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Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Homework
Reading: Chap. 32 and Chap. 33
Suggested exercises: 32.1, 32.3, 32.5, 32.7, 32.9, 32.11,
32.13, 32.15, 32.18, 32.20, 32.24, 32.28, 32.32,
32.33, 32.35, 32.37, 32.39
Problems: 32.46, 32.48, 32.52, 32.53, 32.56, 32.57,
32.60, 32.63, 32.65, 32.67, 32.68, 32.71 (due Fri.,
Nov. 20)
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20 40 60 80 1000
5
10
15
20
25
30
Nu
mbe
r of S
tud
ents
Grade
Average = 47.4
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Chapter 32. The Magnetic Field
Digital information is stored
on a hard disk as
microscopic patches of
magnetism. Just what is
magnetism? How are
magnetic fields created?
What are their properties?
These are the questions we
will address.
Chapter Goal: To learn how
to calculate and use the
magnetic field.
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Topics:
• Magnetism
• The Discovery of the Magnetic Field
• The Source of the Magnetic Field: Moving
Charges
• The Magnetic Field of a Current
• Magnetic Dipoles
• Ampère’s Law and Solenoids
• The Magnetic Force on a Moving Charge
• Magnetic Forces on Current-Carrying Wires
• Forces and Torques on Current Loops
• Magnetic Properties of Matter
Chapter 32. The Magnetic Field
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Chapter 32. Basic Content and Examples
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Source of Magnetic Forces
Permanent magnet
Electromagnet
Moving charged particles
Elementary particles (electron)
S N
All the magnetic sources are presented as a dipole,
having two poles, North pole and South pole
No magnetic monopole has been founded yet!
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Magnetic Forces
Unlike poles attract
S N S N
Like poles repel
S N SN
SN S N
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Magnetic Field
Magnetic field B A vector Unit: Tesla
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Magnetic Field
Practical definition:
Fire a positive charge
perpendicular to the magnetic
field, the magnitude of the
magnetic field is given by:
The direction of the magnetic
field is given by:
Not that the force, field, and velocity are not in the same direction!
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Magnetic Field Lines
Magnetic field can be described by magnetic field lines.
1. Magnetic field lines show the direction of B at any
point (tangent)
2. Magnetic field lines for a bar magnet come out of
the North pole and enter into the South pole
Unlike electric field lines, magnetic field lines do not begin or end.
They either form closed loops or extend out to infinity.
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Magnetic Field Lines
S N
Small magnet
All the small magnets aligned S N along the direction of magnetic field
lines: field line direction is the same as the magnetic dipole direction.
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Magnetic Field Lines
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Magnetic Field Lines
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Magnetic Field Lines
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Magnet
When you cut bar magnet into two, are you left with one
North pole and one South pole?
No, since magnetic field lines are continuous both inside and
outside the magnet, you get two North poles and two South
poles.
N
S
N N
S S+
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Earth Magnetic Field
The earth’s core is made of molten metal such as ion or nickel.
Iron and nickel are very good conductors of electricity and electric
currents can flow easily in them. As the earth rotates, large electric
currents are built up and these produce the earth’s magnetic field.
The magnetic South
pole is actually
geographic North pole.
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Magnetic Field versus Electric Field
Electric sources are
inherently “monopole” or
point charge sources.
B-field E-field
Magnetic sources are
inherently dipole sources –
you cannot isolate North and
South “monopoles”.
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Magnetic Force on Moving Charges
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Magnetic Force on Moving Charges
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The Electromagnetic Force
If both the electric field and magnetic field exist, the force on a
charge is defined as the Lorentz force:
The electric force is straightforward, being in the direction of the
electric field if the charge q is positive, but the direction of the
magnetic part of the force is given by the right hand rule.
