Exam 2 Practice Problems_1
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21.1
Resistors
in
Series
and
Parallel
Note: Data taken from figures can be assumed to be accurate to
three
significant digits.
1.
(a) What is the resistance of ten
275-Ω
resistors connected in
series? (b) In
p
a
r
a
ll
e
l
?
2.
(a)
What
is
the
resistance
of
a
1.00
×
10
2
resistor connected in series? (b) In
p
a
r
a
ll
e
l
?
−Ω
,
a
2.50-kΩ
,
and
a
4.00-kΩ
3.
What are the largest and smallest resistances you can obtain by
connecting
a
36.0-Ω
, a
50.0-Ω
, and a
700-Ω
resistor together?
4.
An 1800-W toaster, a 1400-W electric frying pan, and a 75-W lamp
are plugged into the same outlet in a 15-A, 120-V circuit. (The three
devices are in parallel when plugged into the same socket.). (a)
What current is drawn by each device? (b) Will this combination blow
the 15-A fuse?
5.
Your car’s 30.0-W headlight and 2.40-kW starter are ordinarily
connected in parallel in a 12.0-V system. What power would one
headlight and the starter consume if connected in series to a 12.0-V
battery? (Neglect any other resistance in the circuit and any change
in resistance in the two devices.)
6.
(a) Given a 48.0-V battery and
24.0-Ω
and
96.0-Ω
resistors,
find the current and power for each when connected in series. (b)
Repeat when the resistances are in parallel.
7.
Referring to the example combining series and parallel circuits and
Fi
gure
21.6
, calculate
3
in the following two different ways: (a) from the known
values of
and
2
; (b) using Ohm’s
law for
3
.
In
both
parts
explicitly
show
how you follow the steps in the
Problem-Solvin
g Strategies for Series
and
Parallel Resistors
.
8.
Referring to
Fi
gure
21.6
: (a)
Calculate
3
and note how it compares with
3
found in the first two example problems in this module. (b) Find the
total
power supplied by the source and compare it with the sum of the
powers dissipated by the resistors.
9.
Refer to
Fi
gure 21.7
and the discussion of lights dimming when
a heavy appliance comes on. (a) Given the voltage source is 120 V,
the wire resistance is
0.400 Ω
, and the bulb is nominally 75.0 W,
what power will the bulb dissipate if a total of 15.0 A passes through
the wires when the motor comes on? Assume negligible change in
bulb resistance. (b) What power is consumed by the motor?
10.
A 240-kV power transmission line carrying
5.00 × 10
2
A
is hung from
grounded metal towers by ceramic insulators, each having a
1.00×10
9
−Ω
resistance.
Fi
gure
21.51
. (a) What is the resistance to ground of
100 of these insulators? (b) Calculate the power dissipated by 100 of
them. (c) What fraction of the power carried by the line is this?
Explicitly show how you follow
the steps in the
Problem-Solvin
g
Strategies for Series and Parallel
Resistors
.
Figure 21.51
High-
voltage (240-kV)
transmission line
carrying
5.00
×
10
2
A
is
hung from a grounded
metal transmission tower.
The row of
ceramic
insulators provide
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1.00 ×
10
9
Ω
of resistance
each.
11.
Show that if two resistors
1
and
2
are combined and one is much
greater
than
the
other
(
1
>>
2
):
(a)
Their
series
resistance
is
very
nearly
equal
to
the
greater
resistance
1
.
(b)
Their
parallel
resistance
is
very
nearly
equal
to
smaller
resistance
2
.
12.
Unreasonable
Results
Two resistors, one having a resistance of
145 Ω
, are connected in
parallel to produce a total resistance of
150 Ω
. (a) What is the value
of the second resistance? (b) What is unreasonable about this
r
e
su
l
t
?
(c) Which assumptions
are unreasonable or inconsistent?
13.
Unreasonable
Results
Two resistors, one having a resistance of
900 kΩ
, are connected in
series to produce a total resistance of
0.500 MΩ
. (a) What is the
value of the second resistance? (b) What is unreasonable about this
r
e
su
l
t
?
(c) Which assumptions are unreasonable or inconsistent?
21.2
Electromotive
Force:
Terminal
Volta
ge
14.
Standard automobile batteries have six lead-acid cells in
series, creating a total emf of 12.0 V. What is the emf of an individual
lead-acid
c
e
ll
?
15.
Carbon-zinc dry cells (sometimes referred to as non-alkaline
cells) have an emf of 1.54 V, and they are produced as single cells or
in various combinations to form other voltages. (a) How many 1.54-V
cells are needed to make the common 9-V battery used in many
small electronic devices? (b) What is the actual emf of the
approximately 9-V battery? (c) Discuss how internal resistance in the
series connection of cells will affect the terminal voltage of this
approximately 9-V battery.
16.
What is the output voltage of a 3.0000-V lithium cell in a
digital wristwatch that draws 0.300 mA, if the cell’s internal
resistance is
2.00 Ω
?
2
bulb
17.
(a) What is the terminal voltage of a large 1.54-V carbon-zinc
dry cell used in a physics lab to supply 2.00 A to a circuit, if the
cell’s internal resistance is
0.100 Ω
? (b) How much electrical power does the cell produce? (c)
What
power goes to its
l
o
a
d
?
18.
What
is
the
internal
resistance
of
an
automobile
battery
that
has
an
emf
of
12.0 V and a terminal voltage of 15.0 V while a current of 8.00 A is
charging it?
19.
(a) Find the terminal voltage of a 12.0-V motorcycle battery
having a
0.600-Ω
internal resistance, if it is being charged by a
current of 10.0 A. (b)
What is the output voltage of the battery
charger?
20.
A car battery with a 12-V emf and an internal resistance of
0.050 Ω
is being charged with a current of 60 A. Note that in this
process the battery is being charged. (a) What is the potential
difference across its terminals? (b) At what rate is thermal energy
being dissipated in the battery? (c) At what rate is electric energy
being converted to chemical energy? (d) What are the answers to (a)
and (b) when the battery is used to supply 60 A to the starter motor?
21.
The hot resistance of a flashlight bulb is
2.30 Ω
, and it is
run by a 1.58-V
alkaline cell having a
0.100-Ω
internal resistance. (a)
What current flows? (b)
Calculate the power supplied to the bulb
using same as calculated using
?
2
bulb
.
(c)
Is
this
power
the
22.
The label on a portable radio recommends the use of
rechargeable nickel- cadmium cells (nicads), although they have a
1.25-V emf while alkaline cells have a 1.58-V emf. The radio has a
3.20-Ω
resistance. (a) Draw a circuit diagram of the radio and its
batteries. Now, calculate the power delivered to the radio. (b) When
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using Nicad cells each having an internal resistance of
0.0400 Ω
.
(c) When using alkaline cells each having an internal resistance of
0.200
Ω
.
(d)
Does
this
difference
seem
significant,
considering
that
the
radio’s
effective resistance is lowered when its volume is turned up?
23.
An automobile starter motor has an equivalent resistance of
0.0500 Ω
and
is supplied by a 12.0-V battery with a
0.0100-Ω
internal resistance. (a) What
is the current to the motor? (b) What voltage is applied to it? (c) What
power is supplied to the motor? (d) Repeat these calculations for when
the battery connections are corroded and add
0.0900 Ω
to the
circuit. (Significant problems are caused by even small amounts of
unwanted resistance in low- voltage, high-current applications.)
24.
A child’s electronic toy is supplied by three 1.58-V alkaline
cells having internal resistances of
0.0200 Ω
in series with a 1.53-
V carbon-zinc dry cell
having a
0.100-Ω
internal resistance. The load
resistance is
10.0 Ω
. (a) Draw
a circuit diagram of the toy and its batteries. (b) What current
flows? (c) How much power is supplied to the
l
o
a
d
?
(d) What is the
internal resistance of the dry cell if it goes bad, resulting in only
0.500 W being supplied to the
l
o
a
d
?
25.
(a) What is the internal resistance of a voltage source if its
terminal voltage
drops by 2.00 V when the current supplied increases
by 5.00 A? (b) Can the emf of the voltage source be found with the
information
su
pp
l
i
e
d
?
26.
A person with body resistance between his hands of
10.0 kΩ
accidentally grasps the terminals of a 20.0-kV power supply. (Do NOT
do this!) (a) Draw a circuit diagram to represent the situation. (b) If
the internal resistance of the power supply is
2000 Ω
, what is the
current through his body? (c) What is the power dissipated in his
body? (d) If the power supply is to be made safe by
increasing its
internal resistance, what should the internal resistance be for the
maximum current in this situation to be 1.00 mA or less? (e) Will this
modification compromise the effectiveness of the power supply for
driving
low-resistance devices? Explain your reasoning.
27.
Electric fish generate current with biological cells called
electroplaques, which are physiological emf devices. The
electroplaques in the South American eel are arranged in 140 rows,
each row stretching horizontally along
the body and each containing 5000 electroplaques. Each
electroplaque has an emf of 0.15 V and internal resistance of
0.25
Ω
. If the water surrounding the fish has resistance of
800 Ω
, how
much current can the eel produce in water from near its head to near
its
t
a
i
l
?
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28.
Integrated
Concepts
A 12.0-V emf automobile battery has a terminal voltage of 16.0 V
when being charged by a current of 10.0 A. (a) What is the battery’s
internal resistance? (b)
What power is dissipated inside the battery? (c)
At what rate (in
ºC/min
) will
its temperature increase if its mass is
20.0 kg and it has a specific heat of
0.300
kcal/kg
⋅
ºC
,
assuming
no
heat
escapes?
29.
Unreasonable
Results
A 1.58-V alkaline cell with a
0.200-Ω
internal resistance is supplying
8.50 A to
a load. (a) What is its terminal vo
l
t
a
g
e
?
(b) What is the
value of the load resistance? (c) What is unreasonable about these
results? (d) Which assumptions are unreasonable or inconsistent?
30.
Unreasonable
Results
(a) What is the internal resistance of a 1.54-V dry cell that supplies
1.00 W of power to a
15.0-Ω
b
u
l
b
?
(b) What is unreasonable about
this
r
e
su
l
t
?
(c) Which assumptions are unreasonable or inconsistent?
21.3
Kirchhoff’s
Rules
31.
Apply
the
loop
rule
to
loop
abcdefgha
in
Fi
gure
21.25
.
32.
Apply
the
loop
rule
to
loop
aedcba
in
Fi
gure
21.25
.
33.
Verify
the
second
equation
in
Exam
ple
21.5
by substituting the values
found
for
the
currents
1
and
2
.
34.
Verify the third equation in
Exam
ple 21.5
by substituting the values
found
for the currents
1
and
3
.
35.
Apply the junction rule at point a in
Fi
gure
21.52
.
Figure 21.52
36.
Apply
the
loop
rule
to
loop
abcdefghija
in
Fi
gure
21.52
.
37.
Apply
the
loop
rule
to
loop
akledcba
in
Fi
gure
21.52
.
38.
Find the currents flowing in the circuit in
Fi
gure
21.52
.
Explicitly show
how you follow the steps in the
Problem-Solvin
g
Strategies for Series and
Parallel Resistors
.
39.
Solve
Exam
ple 21.5
, but use loop abcdefgha instead of
loop akledcba.
Explicitly show how you follow the steps in the
Problem-Solvin
g Strategies
for
Series
and
Parallel
Resistors
.
40.
Find the currents flowing in the circuit in
Fi
gure
21.47
.
41.
Unreasonable
Results
Consider the circuit in
Fi
gure
21.53
, and suppose that the emfs are
unknown
and the currents are given to
be
1
=
5.00 A
,
2
=
3.0 A
, and
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3
= –2.00 A
. (a) Could you find the emfs? (b) What is wrong with the
assumptions?
Figure 21.53
21.4
DC
Voltmeters
and
Ammeters
42.
What is the sensitivity of the galvanometer (that is, what
current gives a
full-scale deflection) inside a voltmeter that has a
1.00-MΩ
resistance on its
30.0-V
sc
a
l
e
?
43.
What is the sensitivity of the galvanometer (that is, what
current gives a
full-scale deflection) inside a voltmeter that has a
25.0-kΩ
resistance on its
100-V
sc
a
l
e
?
44.
Find the resistance that must be placed in series with a
25.0-
Ω
galvanometer having a
50.0−μA
sensitivity (the same as the
one discussed in the text) to allow it to be used as a voltmeter with a
0.100-V full-scale reading.
45.
Find the resistance that must be placed in series with a
25.0-
Ω
galvanometer having a
50.0-μA
sensitivity (the same as the one
discussed in the text) to allow it to be used as a voltmeter with a
3000-V full-scale reading. Include a circuit diagram with your
solution.
46.
Find the resistance that must be placed in parallel with a
25.0-Ω
galvanometer having a
50.0-μA
sensitivity (the same as
the one discussed in the text) to allow it to be used as an ammeter
with a 10.0-A full-scale reading. Include a circuit diagram with your
solution.
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47.
Find the resistance that must be placed in parallel with a
25.0-Ω
galvanometer having a
50.0-μA
sensitivity (the same as
the one discussed in the text) to allow it to be used as an ammeter
with a 300-mA full-scale reading.
48.
Find the resistance that must be placed in series with a
10.0-
Ω
galvanometer having a
100-μA
sensitivity to allow it to be used
as a voltmeter with: (a) a 300-V full-scale reading, and (b) a 0.300-V
full-scale reading.
49.
Find the resistance that must be placed in parallel with a
10.0-Ω
galvanometer having a
100-μA
sensitivity to allow it
to be used as an ammeter with: (a) a 20.0-A full-scale reading,
and (b) a 100-mA full-scale reading.
