Eletromagnetismoeletro1-0219/lib/exe/fetch.php?...β’Linhas de Carga πΈπΏ= 1 4 π0 πΏ π 2...
Transcript of Eletromagnetismoeletro1-0219/lib/exe/fetch.php?...β’Linhas de Carga πΈπΏ= 1 4 π0 πΏ π 2...
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Eletromagnetismo Newton Mansur
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O campo ElΓ©trico
E
1
q1
Q
EQF
O campo elΓ©trico obedece o princΓpio de superposição, isto Γ©, se dois
campos elΓ©tricos, de duas cargas diferentes, forem aplicados no mesmo
ponto, o campo elΓ©trico total serΓ‘ a soma vetorial dos dois.
rr
qkE Λ
2
1
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Lei de
Coulomb
x
y
z
π 1
π 2
π₯2 π₯1
π¦1
π¦2
π§2
π§1
π 21
π 21 π 21 = π 2 β π 1
π 1 = π₯1π₯ + π¦1π¦ + π§1π§ π 2 = π₯2π₯ + π¦2π¦ + π§2π§
= π₯2 β π₯1 π₯ + π¦2 β π¦1 π¦ + π§2 β π§1 π§
π 21 =π 21π 21
=π₯2 β π₯1 π₯ + π¦2 β π¦1 π¦ + π§2 β π§1 π§
π₯2 β π₯12 + π¦2 β π¦1
2 + π§2 β π§12
πΈ21 =1
4ππ0
π1π 2 β π 1 2
π 21
Se temos N cargas aplicando Campo ElΓ©trico
no ponto em r2
πΈπ = 1
4ππ0
πππ 2 β π π 2
π 2π
π
π=1
π 2π π 2π
π π
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β’ CΓ‘lculo de Campo ElΓ©trico Elementos ContΓnuos
23
22322
cos
z+R
dqzk=
r
dqzk=
r
z
r
dqk=ΞΈ
r
dqk=dEz
q r z
R
2r
dqk=dE
23
222
322 z+R
Qzk=E
z+R
dqzk=dE zz
x
y
z
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β’ Linhas de Carga
ππΈπΏ =1
4ππ0
ππΏππ
π 2π π
π = ππΏ π
π π
ππ= ππΏππ
ππΈπΏ
ππΈπΏ π π
π
ππ= ππΏππ
πΈπΏ =1
4ππ0
ππΏππ
π 2π π
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z
r
R r
z
dz
Campo elΓ©trico de um fio fino
q
zβ
q
ππΈ =1
4ππ0
ππ
π 2 ππ = πππ§ π = π2 + π§ β π§β² 2
ππΈ =1
4ππ0
πππ§
π2 + π§ β π§β² 22
ππΈπ = ππΈπππ π =1
4ππ0
ππππ§
π2 + π§ β π§β² 23
2 =
1
4ππ0
πππ§
π2 + π§ β π§β² 22
π
π
ππΈπ§ = ππΈπ πππ =1
4ππ0
π§ β π§β² πππ§
π2 + π§ β π§β² 23
2 =
1
4ππ0
πππ§
π2 + π§ β π§β² 22
π§ β π§β²
π
ππΈπ
ππΈπ§
ππΈ
πΈπ§ =π
4ππ0
π§ β π§β² ππ§
π2 + π§ β π§β² 23
2
π§2
π§1
=π
4ππ0[β
1
π2 + π§ β π§β² 2]π§2π§1
πΈπ =ππ
4ππ0
ππ§
π2 + π§ β π§β² 23
2
π§2
π§1
=ππ
4ππ0[β
π§ β π§β²
π2 π2 + π§ β π§β² 2]π§2π§1
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z
Ο
r Ο
z
dz
Campo elΓ©trico de um fio
fino
q
zβ
q
ππΈ =1
4ππ0
ππ
π2
ππ = πππ§
π = π2 + π§ β π§β² 2
ππΈπ
ππΈπ§
ππΈ
ππΈ =1
4ππ0
πππ 2π
π2ππππ
πππ 2π=
1
4ππ0
πππ
π
ππΈπ = ππΈπππ π =1
4ππ0
π
ππππ πππ
ππΈπ§ = ππΈπ πππ
π‘πππ =π§ β π§β²
