Download POP4e: Ch. 20 Problems

Chapter 20 Problems
1, 2, 3 = straightforward, intermediate,
challenging
= full solution available in Student
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= computer useful in solving problem
= paired numerical and symbolic
problems
= biomedical application
Section 20.1 Potential Difference and
Electric Potential
1.
(a) Calculate the speed of a proton
that is accelerated from rest through a
potential difference of 120 V. (b) Calculate
the speed of an electron that is accelerated
through the same potential difference.
2.
How much work is done (by a
battery, generator, or some other source of
potential difference) in moving Avogadro’s
number of electrons from an initial point
where the electric potential is 9.00 V to a
point where the potential is –5.00 V? (The
potential in each case is measured relative
to a common reference point.)
Section 20.2 Potential Differences in a
Uniform Electric Field
3.
A uniform electric field of
magnitude 250 V/m is directed in the
positive x direction. A +12.0-μC charge
moves from the origin to the point (x, y) =
(20.0 cm, 50.0 cm). (a) What is the change in
the potential energy of the charge–field
system? (b) Through what potential
difference does the charge move?
4.
The difference in potential between
the accelerating plates in the electron gun of
a TV picture tube is about 25 000 V. If the
distance between these plates is 1.50 cm,
what is the magnitude of the uniform
electric field in this region?
5.
An electron moving
parallel to the x axis has an initial speed of
3.70 × 106 m/s at the origin. Its speed is
reduced to 1.40 × 105 m/s at the point x =
2.00 cm. Calculate the potential difference
between the origin and that point. Which
point is at the higher potential?
6.
Review problem. A block having
mass m and positive charge Q is connected
to an insulating spring having constant k.
The block lies on a frictionless, insulating
horizontal track, and the system is
immersed in a uniform electric field of
magnitude E, directed as shown in Figure
P20.6. If the block is released from rest
when the spring is unstretched (at x = 0), (a)
by what maximum amount does the spring
expand? (b) What is the equilibrium
position of the block? (c) Show that the
block’s motion is simple harmonic and
determine its period. (d) Repeat part (a),
assuming that the coefficient of kinetic
friction between block and surface is μk.
what finite value(s) of x is the electric field
zero? (b) For what finite value(s) of x is the
electric potential zero?
Figure P20.6
Section 20.3 Electric Potential and Electric
Potential Energy Due to Point Charges
Note: Unless stated otherwise, assume that
the reference level of potential is V = 0 at r =
∞.
7.
(a) Find the potential at a distance of
1.00 cm from a proton. (b) What is the
potential difference between two points
that are 1.00 cm and 2.00 cm from a proton?
(c) Repeat parts (a) and (b) for an electron.
10.
Compare this problem with Problem
19.55. Four identical point charges (q = +10.0
μC) are located on the corners of a rectangle
as shown in Figure P19.55. The dimensions
of the rectangle are L = 60.0 cm and W = 15.0
cm. Calculate the change in electric
potential energy of the system as the charge
at the lower left corner in Figure P19.55 is
brought to this position from infinitely far
away. Assume that the other three charges
remain fixed in position.
11.
The three charges in Figure P20.11
are at the vertices of an isosceles triangle.
Calculate the electric potential at the
midpoint of the base, taking q = 7.00 μC.
8.
Given two 2.00-μC charges as shown
in Figure P20.8 and a positive test charge q
= 1.28 × 10–18 C at the origin, (a) what is the
net force exerted by the two 2.00-μC
charges on the test charge q? (b) What is the
electric field at the origin due to the two
2.00-μC charges? (c) What is the electrical
potential at the origin due to the two 2.00μC charges?
Figure P20.8
9.
A charge +q is at the origin. A charge
–2q is at x = 2.00 m on the x axis. (a) For
Figure P20.11
12.