50.
Suppose you measure the terminal voltage of a 1.585-V
alkaline cell having an internal resistance of
0.100 Ω
by placing a
1.00-kΩ
voltmeter across its terminals. (See
Fi
gure
21.54
.) (a)
What current flows? (b) Find the terminal voltage. (c) To see how
close the measured terminal voltage is to the emf, calculate their
ratio.
Figure 21.54
51.
Suppose you measure the terminal voltage of a 3.200-V
lithium cell having an internal resistance of
5.00 Ω
by placing a
1.00-kΩ
voltmeter across its
terminals. (a) What current flows? (b)
Find the terminal voltage. (c) To see how
close the measured terminal
voltage is to the emf, calculate their ratio.
52.
A certain ammeter has a resistance of
5.00×
10
−5
Ω
on its 3.00-A scale
and contains a
10.0-Ω
galvanometer. What is the sensitivity of the
g
a
l
va
n
o
m
e
t
e
r
?
53.
A
1.00-MΩ
voltmeter is placed in parallel with a
75.0-kΩ
resistor in a
circuit. (a) Draw a circuit diagram of the connection. (b)
What is the resistance
of the combination? (c) If the voltage across
the combination is kept the same as it was across the
75.0-kΩ
resistor alone, what is the percent increase in current? (d) If the
current through the combination is kept the same as it was through
the
75.0-kΩ
resistor alone, what is the percentage decrease in
vo
l
t
a
g
e
?
(e) Are the changes found in parts (c) and (d) significant?
Discuss.
54.
A
0.0200-Ω
ammeter is placed in series with a
10.00-Ω
resistor in a circuit. (a) Draw a circuit diagram of the connection. (b)
Calculate the resistance of the combination. (c) If the voltage is kept
the same across the
combination as it was through the
10.00-Ω
resistor alone, what is the percent
decrease in current? (d) If the
current is kept the same through the
combination as it was through
the
10.00-Ω
resistor alone, what is the percent
increase in vo
l
t
a
g
e
?
(e) Are the changes found in parts (c) and (d) significant? Discuss.
55.
Unreasonable
Results
Suppose you have a
40.0-Ω
galvanometer with a
25.0-μA
sensitivity. (a)
What resistance would you put in series with it to
allow it to be used as a voltmeter that has a full-scale deflection
for 0.500 mV? (b) What is unreasonable about this
r
e
su
l
t
?
(c)
Which assumptions are
r
e
sp
o
n
si
b
l
e
?
56.
Unreasonable
Results
(a)
What resistance would you put in parallel with a
40.0-Ω
galvanometer having a
25.0-μA
sensitivity to allow it to be used
as an ammeter that has a full-scale deflection for
10.0-μA
? (b)
What is unreasonable about this
r
e
su
l
t
?
(c) Which assumptions are
r
e
sp
o
n
si
b
l
e
?
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21.5
Null
Measurements
57.
What is the
emf
x
of
a
cell
being
measured
in
a
potentiometer,
if
the
standard cell’s emf is 12.0 V and the potentiometer balances for
x
=
5.000 Ω
and
s
=
2.500 Ω
?
58.
Calculate
the
emf
x
of
a
dry
cell
for
which
a
potentiometer
is
balanced
when
x
=
1.200
Ω
,
while
an
alkaline
standard
cell
with
an
emf
of
1.600
V
requires
s
= 1.247 Ω
to balance the potentiometer.
59.
When an unknown
resistance
x
is placed in a Wheatstone bridge, it is
possible to balance the bridge by adjusting
3
to
be
2500
Ω
. What is
x
if
2
= 0.625
?
1
60.
To what value must you
adjust
3
to balance a Wheatstone bridge, if
the
unknown resistance
x
is
100 Ω
,
1
is
50.0 Ω
,
and
2
is
175 Ω
?
61.
(a)
What
is
the
unknown
emf
x
in
a
potentiometer
that
balances
when
x
is
10.0 Ω
, and balances when
s
is
15.0 Ω
for a standard 3.000-V
emf? (b)
The same
emf
x
is placed in the same potentiometer, which now
balances
when
s
is
15.0 Ω
for a standard emf of 3.100 V. At what
resistance
x
will
the
potentiometer
b
a
l
a
n
c
e
?
62.
Suppose you want to measure resistances in the range from
10.0 Ω
to
10.0 kΩ
using a Wheatstone bridge that
has
2
1
=
2.000
.
Over
what
range
should
3
be
a
d
j
ust
a
b
l
e
?
21.6
DC Circuits Containin
g
Resistors and
Ca
pacitors
63.
The timing device in an automobile’s intermittent wiper
system is based on an time constant and utilizes a
0.500-μF
capacitor and a variable resistor. Over what range must be made to
vary to achieve time constants from 2.00 to 15.0
s?
64.
A heart pacemaker fires 72 times a minute, each time a 25.0-
nF capacitor is charged (by a battery in series with a resistor) to
0.632 of its full voltage.
What is the value of the resistance?
65.
The duration of
a photographic flash is related to an
time constant,
which is
0.100 μs
for a certain camera. (a) If the resistance of the flash
lamp is
0.0400 Ω
during discharge, what is the size of the capacitor
supplying its energy? (b) What is the time constant for charging the
capacitor, if the charging resistance is
800 kΩ
?
66.
A 2.00- and a
7.50-μF
capacitor can be connected in series or parallel, as can a
25.0- and a
100-kΩ
resistor. Calculate the four-time constants
possible from connecting the resulting capacitance and resistance in
series.
67.
After two
time constants, what percentage of the final voltage, emf, is on
an
initially uncharged capacitor
, charged through a resistance
?
68.
A
500-Ω
resistor, an uncharged
1.50-μF
capacitor, and a
6.16-V emf are connected in series. (a) What is the initial current?
(b) What is the
time constant? (c) What is the current after one
time constant? (d) What is the voltage on the capacitor after one
time constant?
69.
A heart
defibrillator being used on a patient has an
time
constant
of
10.0 ms due to the resistance of the patient and the capacitance of
the
defibrillator. (a) If the defibrillator has an
8.00-μF
capacitance,
what is the
resistance of the path through the patient? (You may
neglect the capacitance
of the patient and the resistance of the
defibrillator.) (b) If the initial voltage is
12.0
kV,
how
long
does
it
take
to
decline
to
6.00
×
10
2
V
?
70.
An ECG monitor must have an
time constant less than
1.00 × 10
2
μs
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to be able to measure variations in voltage over small time intervals.
(a) If the
resistance of the circuit (due mostly to that of the patient’s
chest) is
1.00 kΩ
,
what is the maximum capacitance of the circuit? (b)
Would it be difficult in practice to limit the capacitance to less than the
value found in
(
a
)
?
71.
Fi
gure 21.55
shows how a bleeder resistor is used
to discharge a capacitor after an electronic device is shut off,
allowing a person to work on the electronics with less risk of shock.
(a) What is the time constant? (b) How
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long will it take to reduce the voltage on the capacitor to
0
.
250
%
(
5
%
of 5%) of its full value once discharge begins? (c) If the
capacitor is charged to a
voltage
0.865
0
0
through
a
100-Ω
resistance,
calculate
the
time
it
takes
to
rise
to
(This is about two time constants.)
Figure 21.55
72.
Using the exact exponential treatment, find how much time is
required to discharge a
250-μF
capacitor through a
500-Ω
resistor
down to
1
.
00
%
of its original voltage.
73.
Using the exact exponential treatment, find how much time is
required to charge an initially uncharged 100-pF capacitor through a
75.0-MΩ
resistor to
90
.
0
%
of its final voltage.
74.
Integrated
Concepts
If you wish to take a picture of a bullet traveling at 500 m/s, then a
very brief flash of light produced by an
discharge through a flash
tube can limit
blurring. Assuming 1.00 mm of motion during one
constant is
acceptable, and given that the flash is driven by a
600-μF
capacitor, what is the resistance in the flash tube?
75.
Integrated
Concepts
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A flashing lamp in a Christmas earring is based on an
discharge of
a capacitor through its resistance. The effective duration of the flash is
0.250 s, during which it produces an average 0.500 W from an average
3.00 V. (a) What
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energy does it dissipate? (b) How much charge moves through the
l
am
p
?
(c) Find the capacitance. (d) What is the resistance of the
l
am
p
?
76.
Integrated
Concepts
A
160-μF
capacitor charged to 450 V is discharged through a
31.2-
kΩ
resistor.
(a)
Find
the
time
constant.
(b)
Calculate
the
temperature
increase
of
the resistor, given that its mass is 2.50 g and its specific heat is
kJ
,
kg
⋅
ºC
noting that most of the thermal energy is retained in the short time
of the discharge. (c) Calculate the new resistance, assuming it is
pure carbon. (d) Does this change in resistance seem significant?
77.
Unreasonable
Results
(a)
Calculate the capacitance needed to get an
time constant of
1.00
×
10
3
s
with
a
0.100-Ω
resistor.
(b)
What
is
unreasonable
about
this
r
e
su
l
t
?
(c) Which assumptions are
r
e
sp
o
n
si
b
l
e
?
78.
Construct Your Own Problem
Consider a camera’s flash unit. Construct a problem in which you
calculate the size of the capacitor that stores energy for the flash
lamp. Among the things to be considered are the voltage applied to
the capacitor, the energy needed in the flash and the associated
charge needed on the capacitor, the resistance of the flash lamp
during discharge, and the desired
time constant.
79.
Construct Your Own Problem
1.6
7
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Consider a rechargeable lithium cell that is to be used to power a
camcorder. Construct a problem in which you calculate the internal
resistance of the cell during normal operation. Also, calculate the
minimum voltage output of a battery charger to be used to recharge
your lithium cell. Among the things to
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be considered are the emf and useful terminal voltage of a lithium cell
and the current it should be able to supply to a camcorder.
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22.4
Ma
gnetic
Field
Strength:
Force
on
a
Moving
Char
ge
in
a
Magnetic
Field
1.
What is the direction of the magnetic force on a positive charge that moves
as shown in each of the six cases shown in
Fi
gure
22.50
?
Figure
22.50
2.
Repeat
[link]
for a negative charge.
3.
What is the direction of the velocity
of a negative charge that experiences the magnetic force shown in each of the
three cases in
Fi
gure
22.51
,
assuming it moves perpendicular to
?
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Figure
22.51
4.
Repeat
[link]
for a positive charge.
5.
What is the direction of the magnetic field that produces the magnetic force
on a positive charge as shown in each of the three cases in the figure below,
assuming
is perpendicular to
?
Figure
22.52
6.
Repeat
[link]
for a negative charge.
7.
What is the maximum magnitude of the force on an aluminum rod with a
0.100-
C
μ
charge that you pass between the poles of a 1.50-T permanent
magnet at a speed of 5.00 m/s? In what direction is the force?
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8.
(a) Aircraft sometimes acquire small static charges. Suppose a supersonic
jet has a
0.500-
C
μ
charge and flies due west at a speed of 660 m/s over the
Earth’s magnetic south pole (near Earth's geographic north
p
o
l
e
)
,
where the
8.00 ×
10
−5
-T
magnetic field points straight down. What are the direction
and the magnitude of the magnetic force on the plane? (b) Discuss whether
the value obtained in part (a) implies this is a significant or negligible e
ect.
ff
9.
(a) A cosmic ray proton moving toward the Earth at
5.00
× 10
7
m/s
experiences a magnetic force of
1.70 ×
10
−16
N
. What is the strength of the
magnetic field if there is a
45º
angle between it and the proton’s velocity? (b)
Is the value obtained in part (a) consistent with the known strength of the
Earth’s magnetic field on its surface? Discuss.
10.
An electron moving at
4.00 ×
10
3
m/s
in a 1.25-T magnetic field
experiences a magnetic force of
1.40 ×
10
−16
N
. What angle does the
velocity of the electron make with the magnetic field? There are two answers.
11.
(a) A physicist performing a sensitive measurement wants to limit
the magnetic force on a moving charge in her equipment to less than
1.00 × 10
−12
N
. What is the greatest the charge can be if it moves at a
maximum speed of 30.0 m/s in the Earth’s field? (b) Discuss whether it would
be di
cult
ffi
to limit the charge to less than the value found in (a) by comparing
it with typical static electricity and noting that static is often absent.
22.5
Force on a Movin
g
Charge in a Magnetic
Field:
Exam
ples and Applications
If you need additional support for these problems, see
More
A
pplications of
Ma
gnetism
.
12.
A cosmic ray electron moves at
7.50 ×
10
6
m/s
perpendicular
to
the
Earth’s magnetic field at an altitude where field strength is
1.00 × 10
−5
T
.
What is the radius of the circular path the electron follows?
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13.
A proton moves at
7.50 ×
10
7
m/s
perpendicular to a magnetic field. The
field causes the proton to travel in a circular path of radius 0.800 m. What is
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the field strength?
14.
(a) Viewers of
Star Trek
hear of an antimatter drive on the Starship
Enterprise
. One possibility for such a futuristic energy source is to store
antimatter charged particles in a vacuum chamber, circulating in a magnetic
field, and then extract them as needed. Antimatter annihilates with normal
matter, producing pure energy. What strength magnetic field is needed to hold
antiprotons, moving at
5.00 × 10
7
m/s
in a circular path 2.00 m in radius?
Antiprotons have the same mass as protons but the opposite (negative)
charge. (b) Is this field strength obtainable with today’s technology or is it a
futuristic possibility?
15.
