20. We imagine a cylindrical Gaussian surface
A
of radius
r
and unit length concentric
with the metal tube. Then by symmetry
enc
0
2
.
A
q
E dA
rE
ε
⋅
=
π
=
³
G
G
v
(a) For
r < R, q
enc
= 0, so
E
= 0.
(b) For
r > R, q
enc
=
λ
, so
0
( )
/ 2
.
E r
r
π ε
= λ
With
8
2.00
10
C/m
λ
−
=
×
and
r
= 2.00
R
=
0.0600 m, we obtain
(
)
(
)
(
)
8
3
12
2
2
2.0
10
C/m
5.99
10
N/C.
2
0.0600 m
8.85
10
C / N m
E
−
−
×
=
=
×
π
×
⋅
(c) The plot of
E
vs.
r
is shown below.
Here, the maximum value is
(
)
(
)
(
)
8
4
max
12
2
2
0
2.0
10
C/m
1.2
10
N/C.
2
2
0.030 m
8.85
10
C / N m
E
r
ε
−
−
×
λ
=
=
=
×
π
π
×
⋅
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uniform, and we neglect fringing effect. Symmetry can be used to show that the electric
field is radial, both between the cylinder and the shell and outside the shell. It is zero, of
course, inside the cylinder and inside the shell.
(a) We take the Gaussian surface to be a cylinder of length
L
, coaxial with the given
cylinders and of larger radius
r
than either of them. The flux through this surface is
2
,
rLE
Φ =
π
where
E
is the magnitude of the field at the Gaussian surface. We may
ignore any flux through the ends. Now, the charge enclosed by the Gaussian surface is
q
enc
=
Q
1 +
Q
2
= –
Q
1
= –3.40
×
10
−
12
C. Consequently, Gauss’ law yields
0
enc
2
,
r
LE
q
π ε
=
or
12
enc
12
2
2
3
0
3.40
10
C
0.214 N/C,
2
2
(8.85
10
C
/ N
m
)(11.0 m)(20.0
1.30
10
m)
q
E
Lr
ε
π
−
−
−
−
×
=
=
= −
π
×
⋅
×
×
or


0.214 N/C.
E
=
(b) The negative sign in
E
indicates that the field points inward.
(c) Next, for
r
= 5.00
R
1
, the charge enclosed by the Gaussian surface is
q
enc
=
Q
1
=
3.40
×
10
−
12
C. Consequently, Gauss’ law yields
0
enc
2
,
r
LE
q
π ε
=
or
12
enc
12
2
2
3
0
3.40
10
C
0.855 N/C.
2
2
(8.85
10
C
/ N
m
)(11.0 m)(5.00
1.30
10
m)
q
E
Lr
πε
π
−
−
−
×
=
=
=
×
⋅
×
×
(d) The positive sign indicates that the field points outward.
(e) we consider a cylindrical Gaussian surface whose radius places it within the shell
itself. The electric field is zero at all points on the surface since any field within a
conducting material would lead to current flow (and thus to a situation other than the
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 Spring '08
 STEPHENIRONS
 Physics, Charge, Electrostatics, Electric charge, gaussian surface, R. Gauss

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