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1P17-
Workshop: Using Visualization in Teaching Introductory E&M
AAPT National Summer Meeting, Edmonton, Alberta, Canada.
Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy
P17- 2
MIT Class: Feeling Magnetic Fields
Magnetic Forces on ChargesMagnetic Dipoles
Experiment: Dipoles in B Fields
P17- 3
The Biot-Savart LawCurrent element of length ds carrying current I produces a magnetic field:
20 ˆ
4 r
dI rsBd
2
ˆx
4o q
r
v r
B
Moving charges are currents too…
P17- 4
=
2 Current Sheets
Ampere’s Law: . encId 0sB
IB
B
X XX
X
X
XX
X
XXX
X
X
XX
X
XXXXXXXXXXXX
B
Long
Circular
Symmetry(Infinite) Current Sheet
Solenoid
Torus
P17- 5
Review:Right Hand Rules
1. Torque: Thumb = torque, fingers show rotation2. Create: Thumb = I, Fingers (curl) = B3. Feel: Thumb = I, Fingers = B, Palm = F4. Moment: Fingers (curl) = I, Thumb = Moment
P17- 6
Demonstration:TV in Field
P17- 7
How a CRT Works: It could…
P17- 8
How a CRT Works: More Typical
P17- 9
How a CRT Works
P17-10
Moving Charges Feel Magnetic Force
B q F v B
Magnetic force perpendicular both to:Velocity v of charge and magnetic field B
P17-11
Reminder: B Field Units
This is called 1 Tesla (T)
41 T = 10 Gauss (G)
Since B q F v B
mA
N1
smC
N1
ndmeter/secocoulomb
newton UnitsB
P17-12
Putting it Together: Lorentz Force
q F E v B
This is the final word on the force on a charge
B q F v B
E qF ECharges Feel…
Electric Fields Magnetic Fields
P17-13
Application: Velocity Selector
What happens here?
P17-14
Velocity Selector
Particle moves in a straight line when
( ) 0net q F E v B
E
vB
P17-15
PRS Question:Hall Effect
P17-16
PRS: Hall EffectA conducting slab has current to the right. A B field is applied out of the page. Due to magnetic forces on the charge carriers, the bottom of the slab is at a higher electric potential than the top of the slab.
IB
V > V(Top)
On the basis of this experiment, the sign of the charge carriers carrying the current in the slab is:
0%
0%
0%
0% 1. Positive2. Negative3. Cannot be determined4. I don’t know
0
P17-17
PRS Answer: Hall Effect
Look at the force on the carriers. If positive, they are flowing to the right, and F will be down. If negative they are flowing to the left and F will be down (don’t forget the sign of q!) So either way the force is down. But we know that the result is a higher potential at the bottom – positive charges are moving down. So the carriers are positive
Answer: 1. Here the charge carriers are positive
IB
V > V(Top)
P17-18
What Kind of Motion in Uniform B Field?
P17-19
Cyclotron Motion
2mvqvB
r
2 2r mT
v qB
2v qB
fr m
(1) r : radius of the circle
(2) T : period of the motion
(3) : cyclotron frequency
mvr
qB
P17-20
Collections of Charges:Current Carrying Wires
P17-21
Demonstration:Jumping Wire
P17-22
Magnetic Force on Current-Carrying Wire
Current is moving charges, and we know that moving charges feel a force in a magnetic field
P17-23
B q F v B
Magnetic Force on Current-Carrying Wire
mcharge
s B
B I F L B
chargem
s B
P17-24
PRS Question:Parallel Current Carrying Wires
P17-25
PRS: Parallel Wires
Consider two parallel current carrying wires. With the currents running in the same direction, the wires are
I1 I2
0%
0%
0%
0%
0% 1. attracted (likes attract?)
2. repelled (likes repel?)
3. pushed another direction
4. not pushed – no net force
5. I don’t know
0
P17-26
PRS Answer: Parallel Wires
I1 creates a field into the page at I2.
That makes a force on I2 to the left.
I2 creates a field out of the page at
I1. That makes a force on I1 to the
right.