π ππ‘πππ
ππππ = π ππ2πππ =
ππ§
π
πππ π =π
π πππ 2π =
π2
π2
πππ 2π
π2=
1
π2
ππ = πππ§ = πππ ππ2πππ =ππππ
πππ 2π
=1
4ππ0
π
ππ πππππ
πΈπ =1
4ππ0
π
ππ πππ
π2π1
πΈπ§ =1
4ππ0
π
πβπππ π
π2π1
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z
r
r
+ + + + + + + + + + + + + + + + + +
Campo elΓ©trico de um fio fino
a b
L
πΈπ =1
4ππ0
π
π πππ πππ
πΌ2
βπΌ1
πΈπ§ =1
4ππ0
π
π π πππππ
πΌ2
βπΌ1
πΈπ =1
4ππ0
ππππ§
π2 + π§23
2
π
βπ
πΈπ§ =1
4ππ0
π§ β π§β² πππ§
π2 + π§23
2
π
βπ
πΌ1 πΌ2
πΈπ
πΈπ§
π + π = πΏ
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z
r
r
+ + + + + + + + + + + + + + + +
Campo elΓ©trico de um fio fino
a b
L
πΈπ =1
4ππ0
π
π πππ πππ
πΌ2
πΌ1
πΈπ§ =1
4ππ0
π
π π πππππ
πΌ2
πΌ1
πΈπ =1
4ππ0
ππππ§
π2 + π§23
2
π
π
πΈπ§ =1
4ππ0
π§ β π§β² πππ§
π2 + π§23
2
π
π
πΌ1
πΌ2
πΈπ
πΈπ§
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+ + + + + + + + + + + + + + + +
Campo elΓ©trico de um fio fino
L/2 L/2
Para o Fio Infinito
z
r
r
πΌ
πΈπ
πΌ
πΈπ =1
4ππ0
π
π πππ πππ
πΌ
βπΌ
πΈπ =1
4ππ0
ππππ§
π2 + π§23
2
πΏ2
βπΏ 2
πΈπ§ = 0
πΈπ =1
4ππ0
π
π πππ πππ
π2
βπ 2
=1
4ππ0
π
π2 =
1
2ππ0
π
π
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β’ SuperfΓcie de Carga
ππΈπ =1
4ππ0
ππππ
π 2π π
π = ππ π
π π
ππ = ππππ
ππΈπ
ππΈπ π π
π
ππ = ππππ
πΈπ =1
4ππ0
ππππ
π 2π π
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Campo elΓ©trico de um
disco
q r z
x
dE dEz
x
y
z
πΈπ =1
4ππ0
ππππ
π2π π
π = ππ
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Campo elΓ©trico de um
disco
x
dx
dΦ
ππ = π₯πΞ¦ππ₯
ππ = π₯2πππ₯
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Campo elΓ©trico de um
disco
23
222222 4
1
4
1
z+x
dqz
ΟΞ΅z+x
z
z+x
dq
ΟΞ΅=dE
00
z
Ο2Οxdx=dq
2
3
22 )(
2
4
1
zx
xdxz
ΟΞ΅=dE
0
z
2200
2200
220
11
2Ξ΅
1
2Ξ΅
2
32Ξ΅ z+Rz
z=
z+x
z=
z+x
xdxz=dE
RR
z
q r z
x
dE dEz
x
y
z
xdx2Ο=dS
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Campo elΓ©trico de um disco muito perto do centro
220
11
2Ξ΅ z+Rz
z=Ez
x
y
z
Campo elΓ©trico de um Plano infinito
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Campo elΓ©trico de um disco muito longe do centro
220
11
2Ξ΅ z+Rz
z=Ez
x y
z
Campo elΓ©trico de uma carga pontual
2
22 2
11
11
z
R
zz+R
2
0 2
11
11
2Ξ΅ z
R
zz
zEz
2
0
2
2
0
2
0
1
Ξ΅44Ξ΅4Ξ΅ z
Q
z
R
R
Q
z
R
-
z
Plano de Carga
y
x
ππΈ
π Ξ¦
π
π π
ππΈ =1
4ππ0
ππ
π 2π π
ππ = ππ ππ π = ππ ππ
π π
π π§ π = βππ π + hπ π§
