11:T T 0 x a≤ ≤ , .

F T, ( )0 ˆ / ext x E E a V m=

. E D

, E

P

.

E 12:

U 3,8

28 / nC m . S , :

) E

;

) P

;

) 2910 C m− ⋅ .


88/167

U F U

F E E

E

P. I N S

88

5.8 C F M D P

V 1ε 2ε

1 2, . E, ,

0 E dL⋅ =∫

,

tan 1 tan 2 0 E w E w∆ − ∆ =

A E

h∆ h∆

. I, ,

tan 1 tan 2 E E =

C ,

w∆ .

C , D

,

tan 1 tan 2tan 1 tan 2

1 2

D D E E

ε ε = = =

tan 1 1

tan 2 2

D

D

ε

ε =


89/167

U F U

F E E

E

P. I N S

89

J , G

, . O

1 2 N N S D S D S Q S ρ ∆ − ∆ = ∆ = ∆

1 2 N N S D D ρ − =

C ,

. C, S ρ

1 2 N N D D=

, D

. D, E

, ,

1 1 2 2 N N E E ε ε =

E D

E

. C

1 D

(

1 E

) 1θ ,

. C D

,

1 1 1 2 2 2cos cos N N D D D Dθ θ = = =

A

tan 1 1 1 1

tan 2 2 2 2

sen

sen

D D

D D

θ ε

θ ε = =

2 1 1 1 2 2sen sen D Dε θ ε θ =

,

1 1

2 2

tan

tan

θ ε

θ ε =


90/167

U F U

F E E

E

P. I N S

90

N 1 2ε ε > , , 1 2θ θ > . A E

D

, D E ε =

.

P , D

2

2

2 222 1 1 1

1

cos sen D D ε θ θ

ε

= +

E

2

2

2 212 1 1 1

2

sen cos E E ε θ θ

ε

= +

E

.

E 13:

D T ( )2,1 Rε = ,

0

ˆext x

E E a=

.


91/167

U F U

F E E

E

P. I N S

91

E 14:

C 0 z < 3,2 Rε = ,

0 z > 2 Rε = . S ( )2

1 ˆ ˆ ˆ30 50 70 / x y z D a a a nC m= − + +

:

) 1 N D

;

) 1t D

;

) 1 D

;

) 1θ ;

) 1

P

.

E 15:A :

) 2 N D

;

) 2t D

;

) 2 D

;

) 2θ ;

) 2P

.

5.9 C

C ,

. O 2 M Q 1 M

, . N , .

C . N, ,

2 M 1 M . N 2 M .


92/167

U F U

F E E

E

P. I N S

92

D 2 M 1 M 0V . P, ,

,

0

QC

V =

E , Q G

0V ,

E dS

Apostila de Eletromagnetismo Verso 2011

Documents

Transcript of Apostila de Eletromagnetismo Verso 2011