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# Magnetism

## 1. Prelude to an Exam Allegro con brio

Next Friday – EXAMINATION #2Watch those WebAssigns .. no more

extensions.

Monday will be

• A Quiz on Circuits

• A review of circuits and some other problems.

Wednesday, more on Magnetism. Only day 1

on the exm. Watch for a new Webassign.

Magnetism

1

## 2. Magnetism

A Whole New TopicMagnetism

2

## 3. DEMO

Magnetism3

## 4. Lodestone (Mineral)

• Lodestones attractediron filings.

• Lodestones seemed to

attract each other.

• Used as a compass.

– One end always

pointed north.

• Lodestone is a natural

magnet.

Magnetism

4

## 5. Magnetism

• Refrigerators are attracted to magnets!Magnetism

5

## 6. Applications

• Motors• Navigation – Compass

• Magnetic Tapes

– Music, Data

• Television

– Beam deflection Coil

• Magnetic Resonance Imaging

• High Energy Physics Research

Magnetism

6

## 7. Magnets

S NShaded End is NORTH Pole

Shaded End of a compass points

to the NORTH.

Magnetism

• Like Poles Repel

• Opposite Poles

Attract

• Magnetic Poles are

only found in pairs.

– No magnetic

monopoles have

ever been

observed.

7

## 8. Observations

+Observations

+

+

• Bring a magnet to a charged electroscope and

nothing happens. No forces.

• Bring a magnet near some metals (Co, Fe, Ni

…) and it will be attracted to the magnet.

– The metal will be attracted to both the N and S

poles independently.

– Some metals are not attracted at all.

– Wood is NOT attracted to a magnet.

– Neither is water.

• A magnet will force a compass needle to align

with it. (No big Surprise.)

Magnetism

8

## 9. Magnets

Cutting a bar magnet in half produces TWO barmagnets, each with N and S poles.

Magnetism

9

## 10. Consider a Permanent Magnet

BN

Magnetism

S

10

## 11. Introduce Another Permanent Magnet

BN

N

S

pivot

S

The bar magnet (a magnetic dipole) wants to align with the B-field.

Magnetism

11

## 12. Field of a Permanent Magnet

BN

N

S

S

The south pole of the small bar magnet is attracted towards

the north pole of the big magnet.

Also, the small bar magnet (a magnetic dipole) wants to align

with the B-field.

The

field attracts and exerts a torque on the small magnet.

Magnetism

12

## 13. Field of a Permanent Magnet

BN

N

S

S

The bar magnet (a magnetic dipole) wants to align with the B-field.

The field exerts a torque on the dipole

Magnetism

13

## 14. The Magnetic Field

• Similar to Electric Field … exists inspace.

– Has Magnitude AND Direction.

• The “stronger” this field, the greater is

the ability of the field to interact with

a magnet.

Magnetism

14

## 15. Convention For Magnetic Fields

XField INTO Paper

Magnetism

B

Field OUT of Paper

15

## 16. Experiments with Magnets Show

• Current carrying wire produces acircular magnetic field around it.

• Force on Compass Needle (or magnet)

increases with current.

Magnetism

16

## 17. Current Carrying Wire

Current intothe page.

B

Right hand RuleThumb in direction of the current

Fingers curl in the direction of B

Magnetism

17

## 18. Current Carrying Wire

• B field is created at ALL POINTS in spacesurrounding the wire.

• The B field had magnitude and direction.

• Force on a magnet increases with the

current.

• Force is found to vary as ~(1/d) from the

wire.

Magnetism

18

## 19. Compass and B Field

• Observations– North Pole of magnets

tend to move toward

the direction of B while

S pole goes the other

way.

– Field exerts a

TORQUE on a

compass needle.

– Compass needle is a

magnetic dipole.

– North Pole of

compass points

toward the NORTH.

