Page 408 - Electromagnetics

The magnetic field components are found using (5.161) and (5.163):
˜
1 1 ∂ A e
˜
H θ = , (5.169)
˜ µ r sin θ ∂φ
˜
˜ 1 1 ∂ A e
H φ =− . (5.170)
˜ µ r ∂θ
2. TE fields. To generate fields TE to r we recall that the electromagnetic fields in
a source-free region may be written in terms of magnetic vector and scalar potentials as
˜
˜
H =− jωA h −∇φ h , (5.171)
˜
˜
D =−∇ × A h . (5.172)
In a source-free region we have from Faraday’s law
1 1
˜ ˜ ˜
H = ∇× D = ∇× (∇× A h ).
− jω ˜µ˜ c jω ˜µ˜ c
˜
˜
Here φ h and A h must satisfy a differential equation that may be derived by examining
˜
˜
c
2 ˜
˜
∇× (∇× H) = jω∇× D = jω˜ (− jω ˜µH) = k H,
2
c
2
where k = ω ˜µ˜ . Substitution from (5.171) gives
˜
˜
˜
˜
2
∇× ∇× [− jωA h −∇φ h ] = k [− jωA h −∇φ h ]
or
k 2
˜ 2 ˜ ˜
∇× (∇× A h ) − k A h = ∇φ h . (5.173)
jω
˜
˜
Choosing A h = ˆ rA h and
˜
˜ jω ∂ A h
φ h =
k 2 ∂r
we find, as with the TM fields,
˜ A h

2
2
(∇ + k ) = 0. (5.174)
r
˜
Thus the quantity A h /r obeys the Helmholtz equation.
We can find the TE fields using (5.171) and (5.172). Substituting we find that
1
∂ 2
˜ 2 ˜
H r = + k A h , (5.175)
jω ˜µ˜ c ∂r 2
2 ˜
1 1 ∂ A h
˜
H θ = , (5.176)
c
jω ˜µ˜ r ∂r∂θ
2 ˜
1 1 ∂ A h
˜
H φ = , (5.177)
c
jω ˜µ˜ r sin θ ∂r∂φ
˜
1 1 ∂ A h
˜
E θ =− , (5.178)
c
˜ r sin θ ∂φ
˜
˜ 1 1 ∂ A h
E φ = . (5.179)
c
˜ r ∂θ
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