Ph1b Analytic Quiz 3 Solution
Chefung Chan, Winter 2008
Problem 1 (6 points total)
A spherically symmetric charge distribution is given by
ρ
(
r
) = 0 ;
0
< r < a
=
K
r
;
a < r < b
= 0 ;
r > b
where
K
is a constant. At the center,
r
= 0, there is a point charge
q
.
(a) (2 points)
Use Gauss’s Law to find the electric field
E
(
r
) in the region
a <
r < b
in terms of
q, K, a,
and
r
.
Solution:
Consider Gauss law with a spherical gaussian surface, we have
E
(4
πr
2
) = 4
πq
enclosed
In this case, charge enclosed has 2 components. One is the center point charge
q
while another one is the charge distribution.
For the distribution one :
q
(
r
) =
r
a
4
πr
2
ρ
(
r
)
dr
=
r
a
4
πKr dr
= 2
πK
(
r
2
−
a
2
)
∴
E
(
r
) =
1
r
2
q
enclosed
=
1
r
2
(
q
+ 2
πK
(
r
2
−
a
2
)
)
=
q
r
2
+ 2
πK
1
−
a
2
r
2
(
a < r < b
)
(b) (1 point)
Find the value of
K
in terms of
q
and
a
for which the electric field
is a
constant
E
0
(independent of
r
) in the region
a < r < b
Solution:
From (a),
E
(
r
) =
q
r
2
+ 2
πK
1
−
a
2
r
2
If
E
a<r<b
=
E
0
= constant, then we have
dE
(
r
)
dr
= 0
−
2
q
r
3
+ 2
πK
2
a
2
r
3
= 0
q
= 2
πKa
2
K
=
Q
2
πa
1
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With
K
=
q/
(2
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 Winter '08
 McKewn
 Charge, Electrostatics, Electric charge

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