4 Divergence and curl
Expressing the total charge
Q
V
contained in a volume
V
as a 3D volume
integral of charge density
ρ
(
r
)
we can express Gauss’s law examined during
the last few lectures in the general form
S
D
·
d
S
=
V
ρ
dV.
This equation asserts that the flux of displacement
D
=
o
E
over any closed
surface
S
equals the net electrical charge contained in the enclosed volume
V
— only the charges included within
V
a
ff
ect the flux of
D
over surface
S
, with charges outside surface
S
making no net contribution to the surface
integral
S
D
·
d
S
.
•
Gauss’s law stated above holds true everywhere in space over all sur
faces
S
and their enclosed volumes
V
, large and small.
•
Application of Gauss’s law to a small volume
Δ
V
=
Δ
x
Δ
y
Δ
z
sur
rounded by a cubic surface
Δ
S
of six faces, leads, in the limit of van
ishing
Δ
x
,
Δ
y
, and
Δ
z
, to the di
ff
erential form of Gauss’s law expressed
in terms of a
divergence operation
to be reviewed next:
(
x, y, z
+
Δ
z
)
x
y
z
(
x, y, z
)
(
x
+
Δ
x, y, z
)
(
x, y
+
Δ
y, z
)
–
Given a su
ffi
ciently small volume
Δ
V
=
Δ
x
Δ
y
Δ
z
, we can assume
that
Δ
V
ρ
dV
≈
ρ
Δ
x
Δ
y
Δ
z.
1
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–
Again under the same assumption
S
D
·
d
S
≈
(
D
x

2

D
x

1
)
Δ
y
Δ
z
+(
D
y

4

D
y

3
)
Δ
x
Δ
z
+(
D
z

6

D
z

5
)
Δ
x
Δ
y
with reference to displacement vector components like
D
x

2
shown
on cubic surfaces depicted in the margin.
Gauss’s law demands
the equality of the two expressions above, namely (after dividing
both sides by
Δ
x
Δ
y
Δ
z
)
x
y
z
(
x, y, z
)
5
2
1
4
3
6
D
x

2

D
x

1
Δ
x
+
D
y

4

D
y

3
Δ
y
+
D
z

6

D
z

5
Δ
z
≈
ρ
,
in the limit of vanishing
Δ
x
,
Δ
y
, and
Δ
z
. In that limit, we obtain
∂
D
x
∂
x
+
∂
D
y
∂
y
+
∂
D
z
∂
z
=
ρ
,
which is known as
di
ff
erential form of Gauss’s law
.
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 Spring '10
 KIM
 Electric charge, Vector field, Gauss’s Law

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