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CHAPTER
19
O U T L I N E
19.1
19.2
19.3
19.4
Magnets
Earth’s Magnetic Field
Magnetic Fields
Magnetic Force on a
Current-Carrying
Conductor
19.5 Torque on a Current Loop
and Electric Motors
19.6 Motion of a Charged
Particle in a Magnetic Field
19.7 Magnetic Field of a Long,
Straight Wire and
Ampère’s Law
19.8 Magnetic Force between
Two Parallel Conductors
19.9 Magnetic Fields of Current
Loops and Solenoids
19.10 Magnetic Domains
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Johnny Johnson/Stone/Getty
Aurora borealis, the Northern Lights.
Displays such as this one are caused
by cosmic ray particles trapped in the
magnetic field of Earth. When the
particles collide with atoms in the
atmosphere, they cause the atoms to
emit visible light.
Magnetism
In terms of applications, magnetism is one of the most important fields in physics. Large electromagnets are used to pick up heavy loads. Magnets are used in such devices as meters,
motors, and loudspeakers. Magnetic tapes and disks are used routinely in sound- and videorecording equipment and to store computer data. Intense magnetic fields are used in magnetic resonance imaging (MRI) devices to explore the human body with better resolution and
greater safety than x-rays can provide. Giant superconducting magnets are used in the
cyclotrons that guide particles into targets at nearly the speed of light, and magnetic bottles
hold antimatter, possibly the key to future space propulsion systems.
Magnetism is closely linked with electricity. Magnetic fields affect moving charges, and
moving charges produce magnetic fields. Changing magnetic fields can even create electric
fields. These phenomena signify an underlying unity of electricity and magnetism, which
James Clerk Maxwell first described in the 19th century. The ultimate source of any magnetic
field is electric current.
19.1
MAGNETS
Most people have had experience with some form of magnet. You are most likely
familiar with the common iron horseshoe magnet that can pick up iron-containing
objects such as paper clips and nails. Several commercially available magnets are
shown in Figure 19.1. In the discussion that follows, we assume the magnet has the
shape of a bar. Iron objects are most strongly attracted to either end of such a bar
magnet, called its poles. One end is called the north pole and the other the south
pole. The names come from the behavior of a magnet in the presence of Earth’s
magnetic field. If a bar magnet is suspended from its midpoint by a piece of string
so that it can swing freely in a horizontal plane, it will rotate until its north pole
points to the north and its south pole points to the south. The same idea is used to
construct a simple compass. Magnetic poles also exert attractive or repulsive forces
on each other similar to the electrical forces between charged objects. In fact, simple
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Courtesy of Central Scientific Company
19.1
Image not Available
Magnets
625
Figure 19.1 An assortment of commercially
available magnets. The four red magnets and
the large black magnet on the left are made of
an alloy of iron, aluminum, and cobalt. The
six horseshoe magnets on the right are made
of different nickel – steel alloys. The rectangular magnets on the lower right are ceramics
made of iron, nickel, and beryllium oxides.
experiments with two bar magnets show that like poles repel each other and unlike
poles attract each other.
Although the force between opposite magnetic poles is similar to the force
between positive and negative electric charges, there is an important difference:
positive and negative electric charges can exist in isolation of each other; north
and south poles don’t. No matter how many times a permanent magnet is cut,
each piece always has a north pole and a south pole. There is some theoretical
basis for the speculation that magnetic monopoles (isolated north or south poles)
exist in nature, and the attempt to detect them is currently an active experimental
field of investigation.
An unmagnetized piece of iron can be magnetized by stroking it with a magnet.
Magnetism can also be induced in iron (and other materials) by other means. For
example, if a piece of unmagnetized iron is placed near a strong permanent
magnet, the piece of iron eventually becomes magnetized. The process can be
accelerated by heating and then cooling the iron.
Naturally occurring magnetic materials such as magnetite are magnetized in
this way because they have been subjected to Earth’s magnetic field for long periods of time. The extent to which a piece of material retains its magnetism depends
on whether it is classified as magnetically hard or soft. Soft magnetic materials,
such as iron, are easily magnetized, but also tend to lose their magnetism easily. In
contrast, hard magnetic materials, such as cobalt and nickel, are difficult to
magnetize, but tend to retain their magnetism.
In earlier chapters we described the interaction between charged objects
in terms of electric fields. Recall that an electric field surrounds any stationary
electric charge. The region of space surrounding a moving charge includes a
magnetic field as well. A magnetic field also surrounds a properly magnetized
magnetic material.
To describe any type of vector field, we must define its magnitude, or strength,
:
and its direction. The direction of a magnetic field B at any location is the
direction in which the north pole of a compass needle points at that location.
Active Figure 19.2a shows how the magnetic field of a bar magnet can be traced
with the aid of a compass, defining a magnetic field line. Several magnetic field
N
S
(a)
N
S
(b)
ACTIVE FIGURE 19.2
(a) Tracing the magnetic field of a
bar magnet. (b) Several magnetic
field lines of a bar magnet.
Log into PhysicsNow at www.cp7e.com
and go to Active Figure 19.2, where
you can move the compass around
and trace the magnetic field for
yourself.
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Chapter 19
Magnetism
Henry Leap and Jim Lehman
626
(a)
(b)
(c)
Figure 19.3 (a) The magnetic field pattern of a bar magnet, as displayed with iron filings on a sheet
of paper. (b) The magnetic field pattern between unlike poles of two bar magnets, as displayed with
iron filings. (c) The magnetic field pattern between two like poles.
A P P L I C AT I O N
Dusting for Fingerprints
lines of a bar magnet traced out in this way appear in the two-dimensional representation in Active Figure19.2b. Magnetic field patterns can be displayed by placing small iron filings in the vicinity of a magnet, as in Figure 19.3.
Forensic scientists use a technique similar to that shown in Figure 19.3 to find
fingerprints at a crime scene. One way to find latent, or invisible, prints is by sprinkling a powder of iron dust on a surface. The iron adheres to any perspiration or
body oils that are present and can be spread around on the surface with a magnetic brush that never comes into contact with the powder or the surface.
19.2
TIP 19.1 The Geographic
North Pole is the Magnetic
South Pole
The north pole of a magnet in a
compass points north because it’s
attracted to the Earth’s magnetic south
pole — located near the Earth’s
geographic north pole.
EARTH’S MAGNETIC FIELD
A small bar magnet is said to have north and south poles, but it’s more accurate
to say it has a “north-seeking” pole and a “south-seeking” pole. By these expressions, we mean that if such a magnet is used as a compass, one end will “seek,” or
point to, the geographic North Pole of Earth and the other end will “seek,” or
point to, the geographic South Pole of Earth. We conclude that the geographic
North Pole of Earth corresponds to a magnetic south pole, and the geographic
South Pole of Earth corresponds to a magnetic north pole. In fact, the configuration of Earth’s magnetic field, pictured in Figure 19.4, very much resembles
Axis of
rotation
Magnetic axis
South
magnetic
pole
North
geographic
pole
11°
Geographic
equator
S
r
c equato
Magneti
N
Figure 19.4 Earth’s magnetic field
lines. Note that magnetic south is at
the north geographic pole and magnetic north is at the south geographic
pole.
South
geographic
pole
North
magnetic
pole
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19.2
what would be observed if a huge bar magnet were buried deep in the Earth’s
interior.
If a compass needle is suspended in bearings that allow it to rotate in the vertical plane as well as in the horizontal plane, the needle is horizontal with respect to
Earth’s surface only near the equator. As the device is moved northward, the needle rotates so that it points more and more toward the surface of Earth. The angle
between the direction of the magnetic field and the horizontal is called the dip
angle. Finally, at a point just north of Hudson Bay in Canada, the north pole of the
needle points directly downward, with a dip angle of 90°. This site, first found in
1832, is considered to be the location of the south magnetic pole of Earth. It is
approximately 1 300 miles from Earth’s geographic North Pole and varies with
time. Similarly, Earth’s magnetic north pole is about 1 200 miles from its geographic South Pole. This means that compass needles point only approximately
north. The difference between true north, defined as the geographic North Pole,
and north indicated by a compass varies from point to point on Earth, a difference
referred to as magnetic declination. For example, along a line through South
Carolina and the Great Lakes a compass indicates true north, whereas in Washington
state it aligns 25° east of true north (Fig. 19.5).
Although the magnetic field pattern of Earth is similar to the pattern that
would be set up by a bar magnet placed at its center the source of Earth’s field
can’t consist of large masses of permanently magnetized material. Earth does have
large deposits of iron ore deep beneath its surface, but the high temperatures in
the core prevent the iron from retaining any permanent magnetization. It’s considered more likely that the true source of Earth’s magnetic field is electric current
in the liquid part of its core. This current, which is not well understood, may be
driven by an interaction between the planet’s rotation and convection in the hot
liquid core. There is some evidence that the strength of a planet’s magnetic field is
related to the planet’s rate of rotation. For example, Jupiter rotates faster than
Earth, and recent space probes indicate that Jupiter’s magnetic field is stronger
than Earth’s, even though Jupiter lacks an iron core. Venus, on the other hand,
rotates more slowly than Earth, and its magnetic field is weaker. Investigation into
the cause of Earth’s magnetism continues.
An interesting fact concerning Earth’s magnetic field is that its direction
reverses every few million years. Evidence for this phenomenon is provided by
basalt (an iron-containing rock) that is sometimes spewed forth by volcanic activity
on the ocean floor. As the lava cools, it solidifies and retains a picture of the direction of Earth’s magnetic field. When the basalt deposits are dated, they provide
evidence for periodic reversals of the magnetic field. The cause of these field
reversals is still not understood.
It has long been speculated that some animals, such as birds, use the magnetic
field of Earth to guide their migrations. Studies have shown that a type of anaerobic bacterium that lives in swamps has a magnetized chain of magnetite as part of
its internal structure. (The term anaerobic means that these bacteria live and grow
20°W
20°E
15°W
10°W
15°E
5°W
10°E
5°E
0°
Figure 19.5 A map of the continental United States showing the declination of a compass from true north.
Earth’s Magnetic Field
A P P L I C AT I O N
Magnetic Bacteria
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Chapter 19
Magnetism
A P P L I C AT I O N
Labeling Airport Runways
without oxygen; in fact, oxygen is toxic to them.) The magnetized chain acts as a
compass needle that enables the bacteria to align with Earth’s magnetic field.
When they find themselves out of the mud on the bottom of the swamp, they
return to their oxygen-free environment by following the magnetic field lines of
Earth. Further evidence for their magnetic sensing ability is the fact that bacteria
found in the Northern Hemisphere have internal magnetite chains that are
opposite in polarity to those of similar bacteria in the Southern Hemisphere. This
is consistent with the fact that in the Northern Hemisphere Earth’s field has a
downward component, whereas in the Southern Hemisphere it has an upward
component. Recently, a meteorite originating on Mars has been found to contain
a chain of magnetite. NASA scientists believe it may be a fossil of ancient Martian
bacterial life.
The magnetic field of Earth is used to label runways at airports according to
their direction. A large number is painted on the end of the runway so that it can
be read by the pilot of an incoming airplane. This number describes the direction
in which the airplane is traveling, expressed as the magnetic heading, in degrees
measured clockwise from magnetic north divided by 10. A runway marked 9 would
be directed toward the east (90° divided by 10), while a runway marked 18 would
be directed toward magnetic south.
Applying Physics 19.1 Compasses Down Under
On a business trip to Australia, you take along your
American-made compass that you may have used on a
camping trip. Does this compass work correctly in
Australia?
Explanation There’s no problem with using the
compass in Australia. The north pole of the magnet in
the compass will be attracted to the south magnetic
19.3
pole near the geographic North Pole, just as it was
in the United States. The only difference in the
magnetic field lines is that they have an upward
component in Australia, whereas they have a downward component in the United States. Held in a
horizontal plane, your compass can’t detect this,
however — it only displays the direction of the
horizontal component of the magnetic field.
MAGNETIC FIELDS
Experiments show that a stationary charged particle doesn’t interact with a static
magnetic field. When a charged particle is moving through a magnetic field, however, a magnetic force acts on it. This force has its maximum value when the
charge moves in a direction perpendicular to the magnetic field lines, decreases in
value at other angles, and becomes zero when the particle moves along the field
lines. This is quite different from the electric force, which exerts a force on a
charged particle whether it’s moving or at rest. Further, the electric force is directed parallel to the electric field while the magnetic force on a moving charge is
directed perpendicular to the magnetic field.
In our discussion of electricity, the electric field at some point in space was
defined as the electric force per unit charge acting on some test charge placed at
:
that point. In a similar way, we can describe the properties of the magnetic field B
at some point in terms of the magnetic force exerted on a test charge at that point.
Our test object is a charge q moving with velocity :
v . It is found experimentally
that the strength of the magnetic force on the particle is proportional to the magnitude of the charge q, the magnitude of the velocity :
v , the strength of the external
:
magnetic field B, and the sine of the angle between the direction of :
v and the
:
direction of B . These observations can be summarized by writing the magnitude of
the magnetic force as
F qvB sin This expression is used to define the magnitude of the magnetic field as
[19.1]
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19.3
B
F
qv sin
629
Magnetic Fields
[19.2]
If F is in newtons, q in coulombs, and v in meters per second, then the SI unit
of magnetic field is the tesla (T), also called the weber (Wb) per square meter
(1 T 1 Wb/m2). If a 1-C charge moves in a direction perpendicular to a magnetic field of magnitude 1 T with a speed of 1 m/s, the magnetic force exerted on
:
the charge is 1 N. We can express the units of B as
[B] T Wb
N
N
2
m
Cm/s
Am
[19.3]
In practice, the cgs unit for magnetic field, the gauss (G), is often used. The gauss
is related to the tesla through the conversion
1 T 10 4 G
Conventional laboratory magnets can produce magnetic fields as large as about
25 000 G, or 2.5 T. Superconducting magnets that can generate magnetic fields as
great as 3 105 G, or 30 T, have been constructed. These values can be compared
with the value of Earth’s magnetic field near its surface, which is about 0.5 G, or
0.5 104 T.
