View
221
Download
0
Category
Preview:
Citation preview
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
8/16/2019 Apostila de Eletromagnetismo Verso 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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 : ,
, ,
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
, .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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= + +
ˆ ˆ ˆ x x y y z z
B B a B a B a= + +
. O
, , A B⋅
, . E,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
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⋅
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
, .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
A B a a A B a a A B a a
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = ),
() .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
ˆ ˆ ˆ x x y y z z
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
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = = = .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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=
( ) .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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, ,
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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− .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
−=
−
( )
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
=
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 , .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
∞
−∞=
+
∫
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ,
, .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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, ,
, ,
, .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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= +
+ +
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = + +
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 (
, ).
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = − .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = ⋅∫
( )
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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= .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 =
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
ˆ ˆ ˆ x x y y z z
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
∆= + ×
∂∆= +
∂
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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− −= − +
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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∆ →
⋅∂ ∂ ∂+ + =
∂ ∂ ∂ ∆
∫
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
∂ ∂ ∂= + +
∂ ∂ ∂
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ;
,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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⋅ = ∇ ⋅∫ ∫
:
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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= .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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= − ⋅
∫
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 , , .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = −
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = +
− −
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = ,
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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⋅ =∫
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
=
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
(
).
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
V V V V E a dx dy dz a a
dy x y z y
∂ ∂ ∂ ∂= − + + = −
∂ ∂ ∂ ∂
8/16/2019 Apostila de Eletromagnetismo Verso 2011
65/167
U F U
F E E
E
P. I N S
65
1ˆ ˆ ˆ
z z z z
V V V V E a dx dy dz a a
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
∂ ∂ ∂∇ = + +
∂ ∂ ∂
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
= − − −
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ,
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 + + + =…
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ∆ ∆ ∆∆∆ = = =
∆ ∆ ∆
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
, ( ) ( ).
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ∂∇ ⋅ ∆ = − ∆
∂
,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 =
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = ⋅ =∫
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 , .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ; ,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
,
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 ∆
, .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 = − ⋅∫
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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− ⋅ .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 =
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 =
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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=
.
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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 .
8/16/2019 Apostila de Eletromagnetismo Verso 2011
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
Recommended