p14 - 1 workshop: using visualization in teaching introductory e&m aapt national summer meeting,...
TRANSCRIPT
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1P14-
Workshop: Using Visualization in Teaching Introductory E&M
AAPT National Summer Meeting, Edmonton, Alberta, Canada.
Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy
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P14- 2
MIT Class: Sources of Magnetic Fields
Creating Fields: Biot-SavartExperiment: Magnetic Fields
Ampere’s Law
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P14- 3
Magnetic Fields
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P14- 4
Gravitational – Electric Fields
2ˆ
mGr
g rMass m Charge q (±)
2ˆ
e
qkr
E r
g mF g
E qF E
Create:
Feel:
Also saw… Dipole p
τ p E
Creates:
Feels:
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P14- 5
Magnetism – Bar Magnet
Like poles repel, opposite poles attract
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Demonstration:Magnetic Field Lines
from Bar Magnet
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P14- 7
Demonstration:Compass (bar magnet) in
Magnetic Field Linesfrom Bar Magnet
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Magnetic Field of Bar Magnet
(1) A magnet has two poles, North (N) and South (S)
(2) Magnetic field lines leave from N, end at S
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P14- 9
Bar Magnets Are Dipoles!
NO! Magnetic monopoles do not exist in isolation
• Create Dipole Field
• Rotate to orient with Field
Is there magnetic “mass” or magnetic “charge?”
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Magnetic Monopoles?
Magnetic monopoles do not exist in isolation
q-qpElectric Dipole
When cut:
2 monopoles (charges)
μ
Magnetic Dipole
When cut: 2 dipoles
Another Maxwell’s Equation! (2 of 4)
0S
inqd
E A
S
0d B A
Gauss’s Law Magnetic Gauss’s Law
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PRS:B Field inside a Magnet
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PRS: Magnetic Field LinesThe picture shows the field lines outside a permanent magnet The field lines inside the magnet point:
0%
0%
0%
0%
0%
0% 1. Up
2. Down
3. Left to right
4. Right to left
5. The field inside is zero
6. I don’t know 15
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PRS Answer: Magnetic Field Lines
Magnetic field lines are continuous.
E field lines begin and end on charges.
There are no magnetic charges (monopoles) so B field lines never begin or end
Answer: 1. They point up inside the magnet
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Magnetic Field of the Earth
North magnetic pole located in southern hemisphere
Also a magnetic
dipole!
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Fields: Grav., Electric, Magnetic
2ˆ
mGr
g rMass m Charge q (±)
2ˆ
e
qkr
E r
g mF g
E qF E
Create:
Feel:
Dipole p Dipole
τ p E
Create:
Feel: τ μ B
No Magnetic
Monopoles!
E
B
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What is B?
B is the magnetic field
It has units of Tesla (T)
N1Tesla 1
A m
This class & next: creating B fields
Next two classes: feeling B fields
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How Big is a Tesla?
• Earth’s Field
• Brain (at scalp)
• Refrigerator Magnet
• Inside MRI
• Good NMR Magnet
• Biggest in Lab
• Biggest in Pulsars
5 x 10-5 T = 0.5 Gauss
~1 fT
3 T
18 T
150 T (pulsed)
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How do we create fields?
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P14- 19
What creates fields?
Magnets – more about this later
The Earth How’s that work?
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Magnetic Field of the Earth
North magnetic pole located in southern hemisphere
(for now)
Also a magnetic
dipole!
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What creates fields?
Magnets – more about this later
The Earth How’s that work?
Moving charges!