From Equation 19.1 we see that the force on a charged particle moving in a
magnetic field has its maximum value when the particle’s motion is perpendicular to
the magnetic field, corresponding to 90°, so that sin 1. The magnitude of
this maximum force has the value
Fmax qvB
B
u
+q
[19.4]
v
:
Also from Equation 19.1, F is zero when v is parallel to B (corresponding to 0°
or 180°), so no magnetic force is exerted on a charged particle when it moves in
the direction of the magnetic field or opposite the field.
Experiments show that the direction of the magnetic force is always perpen:
dicular to both :
v and B, as shown in Figure 19.6 for a positively charged particle.
To determine the direction of the force, we employ the right-hand rule number 1:
:
F
Figure 19.6 The direction of the
magnetic force on a positively
charged particle moving with a velocity :
v in the presence of a magnetic
:
field. When
v is at an angle with
:
respect to B, the magnetic force
is
:
perpendicular to both :
v and B.
1. Point the fingers of your right hand in the direction of the velocity :
v.
:
2. Curl the fingers in the direction of the magnetic field B, moving through
the smallest angle (as in Fig. 19.7).
:
3. Your thumb is now pointing in the direction of the magnetic force F
exerted on a positive charge.
F
:
If the charge is negative rather than positive, the force F is directed opposite that
shown in Figures 19.6 and 19.7. So if q is negative, simply use the right-hand rule
to find the direction for positive q, and then reverse that direction for the negative
charge.
v
Quick Quiz 19.1
A charged particle moves in a straight line through a region of space. Which of the
following answers must be true? (Assume any other fields are negligible.) The magnetic field (a) has a magnitude of zero (b) has a zero component perpendicular to
the particle’s velocity (c) has a zero component parallel to the particle’s velocity in
that region.
Quick Quiz 19.2
The north-pole end of a bar magnet is held near a stationary positively charged
piece of plastic. Is the plastic (a) attracted, (b) repelled, or (c) unaffected by the
magnet?
B
Figure 19.7 Right-hand rule number 1 for determining the direction
of the magnetic force on a positive
v in a
charge moving :with a velocity :
magnetic field B. Point the fingers of
v,
your right hand in the direction of :
and
then curl them in the:direction
:
of B . The magnetic force F points in
the direction of your right thumb.
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Chapter 19
Magnetism
EXAMPLE 19.1 A Proton Traveling in Earth’s Magnetic Field
Goal
Calculate the magnitude and direction of a magnetic force.
Problem A proton moves with a speed of 1.00 10 5 m/s through Earth’s magnetic field, which has a value of 55.0 T
at a particular location. When the proton moves eastward, the magnetic force acting on it is directed straight upward,
and when it moves northward, no magnetic force acts on it. (a) What is the direction of the magnetic field, and
(b) what is the strength of the magnetic force when the proton moves eastward? (c) Calculate the gravitational force
on the proton and compare it with the magnetic force. Compare it also with the electric force if there were an electric field with magnitude E 1.50 10 2 N/C at that location, a common value at Earth’s surface. Note that the mass
of the proton is 1.67 1027 kg.
Strategy The direction of the magnetic field can be found from an application of the right-hand rule, together
with the fact that no force is exerted on the proton when it’s traveling north. Substituting into Equation 19.1 yields
the magnitude of the magnetic field.
Solution
(a) Find the direction of the magnetic field.
No magnetic force acts on the proton when it’s going
north, so the angle such a proton makes with the magnetic field direction must be either 0° or 180°. There:
fore, the magnetic field B must point either north or
south. Now apply the right-hand rule. When the particle
travels east, the magnetic force is directed upward.
Point your thumb in the direction of the force and your
fingers in the direction of the velocity eastward. When
you curl your fingers, they point north, which must
therefore be the direction of the magnetic field.
(b) Find the magnitude of the magnetic force.
Substitute the given values and the charge of a proton
into Equation 19.1. From part (a), the angle between
:
the velocity :
v of the proton and the magnetic field B
is 90.0°.
F qvB sin (1.60 1019 C)(1.00 105 m/s)
(55.0 106 T)sin(90.0°)
(c) Calculate the gravitational force on the proton and
compare it with the magnetic force, and also with the
electric force if E 1.50 102 N/C:
Fgrav mg (1.67 1027 kg)(9.80 m/s2)
8.80 1019 N
1.64 1026 N
Felec qE (1.60 1019 C)(1.50 102 N/C)
2.40 1017 N
Remarks The information regarding a proton moving north was necessary to fix the direction of the magnetic field.
Otherwise, an upward magnetic force on an eastward-moving proton could be caused by a magnetic field pointing
anywhere northeast or northwest. Notice in part (c) the relative strengths of the forces, with the electric force larger
than the magnetic force and both much larger than the gravitational force, all for typical field values found in nature.
Exercise 19.1
Suppose an electron is moving due west in the same magnetic field as in Example 19.1 at a speed of 2.50 105 m/s.
Find the magnitude and direction of the magnetic force on the electron.
Answer 2.20 1018 N, straight up. (Don’t forget, the electron is negatively charged!)
EXAMPLE 19.2 A Proton Moving in a Magnetic Field
Goal Calculate the magnetic force and acceleration when a particle moves at an angle other than ninety degrees to
the field.
Problem A proton moves at 8.00 106 m/s along the x-axis. It enters a region in which there is a magnetic
field of magnitude 2.50 T, directed at an angle of 60.0° with the x-axis and lying in the xy - plane (Fig. 19.8).
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19.4
Magnetic Force on a Current-Carrying Conductor
z
(a) Find the initial magnitude and direction of
the magnetic force on the proton. (b) Calculate
the proton’s initial acceleration.
631
Figure 19.8 (Example
19.2)
:
The magnetic force F on a proton is :
in
the positive z-direction when :
v and B
lie in the xy -plane.
F
Strategy Finding the magnitude and direction
of the magnetic force requires substituting values
into the equation for magnetic force, Equation
19.1, and using the right-hand rule. Applying
Newton’s second law solves part (b).
Solution
(a) Find the magnitude and direction of the
magnetic force on the proton.
Substitute v 8.00 106 m/s, the magnetic field strength
B 2.50 T, the angle, and the charge of a proton into
Equation 19.1:
Apply the right-hand rule number 1 to find the initial
direction of the magnetic force:
+e
y
60°
B
v
x
F qvB sin (1.60 1019 C)(8.00 106 m/s)(2.50 T)(sin 60°)
F 2.77 1012 N
Point the fingers of the right hand in the x-direction
:
v ), and then curl them towards B. The
(the direction of :
thumb points upwards, in the positive z-direction.
(b) Calculate the proton’s initial acceleration.
Substitute the force and the mass of a proton into
Newton’s second law:
ma F
:
(1.67 1027 kg)a 2.77 1012 N
a 1.66 1015 m/s2
Remarks The initial acceleration is also in the positive z-direction. Because the direction of :
v changes, however, the
subsequent direction of the magnetic force also changes. In applying right-hand rule number 1 to find the direction, it
was important to take into consideration the charge. A negatively charged particle accelerates in the opposite direction.
Exercise 19.2
Calculate the acceleration of an electron that moves through the same magnetic field as in Example 19.2, at the same
velocity as the proton. The mass of an electron is 9.11 1031 kg.
Answer 3.04 1018 m/s2 in the negative z-direction
If a magnetic field exerts a force on a single charged particle when it moves
through a magnetic field, it should be no surprise that magnetic forces are exerted
on a current-carrying wire, as well (see Fig. 19.9). This follows from the fact that
the current is a collection of many charged particles in motion; hence, the resultant force on the wire is due to the sum of the individual forces on the charged
particles. The force on the particles is transmitted to the “bulk” of the wire
through collisions with the atoms making up the wire.
Some explanation is in order concerning notation in many of the figures. To
:
indicate the direction of B, we use the following conventions:
:
If B is directed into the page, as in Figure 19.10 (page 632), we use a series
:
of blue crosses, representing the tails of arrows. If B is directed out of
:
the page, we use a series of blue dots, representing the tips of arrows. If B lies
in the plane of the page, we use a series of blue field lines with arrowheads.
Courtesy of Henry Leap and Jim Lehman
19.4 MAGNETIC FORCE ON A CURRENT-CARRYING
CONDUCTOR
Figure 19.9 This apparatus
demonstrates the force on a currentcarrying conductor in an external
magnetic field. Why does the bar
swing away from the magnet after the
switch is closed?
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Chapter 19
Magnetism
Figure 19.10 A segment of a flexible vertical wire partially stretched
between the poles of a magnet, with
the field (blue crosses) directed into
the page. (a) When there is no current in the wire, it remains vertical.
(b) When the current is upward, the
wire deflects to the left. (c) When the
current is downward, the wire
deflects to the right.
Bin
×
×
×
×
×
×
×
×
×
×
×
×
I=0
(a)
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
Bin
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
Bin
×
×
×
×
×
×
×
×
×
×
×
×
I
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
I
(b)
(c)
Fmax
×
×
A
Bin
×
×
×
×
×
×
×
vd
q +
×
×
×
Figure 19.11 A section of a
wire containing moving :
charges in an
external magnetic field B.
TIP 19.2 The Origin of the
Magnetic Force on a Wire
When a magnetic field is applied at
some angle to a wire carrying a current, a magnetic force is exerted on
each moving charge in the wire. The
total magnetic force on the wire is the
sum of all the magnetic forces on the
individual charges producing the
current.
The force on a current-carrying conductor can be demonstrated by hanging a
wire between the poles of a magnet, as in Figure 19.10. In this figure, the magnetic
field is directed into the page and covers the region within the shaded area. The
wire deflects to the right or left when it carries a current.
We can quantify this discussion by considering a straight segment of wire of
length and cross-sectional area A carrying current I in a uniform external mag:
netic field B, as in Figure 19.11. We assume that the magnetic field is perpendicular to the wire and is directed into the page. A force of magnitude Fmax qvd B is
exerted on each charge carrier in the wire, where vd is the drift velocity of the
charge. To find the total force on the wire, we multiply the force on one charge
carrier by the number of carriers in the segment. Because the volume of the segment is A, the number of carriers is nA, where n is the number of carriers per
unit volume. Hence, the magnitude of the total magnetic force on the wire of
length is as follows:
Total force force on each charge carrier total number of carriers
Fmax (qvdB)(nA)
From Chapter 17, however, we know that the current in the wire is given by
I nqvd A, so
Fmax BI
This equation can be used only when the current and the magnetic field are at
right angles to each other.
If the wire is not perpendicular to the field, but is at some arbitrary angle, as in
Figure 19.12, the magnitude of the magnetic force on the wire is
B
u
F BI sin I
Figure 19.12 A wire carrying a
current I in the presence
of an exter:
nal magnetic field B that makes an
angle with the wire.
Current
S
Paper
cone
B
N
F
S Voice
coil
Figure 19.13
loudspeaker.
A diagram of a
[19.5]
:
[19.6]
where is the angle between B and the direction of the current. The direction of
this force can be obtained by the use of right-hand rule number 1. However, in
this case you must place your fingers in the direction of the positive current I,
v . The current, naturally, is made up of charges
rather than in the direction of :
moving at some velocity, so this really isn’t a separate rule. In Figure 19.12, the
direction of the magnetic force on the wire is out of the page.
Finally, when the current is either in the direction of the field or opposite the
direction of the field, the magnetic force on the wire is zero.
The fact that a magnetic force acts on a current-carrying wire in a magnetic
field is the operating principle of most speakers in sound systems. One speaker
design, shown in Figure 19.13, consists of a coil of wire called the voice coil, a flexible paper cone that acts as the speaker, and a permanent magnet. The coil of wire
surrounding the north pole of the magnet is shaped so that the magnetic field
44920_19_p624-659 1/5/05 1:47 PM Page 633
19.4
Magnetic Force on a Current-Carrying Conductor
lines are directed radially outward from the coil’s axis. When an electrical signal is
sent to the coil, producing a current in the coil as in Figure 19.13, a magnetic
force to the left acts on the coil. (This can be seen by applying right-hand rule
number 1 to each turn of wire.) When the current reverses direction, as it would
for a current that varied sinusoidally, the magnetic force on the coil also reverses
direction, and the cone, which is attached to the coil, accelerates to the right. An
alternating current through the coil causes an alternating force on the coil, which
results in vibrations of the cone. The vibrating cone creates sound waves as it
pushes and pulls on the air in front of it. In this way, a 1-kHz electrical signal is
converted to a 1-kHz sound wave.
An unusual application of the force on a current-carrying conductor is illustrated by the electromagnetic pump shown in Figure 19.14. Artificial hearts
require a pump to keep the blood flowing, and kidney dialysis machines also
require a pump to assist the heart in pumping blood that is to be cleansed.
Ordinary mechanical pumps create problems because they damage the blood
cells as they move through the pump. The mechanism shown in the figure
has demonstrated some promise in such applications. A magnetic field is
established across a segment of the tube containing the blood, flowing in the
direction of the velocity :
v . An electric current passing through the fluid in the
direction shown has a magnetic force acting on it in the direction of :
v , as
applying the right-hand rule shows. This force helps to keep the blood in
motion.
