Reading: Chap. 32 and Chap. 33 Suggested exercises: 32.1

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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)


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.


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


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


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!


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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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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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+


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”.


Magnetic Force on Moving Charges

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Magnetic Force on Moving Charges


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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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



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

http://scitec.uwichill.edu.bb/cmp/online/P10D/lecture13/motion.htm

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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


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

http://scitec.uwichill.edu.bb/cmp/online/P10D/lecture13/motor.htm

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Magnetic Dipole Moment

Aμ iDefinition:

Torque: Bμτ


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


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).


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


Activity

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Activity


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


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


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.


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!


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


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.


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


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


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


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.


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


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


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


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


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


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.

Documents

Reading: Chap. 32 and Chap. 33 Suggested exercises: 32.1