27. (a) The linear charge density is the charge per unit length of rod. Since the charge is
uniformly distributed on the rod,
15
14
4.23 10
C
5.19 10
C/m.
0.0815 m
q
L
λ
−
−
−−
×
==
=
−
×
.
(b) We position the
x
axis along the rod with the origin at the left end of the rod, as shown
in the diagram.
Let
dx
be an infinitesimal length of rod at
x
. The charge in this segment is
dq
dx
=λ
. The
charge
dq
may be considered to be a point charge. The electric field it produces at point
P
has only an
x
component and this component is given by
dE
dx
Lax
x
=
+−
1
4
0
2
π
ε
λ
b
g
.
The total electric field produced at
P
by the whole rod is the integral
()
2
0
0
00
0
11
1
44
4
,
L
L
x
dx
E
a La
Lq
aL a
εε
λλ
λ
§·
=
−
¨¸
ππ
+
−
π
+
©¹
λ1
−
π+
³
upon substituting
qL
−=
. With
q
= 4.23
×
10
−
15
C,
L
=0.0815 m and
a
= 0.120 m, we
obtain
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 Fall '10
 Rebello,NobelS
 Vector Space, Charge, Electric charge, Fundamental physics concepts, Standard basis

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