633
A P P L I C AT I O N
Loudspeaker Operation
B
v
I
Figure 19.14 A simple electromagnetic pump has no moving parts to
damage a conducting fluid, such as
blood, passing through. Application
of the right-hand rule #1 (right fingers in the direction of the current
I,
:
curl them in the direction of B,
thumb points in the direction of the
force) shows that the force on the
current-carrying segment of the fluid
is in the direction of the velocity.
A P P L I C AT I O N
Electromagnetic Pumps for
Artificial Hearts and Kidneys
Applying Physics 19.2 Lightning Strikes
In a lightning strike there is a rapid movement of
negative charge from a cloud to the ground. In what
direction is a lightning strike deflected by Earth’s
magnetic field?
Explanation The downward flow of negative charge
in a lightning stroke is equivalent to a current moving
upward. Consequently, we have an upward- moving
current in a northward-directed magnetic field.
According to right-hand rule number 1, the lightning
strike would be deflected toward the west.
EXAMPLE 19.3 A Current-Carrying Wire in Earth’s Magnetic Field
Goal
Compare the magnetic force on a current-carrying wire with the gravitational force exerted on the wire.
Problem A wire carries a current of 22.0 A from west to east. Assume that at this location the magnetic field of
Earth is horizontal and directed from south to north and that it has a magnitude of 0.500 104 T. (a) Find the
magnitude and direction of the magnetic force on a 36.0-m length of wire. (b) Calculate the gravitational force on
the same length of wire if it’s made of copper and has a cross-sectional area of 2.50 106 m2.
Solution
(a) Calculate the magnetic force on the wire.
Substitute into Equation 19.6, using the fact that the
magnetic field and the current are at right angles to
each other:
F BI sin (0.500 104 T)(22.0 A)(36.0 m) sin 90.0°
Apply right-hand rule number 1 to find the direction of
the magnetic force:
With the fingers of your right hand pointing west to east
in the direction of the current, curl them north in the
direction of the magnetic field. Your thumb points
upward.
3.96 102 N
(b) Calculate the gravitational force on the wire segment.
First, obtain the mass of the wire from the density of copper, the length, and cross-sectional area of the wire:
m V (A)
(8.92 103 kg/m3)(2.50 106 m2 36.0 m)
0.803 kg
44920_19_p624-659 1/5/05 1:48 PM Page 634
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Chapter 19
Magnetism
Fgrav mg 7.87 N
To get the gravitational force, multiply the mass by the
acceleration of gravity:
Remarks This calculation demonstrates that under normal circumstances, the gravitational force on a current-carrying
conductor is much greater than the magnetic force due to the Earth’s magnetic field.
Exercise 19.3
What current would make the magnetic force in the example equal in magnitude to the gravitational force?
Answer 4.37 103 A, a large current that would very rapidly melt the wire.
19.5 TORQUE ON A CURRENT LOOP
AND ELECTRIC MOTORS
In the preceding section we showed how a magnetic force is exerted on a currentcarrying conductor when the conductor is placed in an external magnetic field.
With this as a starting point, we now show that a torque is exerted on a current
loop placed in a magnetic field. The results of this analysis will be of great practical
value when we discuss generators and motors in Chapter 20.
Consider a rectangular loop carrying current I in the presence of an external
uniform magnetic field in the plane of the loop, as shown in Figure 19.15a. The
forces on the sides of length a are zero because these wires are parallel to the field.
The magnitudes of the magnetic forces on the sides of length b, however, are
F1 F2 BIb
:
The direction of F1, the force on the left side of the loop, is out of the page, and
:
that of F2, the force on the right side of the loop, is into the page. If we view the
loop from the side, as in Figure 19.15b, the forces are directed as shown. If we
assume that the loop is pivoted so that it can rotate about point O, we see that
these two forces produce a torque about O that rotates the loop clockwise. The
magnitude of this torque, max, is
max F1
a
a
a
a
F 2 (BIb) (BIb) BIab
2
2
2
2
where the moment arm about O is a/2 for both forces. Because the area of the
loop is A ab, the torque can be expressed as
max BIA
[19.7]
This result is valid only when the magnetic field is parallel to the plane of the loop,
as in Figure 19.15b. If the field makes an angle with a line perpendicular to the
plane of the loop, as in Figure 19.15c, the moment arm for each force is given by
(a/2) sin . An analysis similar to the previous gives, for the magnitude of the
torque,
I
Figure 19.15 (a) Top view of a rectangular
loop in a uniform magnetic
:
field B. No magnetic forces act on
:
the sides of length a parallel to B, but
forces do act on the sides of length b.
(b) A side view of the rectangular
:
:
loop shows that the forces F1 and F2
on the sides of length b create a
torque that tends:to twist the loop
clockwise. (c) If B is at an angle with a line perpendicular to the plane
of the loop, the torque is given by
BIA sin .
B
F1
F1
b
a
2
a/2
O
B
(a)
a
(b)
u
B
a sin u
2
B
F2
(c)
F2
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19.5
635
Torque on a Current Loop and Electric Motors
BIA sin [19.8]
This result shows that the torque has the maximum value BIA when the field is parallel to the plane of the loop ( 90°) and is zero when the field is perpendicular
to the plane of the loop ( 0). As seen in Figure 19.15c, the loop tends to rotate
to smaller values of (so that the normal to the plane of the loop rotates toward
the direction of the magnetic field).
Although the foregoing analysis was for a rectangular loop, a more general
derivation shows that Equation 19.8 applies regardless of the shape of the loop.
Further, the torque on a coil with N turns is
BIAN sin [19.9a]
:
The quantity IAN is defined as the magnitude of a vector called the magnetic
:
moment of the coil. The vector always points perpendicular to the plane of the
loop(s). The angle in Equations 19.8 and 19.9 lies between the directions of the
:
:
magnetic moment and the magnetic field B. The magnetic torque can then be
written
B sin [19.9b]
Quick Quiz 19.3
A square and a circular loop with the same area lie in the xy-plane, where there is a
:
uniform magnetic field B pointing at some angle with respect to the positive
z-direction. Each loop carries the same current, in the same direction. Which magnetic torque is larger? (a) the torque on the square loop (b) the torque on the circular loop (c) the torques are the same (d) more information is needed
EXAMPLE 19.4 The Torque on a Circular Loop in a Magnetic Field
Goal
B
Problem A circular wire loop of radius 1.00 m is placed
in a magnetic field of magnitude 0.500 T. The normal to
the plane of the loop makes an angle of 30.0° with the
magnetic field (Fig. 19.16a). The current in the loop is
2.00 A in the direction shown. (a) Find the magnetic
moment of the loop and the magnitude of the torque at
this instant. (b) The same current is carried by the rectangular 2.00-m by 3.00-m coil with three loops shown in Figure
19.16b. Find the magnetic moment of the coil and the
magnitude of the torque acting on the coil at that instant.
B
30°
30°
2.00 m
r 1.00 m
x
y
3.00 m
y
x
I
I
(a)
(b)
z
Strategy For each part, we just have to calculate the
area, use it in the calculation of the magnetic moment,
and multiply the result by B sin . Altogether, this
amounts to substituting values into Equation 19.9.
Solution
(a) Find the magnetic moment of the circular loop
and the magnetic torque exerted on it.
z
z
Calculate a magnetic torque on a loop of current.
B
u
2.00 m
x
3.00 m
y
I
(c)
Figure 19.16 (Example 19.4)
(a) A circular current loop in
:
an external magnetic field B. (b) A rectangular current loop in the
same field. (c) (Exercise 19.4)
First, calculate the enclosed area of the circular loop:
A r 2 (1.00 m)2 3.14 m2
Calculate the magnetic moment of the loop:
IAN (2.00 A)(3.14 m2)(1) 6.28 Am2
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Chapter 19
Magnetism
Now substitute values for the magnetic moment,
magnetic field, and into Equation 19.9b:
B sin (6.28 A m2)(0.500 T)(sin 30.0°)
1.57 Nm
(b) Find the magnetic moment of the rectangular coil
and the magnetic torque exerted on it.
Calculate the area of the coil:
A L H (2.00 m)(3.00 m) 6.00 m2
Calculate the magnetic moment of the coil:
IAN (2.00 A)(6.00 m2)(3) 36.0 Am2
Substitute values into Equation 19.9b:
B sin (0.500 T)(36.0 A m2)(sin 30.0°)
9.00 Nm
Remarks In calculating a magnetic torque, it’s not strictly necessary to calculate the magnetic moment. Instead,
Equation 19.9a can be used directly.
Exercise 19.4
Suppose a right triangular coil with base of 2.00 m and height 3.00 m having two loops carries a current of 2.00 A as
shown in Figure 19.16c. Find the magnetic moment and the torque on the coil. The magnetic field is again 0.500 T
and makes an angle of 30.0° with respect to the normal direction.
Answer 12.0 A m2, 3.00 N m
Electric Motors
A P P L I C AT I O N
Electric Motors
It’s hard to imagine life in the 21st century without electric motors. Some appliances that contain motors include computer disk drives, CD players, VCR and
DVD players, food processors and blenders, car starters, furnaces, and air conditioners. The motors convert electrical energy to kinetic energy of rotation, and
consist basically of a rigid current-carrying loop that rotates when placed in the
field of a magnet.
As we have just seen (Fig. 19.15), the torque on such a loop rotates the loop to
smaller values of until the torque becomes zero, when the magnetic field is
perpendicular to the plane of the loop and 0. If the loop turns past this
angle and the current remains in the direction shown in the figure, the torque
reverses direction and turns the loop in the opposite direction — that is, counterclockwise. To overcome this difficulty and provide continuous rotation in one
direction, the current in the loop must periodically reverse direction. In
alternating current (AC) motors, such a reversal occurs naturally 120 times
each second. In direct current (DC) motors, the reversal is accomplished mechanically with split-ring contacts (commutators) and brushes, as shown in Active
Figure 19.17.
N
Axis of
rotation
of loop
I
B
S
ACTIVE FIGURE 19.17
Simplified sketch of a DC electric motor.
Brushes
Log into PhysicsNow at www.cp7e.com and go
to Active Figure 19.17, where you can adjust
the speed of rotation and the strength of the
field and see the effects on the generated emf.
–
+
DC Power
source
Split-ring
commutators
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Motion of a Charged Particle in a Magnetic Field
Figure 19.18 The Honda Insight combines a three-cylinder gasoline automobile
engine with a thin electric motor for
improved efficiency and added power
when needed. The electric motor (circled)
also acts as a generator during braking or
coasting downhill to recharge the batteries,
with the result that they never need to be
recharged by the owner.
Image not Available
B in
19.6 MOTION OF A CHARGED PARTICLE
IN A MAGNETIC FIELD
Consider the case of a positively charged particle moving in a uniform magnetic
field so that the direction of the particle’s velocity is perpendicular to the field, as in
:
:
Active Figure 19.19. The label Bin and the crosses indicate that B is directed into
the page. Application of the right-hand rule at point P shows that the direction of
:
the magnetic force F at that location is upward. The force causes the particle to
alter its direction of travel and to follow a curved path. Application of the right-hand
rule at any point shows that the magnetic force is always directed toward the center
of the circular path; therefore, the magnetic force causes a centripetal accelera:
tion, which changes only the direction of :
v and not its magnitude. Because F produces the centripetal acceleration, we can equate its magnitude, qvB in this case,
to the mass of the particle multiplied by the centripetal acceleration v 2/r. From
Newton’s second law, we find that
F qvB v
q
F
r
F
+
v
q
F
P
v
+
Although actual motors contain many current loops and commutators, for simplicity Active Figure 19.17 shows only a single loop and a single set of split-ring
contacts rigidly attached to and rotating with the loop. Electrical stationary contacts called brushes are maintained in electrical contact with the rotating split ring.
These brushes are usually made of graphite, because graphite is a good electrical
conductor as well as a good lubricant. Just as the loop becomes perpendicular to
the magnetic field and the torque becomes zero, inertia carries the loop forward
in the clockwise direction and the brushes cross the gaps in the ring, causing
the loop current to reverse its direction. This provides another pulse of torque in
the clockwise direction for another 180°, the current reverses, and the process
repeats itself. Figure 19.18 shows a modern motor used to power a hybrid
gas – electric car.
+
Courtesy of American Honda Motor Co., Inc.
19.6
q
ACTIVE FIGURE 19.19
When the velocity of a charged particle is perpendicular to a uniform
magnetic field, the particle moves in
a circle
whose plane is perpendicular
:
to B, which is directed into the page.
(The crosses represent the tails of the
magnetic
field vectors.) The magnetic
:
force F on the charge is always directed toward the center of the circle.
Log into PhysicsNow at
www.cp7e.com and go to Active
Figure 19.19, where you can adjust
the mass, speed, charge of the particle, and the magnitude of the magnetic field, and observe the resulting
circular motion.
y
+q
mv 2
r
Helical
path
+
which gives
B
r
mv
qB
[19.10]
This equation says that the radius of the path is proportional to the momentum
mv of the particle and is inversely proportional to the charge and the magnetic
field. Equation 19.10 is often called the cyclotron equation, because it’s used in the
design of these instruments (popularly known as atom smashers).
If the initial direction of the velocity of the charged particle is not perpendicular to the magnetic field, as shown in Active Figure 19.20, the path followed by the
particle is a spiral (called a helix) along the magnetic field lines.
z
x
ACTIVE FIGURE 19.20
A charged particle having a velocity directed at an angle with a uniform magnetic field moves in a helical path.
