Page 528 - Electromagnetics

Differential operations

ˆ
ˆ
dl = ˆ r dr + θrdθ + φr sin θ dφ (D.102)
2
dV = r sin θ dr dθ dφ (D.103)
2
dS r = r sin θ dθ dφ (D.104)
dS θ = r sin θ dr dφ (D.105)
dS φ = rdrdθ (D.106)

∂ f 1 ∂ f 1 ∂ f
ˆ
∇ f = ˆ r + θ ˆ + φ (D.107)
∂r r ∂θ r sin θ ∂φ
1 ∂ 1
1 ∂ 2
∂ F φ
∇· F = r F r + (sin θ F θ ) + (D.108)
2
r ∂r r sin θ ∂θ r sin θ ∂φ
ˆ ˆ

1 ˆ r rθ r sin θφ
∂
∂

∇× F = ∂ (D.109)
∂r
2
r sin θ ∂θ ∂φ

F r rF θ r sin θ F φ
2
1 ∂ ∂ f 1 ∂ ∂ f 1 ∂ f
2 2
∇ f = r + sin θ + (D.110)
2
2
2
r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ 2
2

2 cos θ 1 ∂ F φ ∂ F θ
2 2
∇ F = ˆ r ∇ F r − F r + F θ + + +
r 2 sin θ sin θ ∂φ ∂θ

ˆ 2 1 1 ∂ F r cos θ ∂ F φ
+ θ ∇ F θ − F θ − 2 + 2 +
2
2
r 2 sin θ ∂θ sin θ ∂φ

ˆ 2 1 1 1 ∂ F r cos θ ∂ F θ
+ φ ∇ F φ − 2 F φ − 2 − 2 2 (D.111)
r 2 sin θ sin θ ∂φ sin θ ∂φ
Separation of the Helmholtz equation

1 ∂ 2 ∂ψ(r,θ,φ) 1 ∂ ∂ψ(r,θ,φ)
r + sin θ +
2
2
r ∂r ∂r r sin θ ∂θ ∂θ
2
1 ∂ ψ(r,θ,φ) 2
+ 2 2 + k ψ(r,θ,φ) = 0 (D.112)
2
r sin θ ∂φ
ψ(r,θ,φ) = R(r) (θ) (φ) (D.113)
η = cos θ (D.114)
1 d 2 dR(r) 2 2
r + k r = n(n + 1) (D.115)
R(r) dr dr
2
d (η) d (η) µ 2
2
(1 − η ) − 2η + n(n + 1) − (η) = 0, −1 ≤ η ≤ 1 (D.116)
dη 2 dη 1 − η 2
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