Page 415 - Electromagnetics

E y (x, y, z) =−E y (−x, −y, z),
E z (x, y, z) = E z (−x, −y, z)?
5.4 Consider an electric Hertzian dipole located on the z-axis at z = h. Showthat
if the dipole is parallel to the plane z = 0, then adding an oppositely-directed dipole of
the same strength at z =−h produces zero electric field tangential to the plane. Also
showthat if the dipole is z-directed, then adding another z-directed dipole at z =−h
produces zero electric field tangential to the z = 0 plane. Since the field for z > 0 is
unaltered in each case if we place a PEC in the z = 0 plane, we establish that tangential
components of electric current image in the opposite direction while vertical components
image in the same direction.
˜
5.5 Consider a z-directed electric line source I 0 located at y = h, x = 0 between con-
ducting planes at y =±d, d > h. The material between the plates has permeability
c
˜ µ(ω) and complex permittivity ˜ (ω). Write the impressed and scattered fields in terms
of Fourier transforms and apply the boundary conditions at z =±d to determine the
electric field between the plates. Show that the result is identical to the expression (5.8)
obtained using symmetry decomposition, which required the boundary condition to be
applied only on the top plate.

˜
5.6 Consider a z-directed electric line source I 0 located at y = h, x = 0 in free space
above a dielectric slab occupying −d < y < d, d < h. The slab has permeability µ 0 and
permittivity . Decompose the source into even and odd constituents and solve for the
electric field everywhere using the Fourier transform approach. Describe how you would
use the even and odd solutions to solve the problem of a dielectric slab located on top of
a PEC ground plane.
5.7 Consider an unbounded, homogeneous, isotropic medium described by permeabil-
c
ity ˜µ(ω) and complex permittivity ˜ (ω). Assuming there are magnetic sources present,
but no electric sources, showthat the fields may be written as

˜
¯
˜ i

H(r) =− jω˜ c G e (r|r ; ω) · J (r ,ω) dV ,
m
V

˜ ¯ ˜ i
E(r) = G m (r|r ; ω) · J (r ,ω) dV ,
m
V
¯
¯
where G e is given by (5.83) and G m is given by (5.84).
¯
¯
5.8 Showthat for a cubical excluding volume the depolarizing dyadic is L = I/3.
5.9 Compute the depolarizing dyadic for a cylindrical excluding volume with height
¯
¯
and diameter both 2a, and with the limit taken as a → 0. Showthat L = 0.293I.
5.10 Showthat the spherical wave function
e − jkr
˜
ψ(r,ω) =
4πr
obeys the radiation conditions (5.96) and (5.97).
5.11 Verify that the transverse component of the Laplacian of A is
2

2 ∂ A t
(∇ A) t = ∇ t (∇ t · A t ) + −∇ t ×∇ t × A t .
∂u 2
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