magnetism1 prelude to an exam allegro con brio next friday – examination #2 next friday –...
TRANSCRIPT
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MagnetismMagnetism 11
Prelude to an ExamPrelude to an ExamAllegro con brio Allegro con brio
Next Friday – EXAMINATION #2Next Friday – EXAMINATION #2 Watch those WebAssigns .. no more Watch those WebAssigns .. no more
extensions.extensions. Monday will beMonday will be
• A Quiz on CircuitsA Quiz on Circuits• A review of circuits and some other problems.A review of circuits and some other problems.
Wednesday, more on Magnetism. Only Wednesday, more on Magnetism. Only day 1 on the exm. Watch for a new day 1 on the exm. Watch for a new Webassign.Webassign.
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MagnetismMagnetism 22
MagnetismMagnetism
A Whole New TopicA Whole New Topic
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Magnetism 3
DEMO
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Magnetism 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.
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Magnetism 5
Magnetism
• Refrigerators are attracted to magnets!
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Magnetism 6
ApplicationsApplications
• Motors• Navigation – Compass• Magnetic Tapes
– Music, Data
• Television– Beam deflection Coil
• Magnetic Resonance Imaging• High Energy Physics Research
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Magnetism 7
Magnets
• Like Poles Repel• Opposite Poles
Attract• Magnetic Poles
are only found in pairs.– No magnetic
monopoles have ever been observed.
Shaded End is NORTH PoleShaded End of a compass points
to the NORTH.
S N
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Magnetism 8
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.)
+
++
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Magnetism 9
Magnets
Cutting a bar magnet in half produces TWO bar magnets, each with N and S poles.
Magnetic Field
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Magnetism 10
Consider a Permanent Magnet
N S
B
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Magnetism 11
Introduce Another Permanent Magnet
N S
B
N
S
The bar magnet (a magnetic dipole) wants to align with the B-field.
pivot
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Magnetism 12
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.
Field of a Permanent Magnet
N S
B
N
S
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Magnetism 13
Field of a Permanent Magnet
N S
B
N S
The field exerts a torque on the dipole
The bar magnet (a magnetic dipole) wants to align with the B-field.
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Magnetism 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.
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Magnetism 15
Convention For Magnetic Fields
X
Field INTO Paper Field OUT of Paper
B
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Magnetism 16
Experiments with Magnets Show
• Current carrying wire produces a circular magnetic field around it.
• Force on Compass Needle (or magnet) increases with current.
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Magnetism 17
Current Carrying Wire
Current intothe page.
B
Right hand Rule-Thumb in direction of the currentFingers curl in the direction of B
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Magnetism 18
Current Carrying WireCurrent 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.
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Magnetism 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.
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MagnetismMagnetism 2020
Planet EarthPlanet Earth
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MagnetismMagnetism 2121
Inside it all.Inside it all.
8000Miles
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MagnetismMagnetism 2222
On the surface it looks like this..On the surface it looks like this..
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MagnetismMagnetism 2323
Inside: Warmer than Inside: Warmer than FloriduhFloriduh
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MagnetismMagnetism 2424
Much Warmer than FloriduhMuch Warmer than Floriduh
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MagnetismMagnetism 2525
FinallyFinally
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MagnetismMagnetism 2626
In BetweenIn Between
The molten iron core exists in a magnetic The molten iron core exists in a magnetic field that had been created from other field that had been created from other sources (sun…).sources (sun…).
The fluid is rotating in this field.The fluid is rotating in this field. This motion causes a current in the molten This motion causes a current in the molten
metal.metal. The current causes a magnetic field.The current causes a magnetic field. The process is self-sustaining.The process is self-sustaining. The driving force is the heat (energy) that The driving force is the heat (energy) that
is generated in the core of the planet.is generated in the core of the planet.
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MagnetismMagnetism 2727
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.
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MagnetismMagnetism 2828
Ancient NavigationAncient Navigation
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MagnetismMagnetism 2929
This planet is really screwed up!This planet is really screwed up!NORTHPOLE
SOUTH POLE
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Magnetism 30
CompassDirection
Repeat
NavigationDIRECTION
N
S
If N directionis pointed to bythe NORTH poleof the Compass Needle, then the
pole at the NORTHof our planet must
be a SOUTH MAGNETICPOLE!
