WOLFRAM|DEMONSTRATIONS PROJECT

Electromagnetic Wave Scattering by Conducting Sphere

​
sphere radius (in m):
a
0.25
frequency (in GHz):
f
0.6
Electromagnetic waves can be scattered by electrical conductors. Consider a linearly polarized plane wave, with electric field in the
x
direction and incident in the
z
direction on a perfectly conducting sphere of radius
a
. The electric and magnetic fields of the incoming wave, expressed in Cartesian coordinates, are
E
in
=(
E
0
,0,0)
ikz
e
and
H
in
=(0,
E
0
/η,0)
ikz
e
, where
k=ω/c
(
c
and
η
are the light velocity and impedance, respectively). The scattered fields
E
s
and
H
s
are best expressed in spherical coordinates
(r,θ,φ)
, using Legendre functions
m
P
n
(kr)
and spherical Hankel functions
(1)
h
n
(kr)
and taking account of the boundary conditions. The total fields are given by
E=
E
in
+
E
s
and
H=
H
in
+
H
s
.
The time-averaged electromagnetic energy density is the sum of the electric and magnetic energy densities:
w
e
=(
ϵ
0
/4)E
2
|
and
w
m
=(
μ
0
/4)H
2
|
. (Note that the fields here represent mean-squared values, hence the additional factors 1/2). The total energy density is given by
w=
w
e
+
w
m
. For a incoming wave with
E
0
=1000V/m
, the total energy density is
w=4.43μJ
3
m
(
w
e
=
w
m
=2.22μJ
3
m
). Taking account of scattering, the energy density fluctuates in space and can locally increase up to double the average value:
w
max
=8.85μJ
3
m
.
This Demonstration shows the three-dimensional distributions of the energy densities
w
e
and
w
m
in the vicinity of the sphere, given the sphere radius
a
and the incident wave frequency
f
. The distributions of
w
e
and
w
m
are displayed in red and blue, respectively, in the
x
-
z
and
y
-
z
planes. The variation in density depends on the relative dimensions of the conductor radius
a
and the electromagnetic wavelength
λ=c/f
. The image refreshes rather slowly, even with the image quality decreased.