Answer: 1. The wires are attracted
I1 I2
X
P17-27
Demonstration:Parallel & Anti-Parallel Currents
P17-28
B
B
B
q
d Id
I
F v B
F s B
F L B
Summary Magnetic Force
P17-29
Can we understand why?Whether they attract or repel can be seen in the shape of the created B field
(Animation) (Animation)
P17-30
Field Pressures and Tensions:A Way To Understand the
qVxB Magnetic Force
P17-31
Tension and Pressures Transmitted by E and B
E & B Fields:• Transmit tension along field direction
(Field lines want to pull straight)• Exert pressure perpendicular to field
(Field lines repel)
P17-32
Example of E Pressure/Tension
Positive charge in uniform (downward) E fieldElectric force on the charge is combination of 1. Pressure pushing down from top 2. Tension pulling down towards bottom
(Animation)
P17-33
Example of B Pressure/Tension
Positive charge moving out of page in uniform (downwards) B field. Magnetic force combines:
1. Pressure pushing from left 2. Tension pulling to right
(Animation)
P17-34
PRS Question:Field Strength
P17-35
PRS: Field StrengthA
B
C
Where is the pictured field the strongest?
0%
0%
0%
0% 1. A
2. B
3. C
4. I don’t know 0
P17-36
PRS Answer: Field Strength
Answer: 3. The field is the strongest at C
Line density is proportional to field strength
A
B
C
P17-37
Example of B Pressure/Tension
Both cases: repelling “pressure” arises from HIGH field strength HIGH energy density
(Animation)
P17-38
Loops of Current
P17-39
Group Problem: Current LoopPlace rectangular current loop in uniform B field
i
j
k
1) What is the net force on this loop?
2) What is the net torque on this loop?
3) Describe the motion the loop makes
P17-40
Torque on Rectangular Loop
ˆ ˆ: area vectorA ab A n n
I τ A B
ˆˆˆ , =Bn k B i
jτ ˆIAB
Familiar? No net force but there is a torque
ijk
x
P17-41
Magnetic Dipole Moment
τ μ B
Analogous to τ p E
tends to align with B
AnμIIA ˆ
Define Magnetic Dipole Moment:
Then:
P17-42
Animation:Another Way To Look At Torque
External field connects to field of magnet and “pulls” the dipole into alignment
P17-43
Demonstration:Galvanometer
P17-44
Magnetic Dipole Moment
AnμIIA ˆ
P17-45
PRS Question:Force on Magnetic Dipole
P17-46
PRS: Dipole in Field
From rest, the coil above will:
0%
0%
0%
0%
0%
0%
0%
0% 1. rotate clockwise, not move
2. rotate counterclockwise, not move
3. move to the right, not rotate
4. move to the left, not rotate
5. move in another direction, without rotating
6. both move and rotate
7. neither rotate nor move
8. I don’t know :00
P17-47
PRS Answer: Dipole in Field
Answer: 1. Coil will rotate clockwise (not move)
No net force so no center of mass motion. BUT Magnetic dipoles rotate to align with external field (think compass)
P17-48
Dipoles don’t move???
This dipole rotates but doesn’t feel a net force
But dipoles CAN feel force due to B. What’s up?
P17-49
Something NewDipoles in Non-Uniform Fields:
Force
P17-50
Force on Magnetic Dipole?
We Want to Know:
What is the force on this dipole?
P17-51
PRS Question:Force on Magnetic Dipole
P17-52
PRS: Dipole in Field
The current carrying coil above will feel a net force
0%
0%
0%
0% 1. upwards
2. downwards
3. of zero
4. I don’t know0
P17-53
PRS Answer: Dipole in Field
Answer: 2. Feels downward force
The I ds x B forces shown produce a net downward force
P17-54
Can just sum I ds x B forces
Is there another way?
P17-55
Energy of Magnetic Dipole
-DipoleU μ B
This equation gives you a general way to think about what dipoles will do in B fields
P17- 56
Magnetic Dipole Moments
AnμIIA ˆ
Generate:
Feel:
1) Torque to align with external field
2) Forces as for bar magnets
-DipoleU μ B
P17-57
Force on Magnetic Dipole
Alternate Thought #1
Where does the dipole want to be?
P17-58
Think Using Energy
-DipoleU μ B
Where does dipole go to reduce its energy?
Aligned dipoles seek high fields!
Force here is down
P17-59
Force on Magnetic Dipole
Alternate Thought #2
What makes the field pictured?
P17-60
Force on Magnetic Dipole
Bar magnet below dipole, with N pole on top It is aligned with the dipole pictured, they attract!