(a) An oxygen-16 ion with a mass of
2.66 ×
10
−26
kg
travels at
5.00 ×
10
6
m/s
perpendicular to a 1.20-T magnetic field, which makes it
move in a circular arc with a 0.231-m radius. What positive charge is on the
ion? (b) What is the ratio of this charge to the charge of an electron? (c)
Discuss why the ratio found in (b) should be an integer.
16.
What radius circular path does an electron travel if it moves at the
same speed and in the same magnetic field as the proton in
[link]
?
17.
A velocity selector in a mass spectrometer uses a 0.100-T magnetic
field.
(a)
What electric field strength is needed to select a speed of
4.00 ×
10
6
m/s
?
(b)
What is the voltage between the plates if they are separated by 1.00 cm?
18.
An electron in a TV CRT moves with a speed of
6.00 ×
10
7
m/s
,
in
a
direction perpendicular to the Earth’s field, which has a strength of
5.00 × 10
−5
T
. (a) What strength electric field must be applied perpendicular
to the Earth’s field to make the electron moves in a straight line? (b) If this is
done between plates separated by 1.00 cm, what is the voltage applied?
(Note that TVs are usually surrounded by a ferromagnetic material to shield
against external magnetic fields and avoid the need for such a
c
o
rr
e
c
t
i
o
n
.
)
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19.
(a) At what speed will a proton move in a circular path of the same
radius as the electron in
[link]
? (b) What would the radius of the path be if
the
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proton had the same speed as the electron? (c) What would the radius be if
the proton had the same kinetic energy as the electron? (d) The same
momentum?
20.
A mass spectrometer is being used to separate common oxygen-16
from the much rarer oxygen-18, taken from a sample of old glacial ice. (The
relative abundance of these oxygen isotopes is related to climatic temperature
at the time the ice was
d
e
p
o
si
t
e
d
.
)
The ratio of the masses of these two ions is
16 to
18, the mass of oxygen-16 is
2.66 ×
10
−26
kg,
and they are singly charged
and travel at
5.00 ×
10
6
m/s
in a 1.20-T magnetic field. What is the
separation between their paths when they hit a target after traversing a
semicircle?
21.
(a) Triply charged uranium-235 and uranium-238 ions are being
separated in a mass spectrometer. (The much rarer uranium-235 is used as
reactor
f
u
e
l
.
)
The masses of the ions are
3.90 × 10
−25
kg
and
3.95 × 10
−25
kg
,
respectively, and they travel at
3.00 × 10
5
m/s
in a 0.250-T field. What is the
separation between their paths when they hit a target after traversing a
semicircle? (b) Discuss whether this distance between their paths seems to be
big enough to be practical in the separation of uranium-235 from uranium-
238.
22.6
The
Hall
Effect
22.
A large water main is 2.50 m in diameter and the average water velocity
is
6.00 m/s. Find the Hall voltage produced if the pipe runs perpendicular to the
Earth’s
5.00 × 10
−5
-T
field.
23.
What Hall voltage is produced by a 0.200-T field applied across a
2.60- cm-diameter aorta when blood velocity is 60.0 cm/s?
24.
(a) What is the speed of a supersonic aircraft with a 17.0-m wingspan,
if it experiences a 1.60-V Hall voltage between its wing tips when in level
flight over the north magnetic pole, where the Earth’s field strength is
8.00 × 10
−5
voltage.
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T?
(b) Explain
why very little
current flows as a result of this Hall
25.
A nonmechanical water meter could utilize the Hall e
ect
ff
by
applying a magnetic field across a metal pipe and measuring the Hall voltage
produced.
What is the average fluid velocity in a 3.00-cm-diameter pipe, if a 0.500-T field
across it creates a 60.0-mV Hall voltage?
26.
Calculate the Hall voltage induced on a patient’s heart while being
scanned by an MRI unit. Approximate the conducting path on the heart wall
by a wire 7.50 cm long that moves at 10.0 cm/s perpendicular to a 1.50-T
magnetic field.
27.
A Hall probe calibrated to read
1.00
V
μ
when placed in a 2.00-T
field is placed in a 0.150-T field. What is its output voltage?
28.
Using information in
Exam
ple 20.6
,
what would the Hall voltage be
if a 2.00-T field is applied across a 10-gauge copper wire
(
2
.
588
mm in
diameter) carrying a 20.0-A current?
29.
Show that the Hall voltage across wires made of the same
material, carrying identical currents, and subjected to the same magnetic
field is inversely proportional to their diameters. (Hint: Consider how drift
velocity depends on wire
d
i
a
m
e
t
e
r
.
)
30.
A patient with a pacemaker is mistakenly being scanned for an MRI
image. A 10.0-cm-long section of pacemaker wire moves at a speed of 10.0
cm/s perpendicular to the MRI unit’s magnetic field and a 20.0-mV Hall
voltage is induced. What is the magnetic field strength?
22.7
Ma
gnetic Force on a Current-
Carrying
Conductor
31.
What is the direction of the magnetic force on the current in each of
the six cases in
Fi
gure
22.53
?
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Figure
22.53
32.
What is the direction of a current that experiences the
magnetic force shown in each of the three cases in
Fi
gure
22.54
,
assuming
the current runs perpendicular to
?
Figure
22.54
33.
What is the direction of the magnetic field that produces the
magnetic force shown on the currents in each of the three cases in
Fi
gure
22.55
,
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assuming
is perpendicular to
?
Figure
22.55
34.
(a) What is the force per meter on a lightning bolt at the equator that
carries 20,000 A perpendicular to the Earth’s
3.00 × 10
−5
-T
field? (b) What is
the direction of the force if the current is straight up and the Earth’s field
direction is due north, parallel to the ground?
35.
(a) A DC power line for a light-rail system carries 1000 A at an angle of
30.0º
to the Earth’s
5.00 × 10
−5
-T
field. What is the force on a 100-m
section of this line? (b) Discuss practical concerns this presents, if any.
36.
What force is exerted on the water in an MHD drive utilizing a 25.0-
cm- diameter tube, if 100-A current is passed across the tube that is
perpendicular to a 2.00-T magnetic field? (The relatively small size of this
force indicates the
need for very large currents and magnetic fields to make practical MHD
drives.)
37.
A wire carrying a 30.0-A current passes between the poles of a strong
magnet that is perpendicular to its field and experiences a 2.16-N force on the
4.00 cm of wire in the field. What is the average field strength?
38.
(a) A 0.750-m-long section of cable carrying current to a car starter
motor
makes an angle of
60º
with the Earth’s
5.50 × 10
−5
T
field. What is the
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current when the wire experiences a force of
7.00 ×
10
−3
N
? (b) If you run
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the wire between the poles of a strong horseshoe magnet, subjecting 5.00 cm
of it to a 1.75-T field, what force is exerted on this segment of wire?
39.
(a) What is the angle between a wire carrying an 8.00-A current and the
1.20-T field it is in if 50.0 cm of the wire experiences a magnetic force of 2.40
N? (b) What is the force on the wire if it is rotated to make an angle of
90º
with
the field?
40.
The force on the rectangular loop of wire in the magnetic field in
Fi
gure
22.56
can be used to measure field strength. The field is uniform, and the
plane of the loop is perpendicular to the field. (a) What is the direction of the
magnetic force on the loop?
J
u
st
i
f
y
the claim that the forces on the sides of
the loop are equal and opposite, independent of how much of the loop is in
the field and do not a
ect
ff
the net force on the loop. (b) If a current of 5.00 A is
used, what is the force per tesla on the 20.0-cm-wide loop?
Figure 22.56
A rectangular loop
of wire carrying a current is
perpendicular to a magnetic field.
The field is uniform in the region
shown and is zero outside that
region.
22.8
Torque
on
a
Current
Loop:
Motors
and
Meters
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41.
(a) By how many percent is the torque of a motor decreased if its
permanent magnets lose 5.0% of their strength? (b) How many percent would
the current need to be increased to return the torque to original values?
42.
(a) What is the maximum torque on a 150-turn square loop of wire 18.0
cm
on a side that carries a 50.0-A current in a 1.60-T field? (b) What is the
torque when
is
10.9º?
43.
Find the current through a loop needed to create a maximum torque of
9.00 N
⋅
m.
The loop has 50 square turns that are 15.0 cm on a side and is in
a uniform 0.800-T magnetic field.
44.
Calculate the magnetic field strength needed on a 200-turn square loop
20.0 cm on a side to create a maximum torque of
300 N
⋅
m
if the loop is
carrying 25.0 A.
45.
Since the equation for torque on a current-carrying loop is
=
sin
, the units of
N
⋅
m
must equal units of
A
⋅
m
2
T
. Verify this.
46.
(a) At what angle
is the torque on a current loop 90.0% of maximum?
(b) 50.0% of maximum? (c) 10.0% of maximum?
47.
A proton has a magnetic field due to its spin on its axis. The field is
similar
to that created by a circular current loop
0.650 × 10
−15
m
in radius with a
current of
1.05 ×
10
4
A
(no
k
i
dd
i
n
g
)
.
Find the maximum torque on a proton in
a 2.50-T field. (This is a significant torque on a small
p
a
r
t
i
c
l
e
.
)
48.
(a) A 200-turn circular loop of radius 50.0 cm is vertical, with its axis on
an
east-west line. A current of 100 A circulates clockwise in the loop when
viewed from the east. The Earth’s field here is due north, parallel to the
ground, with a strength of
3.00 × 10
−5
T
. What are the direction and
magnitude of the torque on the loop? (b) Does this device have any practical
applications as a motor?
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49.
Repeat
[link]
,
but with the loop lying flat on the ground with its current
circulating counterclockwise (when viewed from above) in a location where the
Earth’s field is north, but at an angle
45.0º
below the horizontal and with a
strength of
6.00 ×
10
−5
T
.
22.10
Ma
gnetic
Force
between
Two
Parallel
Conductors
50.
(a) The hot and neutral wires supplying DC power to a light-rail
commuter train carry 800 A and are separated by 75.0 cm. What is the
magnitude and
direction of the force between 50.0 m of these wires? (b) Discuss the practical
consequences of this force, if any.
51.
The force per meter between the two wires of a jumper cable being
used to start a stalled car is 0.225
N
/
m
.
(a) What is the current in the wires,
given
they are separated by 2.00 cm? (b) Is the force attractive or repulsive?
52.
A 2.50-m segment of wire supplying current to the motor of a
submerged submarine carries 1000 A and feels a 4.00-N repulsive force from
a parallel wire 5.00 cm away. What is the direction and magnitude of the
current in the
other wire?
53.
The wire carrying 400 A to the motor of a commuter train feels an
attractive force of
4.00 × 10
−3
N/m
due to a parallel wire carrying 5.00 A to a
headlight. (a) How far apart are the wires? (b) Are the currents in the same
direction?
54.
An AC appliance cord has its hot and neutral wires separated by 3.00 mm
and carries a 5.00-A current. (a) What is the average force per meter between
the wires in the cord? (b) What is the maximum force per meter between the
wires? (c) Are the forces attractive or repulsive? (d) Do appliance cords need
any special design features to compensate for these forces?
55.
Fi
gure
22.57
shows a long straight wire near a rectangular current
loop. What is the direction and magnitude of the total force on the loop?
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Figure
22.57
56.
Find the direction and magnitude of the force that each wire
experiences in
Fi
gure
22.58
(a) by, using vector addition.
Figure
22.58
57.
Find the direction and magnitude of the force that each wire
experiences in
Fi
gure
22.58
(
b
)
,
using vector addition.
22.11
More
A
pplications of Magnetism
58.
Indicate whether the magnetic field created in each of the three
situations shown in
Fi
gure
22.59
is into or out of the page on the left and
right of the current.
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Figure
22.59
59.
What are the directions of the fields in the center of the loop and
coils shown in
Fi
gure
22.60
?
Figure
22.60
60.
What are the directions of the currents in the loop and coils shown in
Fi
gure
22.61
?
Figure
22.61
61.
To see why an MRI utilizes iron to increase the magnetic field created
by a coil, calculate the current needed in a 400-loop-per-meter circular coil
0.660 m in radius to create a 1.20-T field (typical of an MRI instrument) at its
center
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with no iron present. The magnetic field of a proton is approximately like that
of a circular current loop
0.650 × 10
−15
m
in radius carrying
1.05 × 10
4
A
.
What is the field at the center of such a loop?
62.
Inside a motor, 30.0 A passes through a 250-turn circular loop that is
10.0 cm in radius. What is the magnetic field strength created at its center?
63.
Nonnuclear submarines use batteries for power when submerged. (a)
Find the magnetic field 50.0 cm from a straight wire carrying 1200 A from the
batteries to the drive mechanism of a submarine. (b) What is the field if the
wires to and from the drive mechanism are side by side? (c) Discuss the
e
ects this could have for a compass on the submarine that is not shielded.
ff
64.
How strong is the magnetic field inside a solenoid with 10,000
turns per meter that carries 20.0 A?
65.
What current is needed in the solenoid described in
[link]
to produce a
magnetic field
10
4
times the Earth’s magnetic field of
5.00 × 10
−5
T
?
66.
How far from the starter cable of a car, carrying 150 A, must you be to
experience a field less than the Earth’s
(5.00 × 10
−5
T)
?
Assume a long
straight wire carries the current. (In practice, the body of your car shields the
dashboard compass.)
67.
Measurements a
ect the system being measured, such as the current
ff
loop in
Fi
gure
22.56
. (a) Estimate the field the loop creates by calculating the
field at the center of a circular loop 20.0 cm in diameter carrying 5.00 A. (b)
What is the smallest field strength this loop can be used to measure, if its field
must alter the measured field by less than 0.0100%?
68.
