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Transcript of Eletromagnetismo I: 2a Avaliação - unespeletromag.com · Eletromagnetismo Formulario_Final_p2...
Eletromagnetismo I: 2a Avaliação Equações e fórmulas
𝜌 = 𝑥! + 𝑦! , 𝜙 = 𝑡𝑔!! !
! , 𝑧 = 𝑧
𝑥 = 𝜌𝑐𝑜𝑠𝜙,𝑦 = 𝜌𝑠𝑒𝑛𝜙, 𝑧 = 𝑧 𝑎! = cos𝜙 𝑎! + sen𝜙 𝑎𝒚 𝑎𝝓 = − sen𝜙 𝑎! + cos𝜙 𝑎𝒚 𝑎𝒛 = 𝑎𝒛 𝑎! = cos𝜙 𝑎𝝆 − sen𝜙 𝑎𝝓 𝑎𝒚 = sen𝜙 𝑎𝝆 + cos𝜙 𝑎𝝓 𝑎𝒛 = 𝑎𝒛 𝐴!𝐴!𝐴!
= cos𝜙 sen𝜙 0− sen𝜙 cos𝜙 0
0 0 1 𝐴!𝐴!𝐴!
𝐴!𝐴!𝐴!
= cos𝜙 − sen𝜙 0sen𝜙 cos𝜙 00 0 1
𝐴!𝐴!𝐴!
𝑟 = 𝑥! + 𝑦! + 𝑧!, 𝜃 = 𝑡𝑔!! !!!!!
!
𝜙 = 𝑡𝑔!!𝑦𝑥
𝑥 = 𝑟 𝑠𝑒𝑛𝜃𝑐𝑜𝑠𝜙 , 𝑦 = 𝑟 𝑠𝑒𝑛𝜃𝑠𝑒𝑛𝜙 , 𝑧 = 𝑟 𝑐𝑜𝑠𝜃 𝐴!𝐴!𝐴!
= sen𝜃 cos𝜙 sen𝜃 sen𝜙 cos𝜃cos𝜃 cos𝜙 cos𝜃 sen𝜙 − sen𝜃− sen𝜙 cos𝜙 0
𝐴!𝐴!𝐴!
𝐴!𝐴!𝐴!
= sen𝜃 cos𝜙 cos𝜃 cos𝜙 −sen𝜃sen𝜃 sen𝜙 cos𝜃 sen𝜙 cos𝜃cos𝜃 −sen𝜃 0
𝐴!𝐴!𝐴!
𝑎! = sen𝜃 cos𝜙 𝑎𝒓 − cos𝜃 cos𝜙 𝑎𝜽 +−sen𝜃 𝑎𝝓 𝑎𝒚 = sen𝜃 sen𝜙 𝑎𝒓 + cos𝜃 sen𝜙 𝑎𝜽 + cos𝜃 𝑎𝝓 𝑎𝒛 = cos𝜃 𝑎𝒓−sen𝜃 𝑎𝝓 𝑎! = sen𝜃 cos𝜙 𝑎! + sen𝜃 sen𝜙 𝑎𝒚 + cos𝜃 𝑎𝒛 𝑎𝜽 = cos𝜃 cos𝜙 𝑎! + cos𝜃 sen𝜙 𝑎𝒚− sen𝜃 𝑎𝒛 𝑎𝝓 = − sen𝜙 𝑎! + cos𝜙 𝑎𝒚 𝑑𝑙 = 𝑑𝑥𝑎! + 𝑑𝑦𝑎𝒚 + 𝑑𝑧𝑎! 𝑑𝑆 = 𝑑𝑦 𝑑𝑧 𝑎! + 𝑑𝑥 𝑑𝑧 𝑎𝒚 + 𝑑𝑥 𝑑𝑦 𝑎𝒛
𝑑𝑣 = 𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝑙 = 𝑑𝜌𝑎! + 𝜌𝑑𝜙𝑎! + 𝑑𝑧𝑎! 𝑑𝑆 = 𝜌 𝑑𝜙 𝑑𝑧 𝑎! + 𝑑𝜌 𝑑𝑧 𝑎! + 𝜌 𝑑𝜙 𝑑𝜌 𝑎𝒛
𝑑𝑣 = 𝜌 𝑑𝜌 𝑑𝜙 𝑑𝑧 𝑑𝑙 = 𝑑𝑟𝑎! + 𝑟 𝑑𝜃 𝑎! + 𝑟 sen𝜃 𝑑𝜙 𝑎! 𝑑𝑆 = 𝑟! sen𝜃 𝑑𝜃 𝑑𝜙 𝑎! + 𝑟 sen𝜃 𝑑𝑟 𝑑𝜙 𝑎𝜽 + 𝑟 𝑑𝑟 𝑑𝜃 𝑎𝝓 𝑑𝑣 = 𝑟! sen𝜃 𝑑𝑟 𝑑𝜃 𝑑𝜙
!E "r( ) = 1
4πε0ρvdv '!r − !r ' 2vol.
