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3 Gauss’s law and static charge densities
We continue with examples illustrating the use of Gauss’s law in macroscopic
Feld calculations:
ρ
S
E
x
(
x
)=
ρ
s
2
±
o
A
z
S
x
y
E
x
(
x
)
ρ
s
2
±
o
sgn(
x
)
x
Example 1:
Point charges
Q
are distributed over
x
=0
plane with an average surface
charge density of
ρ
s
C/m
2
. Determine the macroscopic electric Feld
E
of this
charge distribution using Gauss’s law.
Solution:
±irst, invoking Coulomb’s law, we convince ourselves that the Feld produced
by surface charge density
ρ
s
C/m
2
on
x
plane will be of the form
E
=ˆ
xE
x
(
x
)
where
E
x
(
x
)
is an odd function of
x
because
y
 and
z
components of the Feld will
cancel out due to the symmetry of the charge distribution. In that case we can
apply Gauss’s law over a cylindrical integration surface
S
having circular caps of
area
A
parallel to
x
, and obtain
±
S
D
·
d
S
=
Q
V
⇒
±
o
E
x
(
x
)
A

±
o
E
x
(

x
)
A
=
Aρ
s
,
which leads, with
E
x
(

x

E
x
(
x
)
, to
E
x
(
x
ρ
s
2
±
o
for
x>
0
.
Hence, in vector form
E
x
ρ
s
2
±
o
sgn
(
x
)
,
where sgn
(
x
)
is the signum function, equal to
±
1
for
x
≷
0
.
Note that the macroscopic Feld calculated above is discontinuous at
x
plane
containing the surface charge
ρ
s
, and points away from the same surface on both
sides.
1
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View Full Documentρ
E
x
(
x
)=
ρx
±
o
A
z
x
W
2

W
2
E
x
(
x
)
ρW
2
±
o

W
2
W
2
x
y
Example 2:
Point charges
Q
are distributed throughout an infnite slab oF width
W
located over

W
2
<x<
W
2
with an average charge density oF
ρ
C/m
3
. Determine
the macroscopic electric feld
E
oF the charged slab inside and outside.
Solution:
Symmetry arguments based on Coulomb’s law once again indicates that we
expect a solution oF the Form
E
=ˆ
xE
x
(
x
)
where
E
x
(
x
)
is an odd Function oF
x
.
In that case, applying Gauss’s law with a cylindrical surFace
S
having circular caps
oF area
A
parallel to
x
=0
extending between

x
and
x<
W
2
, we obtain
±
S
D
·
d
S
=
Q
V
⇒
±
o
E
x
(
x
)
A

±
o
E
x
(

x
)
A
=
ρ
2
xA,
which leads, with
E
x
(

x

E
x
(
x
)
, to
E
x
(
x
ρx
±
o
For
0
W
2
.
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 Spring '10
 KIM

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