Fi
gure
22.62
shows a long straight wire just touching a loop carrying a
current
1.
Both lie in the same plane. (a) What direction must the current
2
in the straight wire have to create a field at the center of the loop in the
direction opposite to that created by the loop? (b) What is the ratio of
1
/
2
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that gives zero field strength at the center of the loop? (c) What is the
direction of the field directly above the loop under this circumstance?
Figure
22.62
69.
Find the magnitude and direction of the magnetic field at the
point equidistant from the wires in
Fi
gure
22.58
(
a
)
,
using the rules of
vector addition to sum the contributions from each wire.
70.
Find the magnitude and direction of the magnetic field at the
point equidistant from the wires in
Fi
gure
22.58
(
b
)
,
using the rules of
vector addition to sum the contributions from each wire.
71.
What current is needed in the top wire in
Fi
gure
22.58
(a) to
produce a field of zero at the point equidistant from the wires, if the currents
in the bottom two wires are both 10.0 A into the page?
72.
Calculate the size of the magnetic field 20 m below a high voltage
power line. The line carries 450 MW at a voltage of 300,000 V.
73.
Integrated
Concepts
(a)A pendulum is set up so that its bob (a thin copper disk) swings between
the poles of a permanent magnet as shown in
Fi
gure
22.63
. What is the
magnitude and direction of the magnetic force on the bob at the lowest point
in its path, if it has a positive
0.250
C
μ
charge and is released from a height
of 30.0 cm above its lowest point? The magnetic field strength is 1.50 T. (b)
What is the acceleration of the bob at the bottom of its swing if its mass is
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30.0 grams and it is hung from a flexible string? Be certain to include a free-
body diagram as part of your analysis.
Figure
22.63
74.
Integrated
Concepts
(a)
What voltage will accelerate electrons to a speed of
6.00 ×
10
6
m/s
?
(b)
Find the radius of curvature of the path of a
proton
accelerated through this
potential in a 0.500-T field and compare this with the radius of curvature of an
electron accelerated through the same potential.
75.
Integrated
Concepts
Find the radius of curvature of the path of a 25.0-MeV proton moving
perpendicularly to the 1.20-T field of a cyclotron.
76.
Integrated
Concepts
To construct a nonmechanical water meter, a 0.500-T magnetic field is placed
across the supply water pipe to a home and the Hall voltage is recorded. (a)
Find the flow rate in liters per second through a 3.00-cm-diameter pipe if the
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Hall voltage is 60.0 mV. (b) What would the Hall voltage be for the same flow
rate through a 10.0-cm-diameter pipe with the same field applied?
77.
Integrated
Concepts
(a)Using the values given for an MHD drive in
[link]
, and assuming the force is
uniformly applied to the fluid, calculate the pressure created in
N/m
2
.
(b) Is
this a significant fraction of an atmosphere?
78.
Integrated
Concepts
(a)Calculate the maximum torque on a 50-turn, 1.50 cm radius circular current
loop carrying
50 A
μ
in a 0.500-T field. (b) If this coil is to be used in a
galvanometer that reads
50
A
μ
full scale, what force constant spring must be
used, if it is attached 1.00 cm from the axis of rotation and is stretched by the
60º
arc moved?
79.
Integrated
Concepts
A current balance used to define the ampere is designed so that the current
through it is constant, as is the distance between wires. Even so, if the wires
change length with temperature, the force between them will change. What
percent change in force per degree will occur if the wires are copper?
80.
Integrated
Concepts
(a)Show that the period of the circular orbit of a charged particle moving
perpendicularly to a uniform magnetic field is
= 2
/(
)
.
(b)
What
is
the
frequency
? (c) What is the angular velocity
? Note that these results are
independent of the velocity and radius of the orbit and, hence, of the energy of
the particle. (
Fi
gure
22
.
6
4
.
)
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Figure 22.64
Cyclotrons accelerate charged particles
orbiting in a magnetic field by placing an AC voltage on the
metal Dees, between which the particles move, so that
energy is added twice each orbit. The frequency is constant,
since it is independent of the particle energy—the radius of
the orbit simply increases with energy until the particles
approach the edge and are extracted for various experiments
and applications.
81.
Integrated
Concepts
A cyclotron accelerates charged particles as shown in
Fi
gure
22.64
. Using the
results of the previous problem, calculate the frequency of the accelerating
voltage needed for a proton in a 1.20-T field.
82.
Integrated
Concepts
(a)A 0.140-kg baseball, pitched at 40.0 m/s horizontally and perpendicular to
the Earth’s horizontal
5.00 × 10
−5
T
field, has a 100-nC charge on it. What
distance is it deflected from its path by the magnetic force, after traveling 30.0
m horizontally? (b) Would you suggest this as a secret technique for a pitcher
to throw curve balls?
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83.
Integrated
Concepts
(a)What is the direction of the force on a wire carrying a current due east in a
location where the Earth’s field is due north? Both are parallel to the ground.
(b)
Calculate the force per meter if the wire carries 20.0 A and the field
strength is
3.00 × 10
−5
T
. (c) What diameter copper wire would have its
weight supported by this force? (d) Calculate the resistance per meter and the
voltage per meter needed.
84.
Integrated
Concepts
One long straight wire is to be held directly above another by repulsion
between their currents. The lower wire carries 100 A and the wire 7.50 cm
above it is 10-gauge
(
2
.
588
mm diameter) copper wire. (a) What current must
flow in the upper wire, neglecting the Earth’s field? (b) What is the smallest
current if the Earth’s
3.00 × 10
−5
T
field is parallel to the ground and is not
neglected? (c) Is the supported wire in a stable or unstable equilibrium if
displaced vertically? If displaced horizontally?
85.
Unreasonable
Results
(a)Find the charge on a baseball, thrown at 35.0 m/s perpendicular to the
Earth’s
5.00 × 10
−5
T
field, that experiences a 1.00-N magnetic force. (b)
What is unreasonable about this result? (c) Which assumption or premise is
responsible?
86.
Unreasonable
Results
A charged particle having mass
6.64 × 10
−27
kg
(that of a helium atom)
moving at
8.70 ×
10
5
m/s
perpendicular to a 1.50-T magnetic field travels in
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a circular path of radius 16.0 mm. (a) What is the charge of the particle? (b)
What is unreasonable about this result? (c) Which assumptions are
responsible?
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87.
Unreasonable
Results
An inventor wants to generate 120-V power by moving a 1.00-m-long wire
perpendicular to the Earth’s
5.00 ×
10
−5
T
field. (a) Find the speed with
which the wire must move. (b) What is unreasonable about this result? (c)
Which assumption is responsible?
88.
Unreasonable
Results
Frustrated by the small Hall voltage obtained in blood flow measurements, a
medical physicist decides to increase the applied magnetic field strength to
get a 0.500-V output for blood moving at 30.0 cm/s in a 1.50-cm-diameter
vessel. (a) What magnetic field strength is needed? (b) What is unreasonable
about this result? (c) Which premise is responsible?
89.
Unreasonable
Results
A surveyor 100 m from a long straight 200-kV DC power line suspects that its
magnetic field may equal that of the Earth and a
ect compass readings. (a)
ff
Calculate the current in the wire needed to create a
5.00 ×
10
−5
T
field at this
distance. (b) What is unreasonable about this result? (c) Which assumption or
premise is responsible?
90.
Construct Your Own Problem
Consider a mass separator that applies a magnetic field perpendicular to the
velocity of ions and separates the ions based on the radius of curvature of
their paths in the field. Construct a problem in which you calculate the
magnetic field strength needed to separate two ions that di
er
ff
in mass, but
not charge, and have the same initial velocity. Among the things to consider
are the types of ions, the velocities they can be given before entering the
magnetic field, and a reasonable value for the radius of curvature of the paths
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they follow. In addition, calculate the separation distance between the ions at
the point where they are detected.
91.
Construct Your Own Problem
Consider using the torque on a current-carrying coil in a magnetic field to
detect relatively small magnetic fields (less than the field of the Earth, for
e
x
a
m
p
l
e
)
.
Construct a problem in which you calculate the maximum torque on
a current-carrying loop in a magnetic field. Among the things to be considered
are the size of the coil, the number of loops it has, the current you pass
through the coil, and the size of the field you wish to detect. Discuss whether
the torque produced is large enough to be e
ectively
ff
measured. Your
instructor may also wish for you to consider the e
ects,
ff
if any, of the field
produced by the coil on the surroundings that could a
ect
ff
detection of the
small field.
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23.1
Induced
Emf
and
Ma
gnetic
Flux
1.
What is the value of the magnetic flux at coil 2 in
Fi
gure
23.56
due to coil
1?
Figure 23.56
(a) The planes of the two coils are
perpendicular. (b) The wire is perpendicular to the plane of
the coil.
2.
What is the value of the magnetic flux through the coil in
Fi
gure
23.56
(b) due to the wire?
23.2
Farada
y’s
Law
of
Induction:
Lenz’s
Law
3.
Referring to
Fi
gure
23.57
(
a
)
,
what is the direction of the current induced
in coil 2: (a) If the current in coil 1 increases? (b) If the current in coil 1
decreases? (c) If the current in coil 1 is constant? Explicitly show how you
follow the
steps
in
the
Problem-Solvin
g Strategy
for
Lenz's
Law
.
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Figure 23.57
(a) The coils lie in the same plane. (b) The wire
is in the plane of the coil
4.
Referring to
Fi
gure
23.57
(
b
)
,
what is the direction of the current induced
in the coil: (a) If the current in the wire increases? (b) If the current in the wire
decreases? (c) If the current in the wire suddenly changes direction? Explicitly
show how you follow the steps in the
Problem-Solvin
g Strategy
for
Lenz’s
Law
.
5.
Referring to
Fi
gure
23.58
,
what are the directions of the currents in coils
1, 2, and 3 (assume that the coils are lying in the plane of the circuit): (a)
When
the switch is first closed? (b) When the switch has been closed for a long
time? (c) Just after the switch is opened?
Figure
23.58
6.
Repeat the previous problem with the battery reversed.
7.
Verify that the units of
Δ
/
Δ
1 T
⋅
m
2
/s = 1 V
.
are volts. That is, show that
8.
Suppose a 50-turn coil lies in the plane of the page in a uniform magnetic
field that is directed into the page. The coil originally has an area of
0.250 m
2
.
It is stretched to have no area in 0.100 s. What is the direction and magnitude
of the induced emf if the uniform magnetic field has a strength of 1.50 T?
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9.
(a) An MRI technician moves his hand from a region of very low magnetic
field strength into an MRI scanner’s 2.00 T field with his fingers pointing in the
direction of the field. Find the average emf induced in his wedding ring, given
its diameter is 2.20 cm and assuming it takes 0.250 s to move it into the field.
(b) Discuss whether this current would significantly change the temperature of
the ring.
10.
Integrated
Concepts
Referring to the situation in the previous problem: (a) What current is induced
in the ring if its resistance is 0.0100
Ω
? (b) What average power is dissipated?
(c) What magnetic field is induced at the center of the ring? (d) What is the
direction of the induced magnetic field relative to the MRI’s field?
11.
An emf is induced by rotating a 1000-turn, 20.0 cm diameter coil in the
Earth’s
5.00 × 10
−5
T
magnetic field. What average emf is induced, given the
plane of the coil is originally perpendicular to the Earth’s field and is rotated to
be parallel to the field in 10.0 ms?
12.
A 0.250 m radius, 500-turn coil is rotated one-fourth of a revolution in 4.17
ms, originally having its plane perpendicular to a uniform magnetic field. (This
is 60 rev/s.) Find the magnetic field strength needed to induce an average emf
of 10,000 V.
13.
Integrated
Concepts
Approximately how does the emf induced in the loop in
Fi
gure
23
.
57
(
b
)
depend on the distance of the center of the loop from the wire?
14.
Integrated
Concepts
(a)A lightning bolt produces a rapidly varying magnetic field. If the bolt strikes
the earth vertically and acts like a current in a long straight wire, it will induce a
voltage in a loop aligned like that in
Fi
gure
23
.
5
7
(
b
)
.
What voltage is induced in
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a 1.00 m diameter loop 50.0 m from a
2.00 × 10
6
A
lightning strike, if the
current falls to zero in
25.0
s
μ
? (b) Discuss circumstances under which such a
voltage would produce noticeable consequences.
23.3
Motional
Emf
15.
Use Faraday’s law, Lenz’s law, and RHR-1 to show that the magnetic force
on the current in the moving rod in
Fi
gure
23.11
is in the opposite direction
of its velocity.
16.
If a current flows in the Satellite Tether shown in
Fi
gure
23.12
,
use
Faraday’s law, Lenz’s law, and RHR-1 to show that there is a magnetic force
on the tether in the direction opposite to its velocity.
17.
(a) A jet airplane with a 75.0 m wingspan is flying at 280 m/s. What emf is
induced between wing tips if the vertical component of the Earth’s field is
3.00 × 10
−5
T
? (b) Is an emf of this magnitude likely to have any
consequences? Explain.
18.
(a) A nonferrous screwdriver is being used in a 2.00 T magnetic field. What
maximum emf can be induced along its 12.0 cm length when it moves at 6.00
m/s? (b) Is it likely that this emf will have any consequences or even be
noticed?
19.
At what speed must the sliding rod in
Fi
gure
23.11
move to produce an
emf of 1.00 V in a 1.50 T field, given the rod’s length is 30.0 cm?
20.
The 12.0 cm long rod in
Fi
gure
23.11
moves at 4.00 m/s. What is the
strength of the magnetic field if a 95.0 V emf is induced?
21.
