Download ADDITIONAL PROBLEMS 52. Three parallel-plate

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566
Chapter 16
Electrical Energy and Capacitance
ADDITIONAL PROBLEMS
52. Three parallel-plate capacitors are constructed, each having the same plate spacing d and with C1 having plate
area A1, C 2 having area A2, and C 3 having area A3. Show
that the total capacitance C of the three capacitors
connected in parallel is the same as that of a capacitor
having plate spacing d and plate area A A1 A2 A 3.
53. Three parallel-plate capacitors are constructed, each having the same plate area A and with C 1 having plate spacing d1, C 2 having plate spacing d 2, and C 3 having plate
spacing d 3. Show that the total capacitance C of the three
capacitors connected in series is the same as a capacitor
of plate area A and with plate spacing d d1 d 2 d 3.
54. Two capacitors give an equivalent capacitance of Cp when
connected in parallel and an equivalent capacitance of
C s when connected in series. What is the capacitance of
each capacitor?
55. An isolated capacitor of unknown capacitance has been
charged to a potential difference of 100 V. When the
charged capacitor is disconnected from the battery and
then connected in parallel to an uncharged 10.0-F capacitor, the voltage across the combination is measured
to be 30.0 V. Calculate the unknown capacitance.
56. Two charges of 1.0 C and 2.0 C are 0.50 m apart at
two vertices of an equilateral triangle as in Figure P16.56.
(a) What is the electric potential due to the 1.0-C
charge at the third vertex, point P ? (b) What is the electric potential due to the 2.0-C charge at P ? (c) Find
the total electric potential at P. (d) What is the work
required to move a 3.0-C charge from infinity to P.
59.
60.
61.
P
62.
0.50 m
0.50 m
0.50 m
1.0 mC
2.0 mC
Figure P16.56
63.
of the outer sphere approaches infinity, the capacitance
approaches the value a/ke 40a.
The immediate cause of many deaths is ventricular fibrillation, an uncoordinated quivering of the heart, as opposed to proper beating. An electric shock to the chest
can cause momentary paralysis of the heart muscle, after
which the heart will sometimes start organized beating
again. A defibrillator is a device that applies a strong electric shock to the chest over a time of a few milliseconds.
The device contains a capacitor of a few microfarads,
charged to several thousand volts. Electrodes called paddles, about 8 cm across and coated with conducting
paste, are held against the chest on both sides of the
heart. Their handles are insulated to prevent injury to
the operator, who calls, “Clear!” and pushes a button on
one paddle to discharge the capacitor through the
patient’s chest. Assume that an energy of 300 W s is to
be delivered from a 30.0-F capacitor. To what potential
difference must it be charged?
When a certain air-filled parallel-plate capacitor is
connected across a battery, it acquires a charge of 150 C
on each plate. While the battery connection is maintained,
a dielectric slab is inserted into, and fills, the region
between the plates. This results in the accumulation of
an additional charge of 200 C on each plate. What is
the dielectric constant of the slab?
Capacitors C1 6.0 F and C 2 2.0 F are charged as a
parallel combination across a 250-V battery. The capacitors 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.
Capacitors C1 4.0 F and C 2 2.0 F are charged as a
series combination across a 100-V battery. The two capacitors are disconnected from the battery and from each
other. They are then connected positive plate to positive
plate and negative plate to negative plate. Calculate the
resulting charge on each capacitor.
The charge distribution shown in Figure P16.63 is referred to as a linear quadrupole. (a) Show that the electric
potential at a point on the x-axis where x d is
57. Find the equivalent capacitance of the group of capacitors shown in Figure P16.57.
5.00 mF
3.00 mF
V
(b) Show that the expression obtained in (a) when x d
reduces to
V
2.00 mF
4.00 mF
7.00 mF
2k e Qd 2
x3
y
3.00 mF
6.00 mF
2k e Qd 2
x 3 xd 2
+Q
–2Q
+Q
x
(–d, 0)
(d, 0)
48.0 V
Figure P16.57
58. A spherical capacitor consists of a spherical conducting
shell of radius b and charge Q concentric with a smaller
conducting sphere of radius a and charge Q. (a) Find the
capacitance of this device. (b) Show that as the radius b
Quadrupole
Figure P16.63
64. The energy stored in a 52.0-F capacitor is used to melt a
6.00-mg sample of lead. To what voltage must the capacitor
be initially charged, assuming that the initial tempera-
44920_16_p531-567 12/27/04 7:47 AM Page 567
Problems
ture of the lead is 20.0°C? Lead has a specific heat of
128 J/kg°C, a melting point of 327.3°C, and a latent heat
of fusion of 24.5 kJ/kg.
65. Consider a parallel-plate capacitor with charge Q and
area A, filled with dielectric material having dielectric
constant . It can be shown that the magnitude of the
attractive force exerted on each plate by the other is
F Q 2/(20A). When a potential difference of 100 V
exists between the plates of an air-filled 20-F parallelplate capacitor, what force does each plate exert on the
other if they are separated by 2.0 mm?
66. An electron is fired at a speed v0 5.6 106 m/s and at
an angle 0 45° between two parallel conducting plates
that are D 2.0 mm apart, as in Figure P16.66. If the voltage difference between the plates is V 100 V, determine
(a) how close, d, the electron will get to the bottom plate
and (b) where the electron will strike the top plate.
y
Path of
the electron
D
0
x
u0
v0
d
Figure P16.66
V
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ACTIVITIES
1. It takes an electric field of about 30 kV/cm to cause a
spark in dry air. Shuffle across a rug and reach toward a
doorknob. By estimating the length of the spark, determine the electric potential difference that existed between your finger and the doorknob just before you
touched the knob. Try this experiment again on a very
humid day, and you will find that the spark is much
shorter or is imperceptible. Why?
2. Suppose you are given a battery, a capacitor, two switches,
a lightbulb, and several pieces of connecting wire. On a
sheet of paper, design a circuit that will do the following:
(1) When switch 1 is closed and switch 2 is open, the capacitor charges, but no current moves through the lightbulb. (2) Then, when switch 1 is opened and switch 2
closed, the lightbulb is connected to the capacitor, but
not to the battery. Describe the motion of charge in the
circuit when switch 1 is closed and switch 2 is open. Is energy being stored in the capacitor? What measurements
would you have to make to determine how much energy,
if any, is stored? What happens to the lightbulb when
switch 1 is opened after the capacitor has charged and
switch 2 is then closed? Will the bulb light and stay lit?
What happens to the charge on the capacitor when
switch 2 is closed in this way?