Class 18: Outline
Hour 1:
Levitation
Experiment 8: Magnetic Forces
Hour 2:
Ampere’s Law
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1. Torque: Thumb = torque, fingers show rotation
2. Feel: Thumb = I, Fingers = B, Palm = F
3. Create: Thumb = I, Fingers (curl) = B
4. Moment: Fingers (curl) = I, Thumb = Moment
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µ
IA
I A
Generate:
Feel:
U
Dipole
= -
G
⋅
G
µ B
1) Torque to align with external field
2) Forces as for bar magnets (seek field)
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Put a Frog in a 16 T Magnet…
For details: http://www.hfml.sci.kun.nl/levitate.html
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Para/Ferromagnetism
Applied external field B
0 tends to align the atomic magnetic moments (unpaired electrons)
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Diamagnetism
Everything is slightly diamagnetic. Why?
More later.
If no magnetic moments (unpaired electrons) then this effect dominates.
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Levitating a Diamagnet
N
S
S
N
S
N
1) Create a strong field
(with a field gradient!)
2) Looks like a 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, its stable:
Restoring force always towards the center
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Using ∇ B to Levitate
•Frog
•Strawberry
•Water Droplets
•Tomatoes
•Crickets
For details: http://www.hfml.ru.nl/levitation-movies.html
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Demonstrating:
Levitating Magnet over
Superconductor
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Perfect Diamagnetism:
“Magnetic Mirrors”
N
S
S
N
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Perfect Diamagnetism:
“Magnetic Mirrors”
N
S
S
N
No matter what the angle, it floats -- STABILITY
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Using ∇ B to Levitate
A Sumo Wrestler
For details: http://www.hfml.sci.kun.nl/levitate.html
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Two PRS Questions Related to
Experiment 8: Magnetic Forces
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Experiment 8: Magnetic Forces
(Calculating µ
0
) http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/16-
MagneticForceRepel/16-MagForceRepel_f65_320.html
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The Biot-Savart Law
Current element of length d s carrying current I produces a magnetic field: d
G
B
0
I d
r
r 2
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rd
Analogous (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
G
B
d s
G
0
I enc
The line integral is around any closed contour bounding an open surface S.
I enc
I enc
= ⋅
S
J d A
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PRS Question:
Ampere’s Law
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Biot-
Savart
Law
Biot-Savart vs. Ampere
G
B =
µ
4
0
π
G
× ˆ r 2 general current source ex: finite wire wire loop
Ampere’s law
G
B ⋅ d = µ
0
I enc symmetric current source ex: infinite wire infinite current sheet
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Applying Ampere’s Law
1. Identify regions in which to calculate B field
Get B direction by right hand rule
2. Choose Amperian Loops S: Symmetry
3. Calculate
B ⋅ d s
4. Calculate current enclosed by loop S
G
B
d s
G
0
I enc
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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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Ampere’s Law Example:
Infinite Wire
I
B
I
Amperian Loop:
B is Constant & Parallel
I Penetrates
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Example: Wire of Radius R
Region 1: Outside wire (r ≥ R)
Cylindrical symmetry Æ
Amperian Circle
B-field counterclockwise v
G
B s
G
=
= µ
v
0
I enc s
G
=
=
B
µ
0
(
I
2 π r
)
G
B =
µ
2 π
0
I r counterclo ckwise
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Example: Wire of Radius R
Region 2: Inside wire (r < R)
v
G
B s =
= µ
v
0
I enc s
G
=
=
B 2 π r
µ
0
I
(
⎛
⎝
π
π r
)
R
2
2
G
B =
2
µ
π
0
Ir
R 2 counterclo ckwise
⎞
⎠
Could also say: J =
I
A
=
I
π R 2
; I enc
= JA enc
=
I
π R 2
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Example: Wire of Radius R
B in
=
µ
π
0
Ir
2 R 2
B out
=
µ
2
0
π r
I
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Group Problem: Non-Uniform
Cylindrical Wire
I A cylindrical conductor has radius R and a nonuniform current density with total current:
G
J = J
0
R r
Find B everywhere
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Applying Ampere’s Law
In 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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Helmholtz Coil
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Closer than Helmholtz Coil
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Multiple Wire Loops
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Multiple Wire Loops –
Solenoid
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Magnetic Field of Solenoid loosely wound tightly wound
For ideal solenoid, B is uniform inside & zero outside
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Magnetic Field of Ideal Solenoid v ∫
G
B ⋅
Using Ampere’s law: Think!
d
⎧ ⊥
⎨
G
=
G
B
1
∫
=
G
Bl d
0 along side 3
⋅ d
G
G
along sides 2 and 4
+
+ ∫
2
G
B
0
⋅ d
G
+
+
3
∫
G
B ⋅ d
0
G
+ ∫
4
+
G
B ⋅ d
0
G
I enc
= nlI v
n: turn density
G
B ⋅ d s = Bl = µ
0 nlI
µ
0 nlI
B = = µ
0 nI l
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Demonstration:
Long Solenoid
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Group Problem: Current Sheet y
A sheet of current (infinite in the y & z directions, of thickness 2d in the x direction) carries a uniform current density:
G
J s
= J
ˆ
Find B everywhere
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Ampere’s Law:
Infinite Current Sheet
B
I
B
Amperian Loops:
B is Constant & Parallel OR Perpendicular OR Zero
I Penetrates
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Solenoid is Two Current Sheets
Field outside current sheet should be half of solenoid, with the substitution: nI = 2 dJ
This is current per unit length
(equivalent of λ , but we don’t have a symbol for it)
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Long
Circular
Symmetry
Solenoid
=
2 Current
Sheets
B
X
X
X
X
X
X
X
X
X
X
X
X
G
B
d s
G
I enc
B
I
B
(Infinite) Current Sheet
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Torus
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Brief Review Thus Far…
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Maxwell’s Equations (So Far)
Gauss's Law: w
S
G
⋅ d
G
E A =
Q in
ε
0
Electric charges make diverging Electric Fields
Magnetic Gauss's Law: w
S
G
⋅ d
G
B A = 0
No Magnetic Monopoles! (No diverging B Fields)
Ampere's Law: v
C
G
B ⋅ d s = µ
0
I enc
Currents make curling Magnetic Fields
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