Magnetism

1. Prelude to an Exam Allegro con brio

Next Friday – EXAMINATION #2
Watch 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 Topic
Magnetism
2

3. DEMO

Magnetism
3

4. Lodestone (Mineral)

• Lodestones attracted
iron 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 N
Shaded 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 bar
magnets, each with N and S poles.
Magnetism
9

10. Consider a Permanent Magnet

B
N
Magnetism
S
10

11. Introduce Another Permanent Magnet

B
N
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

B
N
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

B
N
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 in
space.
– 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

X
Field INTO Paper
Magnetism
B
Field OUT of Paper
15

16. Experiments with Magnets Show

• Current carrying wire produces a
circular magnetic field around it.
• Force on Compass Needle (or magnet)
increases with current.
Magnetism
16

17. Current Carrying Wire

Current into
the 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 space
surrounding 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

Magnetism
20

21. Inside it all.

8000
Miles
Magnetism
21

22. On the surface it looks like this..

Magnetism
22

23. Inside: Warmer than Floriduh

Magnetism
23

24. Much Warmer than Floriduh

Magnetism
24

25. Finally

Magnetism
25

26. In Between

The molten iron core exists in a magnetic
field 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 a
rock. 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

Magnetism
28

29. This planet is really screwed up!

NORTH
POLE
Magnetism
SOUTH POLE
29

30. Repeat

Navigation
DIRECTION
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.

Magnetism
31

32. Rowland’s Experiment

Field is created by
any 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

B
q
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 Earth
3 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 force
acts 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 magnetic
field.
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 = r
s
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.

Magnetism
45

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

Magnetism
48

49. Mass Spectrometer

Smaller Mass
Magnetism
49

50.

Magnetism
50

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 Before
r
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 Forces
Magnetism
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 and
magnetic 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/B
Magnetism
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.

Magnetism
60

61. VECTOR CALCULATIONS

Magnetism
i
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 uniform
magnetic 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

eB
Let
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.

Magnetism
64

65. Wires

• A wire with a current
contains 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 moving
with 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 Force
mg
Where does B point????
Magnetism
Into the paper.
Current = i
iLB mg
mg
B
iL
67

68. MagLev

Magnetism
68

69. Magnetic Repulsion

Magnetism
69

70. Detail

Magnetism
70

71. Moving Right Along ….

Magnetism
71

72. Acceleration

Magnetism
72

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 force
on the ends??
Loop will tend to rotate due to the torque the field applies to the loop.
Magnetism
74

75. The Loop

OBSERVATION
Force 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

Magnetism
76

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 Today
Return to Magnetic Material
Exams not yet returned. Sorry.
Magnetism
78

79. Wires

• A wire with a current
contains 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 moving
with 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 force
on 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 net
torque 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 magnetic
dipole 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 sin
Consider 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

given
magnetic 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 maximum
when 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 dipole
U( ) -m B
m and B want to be aligned!
Magnetism
88

89. The Hall Effect

Magnetism
89

90. What Does it Do?

• Allows the measurement of
Magnetic 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 moving
to 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

Magnetism
Carrier 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, the
field 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

Magnetism
94

95. Try to remember…

1
rdq
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 of
Biot-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 the
Magnetic field associated with a
straight wire seemed to vary with 1/d.
• We can now PROVE this!
Magnetism
97

98. From the Past

Using Magnets
Magnetism
98

99.

Right-hand rule: Grasp the
element 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 R
2
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

Magnetism
103

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/2
m 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 0iR
ds
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

Magnetism
108

109. More arc…

ds
ds 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 0
No Field at C
Magnetism
110

111. Force Between Two Current Carrying Straight Parallel Conductors

Wire “a” creates
a 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 to
wire " 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 parallel
wires 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

AMPERE
Magnetism
114

115. Welcome to Andre’ Marie Ampere’s Law

Normally written as a “circulation” vector
equation.
We will look at another form, but first…
Magnetism
115

116. Remember GAUSS’S LAW??

E
d
A
Surface
Integral
Magnetism
qenclosed
0
116

117. Gauss’s Law

• Made calculations easier than
integration 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 been
lifted 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 Law
m0 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 Law
r
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
Magnetism
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 0
i
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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 INTEGRATION
Around a closed path or LOOP.
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124

125. Ampere’s Law

B
d
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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126

127. Another Right Hand Rule

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127

128. COMPARE

B
d
s
m
i
0
enclosed
Line Integral
E
d
A
Surface Integral
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qenclosed
0
128

129. Simple Example

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129

130. Field Around a Long Straight Wire

B
d
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

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131

132. The Calculation

B ds B ds 2 rB m i
0 enclosed
ienclosed
r 2
i 2
R
and
m 0i
B
r
2
2 R
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132

133.

B
m 0i
2 R
R
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r
133

134. Procedure

• Apply Ampere’s law only to highly symmetrical
situations.
• 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

emerges
perpendicularly 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 Components
Cancel
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136

137. Apply Ampere to Circuit

L
B
Infinite Extent
B
current per unit length
Current inside the loop is therefore :
i L
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137

138. The “Math”

B
Infinite 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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139

140. Inside the Solenoid

For an “INFINITE” (long) solenoid the previous
problem 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 Length
Weak 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 Field
Region
• Minimal field outside
– except at the ends!
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144

145. Two Coils

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145

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 less
dense than
inner portion
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147

148. The Toroid

Ampere again. We need only worry
about 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