Lista de equações do eletromagnetismo

Prévia do material em texto

LISTA DE EQUAC¸O˜ES U´TEIS
x = ρ cos θ y = ρ sen θ ρˆ = cos θ ıˆ+ sen θ jˆ
x = r sen θ cosφ y = r sen θ senφ z = r cos θ
∇ = rˆ
∂
∂r
+
θˆ
r
∂
∂θ
+
φˆ
r sen θ
∂
∂φ
∇ = ρˆ
∂
∂ρ
+
θˆ
ρ
∂
∂θ
+ kˆ
∂
∂z
rˆ = sen θ cosφ ıˆ+ sen θ senφ jˆ+ cos θ kˆ
∇ · ~V =
1
ρ
∂
∂ρ
(ρVρ) +
1
ρ
∂Vθ
∂θ
+
∂Vz
∂z
∇ · ~V =
1
r2
∂
∂r
(r2Vr) +
1
r sen θ
∂
∂θ
(sen θVθ) +
1
r sen θ
∂Vφ
∂φ
∇× ~V =
(1
ρ
∂Vz
∂θ
−
∂Vθ
∂z
)
ρˆ+
(∂Vρ
∂z
−
∂Vz
∂ρ
)
θˆ +
( ∂
∂ρ
(ρVθ)−
∂Vρ
∂θ
) kˆ
ρ
∇× ~V =
[ ∂
∂θ
(sen θVφ)−
∂Vθ
∂φ
] rˆ
r sen θ
+
[ 1
sen θ
∂Vr
∂φ
−
∂(rVφ)
∂r
] θˆ
r
+
[ ∂
∂r
(rVθ)−
∂Vr
∂θ
] φˆ
r
∇2f =
∂2f
∂x2
+
∂2f
∂y2
+
∂2f
∂z2
∇2f =
1
ρ
∂
∂ρ
(
ρ
∂f
∂ρ
)
+
1
ρ2
∂2f
∂θ2
+
∂2f
∂z2
∇2f =
1
r
∂2
∂r2
(rf) +
1
r2 sen θ
∂
∂θ
(
sen θ
∂f
∂θ
)
+
1
r2 sen2 θ
∂2f
∂φ2
d~ℓ = dx ıˆ+ dy jˆ+ dz kˆ d~ℓ = dρ ρˆ+ ρ dθ θˆ + dz kˆ d~ℓ = dr rˆ + r dθ θˆ + r sen θ dφ φˆ
dv = dxdydz dv = ρ dρdθdz dv = r2 sen θ drdθdφ
dAz = dxdy dAy = dxdz dAx = dydz
dAz = ρ dρdθ dAθ = dρdz dAρ = ρ dθdz
dAφ = r drdθ dAθ = r sen θ drdφ dAr = r
2 sen θ dθdφ
∮
C
M dx+N dy =
∫
S
(∂N
∂x
−
∂M
∂y
)
dS A =
1
2
∮
C
x dy − y dx∫
V
∇ · ~V dv =
∮
S
~V · nˆ dS
∫
V
∇× ~V dv =
∮
S
nˆ× ~V dS∫
V
∇f dv =
∮
S
fnˆ dS
∫
S
∇× ~V · nˆ dS =
∮
C
~V · d~ℓ∫
S
(nˆ×∇)× ~V dS =
∮
C
d~ℓ× ~V
∫
S
nˆ×∇f dS =
∮
C
f d~ℓ
1
∇(φ + ψ) = ∇φ+∇ψ (1a)
∇ · ( ~V + ~U ) = ∇ · ~V +∇ · ~U (1b)
∇× ( ~V + ~U ) = ∇× ~V +∇× ~U (1c)
∇ · (φ ~V ) = ∇φ · ~V + φ∇ · ~V (1d)
∇× (φ ~V ) = ∇φ× ~V + φ∇× ~V (1e)
∇ · ( ~V × ~U ) = ~U · (∇× ~V )− ~V · (∇× ~U ) (1f)
∇× ( ~V × ~U ) = ( ~U ·∇) ~V − ~U (∇ · ~V )− ( ~V ·∇) ~U + ~V (∇ · ~U ) (1g)
∇( ~V · ~U ) = ( ~U ·∇) ~V + ~U × (∇× ~V ) + ( ~V ·∇) ~U + ~V × (∇× ~U ) (1h)
∇×∇φ = 0 (1i)
∇ · (∇× ~V ) = 0 (1j)
∇2 = ∇ ·∇ (1k)
∇× (∇× ~V ) = ∇(∇ · ~V )−∇2 ~V (1l)
∇ ·
~r − ~r ′
|~r − ~r ′|3
= 4πδ(~r − ~r ′) ∇2
1
|~r − ~r ′|
= −4πδ(~r − ~r ′) ∇
1
|~r − ~r ′|
= −
~r − ~r ′
|~r − ~r ′|3
∫ b
a
δ(x− x0) dx =
{
1 , x0 ∈ [a, b]
0 , x0 6∈ [a, b]
∫ b
a
f(x)δ(x − x0) dx =
{
f(x0) , x0 ∈ [a, b]
0 , x0 6∈ [a, b]
δ(x − x′) = δ(x′ − x)
2