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A Moving Charge in Magnetic Field
In a uniform magnetic field, a constant magnetic force acts
on a moving charge with speed v. The direction of the
force is perpendicular to the direction of the velocity. So
the particle performs a circular movement:
r
mvF
2
qvBF
B
v
q
m
qB
mvr
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A Moving Charge in Magnetic Field
Other related parameters:
q
m
Bv
rT
22
m
qB
Tf
2
1
m
qBf 2
The period:
The frequency:
The angular frequency:
Only depend
on magnetic
filed and
charge/mass
ratio
Do not depend
on the velocity
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A Moving Charge in Magnetic Field
B
v v||
v┴q
B
v
q
mr
sin
q
m
BT
2 Do not depend
on the speed
q
m
Bvp
2cospitch
sin
cos||
vv
vv
Helical
motion
Charge-B field
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A Moving Charge in Magnetic Field
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A Moving Charge in Magnetic Field
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Magnetic Force on a Current-Carrying Wire
BvF qBFor a moving charged particle
For a current (a flux of moving charged particles) (Demo)
BLF iB
BvF nALeB
b
a
B di BlF
ds
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Torque
Motor
B
B
F1
F2
τ
θ
a
b F3
F4
iaBF
iaBF
iB
2
1
BLF
sin
sin
sin2
sin2
21
iAB
iabB
bF
bF
BAτ i
IF1
F2
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Magnetic Dipole Moment
Aμ iDefinition:
Torque: Bμτ
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Example
The following figure shows a wire carrying a current i = 6.0 A in the
positive direction of the x axis and lying in a nonuniform magnetic
field given by jiB ˆ)/0.2(ˆ)/3.2( xmTxmT
with B in Teslas and x in meters. What is the net magnetic force
FB on the section of the wire between x = 0 and x = 2.0 m?
i
BB
By
xx = 2.0 m
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Example (continued)
What we know: a current wire (i) in a magnetic field (B)
What we expect: a magnetic force will be exerted on the wire
Can we solve the problem directly using the following equation?
BLF iB
No, since B is not uniform!
We must mentally divide the wire into differential lengths and
using above equation to find the differential force dFB on each
length, then sum these differential forces to find the net force FB.
b
a
B di BlF
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Example (continued)
i
BB
By
xx = 2.0 m
dx
B
dL dF
iL ˆdxd
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Example (continued)
The differential force dFB on the length dx of the wire is
k
k
jiii
jii
BLF
ˆ0..2
]ˆ0.20[
)]ˆˆ(0.2)ˆˆ(3.2[
)ˆ0.2ˆ3.2(ˆ
ixdx
xidx
xxidx
xxidx
idd B
The constant 2.0 has the unit of Teslas per meter.
Clearly the magnetic force does not depend on the x-component of B
(this component is parallel to the current).
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Example (continued)
Since we have obtained the expression for dFB, the total magnetic
force will be integrating dFB from x = 0 to x = 2.0 m:
kk
k
k
kFF
ˆ24ˆ24
ˆ)0.2)(2
1)(0.6)(/0.2(
ˆ)/0.2(
ˆ)/0.2(
2
0.2
0
0.2
0
NmAT
mAmT
xdximT
dxixmTd
m
m
BB
The magnitude of the net force is 24N, and the direction is along
the positive direction of z-axis.
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Activity
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Activity
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Activity
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Activity
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Example
A metal rod having a mass per unit length of 0.01 kg/m carries a
current of I = 5.00 A. The rod hangs from two wires in a uniform
vertical magnetic field, as shown in the following figure. If the wire
makes an angle θ = 45.0o with the vertical when in equilibrium,
determine the magnitude of the magnetic field.
θ
θB
I
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Example (count.)
Let’s first think what causes the equilibrium in this system.
We need to focus on the rod (the main object!!!).
θ
θ
B
Usually the equilibrium are caused by force equilibrium, torque
equilibrium, etc. We have to do a force analysis on the system.
I
What are the forces
exerted on the rod?
N
N
FB
Gravity
Tension
Magnetic force
Magnetic force is directly related to the magnetic field.mg
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Example (count.)
Solving the problem in two ways: I. Force equilibrium
45sin
45cos
Nmg
NFB
Since N is unknown, dividing these
two equations,
45tan
45tan
1
mgF
mg
F
B
B
Since we know the mass of the rod, the magnitude of the magnetic
force can be obtained.
mg
FB
N
45o
B
I
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Example (count.)
mg
FB
N
45o
B
I
Since the magnetic force is caused by the magnetic field,
BLF IB
Since L is perpendicular to B,
ILBB F
Therefore,
45tan
)/(
45tan
45tan
L
gLm
IL
mgB
mgILB
mTB 6.19˚0.45tan
A 00.5
sm 80.9mkg 0100.0 2
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Example (count.)
Both magnetic force and gravity induce
torques. The torque caused by the tension
force is zero because the tension force points
to the rotating axis O.
II. Torque equilibriumSecond method:
mg
FB
N
45o
O
45tan
45cos45sin
mgF
mgddF
B
B
The torque caused by the gravity should be
balanced by the torque caused by the
magnetic force.