Compare this problem with Problem
19.14. Two point charges each of magnitude
2.00 μC are located on the x axis. One is at x
= 1.00 m and the other is at x = –1.00 m. (a)
Determine the electric potential on the y
axis at y = 0.500 m. (b) Calculate the change
in electric potential energy of the system as
a third charge of –3.00 μC is brought from
infinitely far away to a position on the y
axis at y = 0.500 m.
13.
Show that the amount
of work required to assemble four identical
point charges of magnitude Q at the corners
of a square of side s is 5.41keQ2/s.
14.
Two charged particles create
influences at the origin, described by the
expressions
 7.00  10 9 C
8.99  10 9 N  m 2 /C 2 
cos 70.0î
2
 0.070 0 m 

9
7.00  10 9 C
ˆj  8.00  10 C ˆj
sin
70
.
0

0.070 0 m 2
0.030 0 m 2 
and
 7.00  10 9 C 8.00  10 9 C 
8.99  10 9 N  m 2 /C 2 

0.030 0 m 
 0.070 0 m
(a) Identify the locations of the particles and
the charges on them. (b) Find the force on a
–16.0 nC charge placed at the origin and (c)
the work required to move this third charge
to the origin from a very distant point.
15.
Review problem. Two insulating
spheres have radii 0.300 cm and 0.500 cm,
masses 0.100 kg and 0.700 kg, and
uniformly distributed charges –2.00 μC and
3.00 μC. They are released from rest when
their centers are separated by 1.00 m. (a)
How fast will each be moving when they
collide? (Suggestion: Consider conservation
of energy and of linear momentum.) (b) If
the spheres were conductors, would the
speeds be greater or less than those
calculated in part (a)? Explain.
16.
Review problem. Two insulating
spheres have radii r1 and r2, masses m1 and
m2, and uniformly distributed charges –q1
and q2. They are released from rest when
their centers are separated by a distance d.
(a) How fast is each moving when they
collide? (Suggestion: Consider conservation
of energy and conservation of linear
momentum.) (b) If the spheres were
conductors, would their speeds be greater
or less than those calculated in part (a)?
Explain.
17.
Compare this problem with Problem
19.26. Three equal positive charges q are at
the corners of an equilateral triangle of side
a as shown in Figure P19.26. (a) At what
point, if any, in the plane of the charges is
the electric potential zero? (b) What is the
electric potential at the point P due to the
two charges at the base of the triangle?
18.
Two particles, with charges of 20.0
nC and –20.0 nC, are placed at the points
with coordinates (0, 4.00 cm) and (0, –4.00
cm) as shown in Figure P20.18. A particle
with charge 10.0 nC is located at the origin.
(a) Find the electric potential energy of the
configuration of the three fixed charges. (b)
A fourth particle, with a mass of 2.00 × 10–13
kg and a charge of 40.0 nC, is released from
rest at the point (3.00 cm, 0). Find its speed
after it has moved freely to a very large
distance away.
Figure P20.18
19.
Review problem. A light, unstressed
spring has length d. Two identical particles,
each with charge q, are connected to the
opposite ends of the spring. The particles
are held stationary a distance d apart and
are then released at the same time. The
system then oscillates on a horizontal
frictionless table. The spring has a bit of
internal kinetic friction, so the oscillation is
damped. The particles eventually stop
vibrating when the distance between them
is 3d. Find the increase in internal energy
that appears in the spring during the
oscillations. Assume that the system of the
spring and two charges is isolated.
20.
In 1911, Ernest Rutherford and his
assistants Hans Geiger and Ernest Marsden
conducted an experiment in which they
scattered alpha particles from thin sheets of
gold. An alpha particle, having charge +2e
and mass 6.64 × 10–27 kg, is a product of
certain radioactive decays. The results of
the experiment led Rutherford to the idea
that most of the mass of an atom is in a very
small nucleus, with electrons in orbit
around it, in his planetary model of the
atom. Assume that an alpha particle,
initially very far from a gold nucleus, is
fired with a velocity of 2.00 × 107 m/s
directly toward the nucleus (charge +79e).
