Digital Signal Processing Reference
In-Depth Information
where
r
ab
denotes the radial distance that the stationary charge moved and
ab
is known as the
electrostatic potential
or voltage between points
a
and
b
(don't
confuse the symbol for potential
with the polar coordinate variable
φ
).
2.4.1 Electrostatic Scalar Potential in Terms of an Electric Field
For electrical engineers, it is often convenient to think of fields in terms of
more familiar circuit concepts, which are usually described in terms of potential
differences (i.e., voltage) between points in a circuit. Therefore, a relationship
between the electrostatic potential and the electric fields that is described in terms
of a scalar function will prove to be useful in the study of signal integrity. To
derive this relationship, it is convenient to sidestep the complexities of spherical
coordinates and think in terms of simple rectangular coordinates. From equations
(2-61) and (2-62) it is obvious that the differential amount of work done in
moving a charge in an electrostatic field is directly proportional to the potential
difference:
dW
=
q(x
+
x, y
+
y, z
+
z)
−
q(x,y,z)
=
q
∂
∂x
z
(2-63)
∂
∂y
y
+
∂
∂x
x
+
However, from equation (2-58),
b
q E
·
dl
→
=−
q E
·
dl
W
=−
dW
a
In rectangular coordinates,
dl
=
a
x
x
+
a
y
y
+
a
z
z
. Thus, (2-63) can be sim-
plified using the definition of the dot product in (2-13).
=
q
∂
∂x
z
∂
∂y
y
+
∂
=−
q E
·
l
dW
∂x
x
+
d
=
q
a
x
∂
∂x
+
a
y
∂
∂y
+
a
z
∂
·
(a
x
x
+
a
y
y
+
a
z
z)
∂z
(2-64)
=−
q E
·
(a
x
x
+
a
y
y
+
a
z
z)
a
x
∂
∂x
+
a
y
∂
∂y
+
a
z
∂
E
=−
∂z
Note that the form of the last term in (2-64) is identical to (2-21), which is the
gradient, which gives a very useful relation between the electric field and the
electrostatic potential:
E
=−∇
(2-65)
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