N
S
N
S
P17-61
PRS Questions:Force on Dipole
P17-62
PRS: Dipole in Field
The current carrying coil above will feel a net force
0%
0%
0%
0% 1. upwards
2. downwards
3. of zero
4. I don’t know
0
P17-63
PRS Answer: Dipole in Field
Answer: 2. The coil feels a force downMany ways to know this:
I ds x B forces Energy (aligned seeks high B) Equivalent bar magnets
S
N
S
N
P17-64
PRS: Free Dipoles
If a number of dipoles are randomly scattered through space, after a while they
0%
0%
0%
0% 1. Attract (move together)2. Repel (move apart)3. Basically stay put4. I don’t know
00
P17-65
PRS Answer: Free Dipoles
Answer: 1. Free Dipoles Attract
• Torque on dipole aligns it with the local field
• Dipole then moves toward stronger field — closer to another dipole
Shockwave
P17-66
Some Fun:Magnetic Levitation
P17-67
Put a Frog in a 16 T Magnet…
For details: http://www.hfml.sci.kun.nl/levitate.html
P17-68
How does that work?First a BRIEF intro to magnetic materials
P17-69
Para/Ferromagnetism
Applied external field B0 tends to align the atomic magnetic moments (unpaired electrons)
P17-70
Diamagnetism
Everything is slightly diamagnetic. Why? More later.
If no unpaired electrons then this effect typically dominates.
P17-71
Back to Levitation
P17-72
Levitating a Diamagnet
1) Create a strong field (with a field gradient!)
2) Looks sort of like dipole field
3) Toss in a frog (diamagnet)
4) Looks like a bar magnet pointing opposite the field
5) Seeks lower field (force up) which balances gravity
S
NN
S
Most importantly, in a certain region it is stable:
Restoring force always towards the center
SN
P17-73
Using B to Levitate
For details: http://www.hfml.sci.kun.nl/levitate.html
P17-74
Using B to Levitate
For details: http://www.hfml.sci.kun.nl/levitate.html
P17-75
Using B to Levitate
For details: http://www.hfml.sci.kun.nl/levitate.html
P17-76
Using B to Levitate
For details: http://www.hfml.sci.kun.nl/levitate.html
P17-77
Demonstration:Levitating Magnet over
Superconductor
P17-78
Perfect Diamagnetism:“Magnetic Mirrors”
N
S
N
S
P17-79
Perfect Diamagnetism:“Magnetic Mirrors”
N
S
N
S
No matter what the angle, it floats -- STABILITY
P17-80
Using B to Levitate
For details: http://www.hfml.sci.kun.nl/levitate.html
P17-81
Levitate Magnet with your Fingers?
P17-82
Well… and a lifting magnet
N
S
Why need diamagnetic stabilization?
1) Magnet seeks STRONG field, wants to snap up to lifter
2) Downward oscillation will move it to region where field gradient is too weak to lift it
Diamagnetic sheets above, below prevent these effects, since they repel the floating magnet
P17-83
Experiment 4:Magnetic Forces on Dipoles
This is a little tricky. We will lead you through with lots
of PRS questions
P17-84
First: Set up current supply
• Open circuit (disconnect a lead)
• Turn current knob full CCW (off)
• Increase voltage to ~12 V§This will act as a protection: V<12 V
• Reconnect leads in Helmholtz mode
• Increase current to ~1 A
P17-85
Field Profiles
Single Coil Helmholtz
VERY
UNIFORM!
Anti-Helmholtz
ZERO
FIELD!
P17-86
PRS Prediction:Dipole in Helmholtz
P17-87
PRS: Dipole in Helmholtz
A randomly aligned dipole at the center of a Helmholtz coil will feel:
0%
0%
0%
0% 1. a force but not a torque2. a torque but not a force3. both a torque and a force4. neither force nor torque
:00
P17-88
PRS Answer: Dipole in Helmholtz
The Helmholtz coil makes a UNIFORM FIELDDipole feels only torque (need gradient for F)
Answer: 2. a torque but not a force
P17-89
Next: Dipole in Helmholtz (Q1-2)
• Set in Helmholtz Mode (~1 A)
• Turn off current
• Put dipole in center (0 on scale)
• Randomly align using bar magnet
• Turn on current
What happens?
P17-90
PRS Prediction:Reverse Helmholtz
P17-91
PRS: Reverse Helmholtz
Using aligned dipole, flip the field. Ideally the dipole will feel:
0%
0%
0%
0% 1. a force but not a torque
2. a torque but not a force
3. both a torque and a force
4. neither force nor torque
0
P17-92
PRS Answer: Reverse Helmholtz
The dipole is exactly anti-aligned, so the torque is 0. Still uniform field means still no force.
Answer: 4. IDEALLY neither force nor torque
P17-93
Next: Reverse Helmholtz (Q3)
Starting from end of previous (aligned dipole at center)
• Turn off current
• VERY CAREFULLY (don’t bump!) Reverse leads at power supply
• Turn on current
What happens?
