Problem Set #8
1. A slab of current density in the region
−∞ <x<∞
,
−∞ <y<∞
, –Δ < z < Δ has
J=ˆ
y J0
.
(a) Use the Green’s function
e−jkzz
2jkz
to find the magnetic vector potential
Ay
as a function of z in the z > Δ region.
What is
kz
?
(b) Find the
Hx
field as a function of z in the z > Δ region.
2. Find equivalent electric and magnetic surface currents (Note #21) located on a
sphere of radius a, centered at the origin, that if radiating in free space reproduce
the fields of a Hertzian dipole at the origin. (Hint: Define these directly in
spherical coordinates.)
3. The fields of a thin, linear dipole are often obtained by approximating the
equivalent electric current density by the “sinusoidal triangle” function considered
in Problem Set #7, Problem 4:
J(x,y,z)=ˆ
z I0sin kh −k z
( )
δ
(x)
δ
(y)p(z;−h,h)
Using the result of Problem 4 on Problem Set #7 to assist you, show that the
ˆ
z
component of the electric field of this source can be expressed without
approximation as
Ez(x,y,z)=−j
η
I0
e−jkR1
4
π
R1
+e−jkR2
4
π
R2
−2 cos(kh)e−jkr
4
π
r
⎧
⎨
⎩
⎫
⎬
⎭
What are R1 and R2 in this expression?
Hint: Use the fact that
