Electricity and Magnetism I (P331)
Homework 6
Due: Wednesday, October 21, 4:00 PM
The next exam will be Friday, October 23 in class.
1. Two concentric conducting spherical shells, with radii
a
and 2
a
, have charge +
Q
and

Q
respectively.
The space between the shells is filled with a linear dielectric with permittivity
(
r
) =
0
a
1
.
5
a

0
.
5
r
,
(1)
which varies with radial distance
r
from
0
at
r
=
a
to 2
0
at
r
= 2
a
.
(a) Use Gauss’s Law to determine the displacement field between the spherical shells.
(b) Determine the bound charge density between the spherical shells.
(c) Determine the total energy of this system from the relation
U
=
1
2
D
·
E
dτ
.
(d) Show that the capacitance is 4
π
0
a/
(0
.
75

0
.
5 ln 2).
2. A point charge
Q
is embedded in a dielectric medium with a dielectric constant
r
.
(a) What is the free charge enclosed by a sphere of radius
R
centered at
Q
?
(b) What is the bound charge enclosed by the sphere? How does the result vary with
R
?
(c) What is the total charge enclosed by the sphere?
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 Fall '09
 STAFF
 Charge, Magnetism, Work, Coaxial cable, Dielectric, bound charge, free charge density, spherical shells, dielectric constant r2

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