Solutions for Homework 6
Problems from Chapter 23
(
a
)
The electric field at the surface of the Earth is the same as that of a point charge,
E
=
Q
4
!"
0
r
0
2
.
The electric potential at the surface, relative to
V
(
!
)
=
0
is given by Eq. 235.
Writing this in terms of the electric field and radius of the earth gives the electric
potential.
V
=
Q
4
0
r
0
=
Er
0
=
#
150V m
( )
6.38
$
10
6
m
( )
=
#
0.96 GV
(
b
) Part (
a
) demonstrated that the potential at the surface of the earth is 0.96 GV lower than
the potential at infinity. Therefore if the potential at the surface of the Earth is taken to
be zero, the potential at infinity must be
V
(
!
)
=
0.96 GV
.
If the charge of the
ionosphere is included in the calculation, the electric field outside the ionosphere is
basically zero. The electric field between the earth and the ionosphere would remain
the same. The electric potential, which would be the integral of the electric field from
infinity to the surface of the earth, would reduce to the integral of the electric field from
the ionosphere to the earth. This would result in a negative potential, but of a smaller
magnitude.
21. We first need to find the electric field. Since the charge distribution is spherically
symmetric, Gauss’s law tells us the electric field everywhere.
r
E
&
d
r
A
&
!
=
E
4
"
r
2
( )
=
Q
encl
#
0
$
E
=
1
4
"#
0
Q
encl
r
2
If
r
<
r
0
,
calculate the charge enclosed in the manner of Example 225.
Q
encl
=
!
E
dV
"
=
0
1
#
r
2
r
0
2
$
%
&
’
(
)
4
*
r
2
dr
0
r
"
=
4
*!
0
r
2
#
r
4
r
0
2
$
%
&
’
(
)
dr
0
r
"
=
4
0
r
3
3
#
r
5
5
r
0
2
$
%
&
’
(
)
The total charge in the sphere is the above expression evaluated at
r
=
r
0
.
Q
total
=
4
0
r
0
3
3
#
r
0
5
5
r
0
2
$
%
&
’
(
)
=
8
0
r
0
3
15
Outside the sphere, we may treat it as a point charge, and so the potential at the surface of
the sphere is given by Eq. 235, evaluated at the surface of the sphere.
V r
=
r
0
( )
=
1
4
0
Q
total
r
0
=
1
4
0
8
!#
0
r
0
3
15
r
0
=
2
0
r
0
2
15
0
The potential inside is found from Eq. 234a. We need the field inside the sphere to use Eq.
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.
 Fall '10
 yildiz
 Physics, Electric Potential, Energy, Potential Energy, Work, Electric charge, el ectric field, ch arge

Click to edit the document details