Log into PhysicsNow at www.cp7e.com
and go to Active Figure 19.20, where
you can adjust the x-component of
the velocity of the particle and observe
the resulting helical motion.
44920_19_p624-659 1/5/05 1:48 PM Page 638
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Chapter 19
Magnetism
Applying Physics 19.3 Trapping Charges
Storing charged particles is important for a variety of
applications. Suppose a uniform magnetic field exists
in a finite region of space. Can a charged particle
be injected into this region from the outside and
remain trapped in the region by magnetic force
alone?
Explanation It’s best to consider separately the
components of the particle velocity parallel and
perpendicular to the field lines in the region. There is
no magnetic force on the particle associated with the
velocity component parallel to the field lines, so that
velocity component remains unchanged.
Now consider the component of velocity perpendicular to the field lines. This component will result
in a magnetic force that is perpendicular to both the
field lines and the velocity component itself. The
path of a particle for which the force is always perpendicular to the velocity is a circle. The particle therefore follows a circular arc and exits the field on the
other side of the circle, as shown in Figure 19.21 for a
particle with constant kinetic energy. On the other
hand, a particle can become trapped if it loses some
kinetic energy in a collision after entering the field, as
in Active Figure 19.20.
Magnetic field
region (out of page)
v
F
Particle
motion
Figure 19.21 (Applying Physics 19.3)
Particles can be injected and contained if, in addition to the magnetic field, electrostatic fields are
involved. These fields are used in the Penning trap. With
these devices, it’s possible to store charged particles for
extended periods. Such traps are useful, for example,
in the storage of antimatter, which disintegrates completely on contact with ordinary matter.
Quick Quiz 19.4
As a charged particle moves freely in a circular path in the presence of a constant
magnetic field applied perpendicular to the particle’s velocity, its kinetic energy
(a) remains constant, (b) increases, or (c) decreases.
EXAMPLE 19.5 The Mass Spectrometer: Identifying Particles
Goal
Use the cyclotron equation to identify a particle.
Problem A charged particle enters the magnetic field of a mass spectrometer at a speed of 1.79 106 m/s. It subsequently moves in a circular orbit with a radius of 16.0 cm in a uniform magnetic field of magnitude 0.350 T having a
direction perpendicular to the particle’s velocity. Find the particle’s mass-to-charge ratio, and identify it based on the
table on page 639.
Strategy After finding the mass-to-charge ratio with Equation 19.10, compare it to the values in the table, identifying the particle.
Solution
Write the cyclotron equation:
Solve this equation for the mass divided by the charge,
m/q and substitute values:
r
mv
qB
m
rB
(0.160 m)(0.350 T)
kg
3.13 108
q
v
1.79 106 m/s
C
44920_19_p624-659 1/5/05 1:48 PM Page 639
19.6
Identify the particle from the table. All particles are
completely ionized.
639
Motion of a Charged Particle in a Magnetic Field
particle
Hydrogen
Deuterium
Tritium
Helium-3
m (kg)
q (C)
m/q (kg/C)
1.67 1027
3.35 1027
5.01 1027
5.01 1027
1.60 1019
1.60 1019
1.60 1019
3.20 1019
1.04 108
2.09 108
3.13 108
1.57 108
The particle is tritium.
Remarks The mass spectrometer is an important tool in both chemistry and physics. Unknown chemicals can be
heated and ionized, and the resulting particles passed through the mass spectrometer and subsequently identified.
Exercise 19.5
Suppose a second charged particle enters the mass spectrometer at the same speed as the particle in Example 19.5. If
it travels in a circle with radius 10.7 cm, find the mass-to-charge ratio and identify the particle from the table above.
Answer 2.09 108 kg/C; deuterium
EXAMPLE 19.6 The Mass Spectrometer: Separating Isotopes
Goal
Apply the cyclotron equation to the process of separating isotopes.
A P P L I C AT I O N
Mass Spectrometers
Problem Two singly ionized atoms move out of a slit at point S in Figure 19.22 and
into a magnetic field of magnitude 0.100 T pointing into the page. Each has a speed of
1.00 106 m/s. The nucleus of the first atom contains one proton and has a mass of
1.67 1027 kg, while the nucleus of the second atom contains a proton and a neutron
and has a mass of 3.34 1027 kg. Atoms with the same number of protons in the
nucleus but different masses are called isotopes. The two isotopes here are hydrogen
and deuterium. Find their distance of separation when they strike a photographic
plate at P.
Strategy Apply the cyclotron equation to each atom, finding the radius of the path of
each. Double the radii to find the path diameters, and then find their difference.
Solution
Use Equation 19.10 to find the radius of the
circular path followed by the lighter isotope, hydrogen.
r1 Use the same equation to calculate the radius of the
path of deuterium, the heavier isotope:
r2 Multiply the radii by 2 to find the diameters, and take the
difference, getting the separation x between the isotopes:
Bin
P
S
2r1
2r2
Figure 19.22 (Example 19.6)
Two isotopes leave the slit at point
S and travel in different circular
paths before striking a photographic plate at P.
m 1v
(1.67 1027 kg)(1.00 106 m/s)
qB
(1.60 1019 C)(0.100 T)
0.104 m
m 2v
(3.34 1027 kg)(1.00 106 m/s)
qB
(1.60 1019 C)(0.100 T)
0.209 m
x 2r 2 2r 1 0.210 m
Remarks During World War II, mass spectrometers were used to separate the highly radioactive uranium isotope
U-235 from its far more common isotope, U-238.
Exercise 19.6
Use the same mass spectrometer as in Example 19.6 to find the separation between two isotopes of helium: normal
helium-4, which has a nucleus consisting of two protons and two neutrons, and helium-3, which has two protons and
a single neutron. Assume both nuclei, doubly ionized (having a charge of 2e 3.20 1019 C), enter the field at
1.00 10 6 m/s. The helium-4 nucleus has a mass of 6.64 1027 kg, and the helium-3 nucleus has a mass of
5.01 1027 kg.
Answer 0.102 m
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Chapter 19
Magnetism
I
I = 0
B
(a)
(b)
North Wind Picture Archives
ACTIVE FIGURE 19.23
(a) When there is no current in the vertical wire, all compass needles point in the same direction.
(b) When the wire carries a strong :
current, the compass needles deflect in directions tangent to the
circle, pointing in the direction of B, due to the current.
Log into PhysicsNow at www.cp7e.com and go to Active Figure 19.23, where you can change the value
of the current and see the effect on the compasses.
19.7 MAGNETIC FIELD OF A LONG, STRAIGHT
WIRE AND AMPÈRE’S LAW
HANS CHRISTIAN OERSTED
(1777–1851), Danish Physicist and
Chemist
Oersted is best known for observing that a
compass needle deflects when placed near
a wire carrying a current. This important
discovery was the first evidence of the connection between electric and magnetic
phenomena. Oersted was also the first to
prepare pure aluminum.
TIP 19.3
Hand!
During a lecture demonstration in 1819, the Danish scientist Hans Oersted
(1777 – 1851) found that an electric current in a wire deflected a nearby compass needle. This momentous discovery, linking a magnetic field with an electric current for
the first time, was the beginning of our understanding of the origin of magnetism.
A simple experiment first carried out by Oersted in 1820 clearly demonstrates
that a current-carrying conductor produces a magnetic field. In this experiment,
several compass needles are placed in a horizontal plane near a long vertical wire,
as in Active Figure 19.23a. When there is no current in the wire, all needles point in
the same direction (that of Earth’s field), as one would expect. However, when the
wire carries a strong, steady current, the needles all deflect in directions tangent to
the circle, as in Active Figure 19.23b. These observations show that the direction of
:
B is consistent with the following convenient rule, right-hand rule number 2:
Raise Your Right
We have introduced two right-hand
rules in this chapter. Be sure to use
only your right hand when applying
these rules.
Point the thumb of your right hand along a wire in the direction of positive
current, as in Figure 19.24a. Your fingers then naturally curl in the direction
:
of the magnetic field B.
When the current is reversed, the filings in Figure 19.24b also reverse.
Figure 19.24 (a) Right-hand rule
number 2 for determining the direction of the magnetic field due to a
long, straight wire carrying a current.
Note that the magnetic field lines form
circles around the wire. (b) Circular
magnetic field lines surrounding a
current-carrying wire, displayed by
iron filings.
© Richard Megna, Fundamental Photographs
I
a
B
(a)
(b)
44920_19_p624-659 1/5/05 1:48 PM Page 641
19.7
:
641
Magnetic Field of a Long, Straight Wire and Ampère’s Law
:
Because the filings point in the direction of B, we conclude that the lines of B
:
form circles about the wire. By symmetry, the magnitude of B is the same everywhere on a circular path centered on the wire and lying in a plane perpendicular
to the wire. By varying the current and distance from the wire, it can be experi:
mentally determined that B is proportional to the current and inversely proportional to the distance from the wire. These observations lead to a mathematical
expression for the strength of the magnetic field due to the current I in a long,
straight wire:
B
0I
2 r
Magnetic field due to a long,
straight wire
[19.11]
The proportionality constant 0, called the permeability of free space, has the
value
0 4 107 Tm/A
I
Arbitrary
closed path
[19.12]
Ampère’s Law and a Long, Straight Wire
Equation 19.11 enables us to calculate the magnetic field due to a long, straight
wire carrying a current. A general procedure for deriving such equations was proposed by the French scientist André-Marie Ampère (1775 – 1836); it provides a
relation between the current in an arbitrarily shaped wire and the magnetic field
produced by the wire.
Consider an arbitrary closed path surrounding a current as in Figure 19.25. The
path consists of many short segments, each of length . Multiply one of these
lengths by the component of the magnetic field parallel to that segment, where
the product is labeled B,. According to Ampère, the sum of all such products
over the closed path is equal to 0 times the net current I that passes through the
surface bounded by the closed path. This statement, known as Ampère’s circuital
law, can be written
B, 0I
B ||
B
Figure 19.25 An arbitrary closed
path around a current is used to calculate the magnetic field of the current by the use of Ampère’s rule.
[19.13]
B, B, B(2r) 0I
Dividing both sides by the circumference 2r, we obtain
B
0I
2r
This is identical to Equation 19.11, which is the magnetic field due to the current
I in a long, straight wire.
Leonard de Selva/CORBIS
:
where B, is the component of B parallel to the segment of length and B,
means that we take the sum over all the products B, around the closed path.
Ampère’s law is the fundamental law describing how electric currents create magnetic fields in the surrounding empty space.
We can use Ampère’s circuital law to derive the magnetic field due to a long,
straight wire carrying a current I. As discussed earlier, each of the magnetic field
lines of this configuration forms a circle with the wire at its centers. The magnetic
field is tangent to this circle at every point, and its magnitude has the same value B
over the entire circumference of a circle of radius r, so that B, B , as shown in
Figure 19.26 (page 642). In calculating the sum B, over the circular path, notice that B, can be removed from the sum (because it has the same value B for
each element on the circle). Equation 19.13 then gives
ANDRÉ-MARIE AMPÈRE
(1775 – 1836)
Ampère, a Frenchman, is credited with the
discovery of electromagnetism — the relationship between electric currents and
magnetic fields. Ampère’s genius, particularly in mathematics, became evident by
the age of 12, but his personal life was
filled with tragedy. His father, a wealthy
city official, was guillotined during the
French Revolution, and his wife died
young, in 1803. Ampère died of pneumonia at the age of 61. His judgment of his
life is clear from the epitaph he chose for his
gravestone: Tandem felix (Happy at last).
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Chapter 19
Magnetism
Ampère’s circuital law provides an elegant and simple method for calculating
the magnetic fields of highly symmetric current configurations. However, it can’t
easily be used to calculate magnetic fields for complex current configurations that
lack symmetry. In addition, Ampère’s circuital law in this form is valid only when
the currents and fields don’t change with time.
r
B
Figure 19.26 A closed circular path
of radius r around a long, straight
current-carrying wire is used to calculate
the magnetic field set up by the wire.
EXAMPLE 19.7 The Magnetic Field of a Long Wire
Goal
Calculate the magnetic field of a long, straight wire and the force that the field exerts on a particle.
Problem A long, straight wire carries a current of 5.00 A. At one
instant, a proton, 4.00 mm from the wire, travels at 1.50 103 m/s
parallel to the wire and in the same direction as the current (Fig.
19.27). (a) Find the magnitude and direction of the magnetic field
created by the wire. (b) Find the magnitude and direction of the
magnetic force the wire’s magnetic field exerts on the proton.
Figure 19.27 (Example 19.7)
The magnetic field due to the current is into the page at the location
of the proton, and the magnetic
force on the proton is to the left.
v
I
+
F
B
Strategy First use Equation 19.11 to find the magnitude of the
magnetic field at the given point. Use right-hand rule number 2
to find the direction of the field. Finally, substitute into Equation
19.1, computing the magnetic force on the proton.
Solution
(a) Find the magnitude and direction of the wire’s
magnetic field.
Use Equation 19.11 to calculate the magnitude of the
magnetic field 4.00 mm from the wire:
B
0I
(4 107 Tm/A)(5.00 A)
2r
2(4.00 103 m)
2.50 104 T
Apply right-hand rule number 2 to find the direction of
:
the magnetic field B :
With the right thumb pointing in the direction of the
current in Figure 19.27, the fingers curl into the page at
the location of the proton. The angle between :
v and
:
B is therefore 90°.
