Transcript Force and Current Powerpoint

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- 1
MIT Class:
Feeling Magnetic Fields
Magnetic Forces on Charges
Magnetic Dipoles
Experiment: Dipoles in B Fields
P17- 2
The Biot-Savart Law
Current element of length ds carrying current I
produces a magnetic field:

  0 I d s  rˆ
dB 
2
4 r
Moving charges are currents too…
o q v x rˆ
B
2
4 r
P17- 3
 
Ampere’s Law:  B  d s   0 I enc
.
B
Long
Circular
Symmetry
I
B
(Infinite) Current Sheet
X
X
X
X
X
X
X
X
X
X
X
X
X
X
B
X
X
Solenoid
=
2 Current
Sheets
X
X
X
X
X
X
X
X
X
X
X
X
Torus
P17- 4
Review:
Right Hand Rules
1.
2.
3.
4.
Torque: Thumb = torque, fingers show rotation
Create: Thumb = I, Fingers (curl) = B
Feel: Thumb = I, Fingers = B, Palm = F
Moment: Fingers (curl) = I, Thumb = Moment
P17- 5
Demonstration:
TV in Field
P17- 6
How a CRT Works: It could…
P17- 7
How a CRT Works: More Typical
P17- 8
How a CRT Works
P17- 9
Moving Charges Feel Magnetic Force
FB  q v  B
Magnetic force perpendicular both to:
Velocity v of charge and magnetic field B
P17- 10
Reminder: B Field Units
Since
FB  q v  B
newton
N
N
B Units 
1
1
coulomb meter/seco nd  C  m s A  m
This is called 1 Tesla (T)
4
1 T = 10 Gauss (G)
P17- 11
Putting it Together: Lorentz Force
Charges Feel…
FE  qE
FB  q v  B
Electric Fields
Magnetic Fields

F  q E  vB

This is the final word on the force on a charge
P17- 12
Application: Velocity Selector
What happens here?
P17- 13
Velocity Selector
Particle moves in a straight line when
Fnet
E
 q(E  v  B)  0  v 
B
P17- 14
PRS Question:
Hall Effect
P17- 15
PRS: Hall Effect
A 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.
B
I
V > V(Top)
On the basis of this experiment, the sign of the
charge carriers carrying the current in the slab is:
0%
1. Positive
0%
2. Negative
0%
3. Cannot be determined
4. I don’t know
0%
0
P17- 16
PRS Answer: Hall Effect
Answer: 1. Here the charge carriers are positive
B
I
V > V(Top)
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
P17- 17
What Kind of Motion in
Uniform B Field?
P17- 18
Cyclotron Motion
(1) r : radius of the circle
mv 2
mv
qvB 
 r
r
qB
(2) T : period of the motion
2 r 2 m
T

v
qB
(3)  : cyclotron frequency
v qB
  2 f  
r m
P17- 19
Collections of Charges:
Current Carrying Wires
P17- 20
Demonstration:
Jumping Wire
P17- 21
Magnetic Force on
Current-Carrying Wire
Current is moving charges, and we know that
moving charges feel a force in a magnetic field
P17- 22
Magnetic Force on
Current-Carrying Wire
FB  qv  B
m
  charge   B
s
charge

mB
s

FB  I L  B

P17- 23
PRS Question:
Parallel Current Carrying Wires
P17- 24
0
PRS: Parallel Wires
Consider two parallel current
carrying wires. With the currents
running in the same direction, the
wires are
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
I
I
1
2
attracted (likes attract?)
repelled (likes repel?)
pushed another direction
not pushed – no net force
I don’t know
P17- 25
PRS Answer: Parallel Wires
Answer: 1. The wires are attracted
I1 creates a field into the page at I2.
That makes a force on I2 to the left.
X
I
I
1
2
I2 creates a field out of the page at I1.
That makes a force on I1 to the right.
P17- 26
Demonstration:
Parallel & Anti-Parallel Currents
P17- 27
Summary Magnetic Force
FB  qv  B
dFB  Id s  B

FB  I L  B

P17- 28
Can we understand why?
Whether they attract or repel can be seen in
the shape of the created B field
(Animation)
(Animation)
P17- 29
Field Pressures and Tensions:
A Way To Understand the
qVxB Magnetic Force
P17- 30
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- 31
Example of E Pressure/Tension
(Animation)
Positive charge in uniform (downward) E field
Electric force on the charge is combination of
1. Pressure pushing down from top
2. Tension pulling down towards bottom
P17- 32
Example of B Pressure/Tension
(Animation)
Positive charge moving out of page in uniform
(downwards) B field. Magnetic force combines:
1. Pressure pushing from left
2. Tension pulling to right
P17- 33
PRS Question:
Field Strength
P17- 34
PRS: Field Strength
A
B
C
0%
0%
0%
0%
Where is the pictured field the strongest?
1. A
2. B
3. C
4. I don’t know
0
P17- 35
PRS Answer: Field Strength
A
B
C
Answer: 3. The field is the strongest at C
Line density is proportional to field strength
P17- 36
Example of B Pressure/Tension
(Animation)
Both cases: repelling “pressure” arises from
HIGH field strength  HIGH energy density
P17- 37
Loops of Current
P17- 38
Group Problem: Current Loop
Place rectangular current loop in uniform B field
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
ĵ
k̂
î
P17- 39
Torque on Rectangular Loop

