MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2007
Problem Set 2 Solutions
Problem 1: Closed Surfaces
Four closed surfaces,
S
1
through
S
4
, together with the charges
–2
Q
,
Q
, and –
Q
are sketched in the figure at right. The
colored lines are the intersections of the surfaces with the
page.
Find the electric flux through each surface.
By Gauss’s Law, the flux through the closed surfaces is equal
to the charge enclosed over
ε
0
.
So,
1
2
3
4
0
0
0
S
S
S
S
2
;
0;
;
Q
Q
ε
ε
Φ
= −
Φ
=
Φ
= −
Φ
=
Problem 2:
Field on Axis of a Line Charge
A wire of length
has a uniform positive linear charge density and a total charge
.
Calculate the electric field at a point
located along the axis of the wire and a distance
from one end:
l
Q
P
a
a.
Give an integral expression for the electric field at point
P
in terms of the
variables used in the above figure
(
)
2
ˆ
a
l
e
dx
k
a
x
λ
+
′
= −
′
∫
E
i
b.
Evaluate this integral.
1
1
1
ˆ
ˆ
ˆ
ˆ
(
)
(
)
a
l
e
e
e
e
l
Q
k
k
k
k
a
x
a
l
a
a a
l
a a
l
λ
λ
λ
+
⎡
⎤
−
⎡
⎤
=
=
−
=
= −
⎢
⎥
⎢
⎥
′
+
+
E
i
i
i
i
+
⎣
⎦
⎣
⎦
c.
In the limit that the length of the rod goes to zero, does your answer reduce to the
right expression?
2
ˆ
In the limit that
goes to zero, our expression becomes
, as we desire.
e
Q
l
k
⎛
⎞
=
−
⎜
⎟
E
i
a
⎝
⎠
G
PS02 Solution - 1

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