Magnetism

19

## 20. Planet Earth

Magnetism20

## 21. Inside it all.

8000Miles

Magnetism

21

## 22. On the surface it looks like this..

Magnetism22

## 23. Inside: Warmer than Floriduh

Magnetism23

## 24. Much Warmer than Floriduh

Magnetism24

## 25. Finally

Magnetism25

## 26. In Between

The molten iron core exists in a magneticfield that had been created from other

sources (sun…).

The fluid is rotating in this field.

This motion causes a current in the molten

metal.

The current causes a magnetic field.

The process is self-sustaining.

The driving force is the heat (energy) that

is generated in the core of the planet.

Magnetism

26

## 27.

After molten lava emerges from a volcano, it solidifies to arock. In most cases it is a black rock known as basalt, which is

faintly magnetic, like iron emerging from a melt. Its

magnetization is in the direction of the local magnetic force

at the time when it cools down.

Instruments can measure the magnetization of basalt.

Therefore, if a volcano has produced many lava flows over a

past period, scientists can analyze the magnetizations of the

various flows and from them get an idea on how the direction

of the local Earth's field varied in the past. Surprisingly, this

procedure suggested that times existed when the

magnetization had the opposite direction from today's. All

sorts of explanation were proposed, but in the end the only

one which passed all tests was that in the distant past,

indeed, the magnetic polarity of the Earth was sometimes

reversed.

Magnetism

27

## 28. Ancient Navigation

Magnetism28

## 29. This planet is really screwed up!

NORTHPOLE

Magnetism

SOUTH POLE

29

## 30. Repeat

NavigationDIRECTION

N

S

If N direction

is pointed to by

the NORTH pole

of the Compass

Needle, then the

pole at the NORTH

of our planet must

be a SOUTH MAGNETIC

POLE!

Compass

Direction

Navigation

DIRECTION

S

N

And it REVERSES from time to time.

Magnetism

30

## 31.

Magnetism31

## 32. Rowland’s Experiment

Field is created byany moving charge.

Rotating

INSULATING

Disk

which is

CHARGED

+ or –

on exterior.

++

Magnetism

+ +

++

xxx

xxx B

xxx

Increases with

charge on the

disk.

Increases with

angular velocity of

the disk.

Electrical curent is a

moving charge.

32

## 33. A Look at the Physics

Bq

v

q B

There is NO force on

a charge placed into a

magnetic field if the

charge is NOT moving.

There is no force if the charge

moves parallel to the field.

• If the charge is moving, there

is a force on the charge,

perpendicular to both v and B.

F=qvxB

Magnetism

33

## 34. WHAT THE HECK IS THAT???

• A WHAT PRODUCT?• A CROSS PRODUCT – Like an

angry one??

• Alas, yes ….

• F=qv X B

Magnetism

34

## 35. The Lorentz Force

This can be summarized as:F qv B

F

or:

F qvBsin

v

B

mq

is the angle between B and V

Magnetism

35

## 36. Note

B is sort of the Force per unit(charge-velocity)

Whatever that is!!

Magnetism

36

## 37. Practice

B and v are parallel.Crossproduct is zero.

So is the force.

Which way is the Force???

Magnetism

37

## 38. Units

F Bqv Sin(θ )Units :

F

N

N

B

qv Cm / s Amp m

Magnetism

1 tesla 1 T 1 N/(A - m)

38

## 39. teslas are

At the Surface of the Earth3 x 10-5 T

Typical Refrigerator Magnet

5 x 10-3 T

Laboratory Magnet

0.1 T

Large Superconducting Magnet

10 T

Magnetism

39

## 40. The Magnetic Force is Different From the Electric Force.

Whereas the electric forceacts in the same direction as

the field:

The magnetic force acts in a

direction orthogonal to the

field:

F qE

F qv B

(Use “Right-Hand” Rule to

determine direction of F)

And

--the

charge

must

be

moving

!!

Magnetism

40

## 41. So…

A moving charge can create a magneticfield.

A moving charge is acted upon by a

magnetic field.

In Magnetism, things move.

In the Electric Field, forces and the

field can be created by stationary

charges.