(b) Compute the magnetic force exerted by the wire on
the proton.
Substitute into Equation 19.1, which gives the
magnitude of the magnetic force on a charged
particle:
F qvB sin (1.60 1019 C)(1.50 103 m/s)
Find the direction of the magnetic force with right-hand
rule number 1:
Point your right fingers in the direction of :
v , curling
:
them into the page toward B. Your thumb points to the
left , which is the direction of the magnetic force.
(2.50 104 T)(sin 90°)
6.00 1020 N
Remarks The location of the proton is important. On the left-hand side, the wire’s magnetic field points outward,
and the magnetic force on the proton is to the right.
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19.8
Magnetic Force between Two Parallel Conductors
643
Exercise 19.7
Find (a) the magnetic field created by the wire and (b) the magnetic force on a helium-3 nucleus located 7.50 mm to
the left of the wire in Figure 19.27, traveling 2.50 103 m/s opposite the direction of the current. (See the data
table presented in Example 19.5 on page 639).
Answers (a) 1.33 104 T (b) 1.07 1019 N, directed to the left in Figure 19.27.
19.8 MAGNETIC FORCE BETWEEN TWO
PARALLEL CONDUCTORS
As we have seen, a magnetic force acts on a current-carrying conductor when the
conductor is placed in an external magnetic field. Because a conductor carrying
a current creates a magnetic field around itself, it is easy to understand that two
current-carrying wires placed close together exert magnetic forces on each other.
Consider two long, straight, parallel wires separated by the distance d and carrying currents I 1 and I 2 in the same direction, as shown in Active Figure 19.28.
Wire 1 is directly above wire 2. What’s the magnetic force on one wire due to a
magnetic field set up by the other wire?
In this calculation, we are finding the force on wire 1 due to the magnetic field
:
:
of wire 2. The current I2, sets up magnetic field B2 at wire 1. The direction of B2 is
perpendicular to the wire, as shown in the figure. Using Equation 19.11, we find
that the magnitude of this magnetic field is
B2 0I 2
2d
According to Equation 19.5, the magnitude of the magnetic force on wire 1 in
:
the presence of field B2 due to I2 is
F1 B 2I1 2Id I 2IId 0 2
1
I1
B2
F1
2
a
I2
d
ACTIVE FIGURE 19.28
Two parallel wires, oriented vertically,
carry steady currents and:exert forces
on each other. The field B2 at wire 1
due to wire 2 produces a force on
wire 1 given by F 1 B 2 I 1. The force
is attractive if the currents have the
same direction, as shown, and repulsive if the two currents have opposite
directions.
0 1 2
We can rewrite this relationship in terms of the force per unit length:
F1
II
012
2d
1
[19.14]
Log into PhysicsNow at
www.cp7e.com and go to Active
Figure 19.28, where you can adjust
the currents in the wires and the
distance between them, and see the
effect on the force.
:
The direction of F1 is downward, toward wire 2, as indicated by right-hand rule
:
number 1. This calculation is completely symmetric, which means that the force F2
:
on wire 2 is equal to and opposite F1, as expected from Newton’s third law of
action – reaction.
We have shown that parallel conductors carrying currents in the same direction
attract each other. You should use the approach indicated by Figure 19.28 and the
steps leading to Equation 19.14 to show that parallel conductors carrying currents
in opposite directions repel each other.
The force between two parallel wires carrying a current is used to define the SI
unit of current, the ampere (A), as follows:
If two long, parallel wires 1 m apart carry the same current, and the magnetic
force per unit length on each wire is 2 107 N/m, then the current is
defined to be 1 A.
Definition
of the ampere
Definition
of the coulomb
The SI unit of charge, the coulomb (C), can now be defined in terms of the ampere
as follows:
If a conductor carries a steady current of 1 A, then the quantity of charge
that flows through any cross section in 1 s is 1 C.
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Chapter 19
Magnetism
Quick Quiz 19.5
If, in Figure 19.28, I 1 2 A and I2 6 A, which of the following is true? (a) F 1 3F 2 (b) F 1 F 2 or (c) F 1 F 2 /3
EXAMPLE 19.8 Levitating a Wire
Goal
Calculate the magnetic force of one current-carrying wire on a parallel current-carrying wire.
Problem Two wires, each having a weight per unit length of 1.00 104 N/m, are parallel with one directly above
the other. Assume that the wires carry currents that are equal in magnitude and opposite in direction. The wires are
0.10 m apart, and the sum of the magnetic force and gravitational force on the upper wire is zero. Find the current
in the wires. (Neglect Earth’s magnetic field.)
Strategy The upper wire must be in equilibrium under the forces of magnetic repulsion and gravity. Set the sum of
the forces equal to zero and solve for the unknown current, I.
Solution
Set the sum of the forces equal to zero, and substitute
the appropriate expressions. Notice that the magnetic
force between the wires is repulsive.
mg The currents are equal, so I1 I 2 I. Make these substitutions and solve for I 2.
0I 2
mg
2d
Substitute given values, finding I 2, then take the square
root. Notice that the weight per unit length, mg/, is
given.
I2 :
:
F grav F mag 0
0I 1I 2
0
2d
:
I2 (2d)(mg/)
0
(2 0.100 m)(1.00 104 N/m)
50.0 A2
(4 107 Tm)
I 7.07 A
Remark Exercise 19.3 showed that using the Earth’s magnetic field to levitate a wire required extremely large currents. Currents in wires can create much stronger magnetic fields than Earth’s in regions near the wire.
Exercise 19.8
If the current in each wire is doubled, how far apart should the wires be placed if the magnitudes of the gravitational
and magnetic forces on the upper wire are to be equal?
Answer 0.400 m
19.9 MAGNETIC FIELDS OF CURRENT LOOPS
AND SOLENOIDS
∆x1
B1
I
∆x2
Figure 19.29 All segments of the
current loop produce a magnetic
field at the center of the loop,
directed out of the page.
The strength of the magnetic field set up by a piece of wire carrying a current can
be enhanced at a specific location if the wire is formed into a loop. You can understand this by considering the effect of several small segments of the current loop,
as in Figure 19.29. The small segment at the top of the loop, labeled x1, produces
a magnetic field of magnitude B 1 at the loop’s center, directed out of the page.
:
The direction of B can be verified using right-hand rule number 2 for a long,
straight wire. Imagine holding the wire with your right hand, with your thumb
pointing in the direction of the current. Your fingers then curl around in the
:
direction of B.
A segment of length x 2 at the bottom of the loop also contributes to the field
at the center, increasing its strength. The field produced at the center by the
segment x 2 has the same magnitude as B 1 and is also directed out of the page.
Similarly, all other such segments of the current loop contribute to the field. The
net effect is a magnetic field for the current loop as pictured in Figure 19.30a.
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19.9
645
Magnetic Fields of Current Loops and Solenoids
N
S
I
S
(a)
© Richard Megna, Fundamental Photographs.
N
(c)
(b)
Figure 19.30 (a) Magnetic field lines for a current loop. Note that the lines resemble those of a bar
magnet. (b) The magnetic field of a bar magnet is similar to that of a current loop. (c) Field lines of a
current loop, displayed by iron filings.
Notice in Figure 19.30a that the magnetic field lines enter at the bottom of the
current loop and exit at the top. Compare this to Figure 19.30b, illustrating the
field of a bar magnet. The two fields are similar. One side of the loop acts as
though it were the north pole of a magnet, and the other acts as a south pole.
The similarity of these two fields will be used to discuss magnetism in matter in an
upcoming section.
Applying Physics 19.4 Twisted Wires
In electrical circuits, it is often the case that insulated
wires carrying currents in opposite directions are
twisted together. What is the advantage of doing this?
Explanation If the wires are not twisted together, the
combination of the two wires forms a current loop,
which produces a relatively strong magnetic field. This
magnetic field generated by the loop could be strong
enough to affect adjacent circuits or components.
When the wires are twisted together, their magnetic
fields tend to cancel.
The magnitude of the magnetic field at the center of a circular loop carrying
current I as in Figure 19.30a is given by
B
0I
2R
This must be derived with calculus. However, it can be shown to be reasonable
by calculating the field at the center of four long wires, each carrying current I
and forming a square, as in Figure 19.31, with a circle of radius R inscribed
within it. Intuitively, this arrangement should give a magnetic field at the center
that is similar in magnitude to the field produced by the circular loop. The current in the circular wire is closer to the center, so that wire would have a magnetic field somewhat stronger than just the four legs of the rectangle, but the
lengths of the straight wires beyond the rectangle compensate for this. Each
wire contributes the same magnetic field at the exact center, so the total field is
given by
B4
0I
4
2R
2RI (1.27) 2RI 0
0
This is approximately the same as the field produced by the circular loop of current.
When the coil has N loops, each carrying current I, the magnetic field at the
I
I
I
R
I
Figure 19.31 The field of a circular loop carrying current I can be
approximated by the field due to
four straight wires, each carrying
current I.
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Chapter 19
Magnetism
center is given by
Exterior
B N
0I
2R
[19.15]
Magnetic Field of a Solenoid
Interior
Figure 19.32 The magnetic field
lines for a loosely wound solenoid.
If a long, straight wire is bent into a coil of several closely spaced loops, the resulting device is a solenoid, often called an electromagnet. This device is important in
many applications because it acts as a magnet only when it carries a current. The
magnetic field inside a solenoid increases with the current and is proportional to
the number of coils per unit length.
Figure 19.32 shows the magnetic field lines of a loosely wound solenoid of
length and total number of turns N. Notice that the field lines inside the solenoid are nearly parallel, uniformly spaced, and close together. As a result, the field
inside the solenoid is strong and approximately uniform. The exterior field at the
sides of the solenoid is nonuniform, much weaker than the interior field, and
opposite in direction to the field inside the solenoid.
If the turns are closely spaced, the field lines are as shown in Figure 19.33a,
entering at one end of the solenoid and emerging at the other. This means that
one end of the solenoid acts as a north pole and the other end acts as a south
pole. If the length of the solenoid is much greater than its radius, the lines that
leave the north end of the solenoid spread out over a wide region before returning to enter the south end. The more widely separated the field lines are, the
weaker the field. This is in contrast to a much stronger field inside the solenoid,
where the lines are close together. Also, the field inside the solenoid has a
constant magnitude at all points far from its ends. As will be shown subsequently,
these considerations allow the application of Ampere’s law to the solenoid,
giving a result of
B 0nI
[19.16]
The magnetic field inside a solenoid for the field inside the solenoid, where n N/ is the number of turns per unit
length of the solenoid.
So-called steering magnets placed along the neck of the picture tube in a televi-
b, Courtesy of Henry Leap and Jim Lehman
N
S
(a)
(b)
Figure 19.33 (a) Magnetic field lines for a tightly wound solenoid of finite length carrying a steady
current. The field inside the solenoid is nearly uniform and strong. Note that the field lines resemble
those of a bar magnet, so the solenoid effectively has north and south poles. (b) The magnetic field
pattern of a bar magnet, displayed by small iron filings on a sheet of paper.
44920_19_p624-659 1/5/05 1:48 PM Page 647
19.9
Magnetic Fields of Current Loops and Solenoids
647
EXAMPLE 19.9 The Magnetic Field inside a Solenoid
Goal Calculate the magnetic field of a solenoid from given data and the momentum of a charged particle in this
field.
Problem A certain solenoid consists of 100 turns of wire and has a length of 10.0 cm. (a) Find the magnitude of the
magnetic field inside the solenoid when it carries a current of 0.500 A. (b) What is the momentum of a proton orbiting inside the solenoid in a circle with a radius of 0.020 m? The axis of the solenoid is perpendicular to the plane of
the orbit. (c) Approximately how much wire would be needed to build this solenoid? Assume the solenoid’s radius is
5.00 cm.
Strategy In part (a), calculate the number of turns per meter and substitute that and given information into Equation 19.16, getting the magnitude of the magnetic field. Part (b) is an application of Newton’s second law.
Solution
(a) Find the magnitude of the magnetic field inside
the solenoid when it carries a current of 0.500 A.
N
100 turns
1.00 103 turns/m
0.100 m
Calculate the number of turns per unit length:
n
Substitute n and I into Equation 19.16 to find the
magnitude of the magnetic field:
B 0nI
(4 107 T m/A)(1.00 103 turns/m)(0.500 A)
6.28 104 T
(b) Find the momentum of a proton orbiting in a circle
of radius 0.020 m near the center of the solenoid.
Write Newton’s second law for the proton:
ma F qvB
Substitute the centripetal acceleration a v 2/r :
m
Cancel one factor of v on both sides and multiply by r,
getting the momentum mv :
mv rqB (0.020 m)(1.60 1019 C)(6.28 104 T)
v2
qvB
r
p mv 2.01 1024 kgm/s
(c) Approximately how much wire would be needed to
build this solenoid?
Multiply the number of turns by the circumference of
one loop:
length of wire (number of turns)(2r)
(1.00 102 turns)(2 0.0500 m)
31.4 m
Remarks An electron in part (b) would have the same momentum as the proton, but a much higher speed. It
would also orbit in the opposite direction. The length of wire in part (c) is only an estimate, because the wire has a
certain thickness, slightly increasing the size of each loop. In addition the wire loops aren’t perfect circles, because
they wind slowly up along the solenoid.
Exercise 19.9
Suppose you have a 32.0-m length of copper wire. If the wire is wrapped into a solenoid 0.240 m long and having a
radius of 0.0400 m, how strong is the resulting magnetic field in its center when the current is 12.0 A?