ˆ
τ  IABj
A  A nˆ  ab nˆ : area vector
nˆ  kˆ , B=B ˆi
k̂
ĵ x
î
τ  IA  B
Familiar? No net force but there is a torque
P17- 40
Magnetic Dipole Moment
Define Magnetic Dipole Moment:


μ  IAnˆ  IA
Then:
τ  μB
Analogous to
τ  pE
t tends to align  with B
P17- 41
Animation:
Another Way To Look At Torque
External field connects to field of magnet
and “pulls” the dipole into alignment
P17- 42
Demonstration:
Galvanometer
P17- 43
Magnetic Dipole Moment


μ  IAnˆ  IA
P17- 44
PRS Question:
Force on Magnetic Dipole
P17- 45
PRS: Dipole in Field

From rest, the coil above will:
0%
0%
0%
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
6.
7.
8.
rotate clockwise, not move
rotate counterclockwise, not move
move to the right, not rotate
move to the left, not rotate
move in another direction, without rotating
both move and rotate
neither rotate nor move
I don’t know
:00P17- 46
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- 47
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- 48
Something New
Dipoles in Non-Uniform Fields:
Force
P17- 49
Force on Magnetic Dipole?
We Want to Know:
What is the force on this dipole?
P17- 50
PRS Question:
Force on Magnetic Dipole
P17- 51
PRS: Dipole in Field
The current carrying coil above will feel a net force
0%
0%
0%
0%
1.
2.
3.
4.
upwards
downwards
of zero
I don’t know
0
P17- 52
PRS Answer: Dipole in Field
Answer: 2. Feels downward force
The I ds x B forces shown produce a net
downward force
P17- 53
Can just sum I ds x B forces
Is there another way?
P17- 54
Energy of Magnetic Dipole
U Dipole  -μ  B
This equation gives you a general way to
think about what dipoles will do in B fields
P17- 55
Magnetic Dipole Moments


μ  IAnˆ  IA
Generate:
Feel:
1) Torque to align with external field
2) Forces as for bar magnets
U Dipole  -μ  B
P17- 56
Force on Magnetic Dipole
Alternate Thought #1
Where does the dipole want to be?
P17- 57
Think Using Energy
U Dipole  -μ  B
Where does dipole go
to reduce its energy?
Aligned dipoles seek high fields!
 Force here is down
P17- 58
Force on Magnetic Dipole
Alternate Thought #2
What makes the field pictured?
P17- 59
Force on Magnetic Dipole

N
S
N
S
Bar magnet below dipole, with N pole on top
It is aligned with the dipole pictured, they attract!
P17- 60
PRS Questions:
Force on Dipole
P17- 61
PRS: Dipole in Field
0
The current carrying coil above will feel a net force
0%
0%
0%
0%
1.
2.
3.
4.
upwards
downwards
of zero
I don’t know
P17- 62
PRS Answer: Dipole in Field
S