NavigationDIRECTION
S
N
And it REVERSES from time to time.
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MagnetismMagnetism 3131
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Magnetism 32
Rowland’s Experiment
RotatingINSULATING
Diskwhich is CHARGED
+ or –on exterior.
xxxxxx Bxxx
Field is created byany moving charge.
Increases with charge on thedisk.
Increases withangular velocity ofthe disk.
Electrical curent is a moving charge.
++ + + ++
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Magnetism 33
A Look at the Physics
B
q
There is NO force ona charge placed into amagnetic field if thecharge is NOT moving.
Bq
v
• If the charge is moving, thereis a force on the charge,perpendicular to both v and B.
F = q v x B
There is no force if the chargemoves parallel to the field.
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Magnetism 34
WHAT THE HECK IS THAT???
• A WHAT PRODUCT?• A CROSS PRODUCT – Like an angry
one??• Alas, yes ….
• F=qv X B
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Magnetism 35
The Lorentz Force
This can be summarized as: F q B v
vF
Bqm
or:
F qvBsin
is the angle between B and V
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Magnetism 36
Note
B is sort of the Force per unit (charge-velocity)
Whatever that is!!
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Magnetism 37
Practice
Which way is the Force???
B and v are parallel.Crossproduct is zero.So is the force.
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Magnetism 38
Units
m)-N/(A 1 T 1 tesla1
mAmp
N
/
:
)
sCm
N
qv
FB
Units
Bqv Sin(θF
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Magnetism 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
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Magnetism 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:
And --- the charge must be moving !!
F q B v
F qE
(Use “Right-Hand” Rule to determine direction of F)
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MagnetismMagnetism 4141
So…So…A moving charge can create a A moving charge can create a magnetic field.magnetic field.A moving charge is acted upon by a A moving charge is acted upon by a magnetic field.magnetic field.
In Magnetism, things move.In Magnetism, things move.In the Electric Field, forces and the In the Electric Field, forces and the field can be created by stationary field can be created by stationary charges.charges.
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Magnetism 42
Trajectory of Charged Particlesin a Magnetic Field
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
vB
F
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
vB
F
(B field points into plane of paper.)
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Magnetism 43
Trajectory of Charged Particlesin a Magnetic Field
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
vvB B
FF
(B field points into plane of paper.)
Magnetic Force is a centripetal force
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Magnetism 44
Review of Rotational Motion
atar
at = r tangential acceleration
ar = v2 / r radial acceleration
The radial acceleration changes the direction of motion,while the tangential acceleration changes the speed.
r s = s / r s = r ds/dt = d/dt r v = r
= angle, = angular speed, = angular acceleration
Uniform Circular Motion
= constant v and ar constant but direction changes
ar = v2/r = 2 rF = mar = mv2/r = m2r
KE = ½ mv2 = ½ mw2r2
v
ar
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Magnetism 45
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Magnetism 46
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
Radius of a Charged ParticleOrbit in a Magnetic Field
vB
F
r
mr
q B
r mqB
v v
v
2
Centripetal Magnetic Force Force =
Note: as , the magneticforce does no work!
Fv
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Magnetism 47
Cyclotron Frequency
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
vB
F
r
The time taken to complete one orbit is:
qB
m
rT
v
v
2v
2
m
qBf
m
qB
Tf
c
2
2
1
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Magnetism 48
More Circular Type Motion in a Magnetic Field
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Magnetism 49
Mass Spectrometer
Smaller Mass
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Magnetism 50
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Magnetism 51
Cyclotron Frequency
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
vB
F
r
The time taken to complete one orbit is:
qB
m
rT
v
v
2v
2
m
qBf
m
qB
Tf
c
2
2
1
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Magnetism 52
An ExampleA 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
22
2
de
mKB
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Magnetism 53
Problem Continued
r
22
22
2
e
2mKB
:Bfor Solvee
22
eB
m
2 so
2
1
qB
mvr
Before From
d
dB
mK
m
Kr
m
KvmvK
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Magnetism 54
Some New Stuff
Magnetism and Forces
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Magnetism 55
Let’s Look at the effect of crossed E and B Fields:
x
x xx
xx
•q , m
B
v
E
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Magnetism 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 xx
xx
•q m
B
v
E
•FEFB
FE = q E and FB = q v B
If FE = FB the particle will movefollowing a straight line trajectory
q E = q v B
v = E / B
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MagnetismMagnetism 5757
What does this mean??What does this mean??
