Digital Signal Processing Reference
In-Depth Information
2.4.3 Capacitance
In circuit terms, the quantity associated with storing energy in an electric field is
capacitance. To define capacitance, imagine two conductors, with a charge of
+
Q
on one and of
−
Q
on the other. If we assume that the voltage is constant over
each conductor, the potential difference (voltage) between them is calculated as
b
E
·
dl
v(b)
−
v(a)
=−
V
(2-58)
a
E
is proportional to
Q
:
We show that
Q
4
πε
0
r
2
E
=
a
r
E
r
=
V
/
m
(2-60)
E
is proportional to both
Q
and
v
, we can define a constant of proportion-
ality that relates
Q
and
v
. The constant of proportionality is defined to be the
capacitance
:
Since
Q
v
C
≡
farads
(2-76)
where
Q
is the total charge in coulombs and
v
is the voltage potential between the
conductors, given in units of farads, defined as 1 coulomb per volt. Capacitance
depends purely on the geometry of the structures and the value of the dielectric
permittivity. Note that
v
is defined as the potential of the positive conductor
minus the negative conductor and that
Q
is the charge on the positive conductor.
Therefore,
capacitance is always a positive value
.
Example 2-3
Consider the case where two conductive plates of area
A
are
oriented parallel to each other separated by a distance
d
. Assume that we place
a charge of
−
Q
on the bottom plate and assume that
the charges will spread out evenly (a reasonable assumption, assuming a good
conductor). Then the surface charge density becomes
+
Q
on the top plate and
(
C
/
m
2
).
Calculate the capacitance.
ρ
=
Q/A
SOLUTION Using the integral form of Gauss's law (2-59), we can calculate
the electric field:
ε E
·
d
s
=
ρ dV
s
V
where
dV
in this case refers to the volume. Since we are considering the charge
distribution on a surface,
ds
=
dV
=
nA
(where
n
is the unit normal vector
to the plate), we can write the electric field in terms of the area and dielectric
permittivity:
Q
A
A
→
E
=
Q
εA
εEA
=
Search WWH ::
Custom Search