Question 1
A very long (you can count it as infinite) ceramic (i.e. nonconducting)
cylindrical pipe of inner radius a and outer radius b has accidentally become
charged with a uniform charge density
ρ
.
You are concerned that the
electric field due to this charge may be affecting sensitive instruments inside
the inner radius of the pipe, embedded in the pipe, and hanging outside the
pipe.
Having learned Gauss’ Law you can calculate the fields that might be
upsetting the instruments.
Consider a length of the pipe L
Inside the pipe
The contained charge is 0.
Therefore by Gauss’ Law,
E=0 from r=0 to r=a
Outside the pipe
draw a Gaussian cylinder parallel to the pipe at a distance
r
Contained charge is
(
)
2
2
b
a
L
π
ρ
−
By Gauss’ Law
0
q
E dS
ε
=
∫
G
G
i
v
a
b
ρ
L
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From symmetry E is perpendicular to the cylinder and constant at radius r,
therefore
2
E dS
E
dS
rLE
π
=
=
∫
∫
G
G
i
v
Then
(
)
(
)
2
2
0
2
2
0
2
2
b
a
L
rLE
b
a
E
r
π
ρ
π
ε
ρ
ε
−
=
−
=
This is the magnitude of E, the vector field is in the direction of r
ˆ
E
Er
=
G
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 Spring '08
 Marshak
 Charge, Electric charge, positive point charge, radius r Charge, charge Q

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