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Electric Field Of Point ChargeAn electric charge produces an electric field:
2
1ˆ
4 o
q
rE r
r : unit vector directed from q to P
r
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Magnetic Field Of Moving Charge
Moving charge with velocity v produces magnetic field:
2
ˆx
4o q
r
v r
B
:r unit vector directed from q to Pr
permeability of free space7
0 4 10 T m/A
P
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Recall:Cross Product
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Notation Demonstration
OUT of page
“Arrow Head”
X X
XX
X X
XX
X X
XX
X X
XX
INTO page
“Arrow Tail”
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Cross Product: MagnitudeComputing magnitude of cross product A x B:
xC A B
sinC A B
area of parallelogram| C |:
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Cross Product: Direction
Right Hand Rule #1:
xC A B
For this method, keep your hand flat!
1) Put Thumb (of right hand) along A
2) Rotate hand so fingers point along B
3) Palm will point along C
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Cross Product: Signs
jkijik
ijkikj
kijkji
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
Cross Product is Cyclic (left column)
Reversing A & B changes sign (right column)
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PRS Questions:Right Hand Rule
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PRS: Cross Product
What is the direction of A x B given the following two vectors?
A B
0%
0%
0%
0%
0%
0%
0% 1. up
2. down
3. left
4. right
5. into page
6. out of page
7. Cross product is zero (so no direction)
15
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P14- 31
PRS Answer: Cross Product
Using your right hand, thumb along A, fingers along B, palm into page
Answer: 5. A x B points into the page
A B
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P14- 32
PRS: Cross Product
What is the direction of A x B given the following two vectors?
A B
0%
0%
0%
0%
0%
0%
0% 1. up
2. down
3. left
4. right
5. into page
6. out of page
7. Cross product is zero (so no direction)
15
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PRS Answer: Cross Product
Using your right hand, thumb along A, fingers along B, palm out of page
Also note from before, one vector flipped so result does too
Answer: 6. A x B points out of the page
A B
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Continuous charge distributions:Currents & Biot-Savart
Moving
^
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d dqB v
From Charges to Currents?
mcharge
s
d IdB s
chargem
s
v
dq
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The Biot-Savart LawCurrent element of length ds carrying current I produces a magnetic field:
20 ˆ
4 r
dI rsBd
(Shockwave)
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The Right-Hand Rule #2
ˆ ˆˆ z ρ φ
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Animation:Field Generated by a Moving Charge
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Demonstration:Field Generated by Wire
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Example : Coil of Radius R
Consider a coil with radius R and current I
II
IP
Find the magnetic field B at the center (P)
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Example : Coil of Radius R
Consider a coil with radius R and current I
II
IP
1) Think about it:• Legs contribute nothing
I parallel to r• Ring makes field into page
2) Choose a ds3) Pick your coordinates4) Write Biot-Savart
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Example : Coil of Radius RIn the circular part of the coil…
ˆd s r
rsd
II
I
02
ˆ
4
dIdB
r
s rBiot-Savart:
ˆ| d | ds s r
024
I ds
r
024
I R d
R
0
4
I d
R
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II
I
Example : Coil of Radius RConsider a coil with radius R and current I
sd
0
4
I ddB
R
20
0 4
I dB dB
R
2
0 0
0
24 4
I Id
R R
0 into page2
I
R
B
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Example : Coil of Radius R
Notes:•This is an EASY Biot-Savart problem:
• No vectors involved•This is what I would expect on exam
II
IP page into
20
R
IB
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PRS Questions:B fields Generated by Currents
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PRS: Biot-Savart
The magnetic field at P points towards the
0%
0%
0%
0%
0%
0%
0% 1. +x direction2. +y direction3. +z direction4. -x direction5. -y direction6. -z direction7. Field is zero (so no direction)
1515
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PRS Answer: Biot-Savart
The vertical line segment contributes nothing to the field at P (it is parallel to the displacement). The horizontal segment makes a field out of the page.