Answer 8.00 103 T
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Chapter 19
Magnetism
Electromagnets
TV
tube
Electron
gun
Path of
beam
Screen
Figure 19.34 Electromagnets are
used to deflect electrons to desired
positions on the screen of a television
tube.
sion set, as in Figure 19.34, are used to make the electron beam move to the
desired locations on the screen, tracing out the images. The rate at which the electron beam sweeps over the screen is so fast that to the eye it looks like a picture
rather than a sequence of dots.
Ampère’s Law Applied to a Solenoid
We can use Ampère’s law to obtain the expression for the magnetic field inside a
solenoid carrying a current I. A cross section taken along the length of part of our
:
solenoid is shown in Figure 19.35. B inside the solenoid is uniform and parallel to
:
the axis, and B outside is approximately zero. Consider a rectangular path of
length L and width w, as shown in the figure. We can apply Ampère’s law to this
path by evaluating the sum of B, over each side of the rectangle. The contribu:
tion along side 3 is clearly zero, because B 0 in this region. The contributions
:
from sides 2 and 4 are both zero, because B is perpendicular to along these
:
paths. Side 1 of length L gives a contribution BL to the sum, because B is uniform
along this path, and parallel to . Therefore, the sum over the closed rectangular
path has the value
B, BL
The right side of Ampère’s law involves the total current that passes through the
area bounded by the path chosen. In this case, the total current through the rectangular path equals the current through each turn of the solenoid, multiplied by
the number of turns. If N is the number of turns in the length L, then the total
current through the rectangular path equals NI. Ampère’s law applied to this path
therefore gives
B
×
×
×
w
2
×
1
B, BL 0NI
×
3
×
×
×
4
×
or
B 0
N
I 0nI
L
where n N/L is the number of turns per unit length.
×
×
19.10 MAGNETIC DOMAINS
Figure 19.35 A cross-sectional
view of a tightly wound solenoid. If
the solenoid is long relative to its
radius, we can assume that the magnetic field inside is uniform and the
field outside is zero. Ampère’s law
applied to the blue dashed rectangular path can then be used to calculate
the field inside the solenoid.
µ spin
Figure 19.36 Classical model of a
spinning electron.
The magnetic field produced by a current in a coil of wire gives us a hint as to
what might cause certain materials to exhibit strong magnetic properties. A single
coil like that in Figure 19.30a has a north pole and a south pole, but if this is true
for a coil of wire, it should also be true for any current confined to a circular path.
In particular, an individual atom should act as a magnet because of the motion of the electrons about the nucleus. Each electron, with its charge of 1.6 1019 C, circles the
atom once in about 1016 s. If we divide the electric charge by this time interval,
we see that the orbiting electron is equivalent to a current of 1.6 103 A. Such a
current produces a magnetic field on the order of 20 T at the center of the circular path. From this we see that a very strong magnetic field would be produced if
several of these atomic magnets could be aligned inside a material. This doesn’t
occur, however, because the simple model we have described is not the complete
story. A thorough analysis of atomic structure shows that the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving
electron in the same atom. The net result is that the magnetic effect produced by
the electrons orbiting the nucleus is either zero or very small for most materials.
The magnetic properties of many materials can be explained by the fact that an
electron not only circles in an orbit, but also spins on its axis like a top, with spin
magnetic moment as shown (Fig. 19.36). (This classical description should not
be taken too literally. The property of electron spin can be understood only in the
context of quantum mechanics, which we will not discuss here.) The spinning elec-
44920_19_p624-659 1/5/05 1:48 PM Page 649
Summary
B
(a)
(b)
(c)
649
B
Figure 19.37 (a) Random
orientation of domains in an unmagnetized substance. (b) When an
:
external magnetic field B is applied, the domains tend to align with the magnetic field. (c) As the field
is made even stronger, the domains not aligned with the external field become very small.
tron represents a charge in motion that produces a magnetic field. The field due to
the spinning is generally stronger than the field due to the orbital motion. In atoms
containing many electrons, the electrons usually pair up with their spins opposite
each other, so that their fields cancel each other. That is why most substances are
not magnets. However, in certain strongly magnetic materials, such as iron, cobalt,
and nickel, the magnetic fields produced by the electron spins don’t cancel completely. Such materials are said to be ferromagnetic. In ferromagnetic materials,
strong coupling occurs between neighboring atoms, forming large groups of atoms
with spins that are aligned. Called domains, the sizes of these groups typically range
from about 104 cm to 0.1 cm. In an unmagnetized substance the domains are randomly oriented, as shown in Figure 19.37a. When an external field is applied, as in
Figure 19.37b, the magnetic field of each domain tends to come nearer to alignment with the external field, resulting in magnetization.
In what are called hard magnetic materials, domains remain aligned even after
the external field is removed; the result is a permanent magnet. In soft magnetic materials, such as iron, once the external field is removed, thermal agitation produces
motion of the domains and the material quickly returns to an unmagnetized state.
The alignment of domains explains why the strength of an electromagnet is
increased dramatically by the insertion of an iron core into the magnet’s center.
The magnetic field produced by the current in the loops causes the domains to
align, thus producing a large net external field. The use of iron as a core is also
advantageous because it is a soft magnetic material that loses its magnetism almost
instantaneously after the current in the coils is turned off.
TIP 19.4 The Electron
Spins — but Doesn’t!
Even though we use the word spin,
the electron, unlike a child’s top, isn’t
physically spinning in this sense. The
electron has an intrinsic angular
momentum that causes it to act as if it
were spinning, but the concept of spin
angular momentum is actually a relativistic quantum effect.
SUMMARY
Take a practice test by logging into PhysicsNow at www.cp7e.com and clicking on the Pre-Test link for
this chapter.
The SI unit of the magnetic field is the tesla (T), or weber
per square meter (Wb/m2). An additional commonly used
unit for the magnetic field is the gauss (G); 1 T 10 4 G.
19.3
19.4 Magnetic Force on a
Current-Carrying Conductor
Magnetic Fields
The magnetic force that acts on a charge q moving with
:
v in a magnetic field B has magnitude
velocity :
F qvB sin [19.1]
:
v and B .
where is the angle between :
To find the direction of this force, use right-hand rule
number 1: point the fingers of your open right hand in the
:
v and then curl them in the direction of B .
direction of :
Your thumb then points in the direction of the magnetic
:
force F .
If the charge is negative rather than positive, the force is
directed opposite the force given by the right-hand rule.
If a straight conductor of length carries current I, the
magnetic force on that conductor when it is placed in a
:
uniform external magnetic field B is
F BI sin [19.6]
where is the angle between the direction of the current
and the direction of the magnetic field.
Right-hand rule number 1 also gives the direction of the
magnetic force on the conductor. In this case, however, you
must point your fingers in the direction of the current
rather than in the direction of :
v.
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Chapter 19
Magnetism
19.5 Torque on a Current Loop
and Electric Motors
The torque on a current-carrying loop of wire in a mag:
netic field B has magnitude
BIA sin [19.8]
where I is the current in the loop and A is its cross-sectional
area. The magnitude of the magnetic moment of a currentcarrying coil is defined by IAN, where N is the number
:
of loops. The magnetic moment is considered a vector, ,
that is perpendicular to the plane of the loop. The angle
:
:
between B and is .
19.6 Motion of a Charged Particle
in a Magnetic Field
If a charged particle moves in a uniform magnetic field so
that its initial velocity is perpendicular to the field, it will
move in a circular path in a plane perpendicular to the
magnetic field. The radius r of the circular path can be
found from Newton’s second law and centripetal acceleration, and is given by
r
mv
qB
[19.10]
where m is the mass of the particle and q is its charge.
19.7 Magnetic Field of a Long, Straight
Wire and Ampère’s Law
The magnetic field at distance r from a long, straight wire
carrying current I has the magnitude
B
0I
2r
where 0 4 107 T m/A is the permeability of free
space. The magnetic field lines around a long, straight wire
are circles concentric with the wire.
Ampère’s law can be used to find the magnetic field
around certain simple current-carrying conductors. It can
be written
B, 0I
[19.13]
:
where B, is the component of B tangent to a small current
element of length that is part of a closed path and I is
the total current that penetrates the closed path.
19.8 Magnetic Force between Two
Parallel Conductors
The force per unit length on each of two parallel wires separated by the distance d and carrying currents I1 and I2 has
the magnitude
I I
F
012
2d
[19.14]
The forces are attractive if the currents are in the same
direction and repulsive if they are in opposite directions.
19.9 Magnetic Field of Current
Loops and Solenoids
The magnetic field at the center of a coil of N circular
loops of radius R, each carrying current I, is given by
0I
[19.15]
2R
The magnetic field inside a solenoid has the magnitude
BN
B 0nI
[19.16]
[19.11]
where n N/ is the number of turns of wire per unit length.
1. In your home television set, a beam of electrons moves
from the back of the picture tube to the screen, where it
strikes a fluorescent dot that glows with a particular color
when hit. The Earth’s magnetic field at the location of the
television set is horizontal and toward the north. In which
direction(s) should the set be oriented so that the beam
undergoes the largest deflection?
2. Can a constant magnetic field set a proton at rest into
motion? Explain your answer.
3. A proton moving horizontally enters a region where a uniform magnetic field is directed perpendicular to the
proton’s velocity, as shown in Figure Q19.3. Describe the
subsequent motion of the proton. How would an electron
behave under the same circumstances.
4. No magnetic force acts upon a current-carrying conductor when it is placed in a certain manner in a uniform
magnetic field. Explain.
5. How can the motion of a charged particle be used to distinguish between a magnetic field and an electric field in
a certain region?
6. Which way would a compass point if you were at Earth’s
north magnetic pole?
7. Why does the picture on a television screen become
distorted when a magnet is brought near the screen as in
Figure Q19.7? [Caution: You should not do this at home
on a color television set, because it may permanently
affect the picture quality.]
+
© Loren Winters/Visuals Unlimited
CONCEPTUAL QUESTIONS
v
Figure Q19.3
Figure Q19.7
44920_19_p624-659 1/5/05 1:49 PM Page 651
Problems
9. A Hindu ruler once suggested that he be entombed in a
magnetic coffin with the polarity arranged so that he
could be forever suspended between heaven and Earth. Is
such magnetic levitation possible? Discuss.
10. Will a nail be attracted to either pole of a magnet?
Explain what is happening inside the nail when it is
placed near the magnet.
11. Suppose you move along a wire at the same speed as the
drift speed of the electrons in the wire. Do you now measure a magnetic field of zero?
12. Describe the change in the magnetic field in the space
enclosed by a solenoid carrying a steady current I if
(a) the length of the solenoid is doubled, but the number
of turns remains the same, and (b) the number of turns is
doubled, but the length remains the same.
13. Can you use a compass to detect the currents in wires in
the walls near light switches in your home?
14. Why do charged particles from outer space, called cosmic
rays, strike Earth more frequently at the poles than at the
equator?
15. Two wires carry currents in opposite directions and are
oriented parallel, with one above the other. The wires
repel each other. Is the upper wire in a stable levitation
over the lower wire? Suppose the current in one wire is
reversed, so that the wires now attract. Is the lower wire
hanging in a stable attraction to the upper wire?
20. Is the magnetic field created by a current loop uniform?
Explain.
21. The electron beam in Figure Q19.21 is projected to the
right. The beam deflects downward in the presence of a
magnetic field produced by a pair of current-carrying
coils. (a) What is the direction of the magnetic field?
(b) What would happen to the beam if the magnetic field
were reversed in direction?
Courtesy of Central Scientific Company
8. A magnet attracts a piece of iron. The iron can then
attract another piece of iron. On the basis of domain
alignment, explain what happens in each piece of iron.
651
Image not Available
Figure Q19.21
22. Figure Q19.22 shows two permanent magnets, each having a hole through its center. Note that the upper magnet
is levitated above the lower one. (a) How does this occur?
(b) What purpose does the pencil serve? (c) What can
you say about the poles of the magnets from this observation? (d) If the upper magnet were inverted, what do you
suppose would happen?
17. A hanging Slinky ® toy is attached to a powerful battery
and a switch. When the switch is closed so that the toy now
carries current, does the Slinky ® compress or expand?
18. Is it possible to orient a current loop in a uniform magnetic field such that the loop will not tend to rotate?
19. Parallel wires exert magnetic forces on each other. What
about perpendicular wires? Imagine two wires oriented
perpendicular to each other and almost touching. Each
wire carries a current. Is there a force between the wires?
Courtesy of Central Scientific Company
16. How can a current loop be used to determine the presence of a magnetic field in a given region of space?
Image not Available
Figure Q19.22
PROBLEMS
1, 2, 3 = straightforward, intermediate, challenging
= full solution available in Student Solutions Manual/Study Guide
= coached problem with hints available at www.cp7e.com
= biomedical application
Section 19.3 Magnetic Fields
1. An electron gun fires electrons into a magnetic
field directed straight downward. Find the direction of the
force exerted by the field on an electron for each of the following directions of the electron’s velocity: (a) horizontal
and due north; (b) horizontal and 30° west of north; (c) due
north, but at 30° below the horizontal; (d) straight upward.
(Remember that an electron has a negative charge.)
2. (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields
in Figure P19.2 (page 652), as shown. (b) Repeat part (a),
assuming the moving particle is an electron.
3. Find the direction of the magnetic field acting on the positively charged particle moving in the various situations
shown in Figure P19.3 (page 652) if the direction of the
magnetic force acting on it is as indicated.
4. Determine the initial direction of the deflection of
charged particles as they enter the magnetic fields, as
shown in Figure P19.4 (page 652).