Magnetism

41

## 42. Trajectory of Charged Particles in a Magnetic Field

(B field points into plane of paper.)+

+B

+

v+

+

+

+

+

+

+

+

+ F

+

+

+

+ F +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

B

+

+

+

+

+

Magnetism

v

42

## 43. Trajectory of Charged Particles in a Magnetic Field

(B field points into plane of paper.)+

+B

+

v+

+

+

+

+

+

+

+

+ F

+

+

+

+ F +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

B

+

+

+

+

+

Magnetism

v

Magnetic Force is a centripetal force

43

## 44. Review of Rotational Motion

= s / r s = r ds/dt = d /dt r v = rs

r

= angle, = angular speed, = angular acceleration

at

ar

at = r

tangential acceleration

ar = v2 / r radial acceleration

The radial acceleration changes the direction of motion,

while the tangential acceleration changes the speed.

Uniform Circular Motion

ar

= constant v and ar constant but direction changes

Magnetism

v

ar = v2/r = 2 r

KE = ½ mv2 = ½ mw2r2

F = mar = mv2/r = m 2r

44

## 45.

Magnetism45

## 46. Radius of a Charged Particle Orbit in a Magnetic Field

+B+

+

v+

+

+

+

+

+

+

+

+

+

r

+

+

+

+

+

+

F

+

Magnetism

Centripetal

Force

=

Magnetic

Force

mv 2

qvB

r

mv

r

qB

Note: as F v , the magnetic

46

force does no work!

## 47. Cyclotron Frequency

+B+

+

v+

+

+

+

+

+

+

+

+

+

r

+

+

+

+

+

+

F

+

Magnetism

The time taken to complete one

orbit is:

2 r

v

2 mv

v qB

T

1

qB

f

T 2 m

qB

c 2 f

m

47

## 48. More Circular Type Motion in a Magnetic Field

Magnetism48

## 49. Mass Spectrometer

Smaller MassMagnetism

49

## 50.

Magnetism50

## 51.

Cyclotron Frequency+B

+

+

v+

+

+

+

+

+

+

+

+

+

r

+

+

+

+

+

+

F

+

Magnetism

The time taken to complete one

orbit is:

2 r

v

2 mv

v qB

T

1

qB

f

T 2 m

qB

c 2 f

m

51

## 52. An Example

A beam of electrons whose kinetic energy is K emerges from a thin-foil“window” at the end of an accelerator tube. There is a metal plate a distance d

from this window and perpendicular to the direction of the emerging beam. Show

that we can prevent the beam from hitting the plate if we apply a uniform

magnetic field B such that

2mK

B 2 2

ed

Magnetism

52

## 53. Problem Continued

From Beforer

mv

r

qB

1 2

2K

K mv so v

2

m

m 2K

2mK

r

d

2 2

eB m

e B

Solve for B :

Magnetism

2mK

B

e2d 2

53

## 54. Some New Stuff

Magnetism and ForcesMagnetism

54

## 55.

Let’s Look at the effect of crossed E and B Fields:x x x B

E

x x x

v

q , m

Magnetism

55

## 56.

What is the relation between the intensities of the electric andmagnetic fields for the particle to move in a straight line ?.

x x x B

E

x x x

v

q• m

FE = q E and FB = q v B

If FE = FB the particle will move

following a straight line trajectory

qE=qvB

v=E/B

FB FE

Magnetism

56

## 57. What does this mean??

v=E/BMagnetism

This equation only

contains the E and

B fields in it.

Mass is missing!

Charge is missing!

This configuration

is a velocity filter!

57

## 58. “Real” Mass Spectrometer

Create ions from injected species.This will contain various masses, charges

and velocities.

These are usually accelerated to a certain

ENERGY (KeV) by an applied electric

field.

The crossed field will only allow a

selected velocity to go forward into the

MS.