N
S
N
Answer: 2. The coil feels a force down
Many ways to know this:
 I ds x B forces
 Energy (aligned seeks high B)
 Equivalent bar magnets
P17- 63
PRS: Free Dipoles
If a number of dipoles are randomly
scattered through space, after a while they
0%
0%
0%
0%
1.
2.
3.
4.
Attract (move together)
Repel (move apart)
Basically stay put
I don’t know
0
P17- 64
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- 65
Some Fun:
Magnetic Levitation
P17- 66
Put a Frog in a 16 T Magnet…
For details: http://www.hfml.sci.kun.nl/levitate.html
P17- 67
How does that work?
First a BRIEF intro to
magnetic materials
P17- 68
Para/Ferromagnetism
Applied external field B0 tends to align the
atomic magnetic moments (unpaired
electrons)
P17- 69
Diamagnetism
Everything is slightly
diamagnetic. Why?
More later.
If no unpaired
electrons then this
effect typically
dominates.
P17- 70
Back to Levitation
P17- 71
Levitating a Diamagnet
N
S
S
N
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
Most importantly, in a certain region it is stable:
Restoring force always towards the center
P17- 72
Using B to Levitate
For details: http://www.hfml.sci.kun.nl/levitate.html
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
Demonstration:
Levitating Magnet over
Superconductor
P17- 77
Perfect Diamagnetism:
“Magnetic Mirrors”
N
S
S
N
P17- 78
Perfect Diamagnetism:
“Magnetic Mirrors”
N
S
S
N
No matter what the angle, it floats -- STABILITY
P17- 79
Using B to
Levitate
For details:
http://www.hfml.sci.
kun.nl/levitate.html
P17- 80
Levitate Magnet with your Fingers?
P17- 81
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- 82
Experiment 4:
Magnetic Forces on Dipoles
This is a little tricky. We will
lead you through with lots
of PRS questions
P17- 83
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- 84
Field Profiles
VERY
UNIFORM!
Single Coil
Helmholtz
Anti-Helmholtz
ZERO
FIELD!
P17- 85
PRS Prediction:
Dipole in Helmholtz
P17- 86
:00
PRS: Dipole in Helmholtz
A randomly aligned dipole at the center of a
Helmholtz coil will feel:
0%
1. a force but not a torque
0%
2. a torque but not a force
0%
3. both a torque and a force
4. neither force nor torque
0%
P17- 87
PRS Answer: Dipole in Helmholtz
Answer: 2. a torque but not a force
The Helmholtz coil makes a UNIFORM FIELD
Dipole feels only torque (need gradient for F)
P17- 88
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- 89
PRS Prediction:
Reverse Helmholtz
P17- 90
0
PRS: Reverse Helmholtz
Using aligned dipole, flip the field. Ideally the dipole
will feel:
0%
0%
0%
0%
1.
2.
3.
4.
a force but not a torque
a torque but not a force
both a torque and a force
neither force nor torque
P17- 91
PRS Answer: Reverse Helmholtz
Answer: 4. IDEALLY neither force nor torque
The dipole is exactly anti-aligned, so the torque
is 0. Still uniform field means still no force.
P17- 92
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- 93
PRS Predictions:
Moving in Helmholtz
P17- 94
PRS: Moving in Helmholtz
0
When moving through the above field profile, a
dipole will:
0%
1. Never rotate
0%
2. Rotate once
3. Rotate twice
0%
P17- 95
PRS Answer: Moving in Helmholtz
Answer: 1. The dipole will never rotate
The dipole is always aligned with the field
so it will never rotate
P17- 96
PRS: Moving in Helmholtz
0
When pulling the dipole through the above field
profile, the spring stretch direction will:
0%
0%
0%
0%
1.
2.
3.
4.
Always be the same
Change once
Change twice
Change three times
P17- 97
PRS Answer: Moving in Helmholtz
Answer: 2. The direction will change once
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.
P17- 98
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- 99
PRS Prediction:
Dipole in Anti-Helmholtz
P17-100
PRS: Anti-Helmholtz
:00
A randomly aligned dipole at the center of an AntiHelmholtz coil will feel:
0%
0%
0%
0%
1.
2.
3.
4.
a force but not a torque
a torque but not a force
both a torque and a force
neither force nor torque
P17-101
PRS Answer: Anti-Helmholtz
Answer: 1. A force but not a torque
No field  no torque
Field gradient  force
P17-102
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?
P17-103
PRS Predictions:
Moving in Anti-Helmholtz
P17-104
PRS: Moving in Anti-Helmholtz
0
When moving through the above field profile, a
dipole will:
0%
1. Never rotate
0%
2. Rotate once (at sign change)
3. Rotate twice (at slope changes)
0%
P17-105
PRS Answer: Moving in Anti-HH
Answer: 2. Dipole rotates once at sign change
The dipole always wants to align with the
field so when it crosses through zero it will
rotate
P17-106
PRS: Moving in Helmholtz
:00
When pulling the dipole through the above field
profile, the spring stretch direction will:
0%
0%
0%
0%
1.
2.
3.
4.
Always be the same
Change once
Change twice
Change three times
P17-107
PRS Answer: Moving in Anti-HH



F
F
F

F
Answer: 4. Force direction changes 3 times
The dipole always wants to seek the
strongest field, so the force reverses 3
times
P17-108
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?)
P17-109
Moving in Anti-Helmholtz (Q8-9)
Force  0, then flips
Where does it
want to be?

F
F
Bottom



F
Field reverses,
so does dipole
Top
F
NOTE:
Field Up/Down

Motion Up/Down 
Force  0, then flips
P17-110
PRS Questions:
Force from Single Coil
Fields from Coils
P17-111
PRS: Single Coil
A
0
B
C
A field-aligned dipole located as pictured feels forces:
0%
0%
0%
0%
0%
1.
2.
3.
4.
5.
FA > F B > F C
FA > F B ~ F C
FB > F A ~ F C
FA ~ F B ~ F C
No force, only a torque
P17-112
PRS Answer: Single Coil
A
B
C
Answer: 3. FB > FA ~ FC
The force goes like the slope of the field.
It is ~ 0 at A & C, non-zero at B.
P17-113
PRS: Current Carrying Coils
:00
The above coils have
0%
1. parallel currents that attract
0%
2. parallel currents that repel
3. opposite currents that attract
0%
4. opposite currents that repel
0%
P17-114
PRS Answer: I Carrying Coils
Answer: 4. Opposite currents that repel
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
P17-115
Force on Dipole from Dipole:
Anti-Parallel Alignment
P17-116
Force on Dipole from Dipole:
Parallel Alignment
P17-117
Applications
P17-118
Speakers
P17-119
Speakers
P17-120
DC Motor
P17-121