3 Gauss’s law and static charge densities
We continue with examples illustrating the use of Gauss’s law in macroscopic
field 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 field
E
of this
charge distribution using Gauss’s law.
Solution:
First, invoking Coulomb’s law, we convince ourselves that the field produced
by surface charge density
ρ
s
C/m
2
on
x
= 0
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 field 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
= 0
, and obtain
S
D
·
d
S
=
Q
V
⇒
o
E
x
(
x
)
A

o
E
x
(

x
)
A
=
A
ρ
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
= ˆ
x
ρ
s
2
o
sgn
(
x
)
,
where sgn
(
x
)
is the signum function, equal to
±
1
for
x
≷
0
.
Note that the macroscopic field calculated above is discontinuous at
x
= 0
plane
containing the surface charge
ρ
s
, and points away from the same surface on both
sides.
1
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ρ
E
x
(
x
) =
ρ
x
o
A
z
x
W
2

W
2
E
x
(
x
)
ρ
W
2
o

W
2
W
2
x
y
Example 2:
Point charges
Q
are distributed throughout an infinite 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 field
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
) =
ρ
x
o
for
0
< x <
W
2
.
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 Summer '10
 KUDEKI
 Electric charge, charge density

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