From before: R=mv/Bq

Magnetism

58

## 59. Components of MS:

Accelerate the ions through a known potential difference .1 2

mv qVapplied

2

So

q 1 2 1

v

m 2 Vapplied

The velocity can be selected via an E x B field and the MS will

separate by:

mv

R

Bq

Magnetism

Unknown is mass to charge ratio

which can be sorted from the spectrum

59

## 60.

Magnetism60

## 61. VECTOR CALCULATIONS

Magnetismi

a b ax

j

ay

k

az

bx

by

bz

61

## 62. Problem: A Vector Example

A proton of charge +e and mass m is projected into a uniformmagnetic field B=Bi with an initial velocity v=v0xi +v0yj. Find

the velocity at a later time.

F ev B

i

F vx

B

j

vy

0

k

vz eB(v z j v y k ) ma

0

dvx

max m

etc. for y and z.

dt

Equating components :

dv y eB

dv x

0

v z and

dt

dt

m

Magnetism

vx is constant

dv z

eB

vy

dt

m

62

## 63. More

eBLet

m

d 2v y

dv z

2

vy

2

dt

dt

d 2v y

2

vy 0

2

dt

Simple circular motion!

v y v0 y cos( t )

Same thing for z.

Magnetism

63

## 64.

Magnetism64

## 65. Wires

• A wire with a currentcontains moving charges.

• A magnetic field will

apply a force to those

moving charges.

• This results in a force

on the wire itself.

– The electron’s sort of

PUSH on the side of the

wire.

F

Remember: Electrons go the “other way”.

Magnetism

65

## 66. The Wire in More Detail

Assume all electrons are movingwith the same velocity vd.

q i t i

L

vd

F qvd B i

L

vd B iLB

vd

vector :

F iL B

L in the direction of the motion

of POSITIVE charge (i).

B out of plane of the paper

Magnetism

66

## 67. Magnetic Levitation

Magnetic Forcemg

Where does B point????

Magnetism

Into the paper.

Current = i

iLB mg

mg

B

iL

67

## 68. MagLev

Magnetism68

## 69. Magnetic Repulsion

Magnetism69

## 70. Detail

Magnetism70

## 71. Moving Right Along ….

Magnetism71

## 72. Acceleration

Magnetism72

## 73. Don’t Buy A Ticket Quite Yet..

This is still experimental.Much development still required.

Some of these attempts have been

abandoned because of the high cost

of building a MagLev train.

Probably 10-20 years out.

Or More.

Magnetism

73

## 74. Current Loop

What is forceon the ends??

Loop will tend to rotate due to the torque the field applies to the loop.

Magnetism

74

## 75. The Loop

OBSERVATIONForce on Side 2 is out

of the paper and that on

the opposite side is into

the paper. No net force

tending to rotate the loop

due to either of these forces.

The net force on the loop is

also zero,

pivot

Magnetism

75

## 76. An Application The Galvanometer

Magnetism76

## 77. The other sides

t1=F1 (b/2)Sin( )=(B i a) x (b/2)Sin( )

total torque on

the loop is: 2t1

Total torque:

t=(iaB) bSin( )

=iABSin( )

(A=Area)

Magnetism

77

## 78. Watcha Gonna Do

Quiz TodayReturn to Magnetic Material

Exams not yet returned. Sorry.

Magnetism

78

## 79. Wires

• A wire with a currentcontains moving charges.

• A magnetic field will

apply a force to those

moving charges.

• This results in a force

on the wire itself.

– The electron’s sort of

PUSH on the side of the

wire.

F

Remember: Electrons go the “other way”.

Magnetism

79

## 80. The Wire in More Detail

Assume all electrons are movingwith the same velocity vd.

q i t i

L

vd

F qvd B i

L

vd B iLB

vd

vector :

F iL B

L in the direction of the motion

of POSITIVE charge (i).

B out of plane of the paper

Magnetism

80

## 81. Current Loop

What is forceon the ends??

Loop will tend to rotate due to the torque the field applies to the loop.