Prove that when
,
ℓ
,
and
are not mutually perpendicular, motional emf
is given by
emf
=
ℓ
sin
.
If
is perpendicular to
,
then
is the angle
between
ℓ
and
.
If
ℓ
is perpendicular to
,
then
is the angle between
and
.
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22.
In the August 1992 space shuttle flight, only 250 m of the conducting
tether considered in
Exam
ple 23.2
could be let out. A 40.0 V motional emf
was generated in the Earth’s
5.00 ×
10
−5
T
field, while moving at
7.80 ×
10
3
m/s
. What was the angle between the shuttle’s velocity and the
Earth’s field, assuming the conductor was perpendicular to the field?
23.
Integrated
Concepts
Derive an expression for the current in a system like that in
Fi
gure 23.11
,
under the following conditions. The resistance between the rails is
, the rails
and the moving rod are identical in cross section
and have the same
resistivity
. The distance between the rails is l, and the rod moves at constant
speed
perpendicular to the uniform field
. At time zero, the moving rod is
next to the resistance
.
24.
Integrated
Concepts
The Tethered Satellite in
Fi
gure
23.12
has a mass of 525 kg and is at the end
of a 20.0 km long, 2.50 mm diameter cable with the tensile strength of steel.
(a) How much does the cable stretch if a 100 N force is exerted to pull the
satellite in? (Assume the satellite and shuttle are at the same altitude above
the
E
a
r
t
h
.
)
(b) What is the e
ective
ff
force constant of the cable? (c) How much
energy is stored in it when stretched by the 100 N force?
25.
Integrated
Concepts
The Tethered Satellite discussed in this module is producing 5.00 kV, and a
current of 10.0 A flows. (a) What magnetic drag force does this produce if the
system is moving at 7.80 km/s? (b) How much kinetic energy is removed from
the system in 1.00 h, neglecting any change in altitude or velocity during that
time? (c) What is the change in velocity if the mass of the system is 100,000
kg? (d) Discuss the long term consequences
(
sa
y
,
a week-long mission) on the
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space shuttle’s orbit, noting what e
ect a decrease in velocity has and
ff
assessing the magnitude of the e
ect.
ff
23.4
Edd
y
Currents
and
Magnetic
Damping
26.
Make a drawing similar to
Fi
gure
23.14
,
but with the pendulum moving
in the opposite direction. Then use Faraday’s law, Lenz’s law, and RHR-1 to
show that magnetic force opposes motion.
27.
Figure 23.59
A coil is moved into and out of a region of
uniform magnetic field.
A coil is moved through a magnetic field as shown in
Fi
gure
23.59
.
The field
is uniform inside the rectangle and zero outside. What is the direction of the
induced current and what is the direction of the magnetic force on the coil at
each position shown?
23.5
Electric
Generators
28.
Calculate the peak voltage of a generator that rotates its 200-turn, 0.100
m diameter coil at 3600 rpm in a 0.800 T field.
29.
At what angular velocity in rpm will the peak voltage of a generator be 480
V, if its 500-turn, 8.00 cm diameter coil rotates in a 0.250 T field?
30.
What is the peak emf generated by rotating a 1000-turn, 20.0 cm diameter
coil in the Earth’s
5.00 ×
10
−5
T
magnetic field, given the plane of the coil is
originally perpendicular to the Earth’s field and is rotated to be parallel to the
field in 10.0 ms?
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31.
What is the peak emf generated by a 0.250 m radius, 500-turn coil is
rotated one-fourth of a revolution in 4.17 ms, originally having its plane
perpendicular to a uniform magnetic field. (This is 60 rev/s.)
32.
(a) A bicycle generator rotates at 1875 rad/s, producing an 18.0 V peak
emf. It has a 1.00 by 3.00 cm rectangular coil in a 0.640 T field. How many
turns are in the coil? (b) Is this number of turns of wire practical for a 1.00 by
3.00 cm coil?
33.
Integrated
Concepts
This problem refers to the bicycle generator considered in the previous
problem. It is driven by a 1.60 cm diameter wheel that rolls on the outside rim
of the bicycle tire. (a) What is the velocity of the bicycle if the generator’s
angular velocity is 1875 rad/s? (b) What is the maximum emf of the generator
when the bicycle moves at 10.0 m/s, noting that it was 18.0 V under the
original conditions? (c) If the sophisticated generator can vary its own
magnetic field, what field strength will it need at 5.00 m/s to produce a 9.00 V
maximum emf?
34.
(a) A car generator turns at 400 rpm when the engine is idling. Its 300-turn,
5.00 by 8.00 cm rectangular coil rotates in an adjustable magnetic field so that
it can produce su
cient
ffi
voltage even at low rpms. What is the field strength
needed to produce a 24.0 V peak emf? (b) Discuss how this required field
strength compares to those available in permanent and electromagnets.
35.
Show that if a coil rotates at an angular velocity
, the period of its AC
output is
2
π
/
ω
.
36.
A 75-turn, 10.0 cm diameter coil rotates at an angular velocity of 8.00
rad/s in a 1.25 T field, starting with the plane of the coil parallel to the field. (a)
What is the peak emf? (b) At what time is the peak emf first reached? (c) At
what time is the emf first at its most negative? (d) What is the period of the AC
voltage output?
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37.
(a) If the emf of a coil rotating in a magnetic field is zero at
= 0
, and
increases to its first peak at
= 0.100 ms
, what is the angular velocity of the
coil? (b) At what time will its next maximum occur? (c) What is the period of
the output? (d) When is the output first one-fourth of its maximum? (e) When is
it next one-fourth of its maximum?
38.
Unreasonable
Results
A 500-turn coil with a
0.250
m
2
area is spun in the Earth’s
5.00 ×
10
−5
T
field, producing a 12.0 kV maximum emf. (a) At what angular velocity must the
coil be spun? (b) What is unreasonable about this result? (c) Which
assumption or premise is responsible?
23.6
Back
Emf
39.
Suppose a motor connected to a 120 V source draws 10.0 A when it first
starts. (a) What is its resistance? (b) What current does it draw at its normal
operating speed when it develops a 100 V back emf?
40.
A motor operating on 240 V electricity has a 180 V back emf at operating
speed and draws a 12.0 A current. (a) What is its resistance? (b) What current
does it draw when it is first started?
41.
What is the back emf of a 120 V motor that draws 8.00 A at its normal
speed and 20.0 A when first starting?
42.
The motor in a toy car operates on 6.00 V, developing a 4.50 V back emf at
normal speed. If it draws 3.00 A at normal speed, what current does it draw
when starting?
43.
Integrated
Concepts
The motor in a toy car is powered by four batteries in series, which produce a
total emf of 6.00 V. The motor draws 3.00 A and develops a 4.50 V back emf
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at normal speed. Each battery has a
0.100
Ω
internal resistance. What is the
resistance of the motor?
23.7
Transformers
44.
A plug-in transformer, like that in
Fi
gure
23.26
,
supplies 9.00 V to a video
game system. (a) How many turns are in its secondary coil, if its input voltage
is 120 V and the primary coil has 400 turns? (b) What is its input current when
its output is 1.30 A?
45.
An American traveler in New Zealand carries a transformer to convert New
Zealand’s standard 240 V to 120 V so that she can use some small appliances
on her trip. (a) What is the ratio of turns in the primary and secondary coils of
her transformer? (b) What is the ratio of input to output current? (c) How could
a New Zealander traveling in the United States use this same transformer to
power her 240 V appliances from 120 V?
46.
A cassette recorder uses a plug-in transformer to convert 120 V to 12.0 V,
with a maximum current output of 200 mA. (a) What is the current input? (b)
What is the power input? (c) Is this amount of power reasonable for a small
appliance?
47.
(a) What is the voltage output of a transformer used for rechargeable
flashlight batteries, if its primary has 500 turns, its secondary 4 turns, and the
input voltage is 120 V? (b) What input current is required to produce a 4.00 A
output? (c) What is the power input?
48.
(a) The plug-in transformer for a laptop computer puts out 7.50 V and can
supply a maximum current of 2.00 A. What is the maximum input current if the
input voltage is 240 V? Assume 100% e
ciency.
ffi
(b) If the actual e
ciency
ffi
is
less than 100%, would the input current need to be greater or smaller?
Explain.
49.
A multipurpose transformer has a secondary coil with several points at
which a voltage can be extracted, giving outputs of 5.60, 12.0, and 480 V. (a)
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The input voltage is 240 V to a primary coil of 280 turns. What are the
numbers of turns in the parts of the secondary used to produce the output
voltages? (b) If the maximum input current is 5.00 A, what are the maximum
output currents (each used alone)?
50.
A large power plant generates electricity at 12.0 kV. Its old transformer
once converted the voltage to 335 kV. The secondary of this transformer is
being replaced so that its output can be 750 kV for more e
cient
ffi
cross-
country transmission on upgraded transmission lines. (a) What is the ratio of
turns in the new secondary compared with the old secondary? (b) What is the
ratio of new current output to old output (at 335 kV) for the same power? (c) If
the upgraded transmission lines have the same resistance, what is the ratio of
new line power loss to old?
51.
If the power output in the previous problem is 1000 MW and line
resistance is
2.00
Ω
, what were the old and new line losses?
52.
Unreasonable
Results
The 335 kV AC electricity from a power transmission line is fed into the
primary coil of a transformer. The ratio of the number of turns in the secondary
to the number in the primary is
s
/
p
= 1000
. (a) What voltage is induced in
the secondary? (b) What is unreasonable about this result? (c) Which
assumption or premise is responsible?
53.
Construct Your Own Problem
Consider a double transformer to be used to create very large voltages. The
device consists of two stages. The first is a transformer that produces a much
larger output voltage than its input. The output of the first transformer is used
as input to a second transformer that further increases the voltage. Construct
a problem in which you calculate the output voltage of the final stage based
on the input voltage of the first stage and the number of turns or loops in both
parts of both transformers (four coils in
a
ll
)
.
Also calculate the maximum
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output current of the final stage based on the input current. Discuss the
possibility of power losses in the devices and the e
ect
ff
on the output current
and power.
23.8
Electrical
Safet
y:
Systems
and
Devices
54.
Integrated
Concepts
A short circuit to the grounded metal case of an appliance occurs as shown in
Fi
gure
23.60
. The person touching the case is wet and only has a
3.00 k
Ω
resistance to
e
a
r
t
h
/
g
r
o
u
n
d
.
(a) What is the voltage on the case if 5.00 mA
flows through the person? (b) What is the current in the short circuit if the
resistance of the earth/ground wire is
0.200
Ω
? (c) Will this trigger the 20.0 A
circuit breaker supplying the appliance?
Figure 23.60
A person can be shocked even when the case
of an appliance is grounded. The large short circuit current
produces a voltage on the case of the appliance, since the
resistance of the earth/ground wire is not zero.
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23.9
Inductance
55.
Two coils are placed close together in a physics lab to demonstrate
Faraday’s law of induction. A current of 5.00 A in one is switched o
ff
in
1.00 ms, inducing a 9.00 V emf in the other. What is their mutual
inductance?
56.
If two coils placed next to one another have a mutual inductance of 5.00
mH, what voltage is induced in one when the 2.00 A current in the other is
switched o
in 30.0 ms?
ff
57.
The 4.00 A current through a 7.50 mH inductor is switched o
ff
in 8.33 ms.
What is the emf induced opposing this?
58.
A device is turned on and 3.00 A flows through it 0.100 ms later. What is
the self-inductance of the device if an induced 150 V emf opposes this?
59.
Starting with
emf
2
(V
s)/A =
s
⋅
Ω ⋅
.
= −
Δ
1
, show that the units of inductance are
60.
Camera flashes charge a capacitor to high voltage by switching the
current through an inductor on and o
ff
rapidly. In what time must the 0.100
A
current through a 2.00 mH inductor be switched on or o
ff
to induce a 500 V
emf?
61.
A large research solenoid has a self-inductance of 25.0 H. (a) What
induced emf opposes shutting it o
when 100 A of current through it is
ff
switched o
ff
in 80.0 ms? (b) How much energy is stored in the inductor at full
current? (c) At what rate in watts must energy be dissipated to switch the
current o
in 80.0 ms? (d) In view of the answer to the last part, is it surprising
ff
that shutting it down this quickly is di
cult?
ffi
62.
(a) Calculate the self-inductance of a 50.0 cm long, 10.0 cm diameter
solenoid having 1000 loops. (b) How much energy is stored in this inductor
when 20.0 A of current flows through it? (c) How fast can it be turned o
ff
if the
induced emf cannot exceed 3.00 V?
Δ
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63.
A precision laboratory resistor is made of a coil of wire 1.50 cm in diameter
and 4.00 cm long, and it has 500 turns. (a) What is its self-inductance? (b)
What average emf is induced if the 12.0 A current through it is turned on in
5.00 ms (one-fourth of a cycle for 50 Hz AC)? (c) What is its inductance if it is
shortened to half its length and counter-wound (two layers of 250 turns in
opposite directions)?
64.
The heating coils in a hair dryer are 0.800 cm in diameter, have a
combined length of 1.00 m, and a total of 400 turns. (a) What is their total self-
inductance assuming they act like a single solenoid? (b) How much energy is
stored in them when 6.00 A flows? (c) What average emf opposes shutting
them o
if this is done in 5.00 ms (one-fourth of a cycle for 50 Hz AC)?
ff
65.
When the 20.0 A current through an inductor is turned o
ff
in 1.50 ms,
an 800 V emf is induced, opposing the change. What is the value of the self-
inductance?
66.
How fast can the 150 A current through a 0.250 H inductor be shut o
ff
if the induced emf cannot exceed 75.0 V?
