Homework 9 Answers, 95.658 Spring 2011, Electromagnetic Theory II
Dr. Christopher S. Baird, UMass Lowell
Problem 1
Jackson 10.3
A solid uniform sphere of radius
R
and conductivity
σ
acts as a scatterer of a planewave beam of
unpolarized radiation of frequency
ω
, with
ωR
/
c
<< 1. The conductivity is large enough that the skin
depth
δ
is small compared to
R
.
(a) Justify and use a magnetostatic scalar potential to determine the magnetic field around the sphere,
assuming the conductivity is infinite. (Remember that
ω
≠ 0.)
(b) Use the technique of Section 8.1 to determine the absorption cross section of the sphere. Show that
it varies as (
ω
)
1/2
provided
σ
is independent of frequency.
SOLUTION:
(a)
The statement
ωR
/
c
<< 1 is equivalent to
R <<
λ. This is the longwavelength region. In this
region, we can safely assume that the fields are constant across the object, so that the problem reduces
to a electrostatics/magnetostatics problem. In this case, we want to find the absorption, which is caused
by induced currents meeting resistance. So we first need to find the currents. So we choose to approach
this problem as a magnetostatics problem, because static magnetic fields are linked to currents.
A perfectly conducting sphere is placed in an originally uniform magnetic field.
The region of interest, outside the sphere, is free space and contains no charges or currents, so that we
immediately know:
∇⋅
B
=
0
,
∇×
B
=
0
(Because
B
=μ
0
H
and
J
= 0)
The curl being zero (the second equation) lets us define the magnetic field in terms of the gradient of a
scalar potential:
B
=−∇ Ψ
M
Plugging this into the first equation leads to:
∇
2
Ψ
M
=
0
This is just the familiar Laplace equation. If we place the sphere at the origin, and set the original field
pointing along the zaxis, the problem is azimuthally symmetric. The solution to the Laplace equation
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Staff
 Conductivity, Mass, Work, Radiation, Magnetic Field, Cos, Dr. Christopher S. Baird, Electromagnetic Theory II

Click to edit the document details