Magnetism

81

## 82. Last Time

t1=F1 (b/2)Sin( )=(B i a) x (b/2)Sin( )

total torque on

the loop is: 2t1

Total torque:

t=(iaB) bSin( )

=iABSin( )

(A=Area)

Magnetism

82

## 83. A Coil

For a COIL of N turns, the nettorque on the coil is therefore :

τ NiABSin(θ )

Normal to the

coil

RIGHT HAND RULE TO FIND NORMAL

TO THE COIL:

“Point or curl you’re the fingers of your right

hand in the direction of the current and your

thumb will point in the direction of the normal

to the coil.

Magnetism

83

## 84. Dipole Moment Definition

Define the magneticdipole moment of

the coil m as:

m=NiA

Magnetism

We can convert this

to a vector with A

as defined as being

normal to the area as

in the previous slide.

84

## 85. Current Loop

t iAB sinConsider a coil with N turns of wire.

Define Magnetic Moment

μ NiA

and

t μ B

Magnetism

85

## 86. A length L of wire carries a current i. Show that if the wire is formed into a circular coil, then the maximum torque in a

givenmagnetic field is developed when the coil has one turn only, and

that maximum torque has the magnitude … well, let’s see.

Circumference = L/N

Magnetism

L

2 r

N

L

r

2 N

86

## 87. Problem continued…

t NiAB since sin( m , B) is maximumwhen the angle is 90 o

A r 2

L

t NiB

2 N

2

2

L N

L2

iB

t iB

(BiA)

2

4 N

2 N

Maximum when N 1 and

Magnetism

iBL2

t

4

87

## 88. Energy

Like the electric dipoleU( ) -m B

m and B want to be aligned!

Magnetism

88

## 89. The Hall Effect

Magnetism89

## 90. What Does it Do?

• Allows the measurement ofMagnetic Field if a material is

known.

• Allows the determination of the

“type” of current carrier in

semiconductors if the magnetic

field is known.

• Electrons

Magnetism

• Holes

90

## 91. Hall Geometry (+ Charge)

Current is movingto the right. (vd)

Magnetic field will

force the charge to

the top.

This leaves a

deficit (-) charge on

the bottom.

This creates an

electric field and a

potential difference.

Magnetism

91

## 92. Negative Carriers

MagnetismCarrier is negative.

Current still to the

right.

Force pushes

negative charges to

the top.

Positive charge

builds up on the

bottom.

Sign of the potential

difference is

reversed.

92

## 93. Hall Math

• Eventually, thefield due to the

Hall effect will

allow the current

to travel undeflected through

the conductor.

balance :

qvd B qE Hall q

VHall

w

or

VHall wvd B

J nevd i / A

vd

i

neA

VHall wvd B wB

i

neA

A wt

VHall wB

Magnetism

iB

i

newt net

93

## 94. Magnetic Fields Due to Currents Chapter 30

Magnetism94

## 95. Try to remember…

1rdq

r dq

dE

2

3

4 0 r r

4 0 r

1

r

UNIT VECTOR

r

Magnetism

95

## 96. For the Magnetic Field, current “elements” create the field.

This is the Law ofBiot-Savart

In a similar fashion to E field :

m 0 id s runit m 0 id s r

B

2

4

r

4 r 3

permeabili ty

m 0 4 10 7 Tm / A 1.26 10 7 Tm

BY DEFINITION

Magnetism

96

## 97. Magnetic Field of a Straight Wire

• We intimated via magnets that theMagnetic field associated with a

straight wire seemed to vary with 1/d.

• We can now PROVE this!

Magnetism

97

## 98. From the Past

Using MagnetsMagnetism

98

## 99.

Right-hand rule: Grasp theelement in your right hand with

your extended thumb pointing

in the direction of the current.

Your fingers will then naturally

curl around in the direction of

the magnetic field lines due to

that element.

