Electromagnetic Wave Scattering by Conducting Sphere
Electromagnetic Wave Scattering by Conducting Sphere
Electromagnetic waves can be scattered by electrical conductors. Consider a linearly polarized plane wave, with electric field in the direction and incident in the direction on a perfectly conducting sphere of radius . The electric and magnetic fields of the incoming wave, expressed in Cartesian coordinates, are =(,0,0) and =(0,/η,0), where ( and are the light velocity and impedance, respectively). The scattered fields and are best expressed in spherical coordinates , using Legendre functions (kr) and spherical Hankel functions (kr) and taking account of the boundary conditions. The total fields are given by and .
x
z
a
E
in
E
0
ikz
e
H
in
E
0
ikz
e
k=ω/c
c
η
E
s
H
s
(r,θ,φ)
m
P
n
(1)
h
n
E=+
E
in
E
s
H=+
H
in
H
s
The timeaveraged electromagnetic energy density is the sum of the electric and magnetic energy densities: =(/4)E and =(/4)H. (Note that the fields here represent meansquared values, hence the additional factors 1/2). The total energy density is given by . For a incoming wave with =1000V/m, the total energy density is (==2.22μJ). Taking account of scattering, the energy density fluctuates in space and can locally increase up to double the average value: =8.85μJ.
w
e
ϵ
0
2

w
m
μ
0
2

w=+
w
e
w
m
E
0
w=4.43μJ
3
m
w
e
w
m
3
m
w
max
3
m
This Demonstration shows the threedimensional distributions of the energy densities and in the vicinity of the sphere, given the sphere radius and the incident wave frequency . The distributions of and are displayed in red and blue, respectively, in the  and  planes. The variation in density depends on the relative dimensions of the conductor radius and the electromagnetic wavelength . The image refreshes rather slowly, even with the image quality decreased.
w
e
w
m
a
f
w
e
w
m
x
z
y
z
a
λ=c/f