Page 416 - Electromagnetics

Verify that the longitudinal component of the Laplacian of A is

2 2
ˆ u ˆ u ·∇ A = ˆ u∇ A u .
5.12 Verify the identities (B.82)–(B.93).
5.13 Verify the identities (B.94)–(B.98).

5.14 Derive the formula (5.112) for the transverse component of the electric field.

5.15 The longitudinal/transverse decomposition can be performed beginning with the
time-domain Maxwell’s equations. Show that for a homogeneous, lossless, isotropic region
described by permittivity and permeability µ the longitudinal fields obey the wave
equations

2 2
∂ 1 ∂ ∂ H u ∂E u ∂J mt ∂J t
− H t =∇ t − ˆ u ×∇ t + − ˆ u × ,
2
∂u 2 v ∂t 2 ∂u ∂t ∂t ∂u

2 2
∂ 1 ∂ ∂E u ∂ H u ∂J mt ∂J t
− E t =∇ t + µˆ u ×∇ t + ˆ u × + µ .
2
∂u 2 v ∂t 2 ∂u ∂t ∂u ∂t
Also showthat the transverse fields may be found from the longitudinal fields by solving

1 ∂ 1 ∂ρ ∂ J u
2
∇ − E u = + µ +∇ t × J mt ,
2
v ∂t 2 ∂u ∂t

2 1 ∂ 1 ∂ρ m ∂ J mu
∇ − H u = + −∇ t × J t .
2
v ∂t 2 µ ∂u ∂t
√
Here v = 1/ µ .
5.16 Consider a homogeneous, lossless, isotropic region of space described by permittiv-
ity and permeability µ. Beginning with the source-free time-domain Maxwell equa-
tions in rectangular coordinates, choose z as the longitudinal direction and showthat the
TE–TM decomposition is given by

2 2 2 2
∂ 1 ∂ ∂ E z ∂ H z
− E y = + µ , (5.194)
2
∂z 2 v ∂t 2 ∂z∂y ∂x∂t
∂ 1 ∂ ∂ E z ∂ H z

2 2 2 2
− E x = − µ , (5.195)
2
∂z 2 v ∂t 2 ∂x∂z ∂y∂t

2 2 2 2
∂ 1 ∂ ∂ E z ∂ H z
− H y =− + , (5.196)
2
∂z 2 v ∂t 2 ∂x∂t ∂y∂z

2 2 2 2
∂ 1 ∂ ∂ E z ∂ H z
− H x = + , (5.197)
2
∂z 2 v ∂t 2 ∂y∂t ∂x∂z
with

2
1 ∂
2
∇ − E z = 0, (5.198)
2
v ∂t 2
1 ∂

2
2
∇ − H z = 0. (5.199)
2
v ∂t 2
√
Here v = 1/ µ .
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