Magnetism

99

## 100. Let’s Calculate the FIELD

Note:For ALL current elements

ds X r

is into the page

Magnetism

100

## 101. The Details

m 0 ids sin( )dB

4

r2

Negative portion of the wire

contribute s an equal amount so we

integrate from 0 to and DOUBLE it.

m 0i sin( )ds

B

2

2 0

r

Magnetism

101

## 102. Moving right along

r s R2

2

sin sin( )

R

s2 R2

So

m 0i

m 0i

rds

B

3

/

2

2 0 s 2 R 2

2 R

Magnetism

1/d

102

## 103. A bit more complicated A finite wire

Magnetism103

## 104. P1

NOTE : sin( ) sin( )ds r ds r sin( )

r

ds

R

sin( )

r

m 0i ds sin( )

dB

2

4

r

r s R

Magnetism

2

2 1/ 2

104

## 105. More P1

L/2m 0i

ds

B

3/ 2

2

2

4 L / 2 s R

and

m 0i

L

B

2 R L2 4 R 2

Magnetism

when L ,

m 0i

B

2 R

105

## 106. P2

m 0iRds

B

3/ 2

2

2

4 L s R

0

or

m 0i

L

B

4 R s 2 R 2

Magnetism

106

## 107.

APPLICATION:Find the magnetic field B at point P in for i = 10 A and a = 8.0

cm.

Magnetism

107

## 108. Circular Arc of Wire

Magnetism108

## 109. More arc…

dsds Rd

m 0 ids m 0 iRd

dB

2

4 R

4 R 2

m 0 iRd m 0i

B dB

d

2

4 R

4 R 0

0

m 0 i

B

at point C

4 R

Magnetism

109

## 110. Howya Do Dat??

ds r 0No Field at C

Magnetism

110

## 111. Force Between Two Current Carrying Straight Parallel Conductors

Wire “a” createsa field at wire “b”

Magnetism

Current in wire “b” sees a

force because it is moving

in the magnetic field of “a”.

111

## 112. The Calculation

The FIELD at wire " b" due towire " a" is what we just calculated :

m 0ia

Bat "b"

2 d

Fon "b" ib L B

Since L and B are at right angles...

m 0 Lia ib

F

2 d

Magnetism

112

## 113. Definition of the Ampere

The force acting between currents in parallelwires is the basis for the definition of the

ampere, which is one of the seven SI base

units. The definition, adopted in 1946, is

this: The ampere is that constant current

which, if maintained in two straight, parallel

conductors of infinite length, of negligible

circular cross section, and placed 1 m apart

in vacuum, would produce on each of these

conductors a force of magnitude 2 x 10-7

newton per meter of length.

Magnetism

113

## 114. TRANSITION

AMPEREMagnetism

114

## 115. Welcome to Andre’ Marie Ampere’s Law

Normally written as a “circulation” vectorequation.

We will look at another form, but first…

Magnetism

115

## 116. Remember GAUSS’S LAW??

Ed

A

Surface

Integral

Magnetism

qenclosed

0

116

## 117. Gauss’s Law

• Made calculations easier thanintegration over a charge distribution.

• Applied to situations of HIGH

SYMMETRY.

• Gaussian SURFACE had to be defined

which was consistent with the geometry.

• AMPERE’S Law is the Gauss’ Law of

Magnetism! (Sorry)

Magnetism

117

## 118.

The next few slides have beenlifted from Seb Oliver

on the internet

Whoever he is!

Magnetism

118

## 119. Biot-Savart

• The “Coulombs Law of Magnetism”m0 ids rˆ

dB

2

4 r

Magnetism

119

## 120. Invisible Summary

• Biot-Savart Lawm0 ids rˆ

dB

2

4 r

m0 I

– (Field produced by wires)

B

2R

– Centre of a wire loop radius R

m NI

B 0

– Centre of a tight Wire Coil with N turns

2R

– Distance a from long straight wire

• Force between two wires

• Definition of Ampere

Magnetism

m0 I

B

2 a

F m 0 I1 I 2

l

2 a

120

## 121. Magnetic Field from a long wire

Using Biot-Savart Lawr

I

Take a short vector

on a circle, ds

B

ds

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B ds B ds cos

0 cos 1

Thus the dot product of B &

the short vector ds is:

m0 I

B

2 r

B ds B ds

m0 I

B ds

ds

2 r

121

## 122. Sum B.ds around a circular path

Sum.

