Biomedical Engineering Reference
In-Depth Information
The integral form of Gauss' law finds application in calculating electric fields
around charged objects. However, Gauss's law in the differential form (i.e., diver-
gence of the vector electric field) is more useful. If
E
is a function of space variables
x
,
y
, and
z
in Cartesian coordinates [Figure 3.7(b)], then
E
E E
Eu
∂
∂
∂
∇•
=
+
v
+
w
(3.36)
∂
x
∂
y
∂
z
where
u
is the unit vector in the
x
direction,
v
is the unit vector in the
y
direction,
and
w
is the unit vector in the
z
direction. The divergence theorem is used to convert
surface integral into a volume integral and the Gauss law is
Q
(
)
∫
∫
EdA
•=∇•
EdV
=
(3.37)
ε
0
S
V
Substituting (3.35) for
Q
ch
and simplification results in the differential form of
Gauss law
ρ
ε
∇•
E
=
ch
(3.38)
0
Since cross production of the electric field is zero
,
E
is represented
(
∇×
E
=
0)
as a gradient of electrical potential, that is,
E
=−∇ΔΦ
A negative sign is used based on the convention that electric field direction is
away from the positive charge source. Then the divergence of the gradient of the
scalar function is
−
ρ
ε
2
∇ΔΦ=
ch
(3.39)
0
or
∂ΔΦ
2
∂ΔΦ
2
∂ΔΦ
2
−
ρ
ε
+
+
=
ch
(3.40)
2
2
2
∂
x
∂
y
∂
z
0
In a region of space where there is no unpaired charge density (
ρ
ch
=
0), (3.39)
2
reduces to (
0), Laplace's equation.
While the area integral of the electric field gives a measure of the net charge en-
closed, the divergence of the electric field gives a measure of the density of sources.
∇
ΔΦ
=
