24-26
Chapter 24
24.77.
I
DENTIFY
:
The object is equivalent to two identical capacitors in parallel, where each has the same area
A
, plate
separation
d
and dielectric with dielectric constant
K.
S
ET
U
P
:
For each capacitor in the parallel combination,
0
A
C
d
=
P
.
E
XECUTE
:
(a)
The charge distribution on the plates is shown in Figure 24.77.
(b)
2
9
0
0
4
2(4.2)
(0.120 m)
2
2.38
10
F
4.5
10
m
A
C
d
−
−
⎛
⎞
=
=
=
×
⎜
⎟
×
⎝
⎠
P
P
.
E
VALUATE
:
If two of the plates are separated by both sheets of paper to form a capacitor,
9
0
2.38
10
F
2
4
A
C
d
−
×
=
=
P
,
smaller by a factor of 4 compared to the capacitor in the problem.
Figure 24.77
24.78.
I
DENTIFY
:
As in Problem 24.72, the system is equivalent to two capacitors in parallel. One of the capacitors has
plate separation
d
, plate area
(
)
w L
h
−
and air between the plates. The other has the same plate separation
d
, plate area
wh
and dielectric constant
K
.
S
ET
U
P
:
Define
eff
K
by
eff
0
eq
K
A
C
d
=
P
, where
A
wL
=
. For two capacitors in parallel,
eq
1
2
C
C
C
=
+
.
E
XECUTE
:
(a)
The capacitors are in parallel, so
0
0
0
(
)
1
w L
h
K
wh
wL
Kh
h
C
d
d
d
L
L
−
⎛
⎞
=
+
=
+
−
⎜
⎟
⎝
⎠
P
P
P
. This gives
eff
1
Kh
h
K
L
L
⎛
⎞
=
+
−
⎜
⎟
⎝
⎠
.
(b)
For gasoline, with
1.95:
K
=
1
4
full:
eff
1.24
4
L
K
h
⎛
⎞
=
=
⎜
⎟
⎝
⎠
;
1
2
full:
eff
1.48
2
L
K
h
⎛
⎞
=
=
⎜
⎟
⎝
⎠
;
3
4
full:
eff
3
1.71.
4
L
K
h
⎛
⎞
=
=
⎜
⎟
⎝
⎠
(c)
For methanol, with
33:
K
=
1
4
full:
eff
9
4
L
K
h
⎛
⎞
=
=
⎜
⎟
⎝
⎠
;
1
2
full:
eff
17
2
L
K
h
⎛
⎞
=
=
⎜
⎟
⎝
⎠
;
3
4
full:
eff
3
25.
4
L
K
h
⎛
⎞
=
=
⎜
⎟
⎝
⎠
(d)
This kind of fuel tank sensor will work best for methanol since it has the greater range of
eff
K
values.
E
VALUATE
:
When
0
h
=
,
eff
1
K
=
. When
h
L
=
,
eff
K
K
=
.

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