Page 375 - Electromagnetics

Summary of potential relations for lossless media. When both electric and mag-
netic sources are present, we may superpose the potential representations derived above.
We assume a homogeneous, lossless medium with time-invariant parameters µ and .For
the scalar/vector potential representation we have
1
∂A e
E =− −∇φ e − ∇× A h , (5.49)
∂t
1 ∂A h
H = ∇× A e − −∇φ h . (5.50)
µ ∂t
Here the potentials satisfy the wave equations

2
i
∂ −µJ
2 A e
∇ − µ = ρ i , (5.51)
∂t 2 φ e −

2
i
∂ − J
2 A h m
∇ − µ = i ,
∂t 2 φ h − ρ m
µ
and are linked by the Lorentz conditions
∂φ e
∇· A e =−µ ,
∂t
∂φ h
∇· A h =−µ .
∂t
We also have the Hertz potential representation
2
∂ Π e ∂Π h
E =∇(∇· Π e ) − µ − µ∇×
∂t 2 ∂t
P i ∂Π h
=∇ × (∇× Π e ) − − µ∇× , (5.52)
∂t
2
∂Π e ∂ Π h
H = ∇× +∇(∇· Π h ) − µ
∂t ∂t 2
∂Π e
= ∇× +∇ × (∇× Π h ) − M i . (5.53)
∂t
The Hertz potentials satisfy the wave equations

2
1 i
∂
2 Π e − P
∇ − µ = i .
∂t 2 Π h −M
Potential functions for the frequency-domain fields. In the frequency domain it
is much easier to handle lossy media. Consider a lossy, isotropic, homogeneous medium
described by the frequency-dependent parameters ˜µ, ˜ , and ˜σ. Maxwell’s curl equations
are

˜
˜ i
˜
∇× E =−J − jω ˜µH, (5.54)
m
c ˜
˜
˜ i
∇× H = J + jω˜ E. (5.55)
˜
˜ i
˜
Here we have separated the primary and secondary currents through J = J + ˜σE, and
c
used the complex permittivity ˜ = ˜ + ˜σ/jω. As with the time-domain equations we
© 2001 by CRC Press LLC