B ds

around a circular path

m0 I

B ds

ds

2 r

r

I

B

Sum this around the whole ring

ds

Circumference

of circle

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B ds

ds 2 r

m0 I

ds

2 r

m0 I

ds

2 r

m0 I

B ds

2 r m 0 Ι

2 r

122

## 123. Consider a different path

B ds 0i

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• Field goes as

1/r

• Path goes as r.

• Integral

independent of

r

123

## 124. SO, AMPERE’S LAW by SUPERPOSITION:

We will do a LINE INTEGRATIONAround a closed path or LOOP.

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124

## 125. Ampere’s Law

Bd

s

m

i

0

enclosed

USE THE RIGHT HAND RULE IN THESE CALCULATIONS

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125

## 126. The Right Hand Rule

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## 127. Another Right Hand Rule

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## 128. COMPARE

Bd

s

m

i

0

enclosed

Line Integral

E

d

A

Surface Integral

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qenclosed

0

128

## 129. Simple Example

Magnetism129

## 130. Field Around a Long Straight Wire

Bd

s

m

i

0

enclosed

B 2 r m 0i

m 0i

B

2 r

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130

## 131. Field INSIDE a Wire Carrying UNIFORM Current

Magnetism131

## 132. The Calculation

B ds B ds 2 rB m i0 enclosed

ienclosed

r 2

i 2

R

and

m 0i

B

r

2

2 R

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132

## 133.

Bm 0i

2 R

R

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r

133

## 134. Procedure

• Apply Ampere’s law only to highly symmetricalsituations.

• Superposition works.

– Two wires can be treated separately and the

results added (VECTORIALLY!)

• The individual parts of the calculation can be

handled (usually) without the use of vector

calculations because of the symmetry.

• THIS IS SORT OF LIKE GAUSS’s LAW

WITH AN ATTITUDE!

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134

## 135. The figure below shows a cross section of an infinite conducting sheet carrying a current per unit x-length of l; the current

emergesperpendicularly out of the page. (a) Use the Biot–Savart law and

symmetry to show that for all points P above the sheet, and all points

P´ below it, the magnetic field B is parallel to the sheet and directed

as shown. (b) Use Ampere's law to find B at all points P and P´.

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135

## 136. FIRST PART

Vertical ComponentsCancel

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136

## 137. Apply Ampere to Circuit

LB

Infinite Extent

B

current per unit length

Current inside the loop is therefore :

i L

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137

## 138. The “Math”

BInfinite Extent

B

B ds m i

0 enclosed

BL BL m 0 L

B

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m 0

2

138

## 139. A Physical Solenoid

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## 140. Inside the Solenoid

For an “INFINITE” (long) solenoid the previousproblem and SUPERPOSITION suggests that the

field OUTSIDE this solenoid is ZERO!

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140

## 141. More on Long Solenoid

Field is ZERO!Field looks UNIFORM

Field is ZERO

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141

## 142. The real thing…..

Finite LengthWeak Field

Stronger - Leakage

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142

## 143. Another Way

Ampere :B ds m i

0 enclosed

0h Bh m 0 nih

B m 0 ni

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143

## 144. Application

• Creation of Uniform Magnetic FieldRegion

• Minimal field outside

– except at the ends!

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144

## 145. Two Coils

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## 146. “Real” Helmholtz Coils

Used for experiments.Can be aligned to cancel

out the Earth’s magnetic

field for critical measurements.

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146

## 147. The Toroid

Slightly lessdense than

inner portion

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147

## 148. The Toroid

Ampere again. We need only worryabout the INNER coil contained in

the path of integratio n :

B ds B 2 r m Ni (N total # turns)

0

so

m 0 Ni

B

2 r

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148