This equation only This equation only contains the E and contains the E and B fields in it.B fields in it.
Mass is missing!Mass is missing!
Charge is missing!Charge is missing!
This configuration This configuration is a velocity filter!is a velocity filter!
v = E / B
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Magnetism 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
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Magnetism 59
Components of MS:
applied
applied
Vv
m
q
So
qVmv
1
2
1
2
1
.difference potentialknown a through ions theAccelerate
2
2
The velocity can be selected via an E x B field and the MS willseparate by:
Bq
mvR Unknown is mass to charge ratio
which can be sorted from the spectrum
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Magnetism 60
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Magnetism 61
VECTOR CALCULATIONS
zyx
zyx
bbb
aaa
kji
ba
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Magnetism 62
Problem: A Vector ExampleA 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.
yz
xx
yzzyx
vm
eBv
m
eB
dt
dvmma
mvveB
B
vvv
kji
e
dt
dv and
dt
dv 0
dt
dv
:components Equating
z. andy for etc.
)(
00
zyx
akjF
BvF
vx is constant
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Magnetism 63
More
z.for thingSame
)cos(v
motion!circular Simple
0
m
eBLet
0y
22
2
22
2
tv
vdt
vd
vdt
dv
dt
vd
y
yy
yzy
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Magnetism 64
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Magnetism 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”.
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Magnetism 66
The Wire in More Detail
B out of plane of the paper
Assume all electrons are moving with the same velocity vd.
(i). charge POSITIVE of
motion theofdirection in the
:
L
BLF
i
vector
iLBBvv
LiBqvF
v
Litiq
dd
d
d
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Magnetism 67
Magnetic Levitation
Current = img
Magnetic Force
Where does B point???? Into the paper.
iL
mgB
mgiLB
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Magnetism 68
MagLev
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Magnetism 69
Magnetic Repulsion
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Magnetism 70
Detail
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Magnetism 71
Moving Right Along ….
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Magnetism 72
Acceleration
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MagnetismMagnetism 7373
Don’t Buy A Ticket Quite Yet..Don’t Buy A Ticket Quite Yet..
This is still experimental.This is still experimental.
Much development still required.Much development still required.
Some of these attempts have been Some of these attempts have been abandoned because of the high cost abandoned because of the high cost of building a MagLev train.of building a MagLev train.
Probably 10-20 years out.Probably 10-20 years out.
Or More.Or More.
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Magnetism 74
Current Loop
Loop will tend to rotate due to the torque the field applies to the loop.
What is forceon the ends??
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Magnetism 75
The Loop
pivot
OBSERVATION
Force on Side 2 is outof the paper and that onthe opposite side is into the paper. No net forcetending to rotate the loopdue to either of these forces.
The net force on the loop is also zero,
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Magnetism 76
An ApplicationThe Galvanometer
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Magnetism 77
The other sides
1=F1 (b/2)Sin() =(B i a) x (b/2)Sin()
total torque on the loop is: 21
Total torque:
=(iaB) bSin() =iABSin()
(A=Area)
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MagnetismMagnetism 7878
Watcha Gonna DoWatcha Gonna Do
Quiz TodayQuiz Today
Return to Magnetic MaterialReturn to Magnetic Material
Exams not yet returned. Sorry.Exams not yet returned. Sorry.
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Magnetism 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”.
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Magnetism 80
The Wire in More Detail
B out of plane of the paper
Assume all electrons are moving with the same velocity vd.
(i). charge POSITIVE of
motion theofdirection in the
:
L
BLF
i
vector
iLBBvv
LiBqvF
v
Litiq
dd
d
d
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Magnetism 81
Current Loop
Loop will tend to rotate due to the torque the field applies to the loop.
What is forceon the ends??
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Magnetism 82
Last Time
1=F1 (b/2)Sin() =(B i a) x (b/2)Sin()
total torque on the loop is: 21
Total torque:
=(iaB) bSin() =iABSin()
(A=Area)
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Magnetism 83
A Coil
)
: thereforeis coil on the torque
net the turns,N of COIL aFor
NiABSin(θτ
Normal to thecoil
RIGHT HAND RULE TO FIND NORMALTO THE COIL:
“Point or curl you’re the fingers of your right hand in the direction of the current and yourthumb will point in the direction of the normalto the coil.