67.
Integrated
Concepts
A very large, superconducting solenoid such as one used in MRI scans, stores
1.00
MJ
of energy in its magnetic field when 100 A flows. (a) Find its self-
inductance. (b) If the coils “go normal,” they gain resistance and start to
dissipate thermal energy. What temperature increase is produced if all the
stored energy goes into heating the 1000 kg magnet, given its average
specific heat is
200
J
/
kg·º
C
?
68.
Unreasonable
Results
A 25.0 H inductor has 100 A of current turned o
ff
in 1.00 ms. (a) What voltage
is induced to oppose this? (b) What is unreasonable about this result? (c)
Which assumption or premise is responsible?
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23.10
RL
Circuits
69.
If you want a characteristic
RL
time constant of 1.00 s, and you have a
500
Ω
resistor, what value of self-inductance is needed?
70.
Your
RL
circuit has a characteristic time constant of 20.0 ns, and a
resistance of
5.00 M
Ω
. (a) What is the inductance of the circuit? (b) What
resistance would give you a 1.00 ns time constant, perhaps needed for quick
response in an oscilloscope?
71.
A large superconducting magnet, used for magnetic resonance imaging,
has a 50.0 H inductance. If you want current through it to be adjustable with a
1.00 s characteristic time constant, what is the minimum resistance of
system?
72.
Verify that after a time of 10.0 ms, the current for the situation considered
in
Exam
ple 23.9
will be 0.183 A as stated.
73.
Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H,
and resistors ranging from
0.100
Ω
to
1.00 M
Ω
. What is the range of
characteristic
RL
time constants you can produce by connecting a single
resistor to a single inductor?
74.
(a) What is the characteristic time constant of a 25.0 mH inductor that has
a resistance of
4.00
Ω
? (b) If it is connected to a 12.0 V battery, what is the
current after 12.5 ms?
75.
What percentage of the final current
0
flows through an inductor
in
series with a resistor
, three time constants after the circuit is completed?
76.
The 5.00 A current through a 1.50 H inductor is dissipated by a
2.00
Ω
resistor in a circuit like that in
Fi
gure
23.44
with the switch in position 2. (a)
What is the initial energy in the inductor? (b) How long will it take the current to
decline to 5.00% of its initial value? (c) Calculate the average power
dissipated, and compare it with the initial power dissipated by the resistor.
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77.
(a) Use the exact exponential treatment to find how much time is
required
to bring the current through an 80.0 mH inductor in series with a
15.0
Ω
resistor to 99.0% of its final value, starting from zero. (b) Compare your
answer to the approximate treatment using integral numbers of
. (c) Discuss
how significant the di
erence is.
ff
78.
(a) Using the exact exponential treatment, find the time required for the
current through a 2.00 H inductor in series with a
0.500
Ω
resistor to be
reduced to 0.100% of its original value. (b) Compare your answer to the
approximate treatment using integral numbers of
. (c) Discuss how
significant the di
erence is.
ff
23.11
Reactance
, Inductive
and
Ca
pacitive
79.
At what frequency will a 30.0 mH inductor have a reactance of
100
Ω
?
80.
What value of inductance should be used if a
20.0 k
Ω
reactance is
needed at a frequency of 500 Hz?
81.
What capacitance should be used to produce a
2.00 M
Ω
reactance at
60.0
Hz?
82.
At what frequency will an 80.0 mF capacitor have a reactance of
0.250
Ω
?
83.
(a) Find the current through a 0.500 H inductor connected to a 60.0 Hz,
480 V AC source. (b) What would the current be at 100 kHz?
84.
(a) What current flows when a 60.0 Hz, 480 V AC source is connected to a
0.250
F
μ
capacitor? (b) What would the current be at 25.0 kHz?
85.
A 20.0 kHz, 16.0 V source connected to an inductor produces a 2.00 A
current. What is the inductance?
86.
A 20.0 Hz, 16.0 V source produces a 2.00 mA current when connected to
a capacitor. What is the capacitance?
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87.
(a) An inductor designed to filter high-frequency noise from power
supplied to a personal computer is placed in series with the computer. What
minimum inductance should it have to produce a
2.00 k
Ω
reactance for 15.0
kHz noise? (b) What is its reactance at 60.0 Hz?
88.
The capacitor in
Fi
gure
23.55
(a) is designed to filter low-frequency
signals, impeding their transmission between circuits. (a) What capacitance is
needed to produce a
100 k
Ω
reactance at a frequency of 120 Hz? (b) What
would its reactance be at 1.00 MHz? (c) Discuss the implications of your
answers to (a) and
(
b
)
.
89.
The capacitor in
Fi
gure
23.55
(b) will filter high-frequency signals by
shorting them to
e
a
r
t
h
/
g
r
o
u
n
d
.
(a) What capacitance is needed to produce a
reactance of
10.0 m
Ω
for a 5.00 kHz signal? (b) What would its reactance be
at 3.00 Hz? (c) Discuss the implications of your answers to (a) and
(
b
)
.
90.
Unreasonable
Results
In a recording of voltages due to brain activity (an
EE
G
)
,
a 10.0 mV signal with
a 0.500 Hz frequency is applied to a capacitor, producing a current of 100 mA.
Resistance is negligible. (a) What is the capacitance? (b) What is unreasonable
about this result? (c) Which assumption or premise is responsible?
91.
Construct Your Own Problem
Consider the use of an inductor in series with a computer operating on 60 Hz
electricity. Construct a problem in which you calculate the relative reduction in
voltage of incoming high frequency noise compared to 60 Hz voltage. Among
the things to consider are the acceptable series reactance of the inductor for
60 Hz power and the likely frequencies of noise coming through the power
lines.
23.12
RLC
Series
AC
Circuits
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92.
An
RL
circuit consists of a
40.0
Ω
resistor and a 3.00 mH inductor. (a)
Find its impedance
at 60.0 Hz and 10.0 kHz. (b) Compare these values of
with those found in
Exam
ple 23.12
in which there was also a capacitor.
93.
An
RC
circuit consists of a
40.0
Ω
resistor and a
5.00
F
μ
capacitor. (a)
Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of
with those found in
Exam
ple 23.12
,
in which there was also an inductor.
94.
An
LC
circuit consists of a
3.00 mH
inductor and a
5.00
capacitor. (a)
Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of
with those found in
Exam
ple 23.12
in which there was also a resistor.
95.
What is the resonant frequency of a 0.500 mH inductor connected to a
40.0 F
μ
capacitor?
96.
To receive AM radio, you want an
RLC
circuit that can be made to
resonate at any frequency between 500 and 1650 kHz. This is accomplished
with a fixed
1.00 H
μ
inductor connected to a variable capacitor. What
range of capacitance is needed?
97.
Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H,
and capacitors ranging from 1.00 pF to 0.100 F. What is the range of resonant
frequencies that can be achieved from combinations of a single inductor and a
single capacitor?
98.
What capacitance do you need to produce a resonant frequency of 1.00
GHz, when using an 8.00 nH inductor?
99.
What inductance do you need to produce a resonant frequency of 60.0 Hz,
when using a
2.00
F
μ
capacitor?
100.
The lowest frequency in the FM radio band is 88.0 MHz. (a) What
inductance is needed to produce this resonant frequency if it is connected to
a 2.50 pF capacitor? (b) The capacitor is variable, to allow the resonant
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frequency to be adjusted to as high as 108 MHz. What must the capacitance
be at this frequency?
101.
An
RLC
series circuit has a
2.50
Ω
resistor, a
100 H
μ
inductor, and an
80.0
F
μ
c
a
p
a
c
i
t
o
r
.
(
a
)
Find the circuit’s impedance at 120 Hz. (b) Find the
circuit’s impedance at 5.00 kHz. (c) If the voltage source has
rms
= 5.60 V
,
what is
rms
at each frequency? (d) What is the resonant frequency of the
circuit? (e) What is
rms
at resonance?
102.
An
RLC
series circuit has a
1.00 k
Ω
resistor, a
150 H
μ
inductor, and a
25.0 nF capacitor. (a) Find the circuit’s impedance at 500 Hz. (b) Find the
circuit’s impedance at 7.50 kHz. (c) If the voltage source has
rms
= 408 V
,
what is
rms
at each frequency? (d) What is the resonant frequency of the
circuit? (e) What is
rms
at resonance?
103.
An
RLC
series circuit has a
2.50
Ω
resistor, a
100 H
μ
inductor, and an
80.0
F
μ
capacitor. (a) Find the power factor at
= 120 Hz
. (b) What is the
phase angle at 120 Hz? (c) What is the average power at 120 Hz? (d) Find the
average power at the circuit’s resonant frequency.
104.
An
RLC
series circuit has a
1.00 k
Ω
resistor, a
150 H
μ
inductor, and a
25.0 nF capacitor. (a) Find the power factor at
= 7.50 Hz
. (b) What is the
phase angle at this frequency? (c) What is the average power at this
frequency? (d) Find the average power at the circuit’s resonant frequency.
105.
An
RLC
series circuit has a
200
Ω
resistor and a 25.0 mH inductor.
At 8000 Hz, the phase angle is
45.0º
. (a) What is the impedance? (b) Find
the
circuit’s capacitance. (c) If
power supplied?
rms
= 408 V
is applied, what is the average
106.
Referring to
Exam
ple 23.14
,
find the average power at 10.0 kHz.
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0 0
24.1
Maxwell’s
Equations:
Electromagnetic
Waves
Predicted
and
Observed
1.
Verify that the correct value for the speed of light
is obtained when
numerical values for the permeability and permittivity of free space (
0
and
0
)
are entered into the equation
=
1
.
2.
Show that, when SI units for
0
and
0
are entered, the units given by the
right-hand side of the equation in the problem above are m/s.
24.2
Production
of
Electroma
gnetic
Waves
3.
What is the maximum electric field strength in an electromagnetic wave that
has a maximum magnetic field strength of
5.00 × 10
−4
Earth’s)?
T
(about 10 times the
4.
The maximum magnetic field strength of an electromagnetic field is
5 × 10
−6
T
. Calculate the maximum electric field strength if the wave is
traveling in a medium in which the speed of the wave is
0.75
.
5.
Verify the units obtained for magnetic field strength
in
Exam
ple 24.1
(using the equation
=
) are in fact teslas
(
T
)
.
24.3
The
Electroma
gnetic
Spectrum
6.
(a) Two microwave frequencies are authorized for use in microwave ovens:
900 and 2560 MHz. Calculate the wavelength of each. (b) Which frequency
would produce smaller hot spots in foods due to interference e
ects?
ff
7.
(a) Calculate the range of wavelengths for AM radio given its frequency
range is 540 to 1600 kHz. (b) Do the same for the FM frequency range of 88.0
to 108 MHz.
8.
A radio station utilizes frequencies between commercial AM and FM. What
is the frequency of a 11.12-m-wavelength channel?
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9.
Find the frequency range of visible light, given that it encompasses
wavelengths from 380 to 760 nm.
10.
Combing your hair leads to excess electrons on the comb. How fast
would you have to move the comb up and down to produce red light?
11.
Electromagnetic radiation
having a
15.0 −
m
wavelength is classified as
infrared radiation. What is its frequency?
12.
Approximately what is the smallest detail observable with a microscope
that uses ultraviolet light of frequency
1.20 ×
10
15
Hz
?
13.
A radar used to detect the presence of aircraft receives a pulse that has
reflected o
ff
an object
6×10
−5
s
after it was transmitted. What is the distance
from the radar station to the reflecting object?
14.
Some radar systems detect the size and shape of objects such as
aircraft and geological terrain. Approximately what is the smallest observable
detail utilizing 500-MHz radar?
15.
Determine the amount of time it takes for X-rays of frequency
3×10
18
Hz
to travel (a) 1 mm and (b) 1 cm.
16.
If you wish to detect details of the size of atoms (about
1×
10
−10
m
)
with
electromagnetic radiation, it must have a wavelength of about this size. (a)
What is its frequency? (b) What type of electromagnetic radiation might this
be?
17.
If the Sun suddenly turned o
,
ff
we would not know it until its light
stopped
coming. How long would that be, given that the Sun is
1.50×10
11
m
away?
18.
Distances in space are often quoted in units of light years, the
distance light travels in one year. (a) How many meters is a light year? (b) How
many meters is it to Andromeda, the nearest large galaxy, given that it is
2.00×10
6
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light years away? (c) The most distant galaxy yet discovered is
12.0×10
9
years away. How far is this in meters?
light
19.
A certain 50.0-Hz AC power line radiates an electromagnetic wave
having a maximum electric field strength of 13.0
k
V
/
m
.
(a) What is the
wavelength of
this very low frequency electromagnetic wave? (b) What is its maximum
magnetic field strength?
20.
During normal beating, the heart creates a maximum 4.00-mV
potential across 0.300 m of a person’s chest, creating a 1.00-Hz
electromagnetic wave.
(a)What is the maximum electric field strength created? (b) What is the
corresponding maximum magnetic field strength in the electromagnetic wave?
(c) What is the wavelength of the electromagnetic wave?
21.
(a) The ideal size (most e
cient)
ffi
for a broadcast antenna with one end
on
the ground is one-fourth the wavelength (
/4
) of the electromagnetic radiation
being sent out. If a new radio station has such an antenna that is 50.0 m high,
what frequency does it broadcast most e
ciently?
ffi
Is this in the AM or FM
band? (b) Discuss the analogy of the fundamental resonant mode of an air
column closed at one end to the resonance of currents on an antenna that is
one-fourth their wavelength.
22.
(a) What is the wavelength of 100-MHz radio waves used in an MRI unit?