How close does the alpha particle get to the
nucleus before turning around? Assume
that the gold nucleus remains stationary.
Section 20.4 Obtaining Electric Field
From Electric Potential
21.
The potential in a region between x =
0 and x = 6.00 m is V = a + bx, where a = 10.0
V and b = –7.00 V/m. Determine (a) the
potential at x = 0, 3.00 m, and 6.00 m; and
(b) the magnitude and direction of the
electric field at x = 0, 3.00 m, and 6.00 m.
22.
The electric potential inside a
charged spherical conductor of radius R is
given by V = keQ/R and outside the potential
is given by V = keQ/r. Using Er = –dV/dr,
derive the electric field (a) inside and (b)
outside this charge distribution.
23.
Over a certain region of
space, the electric potential is V = 5x – 3x2y +
2yz2. Find the expressions for the x, y, and z
components of the electric field over this
region. What is the magnitude of the field
at the point P that has coordinates (1, 0, –2)
m?
Section 20.5 Electric Potential Due to
Continuous Charge Distributions
24.
Consider a ring of radius R with the
total charge Q spread uniformly over its
perimeter. What is the potential difference
between the point at the center of the ring
and a point on its axis a distance 2R from
the center?
25.
A rod of length L (Fig. P20.25) lies
along the x axis with its left end at the
origin. It has a nonuniform charge density
λ = αx, where α is a positive constant. (a)
What are the units of α? (b) Calculate the
electric potential at A.
Figure P20.25 Problems 20.25 and 20.26.
26.
For the arrangement described in
Problem 20.25, calculate the electric
potential at point B that lies on the
perpendicular bisector of the rod a distance
b above the x axis.
27.
Compare this problem with Problem
19.21. A uniformly charged insulating rod
of length 14.0 cm is bent into the shape of a
semicircle as shown in Figure P19.21. The
rod has a total charge of –7.50 μC. Find the
electric potential at O, the center of the
semicircle.
Section 20.6 Electric Potential of a
Charged Conductor
28.
How many electrons should be
removed from an initially uncharged
spherical conductor of radius 0.300 m to
produce a potential of 7.50 kV at the
surface?
29.
A spherical conductor
has a radius of 14.0 cm and charge of 26.0
μC. Calculate the electric field and the
electric potential (a) r = 10.0 cm, (b) r = 20.0
cm, and (c) r = 14.0 cm from the center.
30.
Electric charge can accumulate on an
airplane in flight. You may have observed
needle-shaped metal extensions on the
wing tips and tail of an airplane. Their
purpose is to allow charge to leak off before
much of it accumulates. The electric field
around the needle is much larger than
around the body of the airplane and can
become large enough to produce dielectric
breakdown of the air, discharging the
airplane. To model this process, assume
that two charged spherical conductors are
connected by a long conducting wire and
that a charge of 1.20 μC is placed on the
combination. One sphere, representing the
body of the airplane, has a radius of 6.00
cm, and the other, representing the tip of
the needle, has a radius of 2.00 cm. (a) What
is the electric potential of each sphere? (b)
What is the electric field at the surface of
each sphere?
642 and modeled in Figure P20.34, a second
identical set of plates is enmeshed with its
plates halfway between those of the first
set. The second set can rotate as a unit.
Determine the capacitance as a function of
the angle of rotation θ, where θ = 0
corresponds to the maximum capacitance.
Section 20.7 Capacitance
31.
(a) How much charge is on each
plate of a 4.00-μF capacitor when it is
connected to a 12.0-V battery? (b) If this
same capacitor is connected to a 1.50-V
battery, what charge is stored?
32.
Two conductors having net charges
of +10.0 μC and –10.0 μC have a potential
difference of 10.0 V between them. (a)
Determine the capacitance of the system.
(b) What is the potential difference between
the two conductors if the charges on each
are increased to +100 μC and –100 μC?