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Magnetism 84
Dipole Moment Definition
Define the magneticdipole moment ofthe coil as:
=NiA
We can convert thisto a vector with Aas defined as being normal to the area asin the previous slide.
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Magnetism 85
Current Loop
Bμ
Aμ
and
Moment Magnetic Define
wire.of turnsN with coil aConsider
sin
Ni
iAB
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Magnetism 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
N
Lr
N
Lr
2
2
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Magnetism 87
Problem continued…
4
and 1N when Maximum
(BiA) 42
2
90 is angle the when
maximum is B),sin( since
2
2
22
2
2
o
iBL
N
LiB
N
NLiB
N
LNiB
rA
NiAB
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Magnetism 88
Energy
aligned! be want toB and
-)U(
dipole electric theLike
B
Similar to Electric Dipole Moment
Similar to Electric Dipole Moment
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Magnetism 89
The Hall Effect
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Magnetism 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• Holes
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Magnetism 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.
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Magnetism 92
Negative Carriers
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.
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Magnetism 93
Hall Math
• Eventually, the field due to the Hall effect will allow the current to travel un-deflected through the conductor.
net
iB
newt
iwBV
wtAneA
iwBBwvV
neA
iv
AinevJ
BwvV
orw
VqqEBqv
balance
Hall
dHall
d
d
dHall
HallHalld
/
:
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Magnetism 94
Magnetic Fields Due to Currents
Chapter 30
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Magnetism 95
Try to remember…
30
20 4
1
4
1
r
dq
r
dq
rd
rrE
VECTOR UNITr
r
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Magnetism 96
For the Magnetic Field,current “elements” create the
field.
DEFINITION BY
1026.1/104
44
:field fashion tosimilar aIn
770
30
20
TmATm
typermeabilir
id
r
id unit
rsrs
B
E
This is the Law ofBiot-Savart
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Magnetism 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!
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Magnetism 98
From the Past
Using Magnets
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Magnetism 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.
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Magnetism 100
Let’s Calculate the FIELD
Note:
For ALL current elements
ds X r
is into the page
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Magnetism 101
The Details
02
0
20
)sin(
2B
it. DOUBLE and to0 from integrate
wesoamount equalan scontribute
wire theofportion Negative
)sin(
4
r
dsi
r
idsdB
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Magnetism 102
Moving right along
R
i
Rs
rdsiB
SoRs
R
Rsr
22
)sin(sin
0
02/322
0
22
22
1/d
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Magnetism 103
A bit more complicatedA finite wire
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Magnetism 104
P1
)sin()sin(: NOTE
r
ds
2/122
20 )sin(
4
)sin(
)sin(
Rsr
r
dsidB
r
R
rdsrds
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Magnetism 105
More P1
R
i
whenRL
L
R
iB
and
Rs
dsiB
L
L
2B
,L 42
4
0
22
0
2/
2/2/322
0
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Magnetism 106
P2
22
0
0
2/322
0
4
4
Rs
L
R
iB
or
Rs
dsiRB
L
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Magnetism 107
APPLICATION:Find the magnetic field B at point P in for i = 10 A and a = 8.0 cm.
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Magnetism 108
Circular Arc of Wire
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Magnetism 109
More arc…
Cpoint at 4
44
44
0
0
0
02
0
20
20
R
iB
dR
i
R
iRddBB
R
iRd
R
idsdB
Rdds ds
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Magnetism 110
Howya Do Dat??
0rsdNo Field at C
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Magnetism 111
Force Between Two Current Carrying Straight Parallel
ConductorsWire “a” createsa field at wire “b”
Current in wire “b” sees aforce because it is movingin the magnetic field of “a”.
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Magnetism 112
The Calculation
d
iLi
iFd
iB
ba
b
a
2F
angles...right at are and Since
2
:calculatedjust what weis a"" wire
todue b"" at wire FIELD The
0
b""on
0b""at
BL
BL
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Magnetism 113
Definition of the AmpereThe 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.