P17-94
PRS Predictions:Moving in Helmholtz
P17-95
PRS: Moving in Helmholtz
When moving through the above field profile, a dipole will:
0%
0%
0% 1. Never rotate
2. Rotate once
3. Rotate twice
0
P17-96
PRS Answer: Moving in Helmholtz
The dipole is always aligned with the field so it will never rotate
Answer: 1. The dipole will never rotate
P17-97
PRS: Moving in Helmholtz
When pulling the dipole through the above field profile, the spring stretch direction will:
0%
0%
0%
0% 1. Always be the same
2. Change once
3. Change twice
4. Change three times
0
P17-98
PRS Answer: Moving in Helmholtz
The dipole always wants to be at the peak field, so when below it the force is up, when above it the force is down.
Answer: 2. The direction will change once
P17-99
Next: Moving in Helmholtz (Q4-5)
• Keep in Helmholtz Mode (~1 A)
• Lower dipole to bottom
• Randomly align (is it possible?)
• Slowly & smoothly raise to well above
What happens (torque? force?)
P17-100
PRS Prediction:Dipole in Anti-Helmholtz
P17-101
PRS: Anti-Helmholtz
A randomly aligned dipole at the center of an Anti- Helmholtz coil will feel:
0%
0%
0%
0% 1. a force but not a torque
2. a torque but not a force
3. both a torque and a force
4. neither force nor torque
:00
P17-102
PRS Answer: Anti-Helmholtz
No field no torque
Field gradient force
Answer: 1. A force but not a torque
P17-103
Next: Dipole in Anti-Helmholtz (Q6-7)
• Set in Anti-Helmholtz Mode (~2 A)
• Turn off current
• Put dipole in center (0 on scale)
• Randomly align using bar magnet
• Turn on current
What happens?
P17-104
PRS Predictions:Moving in Anti-Helmholtz
P17-105
PRS: Moving in Anti-Helmholtz
When moving through the above field profile, a dipole will:
0%
0%
0% 1. Never rotate
2. Rotate once (at sign change)
3. Rotate twice (at slope changes)
0
P17-106
PRS Answer: Moving in Anti-HH
The dipole always wants to align with the field so when it crosses through zero it will rotate
Answer: 2. Dipole rotates once at sign change
P17-107
PRS: Moving in Helmholtz
When pulling the dipole through the above field profile, the spring stretch direction will:
0%
0%
0%
0% 1. Always be the same
2. Change once
3. Change twice
4. Change three times
:00
P17-108
PRS Answer: Moving in Anti-HH
The dipole always wants to seek the strongest field, so the force reverses 3 times
Answer: 4. Force direction changes 3 times
FF
FF
FF
FF
P17-109
Next: Moving in Anti-Helmholtz (Q8-9)
• Keep in Anti-Helmholtz Mode (~2 A)
• Lower dipole to bottom
• Randomly align (is it possible?)
• Slowly & smoothly raise to well above
What happens (torque? force?)
P17-110
Moving in Anti-Helmholtz (Q8-9)
Where does it want to be?
F
F
Force 0, then flips
Field reverses, so does dipole
F
Force 0, then flips
F
NOTE:
Field Up/Down Motion Up/Down
Bottom Top
P17-111
PRS Questions:Force from Single Coil
Fields from Coils
P17-112
PRS: Single Coil
A field-aligned dipole located as pictured feels forces:
A
B
C
0%
0%
0%
0%
0% 1. FA > FB > FC
2. FA > FB ~ FC
3. FB > FA ~ FC
4. FA ~ FB ~ FC
5. No force, only a torque
0
P17-113
PRS Answer: Single Coil
The force goes like the slope of the field. It is ~ 0 at A & C, non-zero at B.
Answer: 3. FB > FA ~ FC
A
B
C
P17-114
PRS: Current Carrying Coils
The above coils have
0%
0%
0%
0% 1. parallel currents that attract2. parallel currents that repel3. opposite currents that attract4. opposite currents that repel
:00
P17-115
PRS Answer: I Carrying Coils
Look at the field lines at the edge between the coils. They are jammed in, want to push out. Also, must be in opposite directions
Answer: 4. Opposite currents that repel
P17-116
Force on Dipole from Dipole:Anti-Parallel Alignment
P17-117
Force on Dipole from Dipole:Parallel Alignment
P17-118
Applications
P17-119
Speakers
P17-120
Speakers
P17-121
DC Motor