33.
An isolated charged conducting
sphere of radius 12.0 cm creates an electric
field of 4.90 × 104 N/C at a distance 21.0 cm
from its center. (a) What is its surface
charge density? (b) What is its capacitance?
34.
A variable air capacitor used in a
radio tuning circuit is made of N
semicircular plates each of radius R and
positioned a distance d from its neighbors,
to which it is electrically connected. As
shown in the opening photograph on page
Figure P20.34
35.
An air-filled capacitor consists of
two parallel plates, each with an area of
7.60 cm2, separated by a distance of 1.80
mm. A 20.0-V potential difference is
applied to these plates. Calculate (a) the
electric field between the plates, (b) the
surface charge density, (c) the capacitance,
and (d) the charge on each plate.
36.
A 50.0-m length of coaxial cable has
an inner conductor that has a diameter of
2.58 mm and carries a charge of 8.10 μC.
The surrounding conductor has an inner
diameter of 7.27 mm and a charge of –8.10
μC. (a) What is the capacitance of this
cable? (b) What is the potential difference
between the two conductors? Assume that
the region between the conductors is air.
37.
A small object of mass m carries a
charge q and is suspended by a thread
between the vertical plates of a parallelplate capacitor. The plate separation is d. If
the thread makes an angle θ with the
vertical, what is the potential difference
between the plates?
38.
A spherical capacitor consists of a
spherical conducting shell of radius b and
charge –Q that is concentric with a smaller
conducting sphere of radius a and charge
+Q (Fig. P20.38). (a) Show that its
capacitance is
C
ab
k e b  a 
39.
Two capacitors, C1 = 5.00 μF and C2 =
12.0 μF, are connected in parallel, and the
resulting combination is connected to a
9.00-V battery. (a) What is the equivalent
capacitance of the combination? What are
(b) the potential difference across each
capacitor and (c) the charge stored on each
capacitor?
40.
The two capacitors of Problem 20.39
are now connected in series and to a 9.00-V
battery. Find (a) the equivalent capacitance
of the combination, (b) the potential
difference across each capacitor, and (c) the
charge on each capacitor.
41.
Four capacitors are
connected as shown in Figure P20.41. (a)
Find the equivalent capacitance between
points a and b. (b) Calculate the charge on
each capacitor, taking ΔVab = 15.0 V.
(b) Show that as b approaches infinity, the
capacitance approaches the value a/ke =
4πε0a.
Figure P20.41
Figure P20.38
Section 20.8 Combinations of Capacitors
42.
Two capacitors when connected in
parallel give an equivalent capacitance of Cp
and an equivalent capacitance of Cs when
connected in series. What is the capacitance
of each capacitor?
43.
Consider the circuit shown in Figure
P20.43, where C1 = 6.00 μF, C2 = 3.00 μF, and
ΔV = 20.0 V. Capacitor C1 is first charged by
the closing of switch S1. Switch S1 is then
opened, and the charged capacitor is
connected to the uncharged capacitor by
the closing of S2. Calculate the initial charge
acquired by C1 and the final charge on each
capacitor.
Figure P20.44
Figure P20.43
44.
Three capacitors are connected to a
battery as shown in Figure P20.44. Their
capacitances are C1 = 3C, C2 = C, and C3 = 5C.
(a) What is the equivalent capacitance of
this set of capacitors? (b) State the ranking
of the capacitors according to the charge
they store, from largest to smallest. (c) Rank
the capacitors according to the potential
differences across them, from largest to
smallest. (d) If now C3 is increased, what
happens to the charge stored by each of the
capacitors?
45.
According to its design specification,
the timer circuit delaying the closing of an
elevator door is to have a capacitance of
32.0 μF between two points A and B. (a)
When one circuit is being constructed, the
inexpensive but durable capacitor installed
between these two points is found to have
capacitance 34.8 μF. To meet the
specification, one additional capacitor can
be placed between the two points. Should it
be in series or in parallel with the 34.8-μF
capacitor? What should be its capacitance?