β
ππ = ππΞ¦ππ = πππ₯ππ¦
ππΈ =1
4ππ0
ππ ππΞ¦ππ
π2 + h2βππ π + hπ π§
π2 + h2
π = π2 + h2
ππΈ =1
4ππ0
ππ ππΞ¦ππ
π2 + h23
2 βππ π + hπ π§
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z
Plano de Carga Infinito
ππΈπ§
y
x
ππΈ
π Ξ¦
π
π π
π π§
β
ππΈ =1
4ππ0
ππ ππΞ¦ππ
π2 + h23
2 βππ π + hπ π§
ππΈπ
πΈπ = 0
ππΈπ
ππΈπ§ =1
4ππ0
ππ πβπΞ¦ππ
π2 + h23
2 π π§
πΈπ§ =1
4ππ0
ππ ππΞ¦ππ
π2 + h23
2
β
0
2π
0
π π§
πΈπ§ =ππ β
2π0
πππ
π2 + h23
2
β
0
π π§
πΈπ§ =ππ β
2π0β
1
π2 + h2β0
π π§
πΈπ§ =ππ 2π0
π π§
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z Plano de Carga
ππΈπ§
y
x
ππΈ
π¦
Ξ¦
π
π π¦ π π§
β
ππΈ =1
4ππ0
ππ ππ₯ππ¦
π₯2 + π¦2 + h23
2 βπ₯π π₯ β π¦π π¦ + hπ π§
ππΈπ¦ ππΈπ₯ = β
1
4ππ0
ππ π₯ππ₯ππ¦
π₯2 + π¦2 + h23
2 π π₯
ππΈπ¦ = β1
4ππ0
ππ π¦ππ₯ππ¦
π₯2 + π¦2 + h23
2 π π¦
ππΈπ§ =1
4ππ0
ππ βππ₯ππ¦
π₯2 + π¦2 + h23
2 π π§
π₯
π π₯
ππΈπ₯
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z Plano de Carga
ππΈπ§
y
x
ππΈ
π¦
Ξ¦
π
π π¦ π π§
β
ππΈπ¦
ππΈπ₯ = β1
4ππ0
ππ π₯ππ¦
π₯2 + π¦2 + h23
2
π₯2
π₯1
ππ₯π π₯
π₯
π π₯
ππΈπ₯
πΈπ₯ = βππ
4ππ0
β1
π₯2 + π¦2 + h2
π¦2
π¦1
ππ¦π₯2π₯1
π π₯
πΈπ₯ =ππ
4ππ0ππ π¦ + π₯2 + π¦2 + h2
π₯2π₯1
π¦2π¦1
π π₯
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z Plano de Carga
ππΈπ§
y
x
ππΈ
π¦
Ξ¦
π
π π¦ π π§
β
ππΈπ¦
π₯
π π₯
ππΈπ₯
ππΈπ₯ = β1
4ππ0
ππ π₯ππ₯
π₯2 + π¦2 + h23
2
π₯2
π₯1
ππ¦π π₯
πΈπ₯ = βππ
4ππ0
π₯π¦
π₯2 + h2 π₯2 + π¦2 + h2
π₯2
π₯1
ππ₯π¦2π¦1
π π₯
LΓ’mina de carga infinita
πΈπ₯ = βππ
2ππ0
1
2ππ π₯2 + h2
π₯2π₯1
π π₯
π¦1 β ββ π¦2 β +β
πΈπ₯ = βππ
2ππ0
π₯
π₯2 + h2
π₯2
π₯1
ππ₯ π π₯
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z Plano de Carga
ππΈπ§
y
x
ππΈ
π¦
Ξ¦
π
π π¦ π π§
β
ππΈπ¦
ππΈπ§ =1
4ππ0
ππ βππ₯ππ¦
π₯2 + π¦2 + h23
2 π π§
π₯
π π₯
ππΈπ₯ ππΈπ§ =
ππ β
4ππ0
1
π₯2 + π¦2 + h23
2
π₯2
π₯1
ππ¦ππ₯π π§
ππΈπ§ =ππ β
4ππ0
π¦
π₯2 + h2 π₯2 + π¦2 + h2
π¦2π¦1
ππ₯π π§
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z Plano de Carga
ππΈπ§
y
x
ππΈ
π¦
Ξ¦
π
π π¦ π π§
β
ππΈπ¦
π₯
π π₯
ππΈπ₯
LΓ’mina de carga infinita
π¦1 β ββ π¦2 β +β
ππΈπ§ =1
4ππ0
ππ βππ₯ππ¦
π₯2 + π¦2 + h23
2 π π§
ππΈπ§ =ππ β
4ππ0
1
π₯2 + π¦2 + h23
2
π₯2
π₯1
ππ¦ππ₯π π§
ππΈπ§ =ππ β
4ππ0
π¦
π₯2 + h2 π₯2 + π¦2 + h2
π¦2π¦1
ππ₯π π§
ππΈπ§ =ππ β
4ππ0
2
π₯2 + h2ππ₯π π§ πΈπ§ =
ππ β
4ππ0
2
π₯2 + h2ππ₯
π₯2
π₯1
π π§ =ππ β
2ππ0
1
βπ‘ππβ1
π₯
β
π₯2π₯1
π π§
π₯1 β ββ π₯2 β +β πΈπ§ =ππ
2ππ0ππ π§ =
ππ 2π0
π π§