∫ aR
!E "r( ) = 1
4πε0ρSdS '!r − !r ' 2S
∫ aR
!E !r( ) = 1
4πε0ρldl '!r − !r ' 2C
∫ aR
!F12 =
14πε0
Q1Q2!R12
2 a12
!R12 =
!r2 −!r1
a12 =!R12!R12
!E =
!FQt
!E(!r ) = Q
4πε0!R2 aR =
Q4πε0
!r − !r '( )!r − !r ' 3
!E(!r ) = Qm
4πε0!r − !rm
2 amm=1
n
∑
ρv =dQdv [C /m3]
Q = ρvvol.∫ !r '( )dv '
C → Caminho
𝑘 =
14𝜋𝜀!
≅ 9×10! 𝑚/𝐹
𝜀! = 8,854 × 10!!" 𝐹/𝑚 𝑒 = 1,602 ×10!!" 𝐶 𝑟! = 3,8 ×10!!" 𝑚 dW = −Q
!E ⋅d!l
𝑉!" = 𝑉! − 𝑉!
∇×!E = 0
!E = − ∂V
∂ρaρ +
1ρ∂V∂φ
aφ +∂V∂z
az⎛
⎝⎜
⎞
⎠⎟
!E = − ∂V
∂rar +
1r∂V∂θ
aθ +1
rsenθ∂V∂φ
aφ⎛
⎝⎜
⎞
⎠⎟
V ≈
Q4πε0
d cosθr2
aR =!r − !r '!r − !r '
ρl =dQdl [C /m]
Q = ρll∫ !r( )dl
ρS =dQdS ' [C /m2 ]
Q = ρsS∫ !r( )dS
!E = ρl
2πε0ρaρ
!E = ρl
2πε0
x − x '( ) ax + y− y '( ) ayx − x '( )2 + y− y '( )2
!E = ρs
2ε0 aN
∇⋅!D =
1ρ
∂ ρDρ( )∂ρ
+1ρ
∂Dφ
∂φ+∂Dz
∂z
∇⋅!D =
1r2∂ r2Dr( )∂r
+1
r.senθ∂ senθ Dθ( )
∂θ+
1r.senθ
∂Dφ
∂φ
ψ =Q [C]!D =
Q4πr2
ar!D = ε0
!E
!D ⋅d!S =
S"∫ ρv dvV∫ψ = d
S!∫ ψ ="D ⋅d!S
S!∫!D ⋅d!S =
S"∫ ∇ ⋅!Ddv
V∫∇⋅!D = ρv
∇⋅!D = lim
Δv→0
!D ⋅d!S
S"∫Δv
W = −Q!E ⋅d!l
inicial
final∫
VAB =Q4πε0
1rA−1rB
⎛
⎝⎜
⎞
⎠⎟
VA =Q4πε0
1rA
VAB =WQ= −
!E ⋅d!l
B
A∫ [V ]
V (!r ) = Q4πε0
1!r − !r '
V (!r ) = Qi
4πε01!r − !rii=1
n
∑
V (!r ) = 14πε0
ρv r '( )dv '!r − !r 'V∫
ρv r '( )dv '→ ρs r '( )dS '→ ρl r '( )dl '!E ⋅d!l"∫ = 0
∇× ∇V( ) = 0!E = −∇V [V /m]
∇V =∂V∂r
ar +1r∂V∂θ
aθ +1
rsenθ∂V∂φ
aφ
QT = ε0
!E ⋅d!S
S"∫
,
!E = Qd
4πε0r3 2cosθ ar + senθ aθ( )
!p =Q !d [C.m]
V =!p ⋅ aR
4πε0!r − !r ' 2
aR =!r − !r '!r − !r '
WE =12 i = 1
N
∑ Qi Vi
WE =12
ρvV dvvol.∫
WE =12
!D ⋅!E( )dvvol.∫ =
ε02
!E
2dv
vol.∫
wE =dWE
dv=12!D ⋅!E [J /m3]
I =!J ⋅d!S
S∫!J = ρv
!v !J ⋅d!S
S"∫ = −∂Qi
∂tQi = ρv dvvol.∫
∇⋅!J = −∂ρv
∂t
∇⋅!J = lim
Δv→0
!J ⋅d!S
S"∫Δv!
F = −e!E
!vd = −µe
!E
!J = −ρeµe
!E
!J = −ρeµe
!E + ρhµh
!E
!J =σ
!E
σ = −ρeµe
σ = −ρeµe + ρhµh
R = 1σLS
1ρv
∂ρv∂t
= −σε
ρv = ρv0e− t τ
τ =εσ!
E × aN S= 0
!D ⋅ aN S
= ρS
!ptotal =!pi
i=1
nΔv
∑!P = lim
Δv→0
1Δv
!pii=1
nΔv
∑⎛
⎝⎜
⎞
⎠⎟ [C /m2 ]
!P = n !pmédio!P = χε0
!E
Qfixas = −!P ⋅d!S
S"∫
QT =Qfixas +Q
!D = ε0
!E +!P
!D = ε0 1+ χ( )
!E = ε0εr
!E
ε = εrε0!E1 −!E2( )× an12 S
= 0
E1t = E2
t D1n = D2
n
!D1 −
!D2( ) ⋅ an12 S
= 0
ε2 tanθ1 = ε1 tanθ2