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MagnetismMagnetism 114114
TRANSITIONTRANSITION
AMPEREAMPERE
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MagnetismMagnetism 115115
Welcome to Welcome to Andre’ Marie Ampere’s LawAndre’ Marie Ampere’s Law
Normally written as a “circulation” vector Normally written as a “circulation” vector equation.equation.
We will look at another form, but first…We will look at another form, but first…
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Magnetism 116
Remember GAUSS’S LAW??
0enclosedq
d AESurfaceIntegral
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Magnetism 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)
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Magnetism 118
The next few slides have been lifted from Seb Oliver
on the internet
Whoever he is!
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Magnetism 119
Biot-Savart
• The “Coulombs Law of Magnetism”
20 ˆ
4 r
rdsB
id
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Magnetism 120
Invisible Summary
• Biot-Savart Law – (Field produced by wires)– Centre of a wire loop radius R– Centre of a tight Wire Coil with N turns– Distance a from long straight wire
• Force between two wires
• Definition of Ampere
a
II
l
F
2
210
a
IB
2
0
R
IB
20
R
NIB
20
20 ˆ
4 r
rdsB
id
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Magnetism 121
Magnetic Field from a long wire
r
IB
2
0
cosdsBdsB
I
B
r
ds1cos0 dsBdsB
dsr
I
2
0 dsB
Using Biot-Savart Law
Take a short vector on a circle, ds
Thus the dot product of B & the short vector ds is:
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Magnetism 122
Sum B.ds around a circular path
I
B
r
ds
dsr
I
2
0 dsB
dsB
Sum this around the whole ring
dsr
I
2
0 dsr
I
2
0
rds 2 Ιrr
I0
0 22
dsB
Circumference of circle
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Magnetism 123
Consider a different path
• Field goes as 1/r
• Path goes as r.• Integral
independent of r
0 dsB
i
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Magnetism 124
SO, AMPERE’S LAWby SUPERPOSITION:
We will do a LINE INTEGRATIONAround a closed path or LOOP.
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Magnetism 125
Ampere’s Law
enclosedid 0 sB
USE THE RIGHT HAND RULE IN THESE CALCULATIONS
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Magnetism 126
The Right Hand Rule
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Magnetism 127
Another Right Hand Rule
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Magnetism 128
COMPARE
enclosedid 0 sB
0enclosedq
d AE
Line Integral
Surface Integral
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Magnetism 129
Simple Example
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Magnetism 130
Field Around a Long Straight Wire
enclosedid 0 sB
r
iB
irB
2
2
0
0
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Magnetism 131
Field INSIDE a WireCarrying UNIFORM Current
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Magnetism 132
The Calculation
rR
iB
andR
rii
irBdsBd
enclosed
enclosed
20
2
2
0
2
2
sB
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Magnetism 133
R r
B
R
i
2
0
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Magnetism 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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Magnetism 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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Magnetism 136
FIRST PART
Vertical ComponentsCancel
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Magnetism 137
Apply Ampere to Circuit
Infinite Extent
B
B
Li
: thereforeis loop theinsideCurrent
lengthunit per current
L
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Magnetism 138
The “Math”
Infinite Extent
B
B
20
0
0
B
LBLBL
id enclosedsB
Bds=0
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Magnetism 139
A Physical Solenoid
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Magnetism 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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Magnetism 141
More on Long Solenoid
Field is ZERO!
Field is ZERO
Field looks UNIFORM
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Magnetism 142
The real thing…..
Weak Field
Stronger - Leakage
Fairly Uniform field
Finite Length
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Magnetism 143
Another Way
0
:
0
0
0
niB
nihBhh
id
Ampere
enclosed
sB
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Magnetism 144
Application
• Creation of Uniform Magnetic Field Region
• Minimal field outside– except at the ends!
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Magnetism 145
Two Coils
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Magnetism 146
“Real” Helmholtz Coils
Used for experiments.
Can be aligned to cancelout the Earth’s magneticfield for critical measurements.
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Magnetism 147
The Toroid
Slightly lessdense than
inner portion
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Magnetism 148
The Toroid
r
NiB
so
totalNirBd
Ampere
2
turns)# (N 2
:nintegratio ofpath the
in contained coil INNER about the
only worry need Weagain.
0
0
sB