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β’ Volume de Carga
πΈπ =1
4ππ0
ππππ
π 2π π
π = ππ
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
ππΈ ππΈπ§ π
π = π β π πΈ =
1
4ππ0
ππππ
π2π π
π = ππ§
π = π₯π₯ + π¦π¦ + π§π§
π = π₯π₯ + π¦π¦ + π β π§ π§
πΈπ§ =ππ
4ππ0
1
π2πππ πππ π§ πππ π =
π β π§
π
πΈπ§ =ππ
4ππ0
1
π2π β π§
πππ π§
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
π = π β π ππΈ ππΈπ§
π πΈπ§ =
ππ4ππ0
1
π2π β π§
πππ π§
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2 ππ₯ππ¦ππ§ π§
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β’ Campo ElΓ©trico de uma esfera
x
y
z
π = π β π
πΈπ§ =ππ
4ππ0
1
π2π β π§
πππ π§
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2 ππ₯ππ¦ππ§ π§
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2
π
βπ
π
βπ
π
βπ
ππ₯ππ¦ππ§
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β’ Campo ElΓ©trico de uma esfera
π = π β π
πΈπ§ =ππ
4ππ0
1
π2π β π§
πππ π§
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2 ππ₯ππ¦ππ§ π§
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2
π2βπ₯2
β π2βπ₯2
π
βπ
π
βπ
ππ¦ππ₯ππ§
x
y
z
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
ππΈ ππΈπ§ π
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2
π2βπ₯2βπ¦2
β π2βπ₯2βπ¦2
π
βπ
π
βπ
ππ§ππ¦ππ₯
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
ππΈ ππΈπ§ π
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2
π2βπ§2βπ¦2
β π2βπ§2βπ¦2
π
βπ
π
βπ
ππ₯ππ¦ππ§
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
ππΈ ππΈπ§ π
πΈπ§ =ππ
4ππ0
π β π§
π2 + π β π§ 23
2
2π
0
π2βπ§2
0
π
βπ
πππππππ§
π
ππ
ππ
πΈπ§ =ππ
4ππ0
π β π§
π₯2 + π¦2 + π β π§ 23
2
π2βπ§2βπ¦2
β π2βπ§2βπ¦2
π
βπ
π
βπ
ππ₯ππ¦ππ§
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
ππΈ ππΈπ§
π
πΈπ =1
4ππ0
ππππ
π2π π
π = π β π π2 = π2 + π 2 β 2pπ πππ π
π πππ π
Ξ±
Ξ±
πΈπ§ =ππ
4ππ0
ππ
π2πππ πΌπ π§
πππ πΌ =π β π πππ π
π
πΈπ§ =ππ
4ππ0
π β π πππ π
π3πππ π§
πΈπ§ =ππ
4ππ0
ππ
π2π β π πππ π
ππ π§
πΈπ§ =ππ
4ππ0
π β π πππ π
π2 + π 2 β 2pπ πππ π3
2 πππ π§
ππ = π 2π πππππππππ πΈπ§ =
ππ4ππ0
π β π πππ π
π2 + π 2 β 2pπ πππ π3
2 π 2π πππππππππ π π§
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
ππΈ ππΈπ§
π π πππ π
Ξ±
Ξ±
ππ = π 2π πππππππππ
πΈπ§ =ππ
4ππ0
π β π πππ π
π2 + π 2 β 2pπ πππ π3
2
π
0
2π
0
π
0
π 2π πππππππππ π π§
π = πππ π ππ = βπ πππππ π = 1 β β1
πΈπ§ =ππ
4ππ0
β π β π π
π2 + π 2 β 2pπ π3
