743
PROB LE M S
58 (a) If C ϭ 50 mF in Fig. 25-52, what
is the equivalent capacitance between
points A and B? (Hint: First imagine
that a battery is connected between
those two points.) (b) Repeat for points
A and D.
59 In Fig. 25-53, V ϭ 12 V, C 1 ϭ C 4 ϭ
2.0 mF, C 2 ϭ 4.0 mF, and C 3 ϭ 1.0 mF.
What is the charge on capacitor 4?
60
The chocolate crumb mystery.
This story begins with Problem 60 in
Chapter 23. As part of the investigation
of the biscuit factory explosion, the electric potentials of the workers were
measured as they emptied sacks of
chocolate crumb powder into the loading bin, stirring up a cloud of the powder
around themselves. Each worker had an
electric potential of about 7.0 kV relative to the ground, which was
taken as zero potential. (a) Assuming that each worker was effectively a capacitor with a typical capacitance of 200 pF, find the energy
stored in that effective capacitor. If a single spark between the
worker and any conducting object connected to the ground neutralized the worker, that energy would be transferred to the spark.
According to measurements, a spark that could ignite a cloud of
chocolate crumb powder, and thus set off an explosion, had to have
an energy of at least 150 mJ. (b) Could a spark from a worker have
set off an explosion in the cloud of
powder in the loading bin? (The
story continues with Problem 60 in
Chapter 26.)
61 Figure 25-54 shows capacitor
1 (C 1 ϭ 8.00 mF), capacitor 2 (C 2
ϭ 6.00 mF), and capacitor 3 (C 3 ϭ
••47
A certain substance has a dielectric constant of
2.8 and a dielectric strength of 18 MV/m. If it is used as the dielectric material in a parallel-plate capacitor, what minimum area
should the plates of the capacitor have to obtain a capacitance of
7.0 ϫ 10
Ϫ2 mF and to ensure that the capacitor will be able to withstand a potential difference of 4.0 kV?
••48 Figure 25-47 shows a parallelplate capacitor with a plate area A
ϭ 5.56 cm
2
and separation d ϭ 5.56
mm. The left half of the gap is filled
with material of dielectric constant
k 1 ϭ 7.00; the right half is filled with
material of dielectric constant k 2 ϭ
12.0. What is the capacitance?
••49 Figure 25-48 shows a parallel-plate capacitor with a plate area A ϭ 7.89 cm
2
and
plate separation d ϭ 4.62 mm. The top half of
the gap is filled with material of dielectric
constant k 1 ϭ 11.0; the bottom half is filled
with material of dielectric constant k 2 ϭ 12.0.
What is the capacitance?
ILW
SSM
material. (b) Determine the magnitude of the charge induced on
each dielectric surface.
••55 The space between two concentric conducting spherical shells
of radii b ϭ 1.70 cm and a ϭ 1.20 cm is filled with
a substance of dielectric constant k 23.5. A
potential difference V ϭ 73.0 V is applied across
the inner and outer shells. Determine (a) the capacitance of the device, (b) the free charge q on
the inner shell, and (c) the charge qЈ induced
along the surface of the inner shell.
Additional Problems
56 In Fig. 25-50, the battery potential
difference V is 10.0 V and each of the seven
capacitors has capacitance 10.0 mF. What is the
charge on (a) capacitor 1 and (b) capacitor 2?
57
In Fig. 25-51, V ϭ 9.0 V, C 1 ϭ C 2 30
mF, and C 3 C 4 15 mF. What is the charge on capacitor 4?
ϭ
ϭ
ϭ
SSM
ϭ
2d
d
d
κ 1
κ 2
κ 3
A/2
A/2
Figure 25-49 Problem 50.
Figure 25-52 Problem 58.
2C
4C
C
6C
A
B
D
d
1
κ
2
κ
A/2
A/2
Figure 25-47 Problem 48.
d
κ 1
κ 2
Figure 25-48
Problem 49.
C 1
C 2
V
– +
Figure 25-50
Problem 56.
C 1
C 3
C 2
C 4
V
Figure 25-53 Problem 59.
C 3
C 2
C 1
C 4
V
Figure 25-51 Problem 57.
