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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
bg
.
The total electric field produced at
P
by the whole rod is the integral
()
()
()
2
0
0
00
0
00
11
1
44
4
,
44
L
L
x
dx
E
Lax
a La
Lax
Lq
aL a
aL a
εε
ε
εε
λλ
λ
⎛⎞
==
=
−
⎜⎟
−
π
+
⎝⎠
+−
λ1
==
−
π+
π+
∫
upon substituting
qL
λ
−=
. With
q
= 4.23
×
10
−
15
C,
L
=0.0815 m and
a
= 0.120 m, we
obtain
3
1.57 10 N/C
x
E
−
=−
×
, or
3

 1.57 10 N/C
x
E
−
=×
.
(c) The negative sign in
x
E
indicates that the field points in the –
x
direction, or
−
180
°
counterclockwise form the +
x
axis.
(d) If
a
is much larger than
L
, the quantity
L
+
a
in the denominator can be approximated
by
a
and the expression for the electric field becomes
E
q
a
x
=−
4
0
2
π
ε
.
Since
50 m
0.0815 m,
aL
==
±
the above approximation applies and we have
8
1.52 10 N/C
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 Spring '10
 BB
 Physics, Charge

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