Page 417 - Electromagnetics

5.17 Consider the case of TM fields in the time domain. Showthat for a homogeneous,
isotropic, lossless medium with permittivity and permeability µ the fields may be
˜
derived from a single Hertzian potential Π e (r, t) = ˆ u e (r, t) that satisfies the wave
equation
1 ∂

2
2
∇ − e = 0
2
v ∂t 2
and that the fields are

2 2
∂ e ∂ 1 ∂ ∂ e
E =∇ t + ˆ u − e , H =− ˆ u ×∇ t .
2
∂u ∂u 2 v ∂t 2 ∂t
5.18 Consider the case of TE fields in the time domain. Showthat for a homogeneous,
isotropic, lossless medium with permittivity and permeability µ the fields may be
˜
derived from a single Hertzian potential Π h (r, t) = ˆ u h (r, t) that satisfies the wave
equation

2
1 ∂
2
∇ − h = 0
2
v ∂t 2
and that the fields are

2 2
∂ h ∂ h ∂ 1 ∂
E = µˆ u ×∇ t , H =∇ t + ˆ u 2 − 2 2 h .
∂t ∂u ∂u v ∂t
5.19 Showthat in the time domain TEM fields may be written for a homogeneous,
isotropic, lossless medium with permittivity and permeability µ in terms of a Hertzian
potential Π e = ˆ u e that satisfies
2
∇ e = 0
t
and that the fields are
∂ e ∂ e
E =∇ t , H =− ˆ u ×∇ t .
∂u ∂t
5.20 Showthat in the time domain TEM fields may be written for a homogeneous,
isotropic, lossless medium with permittivity and permeability µ in terms of a Hertzian
potential Π h = ˆ u h that satisfies

2
∇ h = 0
t
and that the fields are
∂ h ∂ h
E = µˆ u ×∇ t , H =∇ t .
∂t ∂u
5.21 Consider a TEM plane-wave field of the form
˜
˜
˜
˜
E = ˆ xE 0 e − jkz , H = ˆ y E 0 − jkz ,
e
η
√ √
where k = ω µ and η = µ/ . Showthat:
˜
˜
(a) E may be obtained from H using the equations for a field that is TE y ;
˜
˜
(b) H may be obtained from E using the equations for a field that is TM x ;
2
˜
˜
˜
˜
(c) E and H may be obtained from the potential Π h = ˆ y(E 0 /k η)e − jkz ;
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