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Origin of Magnetic Field
Oersted’s Experiment
A current carrying wire generates magnetic field!
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Example
The following figure shows in horizontal cross section, three wires
that are meant to carry current from a lightening rod on top of a
house to the ground in case of lightening strikes the rod. The wires
are parallel, have a length L = 4.0 m, and are spaced r = 5.0 mm
apart. Assume that, during a strike, the current in each wire is I =
5000 A. What are the magnitude and direction of the net force on
each wire due to the currents in the other two wires?
r r
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Gauss’s Law of Magnetic Field
The net magnetic flux through a closed surface is zero.
Magnetic monopole has not been discovered yet!
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Chapter 32. Clicker Questions
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Does the compass needle rotate clockwise
(cw), counterclockwise (ccw) or not at all?
A. Clockwise
B. Counterclockwise
C. Not at all
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A. Clockwise
B. Counterclockwise
C. Not at all
Does the compass needle rotate clockwise
(cw), counterclockwise (ccw) or not at all?
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The magnetic field at the position P points
A. Into the page.
B. Up.
C. Down.
D. Out of the page.
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A. Into the page.
B. Up.
C. Down.
D. Out of the page.
The magnetic field at the position P points
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The positive charge is
moving straight out of the
page. What is the direction
of the magnetic field at the
position of the dot?
A. Left
B. Right
C. Down
D. Up
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A. Left
B. Right
C. Down
D. Up
The positive charge is
moving straight out of the
page. What is the direction
of the magnetic field at the
position of the dot?
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What is the current
direction in this loop?
And which side of the
loop is the north pole?
A. Current counterclockwise, north pole on bottom
B. Current clockwise; north pole on bottom
C. Current counterclockwise, north pole on top
D. Current clockwise; north pole on top
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A. Current counterclockwise, north pole on bottom
B. Current clockwise; north pole on bottom
C. Current counterclockwise, north pole on top
D. Current clockwise; north pole on top
What is the current
direction in this loop?
And which side of the
loop is the north pole?
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A. Left
B. Into the page
C. Out of the page
D. Up
E. Down
An electron moves perpendicular to a
magnetic field. What is the direction
of ?
B
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A. Left
B. Into the page
C. Out of the page
D. Up
E. Down
B
An electron moves perpendicular to a
magnetic field. What is the direction
of ?
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What is the current direction in the loop?
A. Out of the page at the top of the
loop, into the page at the bottom.
B. Out of the page at the bottom of the
loop, into the page at the top.
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A. Out of the page at the top of the
loop, into the page at the bottom.
B. Out of the page at the bottom of
the loop, into the page at the top.
What is the current direction in the loop?
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Which magnet or magnets
produced this induced
magnetic dipole?
A. a or d
B. a or c
C. b or d
D. b or c
E. any of a, b, c or d
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A. a or d
B. a or c
C. b or d
D. b or c
E. any of a, b, c or d
Which magnet or magnets
produced this induced
magnetic dipole?
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Chapter 32. Reading Quizzes
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What is the SI unit for the strength
of the magnetic field?
A. Gauss
B. Henry
C. Tesla
D. Becquerel
E. Bohr magneton
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What is the SI unit for the strength
of the magnetic field?
A. Gauss
B. Henry
C. Tesla
D. Becquerel
E. Bohr magneton
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What is the shape of the trajectory that a
charged particle follows in a uniform
magnetic field?
A. Helix
B. Parabola
C. Circle
D. Ellipse
E. Hyperbola
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What is the shape of the trajectory that a
charged particle follows in a uniform
magnetic field?
A. Helix
B. Parabola
C. Circle
D. Ellipse
E. Hyperbola
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The magnetic field of a point current source
is given by
A. Biot-Savart’s law.
B. Faraday’s law.
C. Gauss’s law.
D. Ampère’s law.
E. Einstein’s law.
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A. Biot-Savart’s law.
B. Faraday’s law.
C. Gauss’s law.
D. Ampère’s law.
E. Einstein’s law.
The magnetic field of a point current source
is given by
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The magnetic field of a straight,
current-carrying wire is
A. parallel to the wire.
B. inside the wire.
C. perpendicular to the wire.
D. around the wire.
E. zero.
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The magnetic field of a straight,
current-carrying wire is
A. parallel to the wire.
B. inside the wire.
C. perpendicular to the wire.
D. around the wire.
E. zero.
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