Physics For Scientists And Engineers 6E - part 208
Answers to Quick Quizzes
829
Example 26.7, in which the battery was removed from the
circuit before the dielectric was introduced.)
70.
A vertical parallel-plate capacitor is half filled with a dielec-
tric for which the dielectric constant is 2.00 (Fig. P26.70a).
When this capacitor is positioned horizontally, what
fraction of it should be filled with the same dielectric (Fig.
P26.70b) in order for the two capacitors to have equal
capacitance?
75.
Determine the equivalent capacitance of the combination
shown in Figure P26.75. (Suggestion: Consider the symme-
try involved.)
(b)
(a)
Figure P26.70
a
b
2.00
µ
F
4.00
µ
F
2.00
µ
F
4.00
µ
F
8.00
µ
F
µ
µ
µ
µ
µ
Figure P26.72
C
C
3C
2C
2C
Figure P26.75
71.
Capacitors C
1
#
6.00 %F and C
2
#
2.00 %F are charged as
a parallel combination across a 250-V battery. The capaci-
tors are disconnected from the battery and from each
other. They are then connected positive plate to negative
plate and negative plate to positive plate. Calculate the
resulting charge on each capacitor.
72.
Calculate the equivalent capacitance between the points a
and b in Figure P26.72. Note that this is not a simple series
or parallel combination. (Suggestion: Assume a potential
difference !V between points a and b. Write expressions
for !V
ab
in terms of the charges and capacitances for the
various possible pathways from a to b, and require conser-
vation of charge for those capacitor plates that are con-
nected to each other.)
The inner conductor of a coaxial cable has a radius of
0.800 mm, and the outer conductor’s inside radius is
3.00 mm. The space between the conductors is filled with
polyethylene, which has a dielectric constant of 2.30 and a
dielectric strength of 18.0 * 10
6
V/m. What is the maxi-
mum potential difference that this cable can withstand?
74.
You are optimizing coaxial cable design for a major manu-
facturer. Show that for a given outer conductor radius b,
maximum potential difference capability is attained when
the radius of the inner conductor is a # b/e where e is the
base of natural logarithms.
73.
76.
Consider two long, parallel, and oppositely charged wires
of radius d with their centers separated by a distance D.
Assuming the charge is distributed uniformly on the
surface of each wire, show that the capacitance per unit
length of this pair of wires is
77.
Example 26.2 explored a cylindrical capacitor of length !
and radii a and b of the two conductors. In the What If?
section, it was claimed that increasing ! by 10% is more
effective in terms of increasing the capacitance than increas-
ing a by 10% if b , 2.85a. Verify this claim mathematically.
Answers to Quick Quizzes
26.1 (d). The capacitance is a property of the physical system
and does not vary with applied voltage. According to
Equation 26.1, if the voltage is doubled, the charge is
doubled.
26.2 (a). When the key is pressed, the plate separation is
decreased and the capacitance increases. Capacitance
depends only on how a capacitor is constructed and not
on the external circuit.
26.3 (a). When connecting capacitors in series, the inverses
of the capacitances add, resulting in a smaller overall
equivalent capacitance.
26.4 (a). When capacitors are connected in series, the volt-
ages add, for a total of 20 V in this case. If they are com-
bined in parallel, the voltage across the combination is
still 10 V.
26.5 (b). For a given voltage, the energy stored in a capacitor
is proportional to C: U # C(!V )
2
/2. Thus, you want to
maximize the equivalent capacitance. You do this by
connecting the three capacitors in parallel, so that the
capacitances add.
26.6 (a) C decreases (Eq. 26.3). (b) Q stays the same because
there is no place for the charge to flow. (c) E remains
constant (see Eq. 24.8 and the paragraph following it).
(d) !V increases because !V # Q /C, Q is constant (part
b), and C decreases (part a). (e) The energy stored in
the capacitor is proportional to both Q and !V (Eq.
C
!
#
()
0
ln[(D $ d)/d]