Answer: 3. B(P) is in the +z direction (out of page)
ikj
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PRS: Bent WireThe magnetic field at P is equal to the field of:
0%
0%
0%
0% 1. a semicircle2. a semicircle plus the field of a long straight wire3. a semicircle minus the field of a long straight wire4. none of the above
15
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P14- 49
PRS Answer: Bent Wire
All of the wire makes B into the page. The two straight parts, if put together, would make an infinite wire. The semicircle is added to this to get the complete field
Answer: 2. Semicircle + infinite wire
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Group Problem:B Field from Coil of Radius R
Consider a coil made of semi-circles of radii R and 2R and carrying a current I
What is B at point P?
P
I
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P14- 51
Group Problem:B Field from Coil of Radius R
Consider a coil with radius R and carrying a current I
What is B at point P? WARNING:
This is much harder than the previous problem. Why??
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Experiment:Magnetic Fields:Bar Magnets &
Wire Coils
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PRS Question:Part I: B Field from Bar Magnet
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PRS: Bar Magnet B FieldThinking of your map of the B field lines from part 1, assume that your magnet and compass were on the table in the orientation shown.
N N S
?
The red end of the compass points:
0%
0%
0%
0%
0%
0%
0%
0% 1. Up
2. Down
3. Right
4. Left
5. Up & right
6. Up & left
7. Down & right
8. Down & left 0
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PRS Answer: Bar Magnet B Field
If you only had to consider the bar magnet (for example, if you were very close to it) the compass would point to the right. But the Earth’s magnetic field (pointing toward geographic North) pulls the field down.
Answer: 7. Down & right
N N S
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Visualization:Bar Magnet &
Earth’s Magnetic Field
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P14- 57
PRS Question:Part 3: B Field from Helmholtz
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PRS: HelmholtzIdentify the three field profiles that you measured as Single (Sgl),
Helmholtz (Hh) or Anti-Helmholtz (A-H):
-2 -1 0 1 2
0Anti-Helmholtz
Single Coil
Helmholtz
Top CoilBottom Coil
Ma
gn
etic
Fie
ld A
mp
litu
de
Distance along the central axis (z/R)
AB
C
0%0%0%0%0%0% 1. Sgl, Hh, A-H
2. Hh, A-H, Sgl
3. A-h, Sgl, Hh
4. Sgl, A-H, Hh
5. A-H, Hh, Sgl
6. Hh, Sgl, A-H
The curves, A, B & C are respectively:
0
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PRS Answer: Helmholtz
Note that the Helmholtz mode creates a very uniform field near the center while the field from the Anti-Helmholtz is zero at the center. The single coil peaks at the coil’s center.
Answer: 6. Helmholtz, Single, Anti-Helmholtz
-2 -1 0 1 2
0Anti-Helmholtz
Single Coil
Helmholtz
Top CoilBottom Coil
Ma
gn
etic
Fie
ld A
mp
litu
de
Distance along the central axis (z/R)
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P14- 60
Last Time:Creating Magnetic Fields:
Biot-Savart
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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…
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Today:3rd Maxwell Equation:
Ampere’s Law
Analog (in use) to Gauss’s Law
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Gauss’s Law – The Idea
The total “flux” of field lines penetrating any of these surfaces is the same and depends only on the amount of charge inside
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Ampere’s Law: The Idea
In order to have a B field around a loop, there must be current punching through the loop
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Ampere’s Law: The Equation
The line integral is around any closed contour bounding an open surface S.
Ienc is current through S:
encId 0sB
enc
S
I d J A
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PRS Questions:Ampere’s Law
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PRS: Ampere’s Law
Integrating B around the loop shown gives us:
0%
0%
0% 1. a positive number
2. a negative number
3. zero :15
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PRS Answer: Ampere’s Law
Answer: 3. Total penetrating current is zero, so
0 0encd I B s
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PRS: Ampere’s Law
Integrating B around the loop shown gives us:
0%
0%
0% 1. a positive number
2. a negative number
3. zero 15
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PRS Answer: Ampere’s Law
Answer: 2.