(b) If the frequencies are swept over a
±1.00
range centered on 100 MHz,
what is the range of wavelengths broadcast?
23.
(a) What is the frequency of the 193-nm ultraviolet radiation used in
laser eye surgery? (b) Assuming the accuracy with which this EM radiation
can ablate the cornea is directly proportional to wavelength, how much more
accurate can this UV be than the shortest visible wavelength of light?
24.
TV-reception antennas for VHF are constructed with cross wires
supported at their centers, as shown in
Fi
gure
24.27
.
The ideal length for the
cross
wires is one-half the wavelength to be received, with the more expensive
antennas having one for each channel. Suppose you measure the lengths of
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the wires for particular channels and find them to be 1.94 and 0.753 m long,
respectively. What are the frequencies for these channels?
Figure 24.27
A television
reception antenna has cross
wires of various lengths to
most e
ciently receive
ffi
di
erent
ff
wavelengths.
25.
Conversations with astronauts on lunar walks had an echo that was
used to estimate the distance to the Moon. The sound spoken by the person
on Earth was transformed into a radio signal sent to the Moon, and
transformed
back into sound on a speaker inside the astronaut’s space suit. This sound
was picked up by the microphone in the space suit (intended for the
astronaut’s voice) and sent back to Earth as a radio echo of sorts. If the
round-trip time was 2.60 s, what was the approximate distance to the Moon,
neglecting any delays in the electronics?
26.
Lunar astronauts placed a reflector on the Moon’s surface, o
ff
which a laser beam is periodically reflected. The distance to the Moon is
calculated from the round-trip time. (a) To what accuracy in meters can the
distance to
the Moon be determined, if this time can be measured to 0.100 ns? (b) What
percent accuracy is this, given the average distance to the Moon is
3.84×10
8
m
?
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27.
Radar is used to determine distances to various objects by measuring
the round-trip time for an echo from the object. (a) How far away is the
planet
Venus if the echo time is 1000 s? (b) What is the echo time for a car 75.0 m
from a Highway Police radar unit? (c) How accurately (in nanoseconds) must
you be able to measure the echo time to an airplane 12.0 km away to
determine its distance within 10.0 m?
28.
Integrated
Concepts
(a)Calculate the ratio of the highest to lowest frequencies of electromagnetic
waves the eye can see, given the wavelength range of visible light is from 380
to 760 nm. (b) Compare this with the ratio of highest to lowest frequencies the
ear can hear.
29.
Integrated
Concepts
(a)Calculate the rate in watts at which heat transfer through radiation occurs
(almost entirely in the infrared) from
1.0
m
2
of the Earth’s surface at night.
Assume the emissivity is 0.90, the temperature of the Earth is
15ºC
, and that
of outer space is 2.7 K. (b) Compare the intensity of this radiation with that
coming to the Earth from the Sun during the day, which averages about
800 W/m
2
, only half of which is absorbed. (c) What is the maximum magnetic
field strength in the outgoing radiation, assuming it is a continuous wave?
24.4
Ener
gy
in
Electromagnetic
Waves
30.
What is the intensity of an electromagnetic wave with a peak electric
field strength of 125 V/m?
31.
Find the intensity of an electromagnetic wave having a peak magnetic
field
strength of
4.00×10
−9
T
.
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32.
Assume the helium-neon lasers commonly used in student
physics laboratories have power outputs of 0.250 mW. (a) If such a laser
beam is
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projected onto a circular spot 1.00 mm in diameter, what is its intensity? (b)
Find the peak magnetic field strength. (c) Find the peak electric field strength.
33.
An AM radio transmitter broadcasts 50.0 kW of power uniformly in
all directions. (a) Assuming all of the radio waves that strike the ground
are
completely absorbed, and that there is no absorption by the atmosphere or
other objects, what is the intensity 30.0 km away? (Hint: Half the power will be
spread over the area of a
h
e
m
i
sp
h
e
r
e
.
)
(b) What is the maximum electric field
strength at this distance?
34.
Suppose the maximum safe intensity of microwaves for human exposure
is taken to be
1.00 W/m
2
. (a) If a radar unit leaks 10.0 W of microwaves
(other than those sent by its antenna) uniformly in all directions, how far away
must you be to be exposed to an intensity considered to be safe? Assume
that the power spreads uniformly over the area of a sphere with no
complications from absorption or reflection. (b) What is the maximum
electric field strength at the safe intensity? (Note that early radar units
leaked more than modern ones do. This caused identifiable health problems,
such as cataracts, for people who worked near
t
h
e
m
.
)
35.
A 2.50-m-diameter university communications satellite dish receives
TV
signals that have a maximum electric field strength (for one channel) of
7.50
V/m
. (See
Fi
gure
24.28
.
)
(a) What is the intensity of this wave?
(b) What is the power received by the antenna? (c) If the orbiting satellite
broadcasts uniformly over an area of
1.50×10
13
m
2
A
m
e
r
i
c
a
)
,
how much power does it radiate?
(a large fraction of North
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2
2
Figure 24.28
Satellite
dishes receive TV signals
sent from orbit. Although
the signals are quite
weak, the receiver can
detect them by being
tuned to resonate at their
frequency.
36.
Lasers can be constructed that produce an extremely high intensity
electromagnetic wave for a brief time—called pulsed lasers. They are used to
ignite nuclear fusion, for example. Such a laser may produce an
electromagnetic wave with a maximum electric field strength of
1.00×10
11
V
/
m
for a time of 1.00 ns. (a) What is the maximum magnetic field
strength in the wave? (b) What is the intensity of the beam? (c) What energy
does it deliver on a
1.00-mm
2
area?
37.
Show that for a continuous sinusoidal electromagnetic wave, the peak
intensity is twice the average intensity (
0
= 2
ave
)
,
using either the fact that
0
=
rms
,
or
0
=
rms
, where rms means average (actually root
mean square, a type of
a
ve
r
a
g
e
)
.
38.
Suppose a source of electromagnetic waves radiates uniformly in
all directions in empty space where there are no absorption or
interference
e
ects.
ff
(a) Show that the intensity is inversely proportional to
2
, the distance
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from the source squared. (b) Show that the magnitudes of the electric and
magnetic fields are inversely proportional to
.
39.
Integrated
Concepts
An
circuit with a 5.00-pF capacitor oscillates in such a manner as to
radiate at a wavelength of 3.30 m. (a) What is the resonant frequency? (b)
What inductance is in series with the capacitor?
40.
Integrated
Concepts
What capacitance is needed in series with an
800 −
H
inductor to form a
circuit that radiates a wavelength of 196 m?
41.
Integrated
Concepts
Police radar determines the speed of motor vehicles using the same Doppler-
shift technique employed for ultrasound in medical diagnostics. Beats are
produced by mixing the double Doppler-shifted echo with the original
frequency. If
1.50×10
9
-Hz
microwaves are used and a beat frequency of 150
Hz is produced, what is the speed of the vehicle? (Assume the same Doppler-
shift formulas are valid with the speed of sound replaced by the speed of
l
i
g
h
t
.
)
42.
Integrated
Concepts
Assume the mostly infrared radiation from a heat lamp acts like a
continuous wave with wavelength
1.50m
. (a) If the lamp’s 200-W output is
focused on a person’s shoulder, over a circular area 25.0 cm in diameter,
what is the intensity in
W/m
2
? (b) What is the peak electric field strength?
(c) Find the peak magnetic field strength. (d) How long will it take to
increase the
temperature of the 4.00-kg shoulder by
2.00º C
, assuming no
other heat
transfer and given that its specific heat is
3.47×
10
3
J
/
kg
⋅
ºC
?
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43.
Integrated
Concepts
On its highest power setting, a microwave oven increases the temperature of
0.400 kg of spaghetti by
45.0ºC
in 120 s. (a) What was the rate of power
absorption by the spaghetti, given that its specific heat is
3.76×10
3
J
/
kg
⋅
ºC
?
(b) Find the average intensity of the microwaves, given that they are absorbed
over a circular area 20.0 cm in diameter. (c) What is the peak electric field
strength of the microwave? (d) What is its peak magnetic field strength?
44.
Integrated
Concepts
Electromagnetic radiation from a 5.00-mW laser is concentrated on a
1.00-mm
2
area. (a) What is the intensity in
W/m
2
? (b) Suppose a 2.00-nC
static charge is in the beam. What is the maximum electric force it
experiences? (c) If the static charge moves at 400 m/s, what maximum
magnetic force can it feel?
45.
Integrated
Concepts
A 200-turn flat coil of wire 30.0 cm in diameter acts as an antenna for FM
radio at a frequency of 100 MHz. The magnetic field of the incoming
electromagnetic wave is perpendicular to the coil and has a maximum
strength of
1.00×
10
−12
T
. (a) What power is incident on the coil? (b) What
average emf is induced in the coil over one-fourth of a cycle? (c) If the radio
receiver has an inductance of
2.50
H
, what capacitance must it have to
resonate at 100 MHz?
46.
Integrated
Concepts
If electric and magnetic field strengths vary sinusoidally in time, being zero at
= 0
, then
=
0
sin
2
π
and
=
0
sin
2
π
. Let
= 1.00 GHz
here. (a) When are the field strengths first zero? (b) When do they reach their
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most negative value? (c) How much time is needed for them to complete one
cycle?
47.
Unreasonable
Results
A researcher measures the wavelength of a 1.20-GHz electromagnetic wave
to be 0.500 m. (a) Calculate the speed at which this wave propagates. (b)
What is unreasonable about this result? (c) Which assumptions are
unreasonable or inconsistent?
48.
Unreasonable
Results
The peak magnetic field strength in a residential microwave oven is
9.20×
10
−5
T
. (a) What is the intensity of the microwave? (b) What is
unreasonable about this result? (c) What is wrong about the premise?
49.
Unreasonable
Results
An
circuit containing a 2.00-H inductor oscillates at such a frequency that
it radiates at a 1.00-m wavelength. (a) What is the capacitance of the circuit?
(b) What is unreasonable about this result? (c) Which assumptions are
unreasonable or inconsistent?
50.
Unreasonable
Results
An
circuit containing a 1.00-pF capacitor oscillates at such a frequency
that it radiates at a 300-nm wavelength. (a) What is the inductance of the
circuit? (b) What is unreasonable about this result? (c) Which assumptions
are unreasonable or inconsistent?
51.
Create Your Own Problem
Consider electromagnetic fields produced by high voltage power lines.
Construct a problem in which you calculate the intensity of this
electromagnetic radiation in
W/m
2
based on the measured magnetic field
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strength of the radiation in a home near the power lines. Assume these
magnetic field strengths are known to average less than a
T
. The intensity is
small enough that it is di
cult to imagine mechanisms for biological damage
ffi
due to it. Discuss how much energy may be radiating from a section of power
line several hundred meters long and compare this to the power likely to be
carried by the lines. An idea of how much power this is can be obtained by
calculating the approximate current responsible for
tens of meters.
T
fields at distances of
52.
Create Your Own Problem
Consider the most recent generation of residential satellite dishes that are a
little less than half a meter in diameter. Construct a problem in which you
calculate the power received by the dish and the maximum electric field
strength of the microwave signals for a single channel received by the dish.
Among the things to be considered are the power broadcast by the satellite
and the area over which the power is spread, as well as the area of the
receiving dish.
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25.1
The
Ra
y
Aspect
of
Light
1.
Suppose a man stands in front of a mirror as shown in
Fi
gure
25.50
.
His eyes are 1.65 m above the floor, and the top of his head is 0.13
m higher. Find the height above the floor of the top and bottom of
the smallest mirror in which he can see both the top of his head and
his feet. How is this distance related to the man’s height?
Figure 25.50
A full-
length mirror is one
in which you can
see all of yourself.
It need not be as
big as you, and its
size is independent
of your distance
from it.
25.2
The
Law
of
Reflection
2.
Show that when light reflects from two mirrors that meet each
other at a
right angle, the outgoing ray is parallel to the incoming ray,
as illustrated in the
following figure.
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Figure 25.51
A corner
reflector sends the
reflected ray back in a
direction parallel to the
incident ray, independent
of incoming direction.
3.
Light shows staged with lasers use moving mirrors to swing
beams and create colorful effects. Show that a light ray reflected
from a mirror changes
direction
by
2
when
the
mirror
is
rotated
by
an
angle
.
4.
A flat mirror is neither converging nor diverging. To prove this,
consider two rays originating from the same point and diverging at
an angle
. Show that after striking a plane mirror, the angle
between their directions remains
.
Figure 25.52
A flat mirror neither
converges nor diverges light
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rays. Two rays continue to
diverge at the same angle after
reflection.
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25.3
The
Law
of
Refraction
5.
What is the speed of light in water? In
g
l
yc
e
r
i
n
e
?
6.
What is the speed of light in air? In crown glass?
7.
Calculate
the
index
of
refraction
for
a
medium
in
which
the
speed
of
light
is
2.012 ×
10
8
25.1
.
m/s
,
and
identify
the
most
likely
substance
based
on
Table
8.
In what substance in
Table
25.1
is the speed of light
2.290
×
10
8
m/s
?
9.
There was a major collision of an asteroid with the Moon in
medieval times. It was described by monks at Canterbury Cathedral
in England as a red glow on and around the Moon. How long after
the asteroid hit the Moon, which is
3.84
×
10
5
km
away, would the light first arrive on Earth?
10.
A scuba diver training in a pool looks at his instructor as shown in
Fi
gure
25.53
. What angle does the ray from the instructor’s face make with
the perpendicular to the water at the point where the ray enters? The
angle
between the ray in the water and the perpendicular to the water is
25.0º
.
