EC 1311 TEORIA ELECTROMAGNETICA FORMULARIO...
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Transcript of EC 1311 TEORIA ELECTROMAGNETICA FORMULARIO...
OJSR - ENERO 1997
EC 1311 TEORIA ELECTROMAGNETICA FORMULARIO Nº 1: ANALISIS DE CAMPOS
Transformación de coordenadas
Rectangulares (x, y, z ) Cilíndricas (ρ, ϕ , z ) Esféricas (r, θ ,ϕ ) x =ρ cos ϕ =rsenθ cos ϕ ρ = x2 + y2 =rsenθ r = x2 + y2 + z2 = ρ 2 + z2 y =ρsenϕ =rsenθsenϕ
ϕ =cos−1 x / x 2 + y2⎛
⎝ ⎜ ⎞
⎠ ⎟ , si y ≥ 0
2π − cos−1 x / x2 + y2⎛ ⎝ ⎜ ⎞
⎠ ⎟ , si y < 0
⎧
⎨ ⎪
⎩ ⎪
θ = cos−1 z
x 2 + y2 + z2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ = cos−1 z
ρ2 + z2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
z=r cosθ z=r cosθ
ϕ =cos−1 x / x 2 + y2⎛
⎝ ⎜ ⎞
⎠ ⎟ , si y ≥ 0
2π − cos−1 x / x2 + y2⎛ ⎝ ⎜ ⎞
⎠ ⎟ , si y < 0
⎧
⎨ ⎪
⎩ ⎪
Transformación de los vectores unitarios Elementos
diferenciales
1x = cos ϕ1ρ − senϕ1ϕ = senθ cos ϕ1r + cosθ cosϕ1θ − senϕ1ϕ dx = dx1x dρ = dρ1ρ dr = dr1 r 1y = senϕ1ρ + cos ϕ1ϕ = senθsenϕ1r + cosθsenϕ1θ + cosϕ1ϕ dy = dy1y dlϕ = ρdϕ1ϕ dlθ = rdθ1θ
1z = cos θ1r + senθ1θ dz = dz1 z dz = dz1 z dlϕ = rsenθ dϕ1ϕ 1ρ = cos ϕ1x + senϕ1y = senθ1 r + cosθ1θ dax = dy dz daρ = ρdϕ dz dar = r2senθ dϕ dθ
1ϕ = −senϕ1x + cosϕ1y day = dx dz daϕ = dρ dz daθ = rsenθdρdϕ 1r = senθ cosϕ1x + senθsenϕ1y + cosθ1 z = senθ1ρ + cos θ1 z daz = dx dy daz = ρ dρ dϕ daϕ = rdrdθ 1θ = cosθ cosϕ1x + cos θsenϕ1y − senθ1z = cos θ1ρ − senθ1z dV = dx dy dz dV = ρ dρ dϕ dz dV = r2senθdrdϕ dθ
Gradiente Coordenadas rectangulares Coordenadas cilíndricas Coordenadas esféricas
∇Φ =∂Φ∂x
1x +∂Φ∂y
1y +∂Φ∂z
1z ∇Φ =∂Φ∂ρ
1ρ +1ρ
∂Φ∂ϕ
1ϕ +∂Φ∂z
1 z ∇Φ =∂Φ∂r
1r +1r
∂Φ∂θ
1θ +1
rsenθ∂Φ∂ϕ
1ϕ
Divergencia Coordenadas rectangulares Coordenadas cilíndricas Coordenadas esféricas
∇ ⋅ F =∂Fx∂x
+∂Fy∂y
+∂Fz∂z
∇ ⋅ F =1ρ
∂ (ρFρ )∂p
+1ρ
∂Fϕ
∂ϕ+
∂Fz∂z
∇ ⋅ F =1r2
∂ (r2Fr )∂r
+1
rsenθ∂ (senθ Fθ )
∂θ+
1rsenθ
∂Fϕ
∂ϕ
Componentes del rotacional Coordenadas rectangulares Coordenadas cilíndricas Coordenadas esféricas
(∇ × F)x =∂Fz∂y
−∂Fy∂z
(∇ × F)ρ =1ρ
∂Fz∂ϕ
−∂Fϕ
∂z (∇ × F)r =
1rsenθ
∂ (senθ Fϕ )∂θ
−1
rsenθ∂ (Fθ )
∂ϕ
(∇ × F)y =∂Fx∂z
−∂Fz∂x
(∇ × F)ϕ =∂Fρ
∂z−
∂Fz∂ρ
(∇ × F)θ =1
rsenθ∂Fr∂ϕ
−1r
∂ (rFϕ )∂r
(∇ × F)z =∂Fy∂x
−∂Fx∂y
(∇ × F)z =1ρ
∂ (ρFϕ )∂ρ
−1ρ
∂Fρ
∂ϕ (∇ × F)ϕ =
1r
∂ (rFθ )∂r
−1r
∂ (Fr )∂θ
Laplaciano (∇2Φ ) Coordenadas rectangulares
Coordenadas cilíndricas Coordenadas esféricas
∂ 2Φ∂x2 +
∂2Φ∂y2 +
∂2Φ∂z2 1
ρ∂
∂ρρ
∂Φ∂ρ
⎛
⎝ ⎜
⎞
⎠ ⎟ +
1ρ2
∂ 2Φ∂ϕ2 +
∂2Φ∂z2 1
r2∂∂r
r2 ∂Φ∂r
⎛ ⎝ ⎜ ⎞
⎠ ⎟ +
1r2senθ
∂∂θ
senθ∂Φ∂θ
⎛ ⎝ ⎜ ⎞
⎠ ⎟ +
1r2sen2θ
∂ 2Φ∂ϕ2
