Page 376 - Electromagnetics

˜ i
introduce the potential functions using superposition. If J ˜ i = 0 and J
= 0 then we
m
may introduce the electric potentials through the relationships
˜
˜
˜
E =−∇φ e − jωA e , (5.56)
1
˜ ˜
H = ∇× A e . (5.57)
˜ µ
Assuming the Lorentz condition
˜
˜
c
∇· A e =− jω ˜µ˜ φ e ,
we find that upon substitution of (5.56)–(5.57) into (5.54)–(5.55) the potentials must
obey the Helmholtz equation
˜

i c
2 2 φ e −˜ρ /˜
∇ + k ˜ = ˜ i .
A e − ˜µJ
˜ i
˜ i
If J
= 0 and J = 0 then we may introduce the magnetic potentials through
m
1
˜ ˜
E =− ∇× A h , (5.58)
˜ c
˜
˜
˜
H =−∇φ h − jωA h . (5.59)
Assuming
˜
˜
c
∇· A h =− jω ˜µ˜ φ h ,
we find that upon substitution of (5.58)–(5.59) into (5.54)–(5.55) the potentials must
obey
˜

i
2 2 φ h −˜ρ / ˜µ
m
∇ + k ˜ = c ˜ i .
A h −˜ J
m
When both electric and magnetic sources are present, we use superposition:
1
˜ ˜ ˜ ˜
E =−∇φ e − jωA e − ∇× A h ,
˜ c
1
˜ ˜ ˜ ˜
H = ∇× A e −∇φ h − jωA h .
˜ µ
Using the Lorentz conditions we can also write the fields in terms of the vector potentials
alone:
jω 1
˜ ˜ ˜ ˜
E =− ∇(∇· A e ) − jωA e − ∇× A h , (5.60)
k 2 ˜ c
1 jω
˜ ˜ ˜ ˜
H = ∇× A e − ∇(∇· A h ) − jωA h . (5.61)
˜ µ k 2
˜ i
We can also define Hertzian potentials for the frequency-domain fields. When J = 0
m
˜ i
and J
= 0 we let
˜ c ˜
A e = jω ˜µ˜ Π e
and find
J ˜ i
˜ ˜ 2 ˜ ˜
E =∇(∇· Π e ) + k Π e =∇ × (∇× Π e ) − (5.62)
jω˜ c
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