2
β1
1
2π
0
π
0
π 2ππππππ π π§
π’ = π β π π ππ’ = βπ ππ π’ = π β π β π + π
πΈπ§ =ππ
4ππ0
π’
π π2 + π 2 β 2pπ π β π’
π
32
π+π
πβπ
2π
0
π
0
π 2ππ’ππππ π π§
π =π β π’
π ππ =
βππ’
π
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β’ Campo ElΓ©trico de uma esfera
x
y
z
x y
z
π
π π
ππΈ ππΈπ§
π π πππ π
Ξ±
Ξ±
ππ = π 2π πππππππππ
πΈπ§ =ππ
4ππ0
π’
π π 2 β π2 + 2pπ’3
2
π+π
πβπ
2π
0
π
0
π 2ππ’ππππ π π§
π’ππ’
π + bπ’3
2 = 2π + ππ’
2
π2 π + bπ’
πΈπ§ =ππ
4ππ0
2 π 2 β π2 + 2ππ’
π 2π2 π 2 β π2 + 2pπ’
2π
0
π
0
π + π π β π
π 2ππππ π π§
πΈπ§ =ππ
4ππ0π2
π + π
π + πβ
π β π
π β π
2π
0
π
0
π 2ππππ π π§
πΈπ§ =ππ
4ππ0π22.2π
π3
3π π§ =
ππ43 ππ
3
4ππ0π2π π§ =
1
4ππ0
π
π2π π§
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Transformação de Coordenadas
βͺ Versor Vetor unitΓ‘rio, adimensional, que define direção e sentido de um vetor
x
y
π΄
π΄ π₯
π΄ π¦ π΄ = π΄π π΄
π π΄
π π₯
π π¦
π΄ π₯ = π΄π₯π π₯
π΄ π¦ = π΄π¦π π¦
π΄ = π΄ π₯ + π΄ π¦
π΄ = π΄π₯π π₯ + π΄π¦π π¦
π΄π π΄ = π΄π₯π π₯ + π΄π¦π π¦
π π΄ =π΄π₯π π₯ + π΄π¦π π¦
π΄
π π΄ =π΄π₯π΄
π π₯ +π΄π¦π΄
π π¦ π
π π΄ = πππ ππ π₯ + π ππππ π¦
π π΄ =π΄
π΄
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βͺ Representação vetorial em coordenadas cartesianas
π = ππ₯π π₯ + ππ¦π π¦ + ππ§π π§
x
y
π
ππ₯
ππ¦
π π₯
π π¦
z
π π§ ππ§
Transformação de Coordenadas
ππ = ππ₯ππ¦ππ§
ππ = ππ¦ππ§π π₯ + ππ₯ππ§π π¦ + ππ₯ππ¦π π§
ππ = ππ₯π π₯ + ππ¦π π¦ + ππ§π π§
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βͺ Representação vetorial em coordenadas cilΓndricas
Transformação de Coordenadas
π = ππ₯π π₯ + ππ¦π π¦
π
π
π§ π§
π π
π§
π₯
π¦
π =ππ₯π π₯ + ππ¦π π¦
π
π = πππ ππ π₯ + π ππππ π¦
π = βπ ππππ π₯ + πππ ππ π¦
π§ = π π§
ππ = πππππππ§
ππ = πππ + ππππ + ππ§π§
ππ = πππππ§π + ππππ§π + ππππππ§
π π₯ = πππ ππ β π ππππ
π π¦ = π ππππ + πππ ππ
π π§ = π§
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βͺ Representação vetorial em coordenadas esfΓ©ricas
Transformação de Coordenadas
π
π
π
π
π = π ππππππ ππ π₯ + π ππππ ππππ π¦ + πππ ππ π§
π =ππ₯π π₯ + ππ¦π π¦ + ππ§π π§
π π = ππ₯π π₯ + ππ¦π π¦ + ππ§π π§
π = πππ ππππ ππ π₯ + πππ ππ ππππ π¦ β π ππππ π§
π = βπ ππππ π₯ + πππ ππ π¦
π π₯ = π ππππππ ππ + πππ ππππ ππ β π ππππ
π = βπ ππππ π₯ + πππ ππ π¦
π π¦ = π ππππ ππππ + πππ ππ ππππ + πππ ππ
ππ = ππ ππππππππππ = π2π πππππππππ
ππ = πππ + ππππ + ππ ππππππ
ππ = π2π ππππππππ + ππ ππππππππ + ππππππ