••50
Figure 25-49 shows a parallelplate capacitor of plate area A ϭ 10.5
cm
2
and plate separation 2d ϭ 7.12 mm.
The left half of the gap is filled with material of dielectric constant k 1 ϭ 21.0;
the top of the right half is filled with material of dielectric constant k 2 ϭ 42.0;
the bottom of the right half is filled
with material of dielectric constant k 3 ϭ
58.0.What is the capacitance?
Module 25-6 Dielectrics and Gauss’ Law
•51
A parallel-plate capacitor has a capacitance of
100 pF, a plate area of 100 cm
2
, and a mica dielectric (k 5.4)
completely filling the space between the plates. At 50 V potential
difference, calculate (a) the electric field magnitude E in the mica,
(b) the magnitude of the free charge on the plates, and (c) the magnitude of the induced surface charge on the mica.
•52 For the arrangement of Fig. 25-17, suppose that the battery
remains connected while the dielectric slab is being introduced.
Calculate (a) the capacitance, (b) the charge on the capacitor
plates, (c) the electric field in the gap, and (d) the electric field in
the slab, after the slab is in place.
••53 A parallel-plate capacitor has plates of area 0.12 m
2
and a
separation of 1.2 cm. A battery charges the plates to a potential difference of 120 V and is then disconnected. A dielectric slab of thickness 4.0 mm and dielectric constant 4.8 is then placed symmetrically
between the plates. (a) What is the capacitance before the slab is inserted? (b) What is the capacitance with the slab in place? What is
the free charge q (c) before and (d) after the slab is inserted? What is
the magnitude of the electric field (e) in the space between the
plates and dielectric and (f) in the dielectric itself? (g) With the slab
in place, what is the potential difference across the plates? (h) How
much external work is involved in inserting the slab?
••54 Two parallel plates of area 100 cm
2
are given charges of
equal magnitudes 8.9 ϫ 10
Ϫ7
C but opposite signs. The electric
field within the dielectric material filling the space between the
plates is 1.4 ϫ 10
6
V/m. (a) Calculate the dielectric constant of the
ϭ
WWW
SSM
V
P
S
C 1
C 3
C 2
C 4
Figure 25-54 Problem 61.
PROB LE M S
58 (a) If C ϭ 50 mF in Fig. 25-52, what
is the equivalent capacitance between
points A and B? (Hint: First imagine
that a battery is connected between
those two points.) (b) Repeat for points
A and D.
59 In Fig. 25-53, V ϭ 12 V, C 1 ϭ C 4 ϭ
2.0 mF, C 2 ϭ 4.0 mF, and C 3 ϭ 1.0 mF.
What is the charge on capacitor 4?
60
The chocolate crumb mystery.
This story begins with Problem 60 in
Chapter 23. As part of the investigation
of the biscuit factory explosion, the electric potentials of the workers were
measured as they emptied sacks of
chocolate crumb powder into the loading bin, stirring up a cloud of the powder
around themselves. Each worker had an
electric potential of about 7.0 kV relative to the ground, which was
taken as zero potential. (a) Assuming that each worker was effectively a capacitor with a typical capacitance of 200 pF, find the energy
stored in that effective capacitor. If a single spark between the
worker and any conducting object connected to the ground neutralized the worker, that energy would be transferred to the spark.
According to measurements, a spark that could ignite a cloud of
chocolate crumb powder, and thus set off an explosion, had to have
an energy of at least 150 mJ. (b) Could a spark from a worker have
set off an explosion in the cloud of
powder in the loading bin? (The
story continues with Problem 60 in
Chapter 26.)
61 Figure 25-54 shows capacitor
1 (C 1 ϭ 8.00 mF), capacitor 2 (C 2
ϭ 6.00 mF), and capacitor 3 (C 3 ϭ
••47
A certain substance has a dielectric constant of
2.8 and a dielectric strength of 18 MV/m. If it is used as the dielectric material in a parallel-plate capacitor, what minimum area
should the plates of the capacitor have to obtain a capacitance of
7.0 ϫ 10
Ϫ2 mF and to ensure that the capacitor will be able to withstand a potential difference of 4.0 kV?