5. At the equator, near the surface of Earth, the magnetic
field is approximately 50.0 T northward, and the electric
field is about 100 N/C downward in fair weather. Find the
gravitational, electric, and magnetic forces on an electron
with an instantaneous velocity of 6.00 106 m/s directed
to the east in this environment.
6. The magnetic field of the Earth at a certain location is
directed vertically downward and has a magnitude of
50.0 T. A proton is moving horizontally toward the west
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Chapter 19
Magnetism
v
9. A proton moves perpendicularly to a uniform magnetic
:
field B at 1.0 107 m/s and exhibits an acceleration of
2.0 1013 m/s2 in the x-direction when its velocity is in
the z-direction. Determine the magnitude and direction of the field.
10. Sodium ions (Na
) move at 0.851 m/s through a bloodstream in the arm of a person standing near a large magnet.
The magnetic field has a strength of 0.254 T and makes an
angle of 51.0° with the motion of the sodium ions. The arm
contains 100 cm3 of blood with 3.00 1020 Na
ions per
cubic centimeter. If no other ions were present in the arm,
what would be the magnetic force on the arm?
v
B
Bin
(a)
(b)
Bout
v
v
B
(c)
(d)
v
B
u
u
v
B
(f)
(e)
Figure P19.2 (Problems 2 and 12) For Problem 12,
replace the velocity vector with a current in that direction.
F
F
F
v (in)
v
v (out)
(a)
(b)
(c)
Figure P19.3 (Problems 3 and 13) For Problem 13,
replace the velocity vector with a current in that direction.
(a)
Bin
(b)
Bup
+
(c)
–
Bright
(d)
+
Bat 45°
45°
+
Figure P19.4
in this field with a speed of 6.20 106 m/s. What are the
direction and magnitude of the magnetic force the field
exerts on the proton?
7.
What velocity would a proton need to
circle Earth 1 000 km above the magnetic equator, where
Earth’s magnetic field is directed horizontally north and
has a magnitude of 4.00 108 T?
8. An electron is accelerated through 2 400 V from rest and
then enters a region where there is a uniform 1.70-T magnetic field. What are (a) the maximum and (b) the minimum magnitudes of the magnetic force acting on this
electron?
Section 19.4 Magnetic Force on a
Current-Carrying Conductor
11. A current I 15 A is directed along the positive x -axis
and perpendicular to a magnetic field. A magnetic force
per unit length of 0.12 N/m acts on the conductor in the
negative y-direction. Calculate the magnitude and direction of the magnetic field in the region through which
the current passes.
12. In Figure P19.2, assume that in each case the velocity vector shown is replaced with a wire carrying a current in the
direction of the velocity vector. For each case, find the
direction of the magnetic force acting on the wire.
13. In Figure P19.3, assume that in each case the velocity
vector shown is replaced with a wire carrying a current in
the direction of the velocity vector. For each case, find the
direction of the magnetic field that will produce the magnetic force shown.
14. A wire having a mass per unit length of 0.500 g/cm
carries a 2.00-A current horizontally to the south.
What are the direction and magnitude of the minimum
magnetic field needed to lift this wire vertically upward?
15. A wire carries a current of 10.0 A in a direction that
makes an angle of 30.0° with the direction of a magnetic
field of strength 0.300 T. Find the magnetic force on a
5.00-m length of the wire.
16. At a certain location, Earth has a magnetic field of
0.60 104 T, pointing 75° below the horizontal in a
north – south plane. A 10.0-m-long straight wire carries a
15-A current. (a) If the current is directed horizontally
toward the east, what are the magnitude and direction of
the magnetic force on the wire? (b) What are the magnitude and direction of the force if the current is directed
vertically upward?
17. A wire with a mass of 1.00 g/cm is placed on a horizontal
surface with a coefficient of friction of 0.200. The wire
carries a current of 1.50 A eastward and moves horizontally to the north. What are the magnitude and the direction of the smallest vertical magnetic field that enables the
wire to move in this fashion?
18. A conductor suspended by two flexible wires as shown in
Figure P19.18 has a mass per unit length of 0.040 0 kg/m.
What current must exist in the conductor for the tension
in the supporting wires to be zero when the magnetic
field is 3.60 T into the page? What is the required direction for the current?
19. An unusual message delivery system is pictured in Figure
P19.19. A 15-cm length of conductor that is free to move is
held in place between two thin conductors. When a 5.0-A
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Problems
653
it. If the coil is in a uniform magnetic field of 2.00 104 T
directed toward the left of the page, what is the magnitude of the torque on the coil? [Hint: The area of an
ellipse is A ab, where a and b are, respectively, the
semimajor and semiminor axes of the ellipse.]
Bin
I
Figure P19.18
40.0 cm
current is directed as shown in the figure, the wire segment
moves upward at a constant velocity. If the mass of the wire
is 15 g, find the magnitude and direction of the minimum
magnetic field that is required to move the wire. (The wire
slides without friction on the two vertical conductors.)
B
30.0 cm
Figure P19.23
15 cm
5.0 A
5.0 A
5.0 A
24. A rectangular loop consists of 100 closely wrapped turns
and has dimensions 0.40 m by 0.30 m. The loop is hinged
along the y-axis, and the plane of the coil makes an angle
of 30.0° with the x-axis (Fig. P19.24). What is the magnitude of the torque exerted on the loop by a uniform magnetic field of 0.80 T directed along the x-axis when the
current in the windings has a value of 1.2 A in the direction shown? What is the expected direction of rotation of
the loop?
Figure P19.19
y
20. A wire 2.80 m in length carries a current of 5.00 A in a
region where a uniform magnetic field has a magnitude
of 0.390 T. Calculate the magnitude of the magnetic force
on the wire, assuming the angle between the magnetic
field and the current is (a) 60.0°, (b) 90.0°, (c) 120°.
21. In Figure P19.21, the cube is 40.0 cm on each edge. Four
straight segments of wire — ab, bc, cd, and da — form a closed
loop that carries a current I 5.00 A in the direction
shown. A uniform magnetic field of magnitude B 0.020 0 T
is in the positive y-direction. Determine the magnitude and
direction of the magnetic force on each segment.
y
B
a
d
I
z
b
x
c
Figure P19.21
Section 19.5 Torque on a Current Loop and Electric Motors
22. A current of 17.0 mA is maintained in a single circular
loop with a circumference of 2.00 m. A magnetic field of
0.800 T is directed parallel to the plane of the loop. What
is the magnitude of the torque exerted by the magnetic
field on the loop?
An eight-turn coil encloses an elliptical
23.
area having a major axis of 40.0 cm and a minor axis of
30.0 cm (Fig. P19.23). The coil lies in the plane of the
page and has a 6.00-A current flowing clockwise around
I = 1.2 A
0.40 m
z
30.0°
x
0.30 m
Figure P19.24
25. A long piece of wire with a mass of 0.100 kg and a total
length of 4.00 m is used to make a square coil with a side
of 0.100 m. The coil is hinged along a horizontal side, carries a 3.40-A current, and is placed in a vertical magnetic
field with a magnitude of 0.010 0 T. (a) Determine the
angle that the plane of the coil makes with the vertical
when the coil is in equilibrium. (b) Find the torque acting
on the coil due to the magnetic force at equilibrium.
26. A copper wire is 8.00 m long and has a cross-sectional
area of 1.00 104 m2. The wire forms a one-turn loop
in the shape of square and is then connected to a battery
that applies a potential difference of 0.100 V. If the loop is
placed in a uniform magnetic field of magnitude 0.400 T,
what is the maximum torque that can act on it? The resistivity of copper is 1.70 108 m.
Section 19.6 Motion of a Charged Particle
in a Magnetic Field
27. A proton moving freely in a circular path perpendicular to a
constant magnetic field takes 1.00 s to complete one
revolution. Determine the magnitude of the magnetic field.
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Chapter 19
Magnetism
28. A cosmic-ray proton in interstellar space has an energy of
10.0 MeV and executes a circular orbit having a radius
equal to that of Mercury’s orbit around the Sun, which is
5.80 1010 m. What is the magnetic field in that region
of space?
29. Figure P19.29a is a diagram of a device called a velocity
selector, in which particles of a specific velocity pass
through undeflected while those with greater or lesser
velocities are deflected either upwards or downwards. An
electric field is directed perpendicular to a magnetic field,
producing an electric force and a magnetic force on the
charged particle that can be equal in magnitude and
opposite in direction (Fig. P19.29b) and hence cancel.
Show that particles with a speed of v E/B will pass
through the velocity selector undeflected.
Bin
+
Source
+
+
+
+
+
+
q vB
magnetic field of 0.600 T directed perpendicularly to the
velocity of the ions. What is the distance between the
impact points formed on the photographic plate by singly
charged ions of 235U and 238U?
33. A proton is at rest at the plane vertical boundary of a
region containing a uniform vertical magnetic field B. An
alpha particle moving horizontally makes a head-on elastic collision with the proton. Immediately after the collision, both particles enter the magnetic field, moving perpendicular to the direction of the field. The radius of the
proton’s trajectory is R . Find the radius of the alpha particle’s trajectory. The mass of the alpha particle is four times
that of the proton, and its charge is twice that of the proton.
Section 19.7 Magnetic Field of a Long, Straight
Wire and Ampère’s Law
34. In each of parts (a), (b), and (c) of Figure P19.34, find
the direction of the current in the wire that would produce a magnetic field directed as shown.
E
+q
v
Slit
–
–
–
–
–
Bin
–
B
qE
–
(a)
Figure P19.29
(b)
Bout
30. Consider the mass spectrometer shown schematically in
Figure P19.30. The electric field between the plates of the
velocity selector is 950 V/m, and the magnetic fields in
both the velocity selector and the deflection chamber
have magnitudes of 0.930 T. Calculate the radius of the
path in the system for a singly charged ion with mass
m 2.18 1026 kg. [Hint: See Problem 29.]
(a)
(b)
Bout
Bin
Photographic
plate
P
r
Bin
(c)
E
q
Figure P19.34
v
Velocity selector
B0, in
Figure P19.30 A mass spectrometer. Charged particles are
first sent through a velocity
selector. They then enter a region
:
where a magnetic field B0 (directed inward) causes positive
ions to move in a semicircular path and strike a photographic
film at P.
31. A singly charged positive ion has a mass of 2.50 1026 kg.
After being accelerated through a potential difference of
250 V, the ion enters a magnetic field of 0.500 T, in a
direction perpendicular to the field. Calculate the radius
of the path of the ion in the field.
32. A mass spectrometer is used to examine the isotopes of
uranium. Ions in the beam emerge from the velocity
selector at a speed of 3.00 105 m/s and enter a uniform
35. A lightning bolt may carry a current of 1.00 104 A for a
short time. What is the resulting magnetic field 100 m
from the bolt? Suppose that the bolt extends far above
and below the point of observation.
36. In 1962, measurements of the magnetic field of a large
tornado were made at the Geophysical Observatory in
Tulsa, Oklahoma. If the magnitude of the tornado’s field
was B 1.50 108 T pointing north when the tornado
was 9.00 km east of the observatory, what current was carried up or down the funnel of the tornado? Model the
vortex as a long, straight wire carrying a current.
37. A cardiac pacemaker can be affected by a static magnetic
field as small as 1.7 mT. How close can a pacemaker
wearer come to a long, straight wire carrying 20 A?
38. The two wires shown in Figure P19.38 carry currents of
5.00 A in opposite directions and are separated by 10.0 cm.
Find the direction and magnitude of the net magnetic
44920_19_p624-659 1/5/05 1:49 PM Page 655
Problems
y
field (a) at a point midway between the wires; (b) at point
P1, 10.0 cm to the right of the wire on the right, and (c) at
point P 2, 20.0 cm to the left of the wire on the left.
5.00 A
P2
(4.00, 3.00) m
6.00 A
5.00 A
655
P
P1
x
10.0 cm
20.0 cm
10.0 cm
7.00 A
Figure P19.41
Figure P19.38
39. Four long, parallel conductors carry equal currents of
I 5.00 A. Figure P19.39 is an end view of the conductors. The direction of the current is into the page at
points A and B (indicated by the crosses) and out of the
page at C and D (indicated by the dots). Calculate the
magnitude and direction of the magnetic field at point P,
located at the center of the square with edge of
length 0.200 m.
A
C
0.200 m
P
B
D
0.200 m
Figure P19.39
40. The two wires in Figure P19.40 carry currents of 3.00 A
and 5.00 A in the direction indicated. (a) Find the direction and magnitude of the magnetic field at a point
midway between the wires. (b) Find the magnitude and
direction of the magnetic field at point P, located 20.0 cm
above the wire carrying the 5.00-A current.
P
20.0 cm
5.00 A
3.00 A
43. The magnetic field 40.0 cm away from a long, straight
wire carrying current 2.00 A is 1.00 T. (a) At what distance is it 0.100 T? (b) At one instant, the two conductors in a long household extension cord carry equal 2.00-A
currents in opposite directions. The two wires are
3.00 mm apart. Find the magnetic field 40.0 cm away
from the middle of the straight cord, in the plane of the
two wires. (c) At what distance is it one-tenth as large?
(d) The center wire in a coaxial cable carries current
2.00 A in one direction, and the sheath around it carries
current 2.00 A in the opposite direction. What magnetic
field does the cable create at points outside?