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Figure 25.53
A scuba diver in a pool and his trainer look
at each other.
11.
Components of some computers communicate with each other
through
optical fibers having an index of
refraction
=
1.55
. What time in
nanoseconds is required for a signal to travel 0.200 m through such a
fiber?
12.
(a) Given that the angle between the ray in the water and the
perpendicular
to the water is
25.0º
, and using information in
Fi
gure
25.53
, find the height of the instructor’s head above the water,
noting that you will first have to calculate the angle of refraction. (b)
Find the apparent depth of the diver’s head below water as seen by
the instructor. Assume the diver and the
d
i
ve
r
'
s
image are the same
horizontal distance from the normal.
13.
Suppose you have an unknown clear substance immersed in
water, and you wish to identify it by finding its index of refraction.
You arrange to have a beam of light enter it at an angle of
45.0º
,
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and you observe the angle of
refraction to be
40.3º
. What is the
index of refraction of the substance and its
likely identity?
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14.
On the Moon’s surface, lunar astronauts placed a corner
reflector, off which a laser beam is periodically reflected. The
distance to the Moon is
calculated from the round-trip time. What
percent correction is needed to
account for the delay in time due
to the slowing of light in Earth’s
atmosphere? Assume the distance to the Moon is precisely
3.84 ×
10
8
m
,
and
Earth’s
atmosphere
(which
varies
in
density
with
altitude)
is
equivalent
to
a
layer 30.0 km thick with a constant index of refraction
= 1.000293
.
15.
Suppose
Fi
gure 25.54
represents a ray of light going from air
through crown glass into water, such as going into a fish tank.
Calculate the amount
the ray is displaced by the glass
(
Δ
and the glass is 1.00 cm thick.
),
given
that
the
incident
angle
is
40.0º
16.
Fi
gure
25.54
shows a ray of light passing from one medium into a
second and then a third. Show
that
3
is the same as it would be if the
second medium were not present (provided total internal reflection
does not occur).
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Figure 25.54
A ray of light passes
from one medium to a third by
traveling through a second. The
final direction is the same as if
the second medium were not
present, but the
ray is displaced by
Δ
exaggerated).
(shown
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17.
Unreasonable
Results
Suppose light travels from water to another substance, with an angle
of
incidence of
10.0º
and an angle of refraction of
14.9º
. (a) What is the
index of
refraction of the other substance? (b) What is unreasonable
about this
r
e
su
l
t
?
(c) Which assumptions are unreasonable or inconsistent?
18.
Construct Your Own Problem
Consider sunlight entering the Earth’s atmosphere at sunrise and
sunset—that is, at a
90º
incident angle. Taking the boundary
between nearly empty space and the atmosphere to be sudden,
calculate the angle of refraction for sunlight. This lengthens the time
the Sun appears to be above the horizon, both at sunrise and sunset.
Now construct a problem in which you determine the angle of
refraction for different models of the atmosphere, such as various
layers of varying density. Your instructor may wish to guide you on
the level of
complexity to consider and on how the index of refraction varies with
air density.
19.
Unreasonable
Results
Light traveling from water to a gemstone strikes the surface at an
angle of
80.0º
and has an angle of refraction of
15.2º
. (a) What is the
speed of light in
the gemstone? (b) What is unreasonable about this
r
e
su
l
t
?
(c) Which assumptions are unreasonable or inconsistent?
25.4
Total
Internal
Reflection
20.
Verify that the critical angle for light going from water to air is
48.6º
, as discussed at the end of
Exam
ple 25.4
, regarding the
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critical angle for light
traveling in a polystyrene (a type of plastic)
pipe surrounded by air.
21.
(a) At the end of
Exam
ple 25.4
, it was stated that the critical
angle for light going from diamond to air is
24.4º
. Verify this. (b)
What is the critical
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angle for light going from zircon to air?
22.
An optical fiber uses flint glass clad with crown glass. What is
the critical
a
n
g
l
e
?
23.
At what minimum angle will you get total internal reflection
of light
traveling in water and reflected from ice?
24.
Suppose you are using total internal reflection to make an
efficient corner reflector. If there is air outside and the incident angle
is
45.0º
, what must be the minimum index of refraction of the
material from which the reflector is made?
25.
You can determine the index of refraction of a substance by
determining its critical angle. (a) What is the index of refraction of a
substance that has a critical angle of
68.4º
when submerged in
water? What is the substance,
based on
Table 25.1
?
(b) What would the critical angle be for this
substance in air?
26.
A ray of light, emitted beneath the surface of an unknown liquid
with air above it, undergoes total internal reflection as shown in
Fi
gure 25.55
. What is the index of refraction for the liquid and its
likely identification?
Figure 25.55
A light ray inside a
liquid strikes the surface at the
critical angle and undergoes
total internal reflection.
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27.
A light ray entering an optical fiber surrounded by air is first
refracted and then reflected as shown in
Fi
gure 25.56
. Show that if
the fiber is made from crown glass, any incident ray will be totally
internally reflected.
Figure 25.56
A light ray enters the
end of a fiber, the surface of
which is perpendicular to its
sides. Examine the conditions
under which it may be totally
internally reflected.
25.5
Dis
persion: The Rainbow and Prisms
28.
(a) What is the ratio of the speed of red light to violet light in
diamond, based on
Table
25.2
?
(b) What is this ratio in
po
l
yst
yr
e
n
e
?
(c) Which is more dispersive?
29.
A beam of white light goes from air into water at an incident
angle of
75.0º
. At what angles are the red (660 nm) and violet (410
nm) parts of the
light refracted?
30.
By how much do the critical angles for red (660 nm) and violet
(410 nm) light differ in a diamond surrounded by air?
31.
(a) A narrow beam of light containing yellow (580 nm) and
green (550 nm) wavelengths goes from polystyrene to air, striking
the surface at a
30.0º
incident angle. What is the angle between
the colors when they emerge? (b) How far would they have to
travel to be separated by 1.00 mm?
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32.
A parallel beam of light containing orange (610 nm) and violet
(410 nm) wavelengths goes from fused quartz to water, striking
the surface between
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them at a
60.0º
incident angle. What is the angle between the two colors
in
water?
33.
A ray of 610 nm light goes from air into fused quartz at an
incident angle of
55.0º
. At what incident angle must 470 nm light
enter flint glass to have the same angle of refraction?
34.
A narrow beam of light containing red (660 nm) and blue (470
nm) wavelengths travels from air through a 1.00 cm thick flat piece
of crown glass and back to air again. The beam strikes at a
30.0º
incident angle. (a) At what angles do the two colors emerge? (b) By
what distance are the red and blue
separated when they emerge?
35.
A narrow beam of white light enters a prism made of crown glass at
a
45.0º
incident angle, as shown in
Fi
gure
25.57
. At what angles,
R
and
V
,
do the red (660 nm) and violet (410 nm) components of the light
emerge from the prism?
Figure 25.57
This prism will disperse
the white light into a rainbow of
colors. The incident angle is
45.0º
, and the angles at which
the red and
violet light emerge
are
R
and
V
.
25.6
Ima
ge
Formation
b
y
Lenses
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36.
What is the power in diopters of a camera lens that has a 50.0
mm focal
l
e
n
g
t
h
?
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37.
Your camera’s zoom lens has an adjustable focal length ranging
from 80.0 to 200 mm. What is its range of powers?
38.
What is the focal length of 1.75 D reading glasses found on the
rack in a pharmacy?
39.
You note that your prescription for new eyeglasses is –4.50 D.
What will their focal length be?
40.
How far from the lens must the film in a camera be, if the lens
has a 35.0 mm focal length and is being used to photograph a
flower 75.0 cm away? Explicitly show how you follow the steps in
the Problem-Solving Strategy for lenses.
41.
A certain slide projector has a 100 mm focal length lens. (a) How
far away is the screen, if a slide is placed 103 mm from the lens and
produces a sharp image? (b) If the slide is 24.0 by 36.0 mm, what
are the dimensions of the image? Explicitly show how you follow the
steps in the Problem-Solving Strategy for lenses.
42.
A doctor examines a mole with a 15.0 cm focal length
magnifying glass held 13.5 cm from the mole (a) Where is the
image? (b) What is its magnification? (c) How big is the image of a
5.00 mm diameter
m
o
l
e
?
43.
How far from a piece of paper must you hold your father’s 2.25
D reading glasses to try to burn a hole in the paper with
su
n
l
i
g
h
t
?
44.
A camera with a 50.0 mm focal length lens is being used to
photograph a person standing 3.00 m away. (a) How far from the
lens must the film be? (b) If the film is 36.0 mm high, what fraction
of a 1.75 m tall person will fit on it? (c) Discuss how reasonable this
seems, based on your experience in taking or posing for
photographs.
45.
A camera lens used for taking close-up photographs has a focal
length of
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22.0
mm.
The
farthest
it
can
be
placed
from
the
film
is
33.0
mm.
(a)
What
is
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the closest object that can be photographed? (b) What is the
magnification of this closest
ob
j
e
c
t
?
46.
Suppose your 50.0 mm focal length camera lens is 51.0 mm
away from the film in the camera. (a) How far away is an object that
is in focus? (b) What is the height of the object if its image is 2.00
cm high?
47.
(a) What is the focal length of a magnifying glass that produces
a magnification of 3.00 when held 5.00 cm from an object, such as a
rare coin?
(b) Calculate the power of the magnifier in diopters. (c) Discuss how
this power compares to those for store-bought reading glasses
(typically 1.0 to 4.0 D). Is the magnifier’s power greater, and should
it be?
48.
What magnification will be produced by a lens of power –4.00 D
(such as might be used to correct myopia) if an object is held 25.0
cm away?
49.
In
Exam
ple 25.7
, the magnification of a book held 7.50 cm from
a 10.0 cm focal length lens was found to be 4.00. (a) Find the
magnification for the book when it is held 8.50 cm from the
magnifier. (b) Do the same for when it is held 9.50 cm from the
magnifier. (c) Comment on the trend in m as the object distance
increases as in these two calculations.
50.
Suppose a 200 mm focal length telephoto lens is being used to
photograph mountains 10.0 km away. (a) Where is the image? (b)
What is the height of the image of a 1000 m high cliff on one of the
mountains?
51.
A camera with a 100 mm focal length lens is used to
photograph the sun and moon. What is the height of the image of
the sun on the film, given the
sun is
1.40 ×
10
6
km
in diameter
and is
1.50
×
10
8
km
away?
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52.
Combine thin lens equations to show that the magnification for a
thin lens is determined by its focal length and the object distance
and is given by
=
/(
−
o
)
.
25.7
Ima
ge
Formation
by
Mirrors
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53.
What is the focal length of a makeup mirror that has a power of
1.50
D?
54.
Some telephoto cameras use a mirror rather than a lens. What
radius of curvature mirror is needed to replace a 800 mm focal
length telephoto lens?
55.
(a) Calculate the focal length of the mirror formed by the shiny
back of a spoon that has a 3.00 cm radius of curvature. (b) What is
its power in diopters?
56.
Find the magnification of the heater element in
Exam
ple 25.9
.
Note that
its large magnitude helps spread out the reflected energy.
57.
What is the focal length of a makeup mirror that produces a
magnification of 1.50 when a person’s face is 12.0 cm away?
Explicitly show how you follow the steps in the
Problem-Solvin
g
Strategy
for
Mirrors
.
58.
A shopper standing 3.00 m from a convex security mirror sees
his image with a magnification of 0.250. (a) Where is his image? (b)
What is the focal length of the mirror? (c) What is its radius of
curvature? Explicitly show how you follow the steps in the
Problem-Solvin
g Strategy
for
Mirrors
.
59.
An object 1.50 cm high is held 3.00 cm from a person’s cornea,
and its reflected image is measured to be 0.167 cm high. (a) What
is the magnification? (b) Where is the image? (c) Find the radius of
curvature of the convex mirror formed by the cornea. (Note that
this technique is used by optometrists to measure the curvature of
the cornea for contact lens fitting. The instrument used is called a
keratometer, or curve measurer.)
60.
Ray tracing for a flat mirror shows that the image is located a
distance
behind the mirror equal to the distance of the object from
the mirror. This is
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stated
i
= –
o
, since this is a negative image distance (it is a virtual
image).
(a)
What
is
the
focal
length
of
a
flat
mirror?
(b)
What
is
its
power?
61.
Show that for a flat mirror
ℎ
i
= ℎ
o
, knowing that the image is a
distance
behind
the
mirror
equal
in
magnitude
to
the
distance
of
the
object
from
the
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mirror.
62.
Use
the
law
of
reflection
to
prove
that
the
focal
length
of
a
mirror
is
half
its
radius
of
curvature.
That
is,
prove
that
=
/2
. Note this is true for a
spherical mirror only if its diameter is small compared with its
radius of curvature.
63.
Referring
to
the
electric
room
heater
considered
in
the
first
example
in
this
section,
calculate
the
intensity
of
IR
radiation
in
W/m
2
projected
by
the
concave mirror on a person 3.00 m away. Assume that the heating
element radiates 1500 W and has an area of
100 cm
2
, and that
half of the radiated power is reflected and focused by the mirror.
64.
Consider a 250-W heat lamp fixed to the ceiling in a bathroom. If
the filament in one light burns out then the remaining three still
work. Construct a problem in which you determine the resistance of
each filament in order to obtain a certain intensity projected on the
bathroom floor. The ceiling is 3.0 m high. The problem will need to
involve concave mirrors behind the filaments. Your instructor may
wish to guide you on the level of complexity to consider in the
electrical components.
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