••48 Figure 25-47 shows a parallelplate capacitor with a plate area A
ϭ 5.56 cm
2
and separation d ϭ 5.56
mm. The left half of the gap is filled
with material of dielectric constant
k 1 ϭ 7.00; the right half is filled with
material of dielectric constant k 2 ϭ
12.0. What is the capacitance?
••49 Figure 25-48 shows a parallel-plate capacitor with a plate area A ϭ 7.89 cm
2
and
plate separation d ϭ 4.62 mm. The top half of
the gap is filled with material of dielectric
constant k 1 ϭ 11.0; the bottom half is filled
with material of dielectric constant k 2 ϭ 12.0.
What is the capacitance?
ILW
SSM
material. (b) Determine the magnitude of the charge induced on
each dielectric surface.
••55 The space between two concentric conducting spherical shells
of radii b ϭ 1.70 cm and a ϭ 1.20 cm is filled with
a substance of dielectric constant k 23.5. A
potential difference V ϭ 73.0 V is applied across
the inner and outer shells. Determine (a) the capacitance of the device, (b) the free charge q on
the inner shell, and (c) the charge qЈ induced
along the surface of the inner shell.
Additional Problems
56 In Fig. 25-50, the battery potential
difference V is 10.0 V and each of the seven
capacitors has capacitance 10.0 mF. What is the
charge on (a) capacitor 1 and (b) capacitor 2?
57
In Fig. 25-51, V ϭ 9.0 V, C 1 ϭ C 2 30
mF, and C 3 C 4 15 mF. What is the charge on capacitor 4?
ϭ
ϭ
ϭ
SSM
ϭ
2d
d
d
κ 1
κ 2
κ 3
A/2
A/2
Figure 25-49 Problem 50.
Figure 25-52 Problem 58.
2C
4C
C
6C
A
B
D
d
1
κ
2
κ
A/2
A/2
Figure 25-47 Problem 48.
d
κ 1
κ 2
Figure 25-48
Problem 49.
C 1
C 2
V
– +
Figure 25-50
Problem 56.
C 1
C 3
C 2
C 4
V
Figure 25-53 Problem 59.
C 3
C 2
C 1
C 4
V
Figure 25-51 Problem 57.
••50
Figure 25-49 shows a parallelplate capacitor of plate area A ϭ 10.5
cm
2
and plate separation 2d ϭ 7.12 mm.
The left half of the gap is filled with material of dielectric constant k 1 ϭ 21.0;
the top of the right half is filled with material of dielectric constant k 2 ϭ 42.0;
the bottom of the right half is filled
with material of dielectric constant k 3 ϭ
58.0.What is the capacitance?
Module 25-6 Dielectrics and Gauss’ Law
•51
A parallel-plate capacitor has a capacitance of
100 pF, a plate area of 100 cm
2
, and a mica dielectric (k 5.4)
completely filling the space between the plates. At 50 V potential
difference, calculate (a) the electric field magnitude E in the mica,
(b) the magnitude of the free charge on the plates, and (c) the magnitude of the induced surface charge on the mica.
•52 For the arrangement of Fig. 25-17, suppose that the battery
remains connected while the dielectric slab is being introduced.
Calculate (a) the capacitance, (b) the charge on the capacitor
plates, (c) the electric field in the gap, and (d) the electric field in
the slab, after the slab is in place.
••53 A parallel-plate capacitor has plates of area 0.12 m
2
and a
separation of 1.2 cm. A battery charges the plates to a potential difference of 120 V and is then disconnected. A dielectric slab of thickness 4.0 mm and dielectric constant 4.8 is then placed symmetrically
between the plates. (a) What is the capacitance before the slab is inserted? (b) What is the capacitance with the slab in place? What is
the free charge q (c) before and (d) after the slab is inserted? What is
the magnitude of the electric field (e) in the space between the
plates and dielectric and (f) in the dielectric itself? (g) With the slab
in place, what is the potential difference across the plates? (h) How
much external work is involved in inserting the slab?
••54 Two parallel plates of area 100 cm
2
are given charges of
equal magnitudes 8.9 ϫ 10
Ϫ7
C but opposite signs. The electric
field within the dielectric material filling the space between the
plates is 1.4 ϫ 10
6
V/m. (a) Calculate the dielectric constant of the
ϭ
WWW
SSM
V
P
S
C 1
C 3
C 2
C 4
Figure 25-54 Problem 61.