Section 19.8 Magnetic Force between
Two Parallel Conductors
44. Two parallel wires are 10.0 cm apart, and each carries a
current of 10.0 A. (a) If the currents are in the same
direction, find the force per unit length exerted on one
of the wires by the other. Are the wires attracted to or
repelled by each other? (b) Repeat the problem with the
currents in opposite directions.
45. A wire with a weight per unit length of 0.080 N/m is suspended directly above a second wire. The top wire carries
a current of 30.0 A and the bottom wire carries a current
of 60.0 A. Find the distance of separation between the
wires so that the top wire will be held in place by magnetic
repulsion.
46. In Figure P19.46, the current in the long, straight wire is
I 1 5.00 A, and the wire lies in the plane of the rectangular loop, which carries 10.0 A. The dimensions shown
are c 0.100 m, a 0.150 m, and 0.450 m. Find the
magnitude and direction of the net force exerted by the
magnetic field due to the straight wire on the loop.
20.0 cm
Figure P19.40
A wire carries a 7.00-A current along
the x - axis, and another wire carries a 6.00-A current along
the y- axis, as shown in Figure P19.41. What is the magnetic field at point P, located at x 4.00 m, y 3.00 m?
42. A long, straight wire lies on a horizontal table and carries
a current of 1.20 A. In a vacuum, a proton moves parallel to the wire (opposite the direction of the current) with
a constant velocity of 2.30 104 m/s at a constant
distance d above the wire. Determine the value of d. (You
may ignore the magnetic field due to Earth.)
41.
I1
I2
c
a
Figure P19.46
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Chapter 19
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Section 19.9 Magnetic Fields of Current Loops
and Solenoids
47. What current is required in the windings of a long solenoid that has 1 000 turns uniformly distributed over a
length of 0.400 m in order to produce a magnetic field of
magnitude 1.00 104 T at the center of the solenoid?
48. It is desired to construct a solenoid that will have a resistance of 5.00 (at 20°C) and produce a magnetic field of
4.00 102 T at its center when it carries a current of
4.00 A. The solenoid is to be constructed from copper
wire having a diameter of 0.500 mm. If the radius of the
solenoid is to be 1.00 cm, determine (a) the number of
turns of wire needed and (b) the length the solenoid
should have.
A single-turn square loop of wire 2.00 cm
49.
on a side carries a counterclockwise current of 0.200 A.
The loop is inside a solenoid, with the plane of the loop
perpendicular to the magnetic field of the solenoid. The
solenoid has 30 turns per centimeter and carries a counterclockwise current of 15.0 A. Find the force on each side
of the loop and the torque acting on the loop.
50. An electron is moving at a speed of 1.0 104 m/s in a
circular path of radius of 2.0 cm inside a solenoid. The
magnetic field of the solenoid is perpendicular to the
plane of the electron’s path. Find (a) the strength of the
magnetic field inside the solenoid and (b) the current in
the solenoid if it has 25 turns per centimeter.
ADDITIONAL PROBLEMS
51. A circular coil consisting of a single loop of wire has a
radius of 30.0 cm and carries a current of 25 A. It is
placed in an external magnetic field of 0.30 T. Find the
torque on the wire when the plane of the coil makes an
angle of 35° with the direction of the field.
52. An electron enters a region of magnetic field of magnitude 0.010 0 T, traveling perpendicular to the linear
boundary of the region. The direction of the field is perpendicular to the velocity of the electron. (a) Determine
the time it takes for the electron to leave the “field-filled”
region, noting that its path is a semicircle. (b) Find the
kinetic energy of the electron if the radius of its semicircular path is 2.00 cm.
53. Two long, straight wires cross each other at right angles,
as shown in Figure P19.53. (a) Find the direction and
magnitude of the magnetic field at point P, which is in the
same plane as the two wires. (b) Find the magnetic field at
a point 30.0 cm above the point of intersection (30.0 cm
out of the page, toward you).
54. A 0.200-kg metal rod carrying a current of 10.0 A glides
on two horizontal rails 0.500 m apart. What vertical magnetic field is required to keep the rod moving at a constant speed if the coefficient of kinetic friction between
the rod and rails is 0.100?
55. Two species of singly charged positive ions of masses
20.0 1027 kg and 23.4 1027 kg enter a magnetic
field at the same location with a speed of 1.00 105 m/s.
If the strength of the field is 0.200 T, and the ions move
perpendicularly to the field, find their distance of separation after they complete one-half of their circular path.
56. Two parallel conductors carry currents in opposite directions, as shown in Figure P19.56. One conductor carries a
current of 10.0 A. Point A is the midpoint between the
wires, and point C is 5.00 cm to the right of the 10.0-A
current. I is adjusted so that the magnetic field at C is
zero. Find (a) the value of the current I and (b) the value
of the magnetic field at A.
I
10.0 A
10.0 cm
Figure P19.56
57. Using an electromagnetic flowmeter (Fig. P19.57), a heart
surgeon monitors the flow rate of blood through an
artery. Electrodes A and B make contact with the outer
surface of the blood vessel, which has interior diameter
3.00 mm. (a) For a magnetic field magnitude of 0.040 0 T,
a potential difference of 160 V appears between the
electrodes. Calculate the speed of the blood. (b) Verify
that electrode A is positive, as shown. Does the sign of the
emf depend on whether the mobile ions in the blood are
predominantly positively or negatively charged? Explain.
Artery
+ A
y
N
3.00 A
30.0 cm
Electrodes
P
C
A
To
voltmeter
S
– B
Blood
flow
40.0 cm
x
5.00 A
Figure P19.53
Figure P19.57
58. Two circular loops are parallel, coaxial, and almost in contact 1.00 mm apart (Fig. P19.58). Each loop is 10.0 cm in
44920_19_p624-659 1/10/05 12:47 PM Page 657
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Problems
radius. The top loop carries a clockwise current of 140 A.
The bottom loop carries a counterclockwise current of
140 A. (a) Calculate the magnetic force that the bottom
loop exerts on the top loop. (b) The upper loop has a
mass of 0.021 0 kg. Calculate its acceleration, assuming
that the only forces acting on it are the force in part
(a) and its weight. [Hint: The distance between the loops
is small in comparison to their radius of curvature, so the
loops may be treated as long, straight parallel wires.]
62. A uniform horizontal wire with a linear mass density of
0.50 g/m carries a 2.0-A current. It is placed in a constant
magnetic field with a strength of 4.0 103 T. The field
is horizontal and perpendicular to the wire. As the wire
moves upward starting from rest, (a) what is its acceleration and (b) how long does it take to rise 50 cm? Neglect
the magnetic field of Earth.
63. Three long parallel conductors carry currents of I 2.0 A.
Figure P19.63 is an end view of the conductors, with each
current coming out of the page. Given that a 1.0 cm,
determine the magnitude and direction of the magnetic
field at points A, B, and C.
140 A
I
140 A
Figure P19.58
a
59. A 1.00-kg ball having net charge Q 5.00 C is thrown
out of a window horizontally at a speed v 20.0 m/s. The
window is at a height h 20.0 m above the ground. A uniform horizontal magnetic field of magnitude B 0.010 0 T
is perpendicular to the plane of the ball’s trajectory. Find
the magnitude of the magnetic force acting on the ball
just before it hits the ground. [Hint: Ignore magnetic
forces in finding the ball’s final velocity.]
60. The idea that static magnetic fields might have a therapeutic value has been around for centuries. A currently
available rare-Earth magnet that is advertised to relieve
joint pain is shown in Figure P19.60. It is 1.0 mm thick
and has a field strength of 5.0 102 T at the center of
the flat surface. The magnetic field strength at points
away from the center of the disk is inversely proportional
to h 3, where h is the distance from the midplane of the
disk. How far from the surface of the disk will the field
strength be reduced to that of Earth (5.0 105 T)?
A
a
B
a
C
I
a
a
I
Figure P19.63
64. Two long parallel wires, each with a mass per unit length of
40 g/m, are supported in a horizontal plane by 6.0-cm-long
strings, as shown in Figure P19.64. Each wire carries the
same current I, causing the wires to repel each other so
that the angle between the supporting strings is 16°.
(a) Are the currents in the same or opposite directions?
(b) Determine the magnitude of each current.
y
6.0 cm
u = 16°
Figure P19.60
61. Two long parallel conductors carry currents I 1 3.00 A
and I 2 3.00 A, both directed into the page in Figure
P19.61. Determine the magnitude and direction of the
resultant magnetic field at P.
z
x
Figure P19.64
I1
5.00 cm
P
13.0 cm
12.0 cm
I
2
Figure P19.61
65. Protons having a kinetic energy of 5.00 MeV are moving
in the positive x-direction and enter a magnetic field of
0.050 0 T in the z-direction, out of the plane of the page,
and extending from x 0 to x 1.00 m as in Figure
P19.65 (page 658). (a) Calculate the y-component of the
protons’ momentum as they leave the magnetic field.
(b) Find the angle between the initial velocity vector of
the proton beam and the velocity vector after the beam
emerges from the field. [Hint: Neglect relativistic effects
and note that 1 eV 1.60 1019 J.]
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Chapter 19
Magnetism
location of I1? (d) What is the force per length exerted
by I2 on I 1?
Figure P19.65
66. A straight wire of mass 10.0 g and length 5.0 cm is suspended from two identical springs that, in turn, form a
closed circuit (Fig. P19.66). The springs stretch a distance
of 0.50 cm under the weight of the wire. The circuit has a
total resistance of 12 . When a magnetic field directed
out of the page (indicated by the dots in the figure is
turned on, the springs are observed to stretch an additional 0.30 cm. What is the strength of the magnetic field?
(The upper portion of the circuit is fixed.)
ACTIVITIES
1. For this activity, you will need a small bar magnet, a small
plastic container, and a bowl of water. Tape the magnet to
the bottom of the container, and float the container and
magnet on the surface of the bowl as in Figure A19.1. The
magnet and the container should rotate and come to
equilibrium, with the magnet pointing along a
north – south line. The compass you have constructed is
similar to the type used by early sailing vessels. How can
you determine which direction is north and which is
south?
Magnet
24 V
Water
Figure A19.01
5.0 cm
Figure P19.66
67. A solenoid 10.0 cm in diameter and 75.0 cm long is made
from copper wire of diameter 0.100 cm with very thin
insulation. The wire is wound onto a cardboard tube in a
single layer, with adjacent turns touching each other. To
produce a field of magnitude 20.0 mT at the center of the
solenoid, what power must be delivered to the solenoid?
68. Assume that the region to the right of a certain vertical plane
contains a vertical magnetic field of magnitude 1.00 mT
and that the field is zero in the region to the left of the
plane. An electron, originally traveling perpendicular to
the boundary plane, passes into the region of the field.
(a) Noting that the path of the electron is a semicircle,
determine the time interval required for the electron to
leave the “field-filled” region. (b) Find the kinetic energy
of the electron if the maximum depth of penetration into
the field is 2.00 cm.
69. Three long wires (wire 1, wire 2, and wire 3) are coplanar
and hang vertically. The distance between wire 1 and wire
2 is 20.0 cm. On the left, wire 1 carries an upward current
of 1.50 A. To the right, wire 2 carries a downward current
of 4.00 A. Wire 3 is located such that when it carries a certain current, no net force acts upon any of the wires. Find
(a) the position of wire 3 and (b) the magnitude and
direction of the current in wire 3.
70. Two long parallel conductors separated by 10.0 cm carry
currents in the same direction. The first wire carries a
current I1 5.00 A and the second carries I2 8.00 A.
(a) What is the magnitude of the magnetic field
created by I1 at the location of I2 ? (b) What is the force
per unit length exerted by I1 on I2 ? (c) What is the
magnitude of the magnetic field created by I 2 at the
2. In the Northern Hemisphere, the direction of Earth’s
magnetic field becomes more and more nearly vertical
the farther north one goes. To find the variation from the
horizontal of the magnetic field in your locale, try the
following: press an unmagnetized needle through a PingPong ® ball, and balance the structure between two drinking glasses that are lined up along an east – west line.
Next, press a magnetized needle through the ball at right
angles to the unmagnetized needle so that the needle
points north. The magnetized needle can now rotate in
the vertical direction and will point in the direction of
Earth’s magnetic field, which is at some angle below the
horizontal. Take several measurements of this dip angle
and obtain an average value.
3. Construct an electromagnet by wrapping about 1 meter of
small-diameter insulated wire around a steel nail. Tape
the ends of the wires to a D-cell battery as in Figure A19.3.
How many staples or paper clips can you pick up with
your electromagnet? How would you increase the magnetic field set up by the nail? Disconnect the wires from
–
+
Figure A19.03
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Problems
the battery and test how much magnetism is retained by
the nail by seeing how many staples it can pick up. A
convenient way to test the strength of a magnet is to
attach a paper clip to a rubber band. Note how far the
rubber band is stretched before the clip comes free of the
magnet. Test your electromagnet in this way. Where is the
magnetic field of the electromagnet strongest, at the ends
of the nail or near its center? When you have your nail
magnetized, bang it against a table or the floor and then
check its magnetism. Why does the nail lose its magnetism
by this procedure?
659
4. You can trace out the field pattern of a magnet with iron
filings. Any machine shop will supply the filings, which
should be soaked in a soap solution to remove grit and oil
and then dried. Scatter them lightly over the surface of a
paper covering the magnet, and then tap the paper gently
to jar the filings into alignment. Explain why the filings
form their pattern. Examine the field pattern set up in the
following situations: (a) Arrange two bar magnets about 4
cm apart, aligned with opposite poles facing each other.
(b) Use two bar magnets about 4 cm apart, aligned with
like poles facing each other. (c) Use a horseshoe magnet.