Net penetrating current is out of the page, so field is counter-clockwise (opposite path)
0d B s
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Biot-Savart vs. Ampere
Biot-Savart Law
general current sourceex: finite wire
wire loop
Ampere’s law
symmetriccurrent source
ex: infinite wireinfinite current sheet
02
ˆ
4
I d
r
s rB
encId 0sB
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P14- 72
Applying Ampere’s Law1. Identify regions in which to calculate B field Get
B direction by right hand rule
2. Choose Amperian Loops S: Symmetry B is 0 or constant on the loop!
3. Calculate
4. Calculate current enclosed by loop S
5. Apply Ampere’s Law to solve for B sB
d
encId 0sB
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Always True, Occasionally Useful
Like Gauss’s Law,
Ampere’s Law is always true
However, it is only useful for calculation in certain specific situations, involving highly symmetric currents.
Here are examples…
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Example: Infinite Wire
I A cylindrical conductor has radius R and a uniform current density with total current I
Find B everywhere
Two regions:
(1) outside wire (r ≥ R)
(2) inside wire (r < R)
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P14- 75
Ampere’s Law Example:Infinite Wire
I
I
B
Amperian Loop:
B is Constant & Parallel
I Penetrates
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Example: Infinite WireRegion 1: Outside wire (r ≥ R)
dB s
ckwisecounterclo2
0
r
I
B
B ds 2B r
0 encI 0I
Cylindrical symmetry Amperian Circle
B-field counterclockwise
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Example: Infinite WireRegion 2: Inside wire (r < R)
2
0 2
rI
R
ckwisecounterclo2 2
0
R
Ir
B
Could also say: 222
; rR
IJAI
R
I
A
IJ encenc
dB s
B ds 2B r
0 encI
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P14- 78
Example: Infinite Wire
20
2 R
IrBin
r
IBout
2
0
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P14- 79
Group Problem: Non-Uniform Cylindrical Wire
I A cylindrical conductor has radius R and a non-uniform current density with total current:
Find B everywhere
0
RJr
J
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Applying Ampere’s LawIn Choosing Amperian Loop:• Study & Follow Symmetry• Determine Field Directions First• Think About Where Field is Zero• Loop Must
• Be Parallel to (Constant) Desired Field• Be Perpendicular to Unknown Fields• Or Be Located in Zero Field
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Other Geometries
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Two Loops
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Two Loops Moved Closer Together
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Multiple Wire Loops
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Multiple Wire Loops –Solenoid
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Demonstration:Long Solenoid
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Magnetic Field of Solenoid
loosely wound tightly wound
For ideal solenoid, B is uniform inside & zero outside
Horiz.
comp.
cancel
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P14- 88
Magnetic Field of Ideal Solenoid
d d d d d 1 2 3 4
B s = B s B s B s B s
Using Ampere’s law: Think!
along sides 2 and 4
0 along side 3
d
B s
B
n: turn densityencI nlI
0d Bl nlI B s
0
0
nlIB nI
l
/ : # turns/unit lengthn N L
0 0 0Bl
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P14- 89
Group Problem: Current Sheet
A sheet of current (infinite in the y & z directions, of thickness 2d in the x direction) carries a uniform current density:
Find B for x > 0
ˆs JJ k
y
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P14- 90
Ampere’s Law:Infinite Current Sheet
I
Amperian Loops:
B is Constant & Parallel OR Perpendicular OR Zero
I Penetrates
B
B
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P14- 91
Solenoid is Two Current SheetsField outside current sheet should be half of solenoid, with the substitution:
2nI dJ
This is current per unit length (equivalent of , but we don’t have a symbol for it)
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P14- 92
=
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
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P14- 93
Brief Review Thus Far…
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P14- 94
Maxwell’s Equations (So Far)
0Ampere's Law:
Currents make curling Magnetic Fields
enc
C
d I B s
Magnetic Gauss's Law: 0
No Magnetic Monopoles! (No diverging B Fields)S
d B A
0
Gauss's Law:
Electric charges make diverging Electric Fields
in
S
Qd
E A