(b) The next circuit comes down the
assembly line with capacitance 29.8 μF
between A and B. What additional
capacitor should be installed in series or in
parallel in that circuit to meet the
specification?
46.
Find the equivalent capacitance
between points a and b in the combination
of capacitors shown in Figure P20.46.
to discharge the capacitor through the
patient’s chest. Assume that an energy of
300 J is to be delivered from a 30.0-μF
capacitor. To what potential difference
must it be charged?
Figure P20.46
Section 20.9 Energy Stored in a Charged
Capacitor
47.
(a) A 3.00-μF capacitor is connected
to a 12.0-V battery. How much energy is
stored in the capacitor? (b) If the capacitor
had been connected to a 6.00-V battery,
how much energy would have been stored?
48.
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 (Fig. P20.48) is a device that
applies a strong electric shock to the chest
over a time interval of a few milliseconds.
The device contains a capacitor of several
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
(Adam Hart-Davis/SPL/Custom Medical Stock)
Figure P20.48 A defibrillator in use.
49.
Two capacitors, C1 = 25.0 μF and C2 =
5.00 μF, are connected in parallel and
charged with a 100-V power supply. (a)
Draw a circuit diagram and calculate the
total energy stored in the two capacitors. (b)
What potential difference would be
required across the same two capacitors
connected in series so that the combination
stores the same energy as in part (a)? Draw
a circuit diagram of this circuit.
50.
As a person moves about in a dry
environment, electric charge accumulates
on the person’s body. Once it is at high
voltage, either positive or negative, the
body can discharge via sometimes
noticeable sparks and shocks. Consider a
human body well separated from ground,
with the typical capacitance 150 pF. (a)
What charge on the body will produce a
potential of 10.0 kV? (b) Sensitive electronic
devices can be destroyed by electrostatic
discharge from a person. A particular
device can be destroyed by a discharge
releasing an energy of 250 μJ. To what
voltage on the body does this energy
correspond?
51.
A parallel-plate capacitor has a
charge Q and plates of area A. What force
acts on one plate to attract it toward the
other plate? Because the electric field
between the plates is E = Q /Aε0, you might
think that the force is F = QE = Q2/Aε0. This
conclusion is wrong because the field E
includes contributions from both plates,
and the field created by the positive plate
cannot exert any force on the positive plate.
Show that the force exerted on each plate is
actually F = Q2/2ε0A. (Suggestion: Let C =
ε0A/x for an arbitrary plate separation x;
then require that the work done in
separating the two charged plates be
W   F dx .) The force exerted by one
charged plate on another is sometimes used
in a machine shop to hold a workpiece
stationary.
52.
A uniform electric field E = 3 000
V/m exists within a certain region. What
volume of space contains an energy equal
to 1.00 × 10–7 J? Express your answer in
cubic meters and in liters.
Section 20.10 Capacitors with Dielectrics
53.
Determine (a) the capacitance and
(b) the maximum potential difference that
can be applied to a Teflon-filled parallel-
plate capacitor having a plate area of 1.75
cm2 and plate separation of 0.040 0 mm.
54.
(a) How much charge can be placed
on a capacitor with air between the plates
before it breaks down if the area of each of
the plates is 5.00 cm2? (b) Find the
maximum charge assuming polystyrene is
used between the plates instead of air.
55.
A commercial capacitor is to be
constructed as shown in Figure 20.29a. This
particular capacitor is made from two strips
of aluminum foil separated by a strip of
paraffin-coated paper. Each strip of foil and
paper is 7.00 cm wide. The foil is 0.004 00
mm thick, and the paper is 0.025 0 mm
thick and has a dielectric constant of 3.70.
What length should the strips have if a
capacitance of 9.50 × 10–8 F is desired before
the capacitor is rolled up? (Adding a
second strip of paper and rolling the
capacitor effectively doubles its capacitance
by allowing charge storage on both sides of
each strip of foil.)
56.
The supermarket sells rolls of
aluminum foil, of plastic wrap, and of
waxed paper. Describe a capacitor made
from supermarket materials. Compute
order-of-magnitude estimates for its
capacitance and its breakdown voltage.
57.
A parallel-plate capacitor in air has a
plate separation of 1.50 cm and a plate area
of 25.0 cm2. The plates are charged to a
potential difference of 250 V and
disconnected from the source. The capacitor
is then immersed in distilled water.
Determine (a) the charge on the plates
before and after immersion, (b) the
capacitance and potential difference after
immersion, and (c) the change in energy of
the capacitor. Assume that the liquid is an
insulator.
Section 20.11 Context Connection—The
Atmosphere as a Capacitor
58.
Lightning can be studied with a Van
de Graaff generator, essentially consisting
of a spherical dome on which charge is
continuously deposited by a moving belt.
Charge can be added until the electric field
at the surface of the dome becomes equal to
the dielectric strength of air (3 × 106 V/m).
Any more charge leaks off in sparks as
shown in Figure P20.58. Assume that the
dome has a diameter of 30.0 cm and is
surrounded by dry air. (a) What is the
maximum potential of the dome? (b) What
is the maximum charge on the dome?
(E. R. Degginer/H. Armstrong Roberts)
Figure P20.58
59.
Review problem. A certain storm
cloud has a potential of 1.00 × 108 V relative
to a tree. If, during a lightning storm, 50.0 C
of charge is transferred through this
potential difference and 1.00% of the energy
is absorbed by the tree, how much sap in
the tree can be boiled away? Model the sap
as water initially at 30.0°C. Water has a
specific heat of 4 186 J/kg · °C, a boiling
point of 100°C, and a latent heat of
vaporization of 2.26 × 106 J/kg.
Additional Problems
60.
Review problem. From a large
distance away, a particle of mass 2.00 g and
charge 15.0 μC is fired at 21.0 î m/s straight
toward a second particle, originally
stationary but free to move, with mass 5.00
g and charge 8.50 μC. (a) At the instant of
closest approach, both particles will be
moving at the same velocity. Explain why.
(b) Find this velocity. (c) Find the distance
of closest approach. (d) Find the velocities
of both particles after they separate again.
61.
The liquid-drop model of the atomic
nucleus suggests that high-energy
oscillations of certain nuclei can split the
nucleus into two unequal fragments plus a
few neutrons. The fission products acquire
kinetic energy from their mutual Coulomb
repulsion. Calculate the electric potential
energy (in electron volts) of two spherical
fragments from a uranium nucleus having
the following charges and radii: 38e and
5.50 × 10–15 m; 54e and 6.20 × 10–15 m.
Assume that the charge is distributed
uniformly throughout the volume of each
spherical fragment and that just before
separating each fragment is at rest and their
surfaces are in contact. The electrons
surrounding the nucleus can be ignored.
62.
The Bohr model of the hydrogen
atom states that the single electron can exist
only in certain allowed orbits around the
proton. The radius of each Bohr orbit is r =
n2(0.052 9 nm) where n = 1, 2, 3, . . . .
Calculate the electric potential energy of a
hydrogen atom when the electron is in the
(a) first allowed orbit, with n = 1; (b) second
allowed orbit, with n = 2; and (c) when the
electron has escaped from the atom, with r
= ∞. Express your answers in electron volts.
63.
Calculate the work that must be
done to charge a spherical shell of radius R
to a total charge Q.
64.
A Geiger–Mueller tube is a radiation
detector that essentially consists of a closed,
hollow metal cylinder (the cathode) of inner
radius ra and a coaxial cylindrical wire (the
anode) of radius rb (Fig. P20.64). The charge
per unit length on the anode is λ, and the
charge per unit length on the cathode is –λ.
A gas fills the space between the electrodes.
When a high-energy elementary particle
passes through this space, it can ionize an
atom of the gas. The strong electric field
makes the resulting ion and electron
accelerate in opposite directions. They
strike other molecules of the gas to ionize
them, producing an avalanche of electrical
discharge. The pulse of electric current
between the wire and the cylinder is
counted by an external circuit. (a) Show
that the magnitude of the potential
difference between the wire and the
cylinder is
r
V  2 k e  ln  a
 rb



(b) Show that the magnitude of the electric
field in the space between cathode and
anode is given by
E
V  1 
 
ln ra / rb   r 
where r is the distance from the axis of the
anode to the point where the field is to be
calculated.
Figure P20.64 Problems 20.64, 20.65, and
20.66.
65.
Assume that the internal diameter of
the Geiger–Mueller tube described in
Problem 20.64 is 2.50 cm and that the wire
along the axis has a diameter of 0.200 mm.
The dielectric strength of the gas between
the central wire and the cylinder is 1.20 ×
106 V/m. Use the result of Problem 20.64 to
calculate the maximum potential difference
that can be applied between the wire and
the cylinder before breakdown occurs in the
gas.
66.
The results of Problem 20.64 apply
also to an electrostatic precipitator (Figs.
P20.64 and P20.66). This pollution-control
device consists of a vertical cylindrical duct
with a wire along its axis at a high negative
voltage. Corona discharge ionizes the air
around the wire to produce free electrons
and positive and negative molecular ions.
The electrons and negative ions accelerate
outward. As air passes through the
cylinder, the dirt particles become
electrically charged by collisions and ion
capture. They are then swept out of the air
by the horizontal electric field between the
wire and the cylinder. In a particular case,
an applied voltage ΔV = Va – Vb = 50.0 kV is
to produce an electric field of magnitude
5.50 MV/m at the surface of the central
wire. Assume that the outer cylindrical wall
has uniform radius ra = 0.850 m. (a) What
should be the radius rb of the central wire?
You will need to solve a transcendental
equation. (b) What is the magnitude of the
electric field at the outer wall?
Figure P20.66
67.
A model of a red blood cell portrays
the cell as a capacitor with two spherical
plates. It is a positively charged conducting
liquid sphere of area A, separated by an
insulating membrane of thickness t from
the surrounding negatively charged
conducting fluid. Tiny electrodes
introduced into the cell show a potential
difference of 100 mV across the membrane.
Take the membrane’s thickness as 100 nm
and its dielectric constant as 5.00. (a)
Assume that a typical red blood cell has a
mass of 1.00 × 10–12 kg and density 1 100
kg/m3. Calculate its volume and its surface
area. (b) Find the capacitance of the cell. (c)
Calculate the charge on the surfaces of the
membrane. How many electronic charges
does this charge represent? (Suggestion: The
chapter text models the Earth’s atmosphere
as a capacitor with two spherical plates.)
68.
Four balls, each with mass m, are
connected by four nonconducting strings to
form a square with side a as shown in
Figure P20.68. The assembly is placed on a
horizontal, nonconducting, frictionless
surface. Balls 1 and 2 each have charge q,
and balls 3 and 4 are uncharged. Find the
maximum speed of balls 3 and 4 after the
string connecting balls 1 and 2 is cut.
Figure P20.69
70.
An electric dipole is located along
the y axis as shown in Figure P20.70. The
magnitude of its electric dipole moment is
defined as p = 2qa. (a) At a point P, which is
far from the dipole (r >> a), show that the
electric potential is
V 
Figure P20.68
69.
The x axis is the symmetry axis of a
stationary, uniformly charged ring of
radius R and charge Q (Fig. P20.69). A
particle with charge Q and mass M is
located at the center of the ring. When it is
displaced slightly, the point charge
accelerates along the x axis to infinity. Show
that the ultimate speed of the point charge
is
 2k Q 2
v   e
 MR



1/ 2
k e p cos 
r2
(b) Calculate the radial component Er and
the perpendicular component Eθ of the
associated electric field. Note that Eθ =
−(1/r)(∂V/∂θ). Do these results seem
reasonable for θ = 90° and 0°? for r = 0? (c)
For the dipole arrangement shown, express
V in terms of Cartesian coordinates using r
= (x2 + y2)1/2 and
cos  
x
y
2
 y2 
1/ 2
Using these results and again taking r >> a,
calculate the field components Ex and Ey.
Figure P20.71
72.
A 2.00-nF parallel-plate capacitor is
charged to an initial potential difference ΔVi
= 100 V and then isolated. The dielectric
material between the plates is mica, with a
dielectric constant of 5.00. (a) How much
work is required to withdraw the mica
sheet? (b) What is the potential difference of
the capacitor after the mica is withdrawn?
Figure P20.70
71.
Two large, parallel metal plates are
oriented horizontally and separated by a
distance 3d. A grounded conducting wire
joins them, and initially each plate carries
no charge. Now a third identical plate
carrying charge Q is inserted between the
two plates, parallel to them and located a
distance d from the upper plate, as shown
in Figure P20.71. (a) What induced charge
appears on each of the two original plates?
(b) What potential difference appears
between the middle plate and each of the
other plates? Let A represent the area of
each plate.
73.
A parallel-plate capacitor is
constructed using a dielectric material
whose dielectric constant is 3.00 and whose
dielectric strength is 2.00 × 108 V/m. The
desired capacitance is 0.250 μF, and the
capacitor must withstand a maximum
potential difference of 4 000 V. Find the
minimum area of the capacitor plates.
74.
A 10.0-μF capacitor is charged to 15.0
V. It is next connected in series with an
uncharged 5.00-μF capacitor. The series
combination is finally connected across a
50.0-V battery as diagrammed in Figure
P20.74. Find the new potential differences
across the 5.00-μF and 10.0-μF capacitors.
Figure P20.74
75.
A capacitor is constructed from two
square plates of sides ℓ and separation d. A
material of dielectric constant κ is inserted a
distance x into the capacitor as shown in
Figure P20.75. Assume that d is much
smaller than x. (a) Find the equivalent
capacitance of the device. (b) Calculate the
energy stored in the capacitor, letting ΔV
represent the potential difference. (c) Find
the direction and magnitude of the force
exerted on the dielectric, assuming a
constant potential difference ΔV. Ignore
friction. (d) Obtain a numerical value for
the force assuming that ℓ = 5.00 cm, ΔV = 2
000 V, d = 2.00 mm, and the dielectric is
glass (κ = 4.50). (Suggestion: The system can
be considered as two capacitors connected
in parallel.)
Figure P20.75 Problems 20.75 and 20.76.
76.
A capacitor is constructed from two
square plates of sides ℓ and separation d as
suggested in Figure P20.75. You may
assume that d is much less than ℓ. The
plates carry charges +Q0 and –Q0. A block of
metal has width ℓ, length ℓ, and thickness
slightly less than d. It is inserted a distance
x into the capacitor. The charges on the
plates are not disturbed as the block slides
in. In a static situation, a metal prevents an
electric field from penetrating inside it. The
metal can be thought of as a perfect
dielectric, with κ → ∞. (a) Calculate the
stored energy as a function of x. (b) Find the
direction and magnitude of the force that
acts on the metallic block. (c) The area of
the advancing front face of the block is
essentially equal to ℓd. Considering the
force on the block as acting on this face,
find the stress (force per area) on it. (d) For
comparison, express the energy density in
the electric field between the capacitor
plates in terms of Q0, ℓ, d, and ε0.
77.
Determine the equivalent
capacitance of the combination shown in
Figure P20.77. (Suggestion: